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Review

Solutions for Some Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 1

Department of Mathematical Methods and Models, Faculty of Applied Sciences, University POLITEHNICA of Bucharest, 060042 Bucharest, Romania
Axioms 2023, 12(6), 532; https://doi.org/10.3390/axioms12060532
Submission received: 8 April 2023 / Revised: 9 May 2023 / Accepted: 18 May 2023 / Published: 29 May 2023
(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)

Abstract

:
This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. This article details three ways to use surjectivity results for a special type of operator involving the duality mapping and a Nemytskii operator, three methods starting from Ekeland’s variational principle and, lastly, one with a generalized variational principle to solve or describe the above-mentioned solutions. The relevance of these operators and the possibility of their involvement in the modeling of an important class of real phenomena determined the author to group these seven procedures together, presented in detail, followed by many applications, accompanied by a general overview of specialty domains. The use of certain variational methods facilitates the complete solution of the problem via appropriate numerical methods and computational algorithms. The exposure of the sequence of theoretical results, together with their demonstration in as much detail as possible has been fulfilled as an opportunity for the complete development of these topics.

1. Introduction

Problems for partial differential equations involving the p-Laplacian and p-pseudo-Laplacian are mathematical models that often occur in studies on the p-Laplace or p-pseudo-Laplace equation, generalized reaction–diffusion theory, non-Newtonian fluid theory, non-Newtonian filtration, the turbulent flow of a gas in a porous medium, glaciology, non-Newtonian rheology, etc. These fractional order operators are very important mathematical models describing a multitude of anomalous dynamic behaviors in applied sciences. In the non-Newtonian fluid theory, the quantity p is a medium characteristic. Media with p > 2 are called dilatant fluids and those with p < 2 are pseudoplastics. If p = 2, they are Newtonian fluids. The p-Laplacian appears in the study of flow through porous media in turbulent regime at Diaz et al. [1,2] or glacier ice when treated as a non-Newtonian fluid with a nonlinear relationship between the rate deformation tensor and the deviatoric stress tensor, as described by Glowinski et al. [3]. It is also used in the Helle-Shaw approximation for a moving boundary problem by King et al. [4] and also for “power-law fluids” at Aronson et al. [5]. The p-Laplacian also appears in the study of flow in porous media (p = 3/2, at Schowalter et al. [6]) or glacial sliding (p ∈ (1, 4/3], at Péllissier [7]). Quasilinear problems with a variable coefficient appear in the mathematical model of the torsional creep (elastic for p = 2, plastic for p → ∞; see in Bhattacharya et al. [8] and at Kawohl [9]). A nonlinear field equation in Quantum Mechanics involving the p-Laplacian for p = 6 was proposed by Benci et al. [10].
Surjectivity methods to solve and/or characterize the solutions for Dirichlet problems involving the p-Laplacian and the p-pseudo-Laplacian have been previously used by the author in [11], together with other variational results, in [12] where two solving methods are displayed, in [13] which discusses some Fredholm alternative types, in [14] with solutions for the p-pseudo-Laplacian treated following two approaches, and in [15] involving surjectivity methods. Solving such problems via results obtained with Ekeland variational principle and other generalized variational principles has been the goal of other works of the author where some Dirichlet or Newmann problems have been studied as in [12,14,16] with the use of a perturbed variational principle, in [17] with variational procedures, as also in [18]. Mountain pass theorem variants and applications involved in modeling real phenomena are performed by the author in [19], while other applied variational methods are capitalized on by the author in [20,21], and several variational principles, together with generalizations and variants, have been compared and analyzed in the monograph [22].
The fractional differential equations are frequently used as modeling tools for processes implied in anomalous diffusion or spatial heterogeneity [23]. Also, in water resources, fractional models have been used to design chemical or contaminant transport in heterogeneous aquifers. In the field of magnetic resonance, fractional models of the Bloch–Torrey equation for drawing anomalous diffusion have been considered. Concerning the domain of cell biology, anomalous diffusion has been measured in fluorescence photo-bleaching recovery and fractional-in-time models have been created for simple types of chemical reaction–diffusion equations and for the simulation of microscale diffusion in the cell wall lining of plants. Similar problems appear in models of chemical reactions, heat transfer, population dynamics and so on [24]. Power law diffusion equations with p-Laplacian having constant and/or variable p are significantly related to researches on non-Newtonian fluids, turbulence modeling phase transitions, data clustering, machine learning and image processing. Many studies devoted to power-law diffusions call for the development of efficient numerical methods for solving elliptic partial differential equations with nonlinearities of p-type gradient [25].
The interest in these kinds of operators is topical both for mathematical approaches and results, as at Mukherjee et al. [26], Zhang et al. [27], Benedikt et al. [28], Lafleche et al. [29], Cellina [30], Khan [31], Xu [32], Akagi et al. [33], Gulsen et al. [34] and Lee et al. [35], and also for applications in many models in various fields, as in Rasouli [36], Yang et al. [37], Elmoataz et al. [38], Gupta et al. [39], Liero et al. [40] and Silva [41].
In this paper, several sequences of results are proposed, starting from the most general and abstract theory, passing step by step through many concrete stages until their applications in models issued from design of real phenomena. The focus falls on a very extended justification, containing all the necessary details and displaying all the theoretical arguments. Some of the results presented in this paper are introduced with an integer justification background and they are used to obtain and/or characterize the solutions of equations of mathematical physics proposed by other authors for similar problems for which they gave solutions via different methods. For such problems resulting from mathematical modeling, we came up with original solving methods following the involved abstract frame. From these original approaches stems the novelty of this work. Our interest is now focused on such kinds of problems and their solutions, and this is one of the initial studies for future developments to find a mathematical model (there is none available) that can be applied for describing diffusion phenomena involved in the micro-emulsification of disperse systems in tight connection with surface properties at the interface in self-organized systems.
This article is the first part of review type of the work under this title containing whole theoretical support detailed in a complete exposition of the arguments, while many applications of the results presented here represent the aim of another original paper under the same title, Part two.

2. Surjectivity for the Operators λJφS: Applications to Partial Differential Equations

2.1. Surjectivity of the Operators of the Form λT − S

In this section, a generalization of a theorem from [42] (Theorem 1.1) is presented and in this result the author used: normed space instead of Banach space, bijection with continuous inverse instead of homeomorphism. Two corollaries of this statement, obtained in [12,14], have also been presented.
Firstly, to have a short expression, we introduce the following.
Definition 2.1.
T: X → Y, where X and Y are normed spaces, is (K, L, a), where K > 0, L > 0, a > 0, if
Kxa ≤ ∥Tx∥ ≤ Lxax from X.
Proposition 2.1.
Let X and Y be real normed spaces, T: X → Y (K, L, a) odd bijection with continuous inverse and S: X → Y odd compact operator. For any λ ≠ 0, if
lim | | x | | + | | λ T x S x | | = + ,
then λT − S is surjective.
Proof. 
Let z0 be from Y; we state that:
x0 in X e.g., λTx0Sx0 = z0.
Take R > 0 with the property (see the hypothesis)
x∥ ≥ R ⇒ ∥λTx − Sx∥ > ∥z0
and the open ball from Y σ : = B(0, r), r : = |λ| L R a . If y ∈ ∂σ and y = λTx, then ∥x∥ ≥ R, and hence,
λ Tx Sx > ( 2.2 ) | | z 0 | |
Let be the operator A: YY,
A y = S T 1 y λ
A is compact, odd and Ayy when y ∈ ∂σ (ad absurdum, put y in the form λTx and take into account (2.3), i.e., 0 ∉ (IA)(∂σ)). Applying Borsuk theorem, the Leray-Schauder degree d(I − A, σ, 0) is odd. However,
H :   [ 0 ,   1 ]   ×   σ ¯     Y ,   H ( t ,   y ) = A y + t z 0
being a homotopy of compact transformations on  σ ¯ , we have
d(I − H(0, ·), σ, 0) = d(I − H(1, ·), σ, 0), i.e.,
d(I − A, σ, 0) = d(I − A − z0, σ, 0),
consequently, d(I − A, σ, 0) is an odd number, particularly different from zero, therefore ∃y0 in σ, e.g., (I − A − z0)(y0) = 0 and it remains only to take x0 in X with y0 = λTx0 to obtain (2.1). □
Corollary 2.1.
Let X, Y be real normed spaces, T: XY odd (K, L, a) bijection with continuous inverse, S: XY odd compact operator and α : =  lim ¯ | | x | | + S x x a < +∞. If
| λ |   >   α K ,   λ R ,
then λTS is surjective.
Explanations [43], Volume 5, V, §5, 11.91, p. 372: f : XY, X and Y normed spaces,
lim ¯ | | x | | +   f   ( x ) = d e f inf ρ > 0 sup x X | | x | | ρ   f   ( x )   =   lim ρ + sup x X | | x | | ρ f   ( x ) .
If α =  lim ¯ | | x | | +   f   ( x ) , then xnXn from N, and ∥xn∥ → +∞ implies  lim ¯ n   f   ( x n ) α.
If α =  lim ¯ n   f   ( x n ) , for any (xn) with xn from X and ∥xn∥ → +∞, then α =  lim ¯ | | x | | + f (x)∥.
Proof. 
It remains to prove:
lim ¯ | | x | | + | | λ T x S x | | = + ,
Assuming, ad absurdum, the contrary, obtain ρ > 0 and a sequence (xn)n≥1, xnX, ∥xn∥ → +∞, e.g.,
∥λTxnSxn∥ ≤ ρ ∀n ≥ 1.
From (2.5),
lim n λ T x n | | x n | | a S x n | | x n | | a = 0 ,
hence,  lim n   λ T x n x n a S ( x n ) x n a = 0, and as  lim ¯ n S ( x n ) x n a ≤ α, it results
lim ¯ n λ T x n x n a α .
But the condition (K, L, a) imposes:
K lim n ¯ T x n x n a .
From (2.6) and (2.7), we obtain K ≤  α | λ | . If α ≠ 0, then |λ| ≤  α K , which contradicts the hypothesis, and if α = 0, then K = 0, also in contradiction with the hypothesis, and consequently, (2.4). □
Corollary 2.2.
Under the conditions of Corollary 2.1, if α = 0, then λT − S is surjective for λ any in R\{0}.

2.2. Surjectivity for Operators of the Form λJφ − S, Jφ Duality Map

2.2.1. Preliminaries—Duality Map

Let X be a real Banach space, X* its dual (), x* an element any in X*, 2M the set of subsets of M.
Definition 2.2.
φ: R+R+ continuous and strictly increasing with φ(0) = 0 and  lim r +
φ (see, for instance [43], Volume 2, p. 27r) = +∞ is, by definition, weight or normalization function.
The multiple-valued map Jφ : X → 2X*, φ weight,
J φ 0 X = { 0 X } , x 0 J φ x = φ ( | | x | | ) { x X :   | | x | | = 1 , x ( x ) =   | | x | | } ,
equivalently,
xXJφ x = {x*X*: ∥x*∥ = φ(∥x∥), x*(x) = φ(∥x∥)∥x∥},
is, by definition, the duality map (on X) relative to φ. (Beurling-Livingstone).
Its name becomes normalized duality map when φ(t) = t (in this case, x* ∈ Jφ xx*(x) = ∥x2 = ∥x*∥2).
Definition 2.3.
Let X be a real normed space. A nonempty set β of bounded subsets of X, having the properties:
1 o A β A = X ,   2 o   A β A β , λ A β λ > 0   a n d
β is filtered to the right related to the inclusion “”, i.e.,
for any A, B in β ∃ C in β, e.g., AC and BC,
is called bornology on X.
Let β be a bornology on X. The function f : X →  R ¯ , which is locally finite in the point a (i.e., there exists a neighborhood of a on which f is finite), is, by definition, β-differentiable in a if there exists φ in the topological dual X* such that for every S in β, we have:
limu t 0 h S f ( a + t h ) f ( a ) t = φ ( h )   ( uniform   limit   on   S   for   t 0 ) .
φ is the β-derivative of f in a, and it is denoted by
β f (a).
If β is the set βG of the finite symmetrical parts of X or the set βF of the bounded symmetrical parts of X, the β-derivative coincides with the Gâteaux derivative and Fréchet derivative, respectively.
Proposition 2.2.
Jφ x ≠ x in X;
For any x in X, Jφ x is a convex, closed, bounded part of X* (it is contained in the sphere of the equationyX* = φ(∥x∥));
Jφ x = ∂ψ (x), the subdifferential in x, where
ψ u = 0 u φ t d t
(apply Asplund theorem; since ψ is continuous and convex, ψ is Gâteaux differentiable in x iff ∂ψ(x) has a single element, and then ψ′(x) = ∂ψ(x); consequently, Jφ is uni-valued iff ψ is Gâteaux differentiable, and in this case, Jφ x = ψ′(x));
4° Jφ is odd (Jφ (−x) =Jφ xx in X);
5°  J φ λ x = φ λ x φ x J x ,   x   i n   X \ 0 ,   λ 0 ;
6° Jφ is monotonous; more precisely: ∀x, y in X andx*, y* from Jφ x, Jφ y,
(x* − y*)(xy) ≥ [φ(∥x∥) − φ(∥y∥)](∥x∥ − ∥y∥) ≥ 0;
The normalized duality map is linear iff X is a Hilbert space;
If x* ∈ Jφ x, then x ∈  J φ 1 (x*) (duality map on X*);
9°  J φ J φ 1  being duality maps on X, there exists μ: [0, +∞] → [0, +∞) such that
J φ   x = μ ( x ) J φ 1   x   in   X ;
10° Let x, y be in X. Then, ∥xx + λy∥ ∀λ ≥ 0 iff there exists x* in Jx, J the normalized duality map on X, with the property x*(y) 0 (Kato).
Furthermore, retain that if Jφ is uni-valued, then it is coercive since:
lim x + J φ x , x x = l i m x + φ x = + .
The Banach space X is, by definition, smooth (Krein) if
x 0   there   exists   x 0 *   unique   in   X ,   e . g . ,   x 0 = 1 ,   x 0 ( x )   =   x .
Thus, any duality map on a smooth space is uni-valued and reciprocal.
Since  x 0 * from X* is sub-gradient in x0 ≠ 0 for x → ∥x∥, iff ∥ x 0 * ∥ = 1 and  x 0 * (x) = ∥x∥ and x → ∥x∥ is convex and continuous.
Proposition 2.3.
X is smooth space iff the norm of X is Gâteaux differentiable on X\{0}.
From this, combined with Proposition 2.2, 3°, we obtain:
X smooth ⇒ Jφ x = φ(∥x∥)∥x∥′, x ≠ 0. (∗)
To validate this formula, we prove the following:
Proposition 2.4.
Let X and Y be real normed spaces and f : X → Y be Gâteaux differentiable, F: YR of Gâteaux C1 class. Then, g : = F o f is Gâteaux differentiable, and g′(x) = F′(f (x)) o f′(x) [43].
Proof. 
We use the formula of finite increases [43], Volume 9, p. 93:
Let β be a bornology on X and f : XR β-differentiable on the segment [a, b] from X. There exists θ in (0,1) such that:
f (b) − f (a) = ∇β f (a + θ(b − a))(b − a).
Standard justification. Take F: [0,1] → R, F(t) = f (a + t(b − a)). With δ being small,
F t + δ F ( t ) δ = 1 δ f a + t b a + δ b a f a + t b a ,
let A be in β with b − aA (property 1° from the definition of the bornology), and take the limit for δ → 0, F′(t) = ∇β f (a + t(b − a))(b − a), F(1) − F(0) = F′(θ), with θ in (0,1), etc. □
We continue the proof of this proposition. Let x0 and h be in X.
1 t g x 0 + t h g x 0 = 1 t F f x 0 + t h F f x 0 = F ( f x 0 + θ t f x 0 + t h f x 0 ) f x 0 + t h f x 0 t , lim t 0 f x 0 + t h f ( x 0 ) t = f x 0 h , l i m t 0 f x 0 + t h f x 0 = 0 , l i m t 0 θ t f x 0 + t h f x 0 = 0
and hence the conclusion, since:
x n a   a n d   y n b F y n x n F b a : F y n x n F b a = F y n x n F y n a + F y n a F b a , F y n x n F y n a = F y n x n a F y n x n a x n a
(the sequence ( F y n ) is bounded, being convergent),
F y n a F b a   =   F y n F b ,   a     a   F y n F b .
Finally, we introduce the following:
Definition 2.4.
F: X → 2X* is upper semicontinuous in x0 if, for any neighborhood V of F(x0in the ∗-weak topology on X*, there exists U neighborhood of x0, e.g., F(U) ⊂ V.
Proposition 2.5.
Any duality map Jφ on X is upper semicontinuous on X (Browder).
Definition 2.5.
A Banach space X is called strictly convex (Clarkson) if one of the following equivalent properties is fulfilled:
xy, ∥x∥ = ∥y ∥ = 1 ⇒ ∥λx + (1 − λ)y ∥ < 1 ∀λ in (0, 1);
x ≠ y,x∥ = ∥y∥ = 1 ⇒ ∥x + y∥ < 2;
3° ∥x + y∥ = ∥x∥ +∥y∥, y0 ⇒ ∃ λ ≥ 0 with x = λy;
4° ∥x∥ = ∥y∥ = 1 andx + y∥ = ∥x∥ + ∥y ∥ ⇒ x = y;
The sphere {xX: ∥x∥ = 1} does not contain any segment;
Any x* from X* attains its inferior upper bound on the unity ball of X in at least one point;
The function x → ∥x2 is strictly convex.
Proposition 2.6.
If X* is smooth (respectively strictly convex), then X is strictly convex (respectively smooth). The reciprocal assertions are true when X is reflexive.
Proposition 2.7.
If the Banach space X is reflexive, there exists a norm on X equivalent strictly convex such that the dual norm is also strictly convex. (Lindenstrauss, Asplund).
Definition 2.6.
A Banach space is uniformly convex (Clarkson) if
∀ε > 0 ∃δ > 0 such that ∥x∥ = ∥y∥ = 1 and ∥xy∥ ≥ ε ⇒ ∥x + y∥ ≤ 2(1 − δ),
equivalently
x n   =   y n   = 1   n in   N   and   x n + y n     2     l i m n ( x n y n ) = 0 .
A Banach space is local uniformly convex (Lovaglia) if
∀ε > 0 and ∀x with ∥x∥ = 1 ∃δ > 0 e.g., ∥y∥ = 1 and ∥xy∥ ≥ ε ⇒ ∥x + y∥ ≤ 2(1 − δ),
equivalently,
x = x n = 1   n in   N   and   x n + x     2     l i m n x n = x .
Proposition 2.8.
X uniformly convex locally uniformly convex X is strictly convex.
Proposition 2.9.
X uniformly convex x is reflexive. (Milman).
Proposition 2.10.
If X is local uniformly convex, then, for any sequence (xn)n≥1, xnX,
x n   w   x   a n d   x n     x     x n     x .
Here is a characterization of the uniform convexity.
Proposition 2.11.
X and X* are uniformly convex iff the norm on X* and X, respectively, is uniformly Fréchet differentiable.
Clarification. f is uniformly Fréchet differentiable  d e f ∀ε > 0 ∃δ > 0 with the property
h∥ ≤ δ ⇒ ∥ f (x + h) − f (x) − f ′(x)(h)∥ ≤ ε∥h∥ ∀x with ∥x∥ < 1;
x →x∥ is uniformly Fréchet differentiable iff it is Fréchet differentiable and the Fréchet derivative is uniformly continuous on the unity ball.
Proposition 2.12.
If X is local uniformly convex and reflexive, then the norm on X* is Fréchet differentiable.
Proposition 2.13.
For any reflexive Banach space, there exists on this an equivalent norm for which it becomes local uniformly convex. (Troianski).
Proposition 2.14.
For any reflexive Banach space X, there exists an equivalent norm on X for which X and X* are locally uniformly convex. (Asplund).
Thus, the equivalent norm and its dual norm are Fréchet differentiable (Proposition 2.12).
Proposition 2.15.
If X is smooth, for any weight φ, Jφ is uni-valued and continuous from X, with the strong topology on X* endowed with the -weak topology.
Proof. 
Take Proposition 2.5 into account. □
Corollary 2.3.
If X is smooth and reflexive, any duality map J on X is semicontinuous.
Proof. 
Let xnx. Setting  x n : = xn, x* : = Jx, we have  x n (u) → x*(u) ∀u from X, i.e., denoting u** : = Φu, Φ is the canonical embedding in bidual, u**( x n ) → u**(x*), hence Jxn  w  Jx. □
The strict convexity can be characterized by the duality map.
Proposition 2.16.
X is strictly convex iff any duality map on X is strictly monotonous. (Petryshin).
Consequently,
Proposition 2.17.
If X is strictly convex and smooth, then any duality map on X is uni-valued and injective.
Proposition 2.18.
Let X be smooth and local uniformly convex, J duality map on X and (xn)n≥1 , xnX. If (Jxn − Jx)(xn − x)0, then xnx.
Proposition 2.19.
Let X be Banach space. A duality map on X is uni-valued and continuous in the topologies of the norms iff the norm on X is Fréchet differentiable.
Proof. 
Necessary. X is smooth, and the norm is Gâteaux differentiable (Proposition 2.3), even of the C1 Gâteaux class in accordance with (∗). Sufficient. This results from a Cudia theorem. □
Proposition 2.20.
Let X be a Banach space and Jφ a duality map on X. X* is uniformly convex iff Jφ is uni-valued and uniformly continuous on the bounded sets of X.
Proof. 
Necessary. X* is uniformly convex  P r o p o s i t i o n   2.11  x → ∥x∥ is Fréchet differentiable ⇒ Jx = φ(∥x∥)∥x∥′, x → ∥x∥′ is uniformly continuous on the closed unity ball. Sufficient. J uni-valued  P r o p o s i t i o n   2.3  x → ∥x∥ is Gâteaux differentiable  P r o p o s i t i o n   2.11  x → ∥x∥ is uniformly Fréchet differentiable  P r o p o s i t i o n   2.11  X* is uniformly convex. □
Place here the characterization of the reflexivity of a Banach space.
Proposition 2.21.
The Banach space X is reflexive iff any x* in X* attains  sup x 1 x ( x ) .
In the following, there is a characterization of the reflexivity of the duality map.
Proposition 2.22.
Let X be a Banach space and Jφ duality map on X. X is reflexive iff
X = x X J φ x .
Proof. 
Necessary. Let  x 0 be in  X . There exists x0 in X with ∥x0∥ = 1 and  x 0 (x0) = ∥ x 0 ∥; take t0 > 0, e.g., φ(t0) = ∥ x 0 ∥, then  x 0 Jφ (t0 x0). Sufficient. Use Proposition 2.21. Let  x 0 be any in X*, ∃ x0 in X, e.g.,  x 0 Jφ x0, i.e., ∥ x 0 ∥ = φ(∥x0∥),  x 0 (x0) = φ(∥x0∥)∥x0∥. For y0: =  x 0 x 0 , we have ∥y0∥ = 1,  x 0 (y0) = ∥ x 0 ∥ =  sup x 1 x 0 ( x ) . □
Corollary 2.4.
Let X be a smooth space and J a duality map on X.
X is reflexive ⟺ J is surjective;
X is reflexive and strictly convex ⟺ J is bijective.
Proof. 
Take Proposition 2.22 and Proposition 2.16 into account. □
Proposition 2.23.
Let X be a reflexive and smooth space and Jφ a duality map on X. Then J−1 : X* → 2X, J−1 x* = {xX: Jφ x = x*} coincides with  J φ 1 , the duality map on X* coincides with the weight φ−1 (identification via canonical embedding Φ in the bidual). If X is also strictly convex, then J−1 is uni-valued, and the formula holds true:
J φ 1 = Φ 1   J φ 1
Proof. 
The statement is, in accordance with Corollary 2.4, correct. First assertion. Let x* be any in X*. xJ−1 x* ⟺ x* = Jφ x ⟺ ∥x*∥ = φ(∥x∥) and x*(x) = φ(∥x∥)∥x∥ ⟺ ∥x∥ = φ−1(∥x*∥) and x(x*) = φ−1 (∥x*∥)∥x*∥ ⟺ x ∈  J φ 1 x*. Second assertion. X* is smooth (Proposition 2.6). □
Pass to continuity properties of the duality map. A previous result is represented by Corollary 2.3.
Definition 2.7.
A Banach space has the property (h) if
x n   w   x   and   x n     x     x n     x .
A Banach space has the property (H) if it is strictly convex, and it has the property (h).
For instance, any local uniformly convex space has the property (H) (Propositions 2.8 and 2.9).
Proposition 2.24.
If X is reflexive and X* has the property (H), then any duality map J on X is uni-valued, surjective and continuous relative to the strong topologies on X and X*.
Proof. 
J is indeed uni-valued (Proposition 2.6) and surjective (Corollary 2.4). Let xnx. Then, Jxn  w Jx (Corollary 2.3); moreover, ∥Jxn∥ → ∥Jx∥ since ∥Jxn∥ = φ(∥xn∥) and ∥Jx∥ = φ(∥x∥). Therefore, JxnJx. □
Combining Proposition 2.24 with Corollary 2.4, one obtains the following.
Proposition 2.25.
If X is reflexive and strictly convex and X* has the property (H), then any duality map on X is uni-valued, bijective and continuous relative to the strong topologies on X and X*.
We next proceed to the continuity of the inverse of the duality map.
Proposition 2.26.
If X is reflexive, smooth and has the (H) property, then any duality map Jφ on X is bijective with  J φ 1  continuous relative to the strong topologies on X and  X .
Proof. 
Jφ is bijective (Corollary 2.4), and X* is reflexive, smooth and strictly convex (Proposition 2.6). Let  x n  →  x 0 . Then,  J φ 1 x n   w   J φ 1 x* (Corollary 2.3), and the formula from Proposition 2.23 imposes  J φ 1 x n  →  J φ 1 x* (see (H)). □
Combining Propositions 2.25 and 2.26, one obtains the following.
Proposition 2.27.
If X is reflexive and X and X* have the property (H), then any duality map on X is a homeomorphism of X in X* relative to the strong topologies.
We finish this subsection with the following result:
Proposition 2.28.
Any uni-valued duality map Jφ on a local uniformly convex space X has the property S+:
x n   w x   a n d   lim ¯ n J φ x n J φ x 0 , x n x 0 0 x n     x .
Proof. 
The hypothesis implies
lim ¯ n J φ x n J φ x 0 , x n x 0 0 ,
but
0 [ φ ( x n ) φ ( x 0 ) ] ( x n x 0     J φ x n J φ x 0 , x n x 0 ,
consequently,
lim n [ φ ( x n ) φ ( x 0 ) ] ( x n x 0 ) = 0 .
We denote tn: =  x n , t0 : =  x 0 . Somehow,
l i m n t n = t 0 ,
we apply Proposition 2.11 and obtain xnx. Assume, ad absurdum, that tnt0. Then there exists ( t k n ) subsequence of (tn) e.g., for instance,  t k n ≥ ρ > t0n. ( t k n ) is bounded, otherwise it has a subsequence ( t i k n ),  t i k n → +∞, which implies φ( t i k n ) → +∞, [φ( t i k n ) − φ(t0)]( t i k n t0) → +∞, in contradiction with (2.8). Thus, ( t k n ) has a convergent subsequence ( t l k n ),  t l k n →  t 0 t0. Consequently, since φ( t l k n ) → φ( t 0 ) ≠ φ(t0), from a rank on, we have, with δ > 0,
| t l k n t 0 |     δ ,   | φ ( t l k n ) φ ( t 0 ) |     δ
and one obtains a final contradiction with (2.8). Here is another justification for (2.9). (tn) is bounded (see above); let t* be an adherence value any for this sequence (a fortiori t* ∈ R) and ( t j n ) subsequence with  l i m n t j n = t*. Then, from (2.8), [φ(t*) − φ(t0)](t* − t0), which implies t* = t0 (ad absurdum !) and, consequently, (2.9). □

2.2.2. Main Results

Proposition 2.29.
Let X be a real reflexive Banach space, smooth and having the property (H), Jφ duality map on X with φ being (K, L, a) function, S: XX* odd compact operator and
α   :   =   lim ¯ | | x | | + S x x a < + .
Then
1° α > 0 ⇒ λJφ − S is surjective ∀λ with |λ| > α K ;
2° α = 0 ⇒ λJφ − S is surjective ∀λ ≠ 0.
Proof. 
Jφ is odd and bijective with a continuous inverse (Proposition 2.26). Moreover, since
Kta ≤ φ(t) ≤ Ltat ≥ 0,
we have
Kxa ≤ φ(∥x∥) = ∥Jφ x∥ ≤ Lxax from X,
i.e., Jφ is (K, L, a). We apply Corollaries 2.1 and 2.2 from Section 2.1. □
Proposition 2.30.
Let X be a real reflexive Banach space, smooth with the property (H), and Jφ  the duality map on X with φ(t) = tp−1, p ∈ (1, +∞). Suppose that X is compactly embedded by the linear injection i in a Banach space Z,
i(u)∥ ≤ c0u∥ ∀u from X
and N: ZZ* is an odd semicontinuous operator with the property
Nx∥ ≤ c1xq–1 + c2x from Z, c1, c2 ≥ 0, q ∈ (1, p).
Then λJφ − N is surjective for any λ ≠ 0.
Explanation. N is the short notation for the operator, which acts from X to X*, i′ o N o i, i′ being the adjoint of i.
Proof. 
It follows to state that
h from X* ∃u in X, e.g., λJφ u − (i ‘ o N o i) u = h.
Apply Proposition 2.1 with T = Jφ (correctly, as φ is (K, L, a) with K = L = 1, a = p − 1), S = i′ o N o i. S is obviously odd and also compact: let (xn)nN, xnX, be a bounded sequence (i(xn))n≥1 has a convergent subsequence  ( x k n ) n 1 ; let i( x k n ) → γ, γ ∈ Z, then N(i( x k n ))  w N(γ) and, consequently, i′(N(i( x k n ))) → i′(N(γ)) since i′ is also compact (Schauder theorem). So, to obtain the conclusion, it remains to prove
lim ¯ | | u | | + ( i N i ) u u p 1 = 0 .
i ‘(N(i(u)))∥ ≤ ∥i ‘∥ ∥N(i(u))∥ ( 2.10 ) , ( 2.11 ) c 0 (c1i(u)∥q–1 + c2) ≤  c 0 ( c 0 q 1 c1uq−1 + c2), from which it results (2.13) (∥i′∥ ≤ ∥i∥ ≤  c 0 has been used) [11,12,15]. □
In the following, we search for the surjectivity of the operator λJφN, when N verifies the growth condition (2.11), where q = p, i.e.,
Nx∥ ≤ c1xp–1 + c2x from Z, c1, c2 ≥ 0.
For this reason, we present the statement:
Proposition 2.31.
Let X be a real reflexive Banach space compactly embedded by the linear injection i in the Banach space Z,
i ( u )     c 0 u   u   f r o m   X .
If
λ 1 : = inf | | u | | p | | i ( u ) | | p : u X \ { 0 } ,   p     ( 1 , + ) ,
then
1° λ1 is attained and nonzero;
2°  λ 1   1 p is optimal for (2.15) (i.e.,  λ 1   1 p   c 0 for any  c 0 );
If X and Z are smooth and JXX* : XX*, JZZ* : ZZ* are duality maps relative to the same weight φ: φ(t) = tp–1, then λ1 is the smallest eigenvalue of the couple (JXX*, JZZ*) [44].
Clarification. λ is, by definition, eigenvalue for the couple (JXX*, JZZ*) if there exists u0 ≠ 0 in Z, e.g., λ (i′ o JXX* o i) u0 = JZZ* u0. In this case, u0 is, by definition, an eigenvector.
Proof. 
The set from the statement is correctly defined: u ≠ 0 ⇒ i(u) ≠ 0.
1° We have
λ1 = inf {∥vp : vX, ∥i(v)∥ = 1}
(the two sets coincide, as    i   u i ( u ) = 1). Let (vn)n≥1, vnX, be with ∥i(vn)∥ = 1 and ∥vn∥ →  λ 1 1 p .
X being reflexive, (vn)n≥1 has a subsequence (we use the same notation for it) that is weakly convergent in X, vn w v (Kakutani theorem [43], Volume 3, p. 155). Then,
v   lim n v n ,
vpλ1.
On the other hand, since i is compact, we have i(vn) → i(v), which implies ∥i(vn)∥ → ∥i(v)∥, ∥i(v)∥ = 1 and hence ∥vp ≥ λ1, ∥vp  = ( 2.16 ) λ1, and λ1 is attained and, a fortiori, nonzero.
2° We take the definitions of λ1 and 10 into account.
3° Firstly, we show that λ1 is an eigenvalue for the couple (JZZ*, JXX*), i.e., ∃u0 ≠ 0 in X e.g.,
λ1 (i′ o JZZ* o i)u0 = JXX* u0.
Taking the functional Φ: XR,
Φ ( u ) = 1 p u p λ 1 p i ( u ) p .
Φ(u) ≥ 0 ∀u in X (see the definition of λ1) and, for u0 ≠ 0 e.g., λ1  = 1 0   u 0 i ( u 0 ) p , Φ(u0) = 0, which imposes (taking into account Proposition 2.3)
Φ′(u0) = 0 (Gâteaux derivative).
Then (the formulae: (∗) and that from Proposition 2.4), ∀u in X,
0 = ( 2.18 )   Φ ( u 0 ) ( u ) = u 0 p 1   ( u 0 ) ,   u λ 1 i ( u 0 ) p 1   ( i ( u 0 ) ) ,   i ( u ) = ( J XX   u 0 ) ( u ) λ 1 ( J ZZ ( i ( u 0 ) ) ( i ( u ) ) ) = J X X u 0 λ 1 ( i   J Z Z i ) ( u 0 ) , u ,   i . e . ,   ( 2.17 ) .
Let λ now be an eigenvalue for the couple (JZZ*, JXX*) and u be a corresponding eigenvector. Then
u p = ( J XX   u ) ( u ) = λ J Z Z ( i ( u ) ) , i ( u ) = λ i ( u ) p ,
hence
λ = u p i ( u ) p   λ 1 .  
We can now state the following.
Proposition 2.32.
Let X be a real reflexive Banach space, smooth and have the property (H), Jφ duality map on X with φ(t) = t p−1, p ∈ (1, +∞). Suppose that X is compactly embedded with the linear injection i in the Banach space Z and let N: ZZ* be an odd semicontinuous operator with:
Nx∥ ≤ c1xp–1 + c2x from Z, c1, c2 ≥ 0.
Then, for any λ, if
| λ |   > lim ¯ u + | | ( i N i ) u | | | | u | | p 1 ,   a   fortiori   if   | λ |   > c 1 λ 1 1 ,
where
λ 1 : = inf u p i ( u ) p : u X \ { 0 } ,
then λJφ − N is surjective (N means i′ o N o i).
Proof. 
The statement is correct, λ1 ≠ 0 (Proposition 2.31, 10). We apply, as for Proposition 2.30, Proposition 2.29 with T = Jφ, S = i′ o N o i. We prove
lim ¯ u + | | ( i N i ) u | | | | u | | p 1 c 1 λ 1 1
using 2° from Proposition 2.31, which will be sufficient to impose the conclusion. ∥(i ‘ o N o i)u∥ ≤ ∥i ‘∥ ∥N(i(u))∥ ≤  λ 1 1 p (c1 λ 1 1 p p   | | u | | p 1 + c2) ( | | i | |   | | i | |   λ 1 1 p ), and (2.19) becomes obvious. □
Remark 2.1.
Propositions 29, 30 and 32 have been briefly presented by the author in [11,12,15].

2.3. Existence of the Solutions of the Problems

Consider the problems
( )   λ div ( | u | p 2 u ) = f (     , u (     ) ) + h ,   x Ω u | Ω = 0 ,   p     ( 1 , + )
and
( )   λ i = 1 n x i u x i p 2 u x i = f (     , u (     ) ) + h ,   x Ω u | Ω = 0 ,   p     [ 2 , + )
In this subsection, we apply (the idea originates in [45]) the results from Section 2.2 to partial differential equations (weak solutions).

2.3.1. Preliminaries for Sobolev Spaces

To prepare the framework for this subsection and to be coherent and understandable, we start with a theoretical recapitulation for Sobolev spaces.
The spaces  L p (Ω) (Lebesgue integral in RN).
Let Ω be a nonempty open set in RN.
p     [ 1 , + )     L p ( Ω )   :   =   { u :   Ω     R :   u measurable ,   | u | p   Lebesgue   integrable } ,
u L p ( Ω ) :   =   Ω | u | p d x 1 p .
This norm is Gâteaux differentiable on  L p (Ω)\{0} for p > 1. It is even of Fréchet C1 class ([43,46]).
p = +     L ( Ω ) :   =   { u :   Ω     R :   u measurable ,     c > 0   e.g. , | u ( x ) |     c ,   on   Ω   a.e. } u L ( Ω ) :   =   inf   c .
If μ(Ω) < +∞, then  u L p ( Ω ) ≤  [ μ ( Ω ) ] 1 p 1 q u L q ( Ω ) , 1 ≤ pq ≤ +∞.
p   [ 1 , + ]   u 0 , p :   =   u L p ( Ω ) .
L p (Ω), modulo the known factorization, is a Banach space for p ∈ [1, +∞],  L p (Ω) is uniformly convex and hence reflexive (Proposition 2.9) for p ∈ (1, +∞), L1(Ω) and L (Ω) are not reflexive,  L p (Ω) is separable for p ∈ [1, +∞), and L (Ω) is not separable.
For p in [1, +∞], p’ is defined by:
1 p + 1 p = 1 .
Then,
p   ( 1 , + )     ( L p ( Ω ) ) * = L p ( Ω ) ,   ( L 1 ( Ω ) ) * = L   ( Ω ) ,   ( L   ( Ω ) ) *     L 1 ( Ω ) .
L l o c 1 (Ω) : = {u: Ω → R: u integrable on any compact part of Ω}.
At the end of this part of the exposure, we prove ([43], Ω is a Lebesgue measurable set):
Proposition 2.33.
When p ∈ (1, +∞), u →  u L p ( Ω ) is of Fréchet C1 class on  L p (Ω)\{0}.
Proof. 
For the following calculus, we use the inequalities [47]:
ξ ,   ζ     R N ,   p   ( 1 , + )       ξ p 2   ξ ζ p 2   ζ   c | | ξ ζ | | ( | ξ | | + | | ζ | | ) p 2 , p > 2 c | | ξ ζ | | p 1 , p ( 1 , 2 ] , c independent   by ξ ,   ζ
Let
Φ ( u ) :   =   u 0 , p ,   Ψ ( u ) :   =   1 p | | u | | 0 , p p ,
hence,
Φ ( u ) = p 1 p [ Ψ ( u ) ] 1 p .
Ψ is Gâteaux differentiable on  L p (Ω) and [48]
Ψ ( u ) ( h ) = Ω | u | p 1 ( sgn u ) h d x ,   h   in   L p ( Ω )
We prove that u → Ψ’(u) is continuous on  L p (Ω) and then (2.21) will impose the conclusion, taking into account Proposition 2.4.
Let
u n     u 0   in   L p ( Ω ) .
It follows to prove
lim n Ψ ( u n ) = Ψ ( u 0 )   in   ( L p ( Ω ) ) * ,   i.e. , lim n sup | | h | | 0 , p = 1 | Ψ ( u n ) Ψ ( u 0 ) ,   h | = 0 .
h0,p = 1 ⇒ |〈Ψ’(un) − Ψ’(u0), h〉| = | Ω ( | u n | p 1 sgn u n | u 0 | p 1 sgn u 0 ) h d x | ≤  Ω | ( | u n | p 1 sgn u n | u 0 | p 1 sgn u 0 ) h | d x   H o ¨ l d e r An, An: = ∥|un|p−1 sgn un − |u0|p−1 sgn u0 | | L p ( Ω ) , since ∥h | | L p ( Ω ) = 1, where  1 p + 1 p = 1 .
However, | |un|p−1 sgn un − |u0|p−1 sgn u0 |p′ = | |un|p−2 un − |u0|p−2 u0 |p′  ( 2.20 )
c 0 | u n u 0 | p ( | u n | + | u 0 | ) p ( p 2 ) , p > 2 c 0 | u n u 0 | p , p ( 1 , 2 ] ,
hence  A n p Ω | | u n | p 1 sgn u n | u 0 | p 1 sgn u 0 | p d x
c 0 Ω | u n u 0 | p ( | u n | + | u 0 | ) p ( p 2 ) d x , p > 2 c 0 Ω | u n u 0 | p d x = c 0 | | u n u 0 | | 0 , p p , p ( 1 , 2 ] ,
therefore, when p ∈ (1, 2],  lim n   A n p = 0 (see (2.22)), i.e.,  lim n An = 0 and hence (2.23), and when p > 2,
lim n A n p = 0 ,
i.e.,  lim n An = 0 and hence (2.23). (2.24) is proved as follows:
Ω | u n u 0 | p ( | u n | + | u 0 | ) p ( p 2 ) d x   H o ¨ l d e r | u n u 0 | p 0 , p p ( | u n | + | u 0 | ) p ( p 2 ) 0 , p p ( p 2 ) = ( | | u n u 0 | | 0 , p ) p ∥|un| + |u0| | | 0 , p p ( p 2 ) lim n unu0 0 , p p = ( 2.22 ) 0, and the second factor is bounded, since ∥ |un| + |u0| ∥0,p ≤ ∥un0,p + ∥u00,p and ∥un0,p ( 2.22 ) u00,p. □
Corollary 2.5.
When p ∈ (1, +∞), any duality map on  L p (Ω) is a homeomorphism of  L p (Ω) on  L p (Ω).
Proof. 
L p (Ω) is smooth (Proposition 2.3) and uniformly convex, hence local uniformly convex (Proposition 2.8) and, in particular, has the (H) property, even  L p (Ω) has the same properties (by applying Proposition 2.27). □
Let Ω be a nonempty open set from RN and p in [1, +∞].
W 1 , p (Ω) designates the real vector space of the functions u from  L p (Ω) for which there exists g1, …, gN in  L p (Ω) e.g.,
φ   in   C c ( Ω )   Ω u φ x i d x = Ω g i φ d x ,   i = 1 , N ¯ .
In the definition,  C c (Ω) can be replaced by  C c 1 (Ω). We denote, for each i, 1 ≤ iN,
u x i g i ,   t h e   w e a k   d e r i v a t i v e .
These are uniquely determined.
Remark 2.2.
By weak differentiation, one remains in  L p (Ω).
Additionally, for u from  W 1 , p (Ω),
u = g r a d   u u x 1 , , u x n , t h e   w e a k   g r a d i e n t | u | i = 1 N u x i 2 1 2 , d i v   u i = 1 N u x i , t h e   w e a k   d i v e r g e n c e . u L p Ω : u g : = i = 1 N u x i L p Ω , | u | p g p .
Moreover, for p ∈ (1, +∞) and p’ the conjugated coefficient,
| u | p 2 u x i L p Ω , i = 1 , N ¯ , x i u x i p 2 u x i L p Ω , i = 1 , N ¯ .
To justify the above two relations, for
p 2 , u p 2 u x i p = u p p 2 u x i p , u p p 2 L p p p 2 Ω , u x i p L p p Ω , p ( p 2 ) p + p p = 1 ,
and for p = 2—obviously since p′ = 2.
We define:
u W 1 , p ( Ω ) u L p Ω + i = 1 N u x i L p Ω ,
a norm on  W 1 , p (Ω).
Sometimes, when p ∈ [1, +∞), one takes the equivalent norm:
u L p Ω p + i = 1 N u x i L p Ω p 1 p .
W 1 , p (Ω) is a Banach space for p ∈ [1, +∞], it is uniformly convex and reflexive for p ∈ (1, +∞) and separable for p ∈ [1, +∞). When p > N, any function from  W 1 , p ( Ω) is Fréchet differentiable on Ω a.e. ([49], Chapter VIII).
We now provide some clarifications related to the weak derivative, p ∈ [1, +∞]. If uC1 (Ω)   L p (Ω) and  u x i ∈  L p (Ω), i =  1 , N ¯ (derivatives in the usual meaning), then u ∈  W 1 , p (Ω) and the weak derivatives coincide with them in the usual sense. In particular, if Ω is bounded, then C1 ( Ω ¯ ) ⊂  W 1 , p (Ω). Reciprocally, if u ∈  W 1 , p (Ω) ∩ C(Ω) and  u x i C(Ω), i =  1 , N ¯ (weak derivatives), then uC1 (Ω).
Let p be in [1, +∞).
W 0 1 , p Ω C c 1 Ω ¯ W 1 , p Ω = C c Ω ¯ W 1 , p Ω .
W 0 1 , p (Ω) with the norm induced is a separable Banach space. It is reflexive if p ∈ (1, +∞).
Since  C c 1 (RN) is dense in  W 1 , p (RN),  W 0 1 , p (RN) =  W 1 , p (RN). However, when Ω ≠ RN, in general,  W 0 1 , p (Ω) ≠  W 1 , p (Ω).
Proposition 2.34.
For any p in [1, +∞),
W 1 , p ( Ω )     C c   ( Ω )   W 0 1 , p ( Ω ) .
The following considerations strongly imply  W 0 1 , p (Ω) in the theory of partial differential equations.
Definition 2.8.
We define, by local charts,  W 1 , p (Γ) with p ∈ [1, +∞), Γ regular manifold, for instance Γ = ∂Ω, Ω open set of C1 class with ∂Ω bounded. In this situation there exists a unique continuous linear operator γ:  W 1 , p (Ω) →  W 1 1 p , p (∂Ω), the trace, such that γ is surjective and
u W 1 , p ( Ω )     C   ( Ω ¯ )     γ ( u ) = u   |   Ω .
By the way,
Proposition 2.35.
Let Ω be of C1 class and u in  W 1 , p (Ω) ∩ C ( Ω ¯ ), p ∈ [1, +∞). Then
u     W 0 1 , p Ω     u   |   Ω = 0 .
Here is another characterization of the spaces  W 0 1 , p (Ω).
Proposition 2.36.
Let Ω be of C1 class and u in  L p (Ω), p ∈ (1, +∞). Then
u W 0 1 , p Ω c > 0   s u c h   t h a t   Ω u φ x i d x c φ L p Ω , i = 1 , N ¯ , φ   i n   C c 1 Ω .
Suppose μ(Ω) < +∞. Then u →  | u | L p ( Ω ) is a norm on  W 0 1 , p (Ω) equivalent (and hence also complete) to u →  u W 1 , p ( Ω ) .
We denote:
u ∈  W 0 1 , p (Ω) ⇒ ∥u1,p: | u | L p ( Ω ) (=  | u | 0 , p , when confusion cannot appear). This norm, when p ∈ (1, +∞), is Gâteaux differentiable on  W 0 1 , p (Ω)\{0} (combine Proposition 2.15 and Proposition 2.4), which assures uni-valued duality maps (Proposition 2.3).
Proposition 2.37.
( W 0 1 , p (Ω), ∥ · ∥1,p), p ∈ (1, +∞), is uniformly convex and hence particularly reflexive.
Proof 
([50]). The case p ∈ [2, +∞). We use the following inequality: ([51], Euclidean norm)
ξ ,   ζ     R n ,   n   1     ξ + η 2 p + ξ η 2 p   1 2   ( ξ p   +   η p )
Let ε be from (0, 2] and u, v from  W 0 1 , p (Ω) with
u1,p = ∥v1,p = 1, ∥u − v1,p ≥ ε.
Then
u + v 2 1 , p p + u v 2 1 , p p = Ω u + v 2 p + u v 2 p d x ( 2.25 ) 1 2 Ω u p + v p d x = 1 2 u 1 , p p + v 1 , p p = 2.26 1 , u + v 2 1 , p p 2.26 1 ε 2 p .
We take δ > 0 defined by
1 ε 2 p 1 p = 1 δ .
The case p ∈ (1, 2). We use
ξ ,   ζ     R n ,   n   1   ξ + η 2 p + ξ η 2 p   1 2 ( ξ p + η p ) 1 p 1 ,
p′ the coefficient conjugated with p ([51]).
We remark that, for u in  W 0 1 , p (Ω),
| u | p     L p 1   ( Ω ) ,   u 1 , p p = | u | p 0 , p 1 .
Let ε be in (0, 2] and u, v in  W 0 1 , p (Ω). Then |∇u|p′, |∇v|p′  ( 2.27 ) L p 1 (Ω) and as
  | u | p 0 , p 1 +   | v | p 0 , p 1       | u | p   + | v | p 0 , p 1 .
Since 0 < p − 1 < 1, it results:
u + v 2 1 , p p + u v 2 1 , p p 2.28 , 2.29 , 2.27 1 2 ( u 1 , p p + v 1 , p p ) 1 p 1 ,
consequently, if ∥u1,p = ∥v1,p = 1 and ∥u − v1,p ≥ ε, one obtains
u + v 2 1 , p p 1 ε 2 p
and hence the conclusion. □
The dual of  W 0 1 , p (Ω), p in [1, +∞), is denoted
W 1 , p Ω ,
p′ the coefficient conjugated to p.
If Ω is bounded,
2 N N + 2 p < +     W 0 1 , p Ω     L 2 Ω     W 1 , p Ω
with continuous injections and dense and, if Ω is not bounded,
2 N N + 2 p 2     W 0 1 , p Ω     L 2 Ω     W 1 , p Ω .
The elements of  W 1 , p (Ω) can be characterized by the following.
Proposition 2.38.
Let F be from  W 1 , p (Ω). There exists f0, …, fN in  L p (Ω) such that
F u = Ω f 0 u d x + i = 1 N Ω f i u x i d x   u   i n   W 0 1 , p Ω
and
F = max 1 i N f i L p ( Ω ) .
When Ω is bounded, one can take f0 = 0.

2.3.2. The Operators −∆p, − Δ p s and Nf

−∆p, p ∈ (1, +∞), the p-Laplacian

Let Ω be an open set, with the finite Lebesgue measure, from RN, N ≥ 2. The norm on  W 0 1 , p (Ω) will be u → ∥u 1 , p .
Consider the operator − ∆p W 0 1 , p (Ω) →  W 1 , p (Ω),
p   u = div ( | u | p 2 u ) .
This acts according to [45]:
p   u ,   v   = Ω | u | p 2 u v dx u ,   v   in   W 0 1 , p ( Ω ) .
Taking into account the following, the next property of the p-Laplacian—the identification with a particular duality map—is the most important.
Proposition 2.39.
Let Ψ:  W 0 1 , p (Ω) → R,
Ψ ( u ) = 1 p | | u | | 1 , p p .
Then Ψ is Gâteaux differentiable on  W 0 1 , p (Ω)\{0}, and
Ψ ( u ) = p   u = J φ   u u from   W 0 1 , p ( Ω ) ,
where φ(t) =  t p 1 ([50]).
Proof. 
Since Ψ(u) =  0 | | u | | 1 , p φ ( t ) d t , we have
J φ   u =   Ψ ( u )   u in   W 0 1 , p ( Ω )   ( Proposition   2.2 ) ,
thus, it remains to prove that Ψ is Gâteaux differentiable and
Ψ ( u ) = p   u   u in   W 0 1 , p ( Ω ) .
Ψ = g o f, where g L p (Ω) → R, g(u) =  1 p   | | u | | 0 , p p , f W 0 1 , p (Ω) →  L p (Ω), f (u) = | u |.
From now on, the proof is continued as in [11] and has been proposed by the author.
g is of Fréchet C1 class on  L p (Ω) (Proposition 2.33). f is Gâteaux differentiable on  W 0 1 , p (Ω)\{0} and f ′(u)(h) =  u h | u | h in  W 0 1 , p (Ω) [50]. Applying Proposition 2.4, u ≠ 0 and h ∈  W 0 1 , p (Ω) ⇒ Ψ′(u)(h) =  Ω | u | p 1 u h | u | d x Ω | u | p 2 u h d x   = ( 2.31 ) 〈− ∆p u, h〉 and the case u = 0 remains to finish with (2.32).
However, Ψ′(0)(h) =  lim t 0 1 t Ψ(th) =  lim t 0 t p 1 p h 1 , p p = 0 = 〈− ∆p 0, h〉. □
Remark 2.3.
Ψ has even the Fréchet C1 class on  W 0 1 , p (Ω) [44,52].
Corollary 2.6.
u → ∥u 1 , p is Gâteaux differentiable on  W 0 1 , p (Ω)\{0} and  W 0 1 , p (Ω) is smooth.
Proof. 
For the first assertion, apply Proposition 2.4 considering φ(u) = ∥u1,p p 1 p [ Ψ ( u ) ] 1 p ; for the second assertion, we use Proposition 2.3. □
Proposition 2.40.
The operator − ∆p:  W 0 1 , p (Ω) →  W 1 , p (Ω) is bijective with monotonous inverse, bounded and continuous.
Proof. 
−∆p = Jφ (Proposition 2.39), and  W 0 1 , p (Ω) is uniformly convex; apply Proposition 2.26 and take into account the formula  J φ 1 = Φ−1 Jφ−1, Φ is the canonical embedding in bidual (Proposition 2.23). □

Δ p s , p ∈ (1, +∞), the p-Pseudo-Laplacian

Let Ω be an open set of finite Lebesgue measure from RN, N ≥ 2, and p in (1, +∞).
u 1 , p : = i = 1 N u x i L p ( Ω ) p 1 p
is a norm on  W 0 1 , p (Ω) since
i = 1 N u x i + v x i 0 , p p 1 p i = 1 N u x i 0 , p p + v x i 0 , p p p 1 p ,
applying Minkovski inequality.
The dual of ( W 0 1 , p (Ω), · 1,p) is also designated by W−1,p′(Ω), where p′ is the exponent conjugated with p.
· 1,p is equivalent to the norm |u|1,p : = i = 1 N u x i L p ( Ω ) :
i = 1 N u x i L p ( Ω ) N u 1 , p N i = 1 N u x i L p ( Ω ) .
However, | · |1,p is equivalent to ∥ · ∥1,p since ∥u1,p|u|1,pNu1,p. Consequently,
Proposition 2.41.
( W 0 1 , p (Ω), · 1,p), p ∈ (1, +∞), is Banach space.
Furthermore,
Proposition 2.42.
( W 0 1 , p (Ω), · 1,p), p ∈ [2, +∞), is uniformly convex.
Proof. 
The following proof was proposed by the author in [11]. Use the inequality (2.25):
ξ, η ∈  R n , n ≥ 1 ⇒ ξ + η 2 p ξ η 2 p ≤  1 2 (∥ξ∥p + ∥η∥p) with the Euclidean norm [51].
Let ε be in (0, 2] and define u, v with u1,p = v1,p = 1, u − v1,p ≥ ε. Suppose p ∈ [2, +∞).  u + v 2 p u v 2 p i = 1 N Ω u x i + v x i 2 p + u x i v x i 2 p d x ≤  i = 1 N Ω 1 2 u x i p + v x i p dx = 1, and hence,  u + v 2 1 , p p ≤ 1 −  ε 2 p , take δ defined by 1 − δ =  1 ε 2 p 1 p . □
Let Ω be an open set in RN, N ≥ 2, of the finite Lebesgue measure and p in (1, +∞). Considering the operator − Δ p s W 0 1 , p (Ω) →  W 1 , p (Ω),
Δ p s u = i = 1 n x i u x i p 2 u x i .
This acts according to [45]:
Δ p s u ,   h = i = 1 N Ω u x i p 2 u x i h x i d x ,   u ,   h   in   W 0 1 , p
Proposition 2.43.
The function Ψ: Ψ(u) =  1 p u 1 , p p , p ∈ (1, +∞), is Gâteaux differentiable on  W 0 1 , p (Ω)\{0}, and
Ψ ( u ) = Δ p s u = J φ   u ,   φ ( t ) : = t p 1 .
Proof 
([11]). Fix the index i, 1 ≤ iN, and let g W 0 1 , p (Ω) → R, g(u) =  u x i 0 , p p . We have g = F o f, f :  W 0 1 , p (Ω) →  L p (Ω), f (u) =  u x i , F L p (Ω) → R, F(v) = ∥v | | 0 , p p . As f ′(u)(h) =  h x i and F′(v)(h) = p Ω | v | p 2 v h dx ([48]), g′(u) = F′(f (u)) o f ′(u) (formula from Proposition 2.4),
g ( u ) ( h ) = p Ω u x i p 2 u x i h x i d x
and hence
Ψ ( u ) ( h ) = i = 1 N u x i 0 , p p ( h ) = ( 2.34 ) , ( 2.33 ) Δ p s u ,   h .
The rest of the proof is the same as for Proposition 2.39. □
Corollary 2.7.
u → u1,p, p ∈ (1, +∞), is Gâteaux differentiable on  W 0 1 , p (Ω)\{0}. Consequently, ( W 0 1 , p (Ω), · 1,p) is a smooth space.
Proof 
([11]). Taking Φ(u) = u1,p, we have
Φ ( u ) = p 1 p ( Ψ ( u ) ) 1 p .
We apply the formula from Proposition 2.4. For the second assertion, we take Proposition 2.3 into account. □

Nemytskii Operator Nf

In the following, some statements from [53] are necessary to develop some results.
Definition 2.9.
Let Ω be a nonempty open Lebesgue measurable (L.m.) set from RN, N ≥ 1, μ the Lebesgue measure in RN and M (Ω) : = {u: Ω → R: u L.m.}. By definition, f : Ω × R  R is a Carathéodory function if:
f (·, s) is L.m. ∀s in R;
f (x, ·) is continuousx in Ω\A, μ(A) = 0.
Proposition 2.44.
If f : Ω × RR is a Carathéodory function, then, for any u in M (Ω), xf (x, u(x)) is L.m.
Proof. 
Let (φn)n≥1 be a sequence of real functions, simple, L.m., with
φ n   ( x )   x Ω u ( x ) .
Fn : Fn (x) : = f (x, φn (x)) is L.m. on A1, …, Ap, where φn | Ak = constant, k 1 , p ¯ (10 from Definition 2.9), hence Fn is L.m. on Ω. ∀x in Ω\A, and since φn (x) → u(x), we have Fn (x) → f (x, u(x)) (20 from Definition 2.9), which implies, since the Lebesgue measure is complete, xf (x, u(x)) L.m. □
Definition 2.10.
Thus, one may consider the Nemytskii operator:
Nf : M (Ω) → M (Ω), (Nf u)x = f (x, u(x)).
Proposition 2.45.
Suppose μ(Ω) < +∞. Then
u n   ( x )   x Ω μ   u 0   ( x )     N f   u ( x )   x Ω μ N f   u 0   ( x ) .
Proof 
([54]). It is sufficient to show the proof in the case f (x, 0) = 0 ∀x in Ω and un (x x Ω μ 0; thus, it remains to prove that:
∀ε, η > 0 ∃ N in N, e.g., nN ⇒ μ({x ∈ Ω: | f (x, un (x))| ≥ ε}) ≤ η.
Set Ω0 : = Ω\A. For kN, Ωk : = {x ∈ Ω0 : |s| 1 k | f(x, s)| < ε} (nonempty set for sufficiently big k; f (x, ·) is continuous in 0), we have Ωk ⊂ Ωk+1k and Ω0 k = 1 Ω k , hence μ(Ω0) =  lim k μ ( Ω k ) and hence ∃ k0 in N, e.g., μ(Ω0\ Ω k 0 ) ≤  η 2 . Let An: = {x ∈ Ω0|un (x)| 1 k 0 }, ∃ N in N, e.g., nN ⇒ μ(Ω0\An) ≤  η 2 . Setting Bn: = {x ∈ Ω0 : |f (x, un (x))|< ε}, we have, since An ∩  Ω k 0 Bn, nN ⇒ μ(Ω0\Bn) ≤ μ(Ω0\An) + μ(Ω0\ Ω k 0 ) ≤ η, which implies (2.35). □
Proposition 2.46.
If the Carathéodory function f verifies the growth condition:
|f(x, s)|c|s|r + β(x), ∀x ∈ Ω\A with μ(A) = 0 ∀sR,
where c ≥ 0, r > 0, β ∈ Lp(Ω), 1 ≤ p ≥ +∞,
then
Nf ( L p r (Ω)) ⊂  L p (Ω);
Nf is continuous (p < +∞) and bounded on  L p r (Ω).
Clarification. A map between metric spaces is bounded if the image of any bounded set is bounded.
Proof. 
1° Let u be in  L p r (Ω).
|f (x, u(x))|c |u(x)|r + β(x), x ∈ Ω\A, sR,
but |u|r ∈  L p (Ω), β ∈  L p (Ω), hence Nf u ∈  L p (Ω) (when p < +∞, taking the power p, the first member is integrable and measurable (Proposition 2.44); when p = +∞, the justification is obvious).
2° From (2.36),
| | N f   u | | L p ( Ω )     | | c   | u | r + β | | L p ( Ω )     c   | |   | u | r   | | L p ( Ω ) + | | β | | L p ( Ω ) = c | | u | | L p ( Ω ) r   +   | | β | | L p ( Ω ) ,
and hence Nf is bounded on  L p r (Ω).
We proceed to the continuity. Suppose f (x, 0) = 0 ∀x in Ω, and let
u n     0   in   L p r ( Ω ) .
We will prove:
N f   u n     0   in   L p ( Ω ) .
For (2.38), it is sufficient to prove that any subsequence (Nf  u k n ) has a subsequence (Nf  u l k n ) convergent to 0 in  L p (Ω) (if (xn), the sequence in the metric space X has the property that any subsequence ( x k n ) has a subsequence ( x l k n ) with  x l k n x0, and then xnx0 − ad absurdum justification).
Since  u k n → 0 in  L p r (Ω), ∃ ( u l k n ) a subsequence with
u l k n ( x )     0 , x     Ω \ B ,   μ ( B ) = 0
and
| u l k n ( x ) |     g ( x ) ,   g   L p r ( Ω ) .
From (2.36),
|   f   ( x ,   u l k n ( x ) ) |     c ( g ( x ) ) r   +   β ( x ) ,   x     Ω \ ( A     B ) .
Taking the power p in (2.39), the second member is integrable, and as x ∈ Ω\(AB) ⇒f (x u l k n (x)) → 0, it results in (Lebesgue theorem of dominated convergence)
Ω | N f u n ( x ) | p d x     0 ,
i.e., (2.38).
Pass to the general case and let unu0 in  L p r (Ω). g: Ω × RR,
g(x, s) = f (x, s + u0 (x)) − f (x, u0 (x)),
is a Carathéodory function. Since g(x, 0) = 0 ∀x in Ω and unu0 → 0 in  L p r (Ω), we obtain Ng (unu0) → 0 in  L p (Ω) and, hence Nf unNf u0 in  L p (Ω). □
Remark 2.4.
Retain the inequality:
| | N f u | | L p ( Ω )   c | | u | | L pr ( Ω ) r + | | β | | L p ( Ω ) .

2.3.3. The Problem

Consider the problem
( ) λ Δ p   u = f (     , u (     ) ) + h ,   x Ω ,   λ R u | Ω = 0
The next two propositions were obtained by the author and are given in [11,12,15].
Proposition 2.47.
Let Ω be an open bounded set of the C1 class from RN, N2, p ∈ (1, +∞), h be from  W 1 , p (Ω) and f : Ω × RR a Carathéodory function with the properties
f (x, − s) = − f (x, s) ∀s from R, ∀x from Ω,
2° |f (x, s)| ≤ c1 |s|q–1 + β(x) ∀s from R, ∀x from Ω\A, μ(A) = 0,
where c1 ≥ 0, q ∈ (1, p), β ∈  L q (Ω),  1 q + 1 q = 1 .
Then, for any λ ≠ 0, the problem (∗) has a solution in  W 0 1 , p (Ω) in the sense of  W 1 , p (Ω).
Explanations. The relationship u | ∂Ω from (∗) is in the sense of the trace (Definition 2.8). Moreover, γ−1(0) =  W 0 1 , p (Ω). f (·, u) = Nf u, where Nf is the Nemytskii operator (see Section 2.3.2 above), and so the equation from (∗) can be written as
− λ ∆p u = Nf u + h.
From 2° of the last assertion, it is determined (via Proposition 2.46) that Nf maps  L q (Ω) on  L q (Ω), and it is continuous and bounded. Moreover (Proposition 2.46),
N f   u 0 , q     c 1   | | u | | 0 , q q 1 + c 2   ,   c 2 : = | | β | | 0 , q   ,   u in   L q ( Ω ) .
Since q ∈ (1, p) and q < p* (the Sobolev conjugated exponent), ( W 0 1 , p (Ω), ∥ ·1, p) is compactly embedded in  L q (Ω). Let i (linear injection) be such an embedding,
i ( u ) 0 , q     c 0 , q   u 1 , q   u in   W 0 1 , p
(using the Rellich-Kondrashev theorem and taking into account that the norms ∥ ·1, p and  W 1 , p ( Ω ) are equivalent).
Let i’ L q (Ω) →  W 1 , p (Ω) be the adjoint of i (as ( L q (Ω))* =  L q (Ω)!).
u0 from  W 0 1 , p (Ω) is a solution for (∗) in the sense of  W 1 , p (Ω) if
− λ ∆p u0 = (i’ o Nf o i)u0 + h.
We proceed to the proof of Proposition 2.47.
Proof. 
−∆p = P r o p o s i t i o n 2.39 Jφ, where Jφ is the duality map with φ(t) = tp−1; the Banach space ( W 0 1 , p (Ω), ∥ · ∥1,p) is uniformly convex (Proposition 2.37) and, consequently, has the (H) property, and it is reflexive (uniformly convex ⇒ reflexive). It is also smooth (its norm being Gâteaux differentiable on  W 0 1 , p (Ω)\{0} (Proposition 2.3)). Thus, one can apply Proposition 2.30 with X W 0 1 , p (Ω), Z L q (Ω), N = Nf − odd continuous operator (Proposition 2.46) and Z* =  L q (Ω) and take (2.41) into account; the operator λ(−∆p) − S:  W 0 1 , p (Ω) →  W 1 , p (Ω), where S = i ′ o Nf o i is surjective, a fortiori the operator −λ∆pSh is surjective (commutative group) and hence ∃u0 in  W 0 1 , p (Ω) which verifies (2.42). □
By replacing q with p in Proposition 2.47, 20, and by applying Proposition 2.32, we obtain the following:
Proposition 2.48.
Let Ω be an open bounded set of the C1 class from RN, N2, p ∈ (1, +∞), h from  W 1 , p (Ω) and f : Ω × RR Carathéodory function having the properties
f (x, −s) = − f (x, s) ∀x from Ω, ∀s from R,
|f (x, s)|c1 |s|p−1 + β(x) ∀s from R, ∀x from Ω\A, μ(A) = 0,
where c1 ≥ 0, β ∈  L p (Ω),  1 p + 1 p = 1.
Finally, let i :  W 0 1 , p (Ω) →  L p (Ω) be linear compact embedding. Then, for any λ, if
| λ |   >   c 1 λ 1 1 ,   λ 1   : = inf   | | u | | 1 , p p | | i ( u ) | | 0 , p p : u W 0 1 , p ( Ω ) \ { 0 }
then the problem (∗) has solution in  W 0 1 , p (Ω) in the sense of  W 1 , p (Ω).
Proof. 
The statement is correct since ( W 0 1 , p (Ω), ∥ · ∥1, p) is compactly embedded in  L p (Ω) (Rellich-Kondrashev theorem). Apply Proposition 2.32. □
Remark 2.5.
The condition from Proposition 2.48 can be replaced (Proposition 2.32 allows this) by:
| λ |   > lim ¯ | | u | | + | | ( i N f i ) u | | | | u | | p 1 .
Attention to λ1 λ 1 1 p is optimal for the inequality from the statement; it is attained and nonzero, and it is the smallest eigenvalue of the couple ( J L q L q , J W 0 1 , p W 1 , p ) (see Proposition 2.31).

2.3.4. The Problem

Consider the problem
( ) λ Δ p s u = f (     , u (     ) ) + h ,   x Ω ,   λ R u | Ω   =   0
The following two statements are obtained by the author and given in [11,12,15].
Proposition 2.49.
Let Ω be an open bounded set of C1 class from RN, N2, p ∈ [2, +∞), h from  W 1 , p (Ω) and f : Ω × RR Carathéodory function with the properties
f (x, − s) = − f (x, s) ∀x from Ω, ∀s from R,
|f (x, s)|c1 |s|q–1 + β(x) ∀s from R, ∀x from Ω\A, μ(A) = 0,
where c1 ≥ 0, q ∈ (1, p), β ∈  L q (Ω),  1 q + 1 q = 1 .
Then, for any λ ≠ 0, the problem (∗∗) has solution in  W 0 1 , p (Ω) in the sense of  W 1 , p (Ω).
Explanations (similar to those for Proposition 2.47): The relationship u | ∂Ω from (∗∗) is in the sense of the trace. f (·, u) = Nf u, Nf Nemytskii operator, and so the equation from (∗∗) can be written as
λ Δ p s u = N f   u + h .
Now, the norm that endows  W 0 1 , p (Ω) is · 1,p, and ( W 0 1 , p (Ω), · 1,p) is a Banach space that is compactly embedded in  L q (Ω) since · 1,p and·1,p, and hence also  | | | | W 1 , p ( Ω ) , are equivalent (see above). Let i be the embedding.
Let i′ L q (Ω) →  W 1 , p (Ω) be the adjoint of i (as ( L q )* =  L q ). u0 from  W 0 1 , p (Ω) is a solution for (∗∗) in the sense of  W 1 , p (Ω) if
λ Δ p s   u 0 = ( i   o   N f   o   i ) u 0 + h .
Proof. 
Δ p s = P r o p . 2.43 Jφ, Jφ duality map with φ(t) = tp−1, the Banach space ( W 0 1 , p (Ω), · 1,p) being uniformly convex (see above, proposition 2.42). It is also smooth (its norm being Gâteaux differentiable on  W 0 1 , p (Ω)\{0}). So, we apply Proposition 2.30 with X W 0 1 , p (Ω), Z L q (Ω), N = Nf − odd continuous operator, Z* =  L q (Ω), take into account ∥Nf u0,q′c1 | | u | | 0 , q q 1 + c2, c2 : = ∥β∥0,q′, ∀u from  L q (Ω) the operator λ (− Δ p s ) − S W 0 1 , p (Ω) →  W 1 , p (Ω), where S = i′ o Nf o i is surjective, a fortiori the operator − λ Δ p s Sh is surjective (commutative group) and hence ∃u0 in  W 0 1 , p (Ω) which verifies (2.44). □
Replacing q with p in 2° from Proposition 2.49 and applying Proposition 2.32, obtain:
Proposition 2.50.
Let Ω be an open bounded set of C1 class from RN, N2, p ∈ [2, +∞), h from  W 1 , p (Ω) and f : Ω × RR Carathéodory function having the properties
f (x, −s) = − f (x, s) ∀x from Ω, ∀s from R,
2° | f (x, s)| ≤ c1 |s|p−1 + β(x) ∀s from R, ∀x from Ω\A, μ(A) = 0,
where c1 ≥ 0, β ∈  L p (Ω), 1 p + 1 p = 1.
Finally, let i W 0 1 , p (Ω) →  L p (Ω) be a linear compact embedding. Then, for any λ, if
| λ |   >   c 1 λ 1 1 ,   λ 1   :   = inf   u 1 , p p | | i ( u ) | | 0 , p p : u W 0 1 , p ( Ω ) \ { 0 } ,
the problem (∗∗) has solution in  W 0 1 , p (Ω) in the sense of  W 1 , p (Ω).
Proof. 
The proof is the same as for Proposition 2.48. □
Remark 2.6.
The condition from Proposition 2.49 can be replaced (see Proposition 2.48) by:
| λ |   > lim ¯ | | u | | + | | ( i N f i ) u | | u p 1 .
Attention to λ1 λ 1 1 p is optimal for the inequality from the statement; it is attained and nonzero, and it is the smallest eigenvalue of the couple ( J L q L q , J W 0 1 , p W 1 , p ) (see Proposition 2.31).
Remark 2.7.
The above results will be used to provide solutions, together with their characterizations, for particular problems from glaciology [55,56,57], for nonlinear elastic membrane [41,58], for the pseudo-torsion problem [9,59] for nonlinear elastic membrane with the p-pseudo-Laplacian as in [60]. They are presented in the second part of this paper.

3. Results of the Fredholm Alternative Type for Operators λJφS

3.1. Important Results

In this section, we continue with results that complete the previous theory provided in Section 2. The statements from this section originate from the generalization due to the author in [11,13] of a theorem of Nečas [42,61] in which normed space is used instead of Banach space and the goal function is a bijection with continuous inverse instead of homeomorphism. The results mentioned above have been obtained based on this theorem and also on propositions of the author and presented in the previous section.
Definition 3.1.
Let X, Y be real normed spaces and F: X → Y, F0 : X→ Y. F is strongly closed and strongly continuous, respectively, if
x n   w   a   and   F ( x n )     α α = F ( a ) ,
and, respectively,
x n   w   a   and   F ( x n )     F ( a ) .
For instance, any linear compact operator between Banach spaces is strongly continuous [62].
Definition 3.2.
Let a be a real, strictly positive number. F is, by definition, a-homogeneous if F(tu) = ta F(u), u in X, t ≥ 0.
F is a-quasi-homogeneous relative to F0, and F0 is a-homogeneous if
t n     0 ,   u n   w   u 0   and   t n a   F u n t n     γ     γ = F 0   ( u 0 ) .
F is a-strongly quasi-homogeneous relative to F0, F0 a-homogeneous if
t n     0 ,   u n   w   u 0   and   t n a   F u n t n     F 0   ( u 0 ) .
Proposition 3.1.
Let F be an a-homogeneous and strongly closed (respectively, strongly continuous), then F is a-quasi-homogeneous (respectively a-strongly quasi-homogeneous) relative to F.
Proof. 
Observe, in both cases, that  t n a F u n t n F0 (u0). □
Proposition 3.2.
If F is a-strongly quasi-homogeneous relative to F0, then F0 is a-homogeneous and strongly continuous.
Proof. 
First assertion. t > 0. Let u0 be arbitrarily fixed from X and tn ↓ 0, un  w u0. Then  t n a F u n t n F0 (u0), hence  ( t t n ) a F   u n t n ta F0 (u0), but  ( t t n ) a F t u n t t n F0 (tu0) since tun  w tu0, ta F0 (u0) = F0 (tu0). t = 0. It should be shown that F0 (0) = 0. Take tn ↓ 0 and (un)n≥1 with un = 0 ∀n. Then,  t n a F u n t n → F0 (0), but F u n t n = 0, etc.
Second assertion. From the hypothesis,
lim t 0 + t a F u t = F 0 u u   i n   X .
Suppose, ad absurdum, that F0 is not strongly continuous. Then there exists un, un  w u0, and
F 0 ( u n )   F 0 u 0 .
ε > 0 being arbitrarily fixed, from (3.2), there exists a subsequence of (un), it is denoted in the same manner, e.g.,
F0 (un) − F0 (u0)∥ ≥ ε ∀n ≥ 1.
Furthermore, from (3.1), for any n from Ntn, 0 < tn ≤  1 n , for which
F 0   ( u n ) t n a   F u n t n     ε 2 .
Then
ε   ( 3.3 )   F 0   ( u 0 ) F 0   ( u n )     F 0   ( u 0 ) t n a   F u n t n   +   t n a   F u n t n F 0   ( u n )   ( 3.4 ) ε 2 + F 0   ( u 0 ) t n a   F u n t n ,
and taking the limit for n → ∞, we obtain ε ≤  ε 2 , which is a contradiction. □
Proposition 3.3.
The uni-valued duality map Jφ is a-homogeneous iff φ is a-homogeneous.
Proof. 
Necessary. ∀u ≠ 0, ∀t ≥ 0, Jφ (tu) =  φ ( t u ) φ ( u ) Jφ u (Proposition 2.2, 5°), Jφ (tu) = ta Jφ u, and by taking the norm, φ(tu∥) = ta φ(∥u∥), taking into account that u → ∥u∥ takes all the values from R+. Sufficient. Use the same formula. □
Thus, for p ∈ (1, +∞), −∆p and − p s are (p − 1)-homogeneous maps on  W 0 1 , p (Ω) (Propositions 2.39 and 2.43).
Proposition 3.4.
If the Banach space is reflexive, smooth and has the property (H), then any duality map Jφ on X is strongly closed.
Proof. 
Let un  w u0 and Jφ un → γ. We have
Jφ unJφ u0, unu0〉 → 0,
but
Jφ unJφ u0, unu0〉 ≥ [φ(∥un∥) − φ(∥u0∥)](∥un∥ − ∥u0∥) ≥ 0 (Proposition 2.2),
hence  lim n [ φ ( u n ) − φ(∥u0∥)](∥un∥ − ∥u0∥) = 0, which implies ∥un∥ → ∥u0∥ (see the proof for Proposition 2.28). With X having the property (H), we obtain unu0, which implies (X is smooth reflexive ⇒ Jφ is semicontinuous, Corollary 2.3) that Jφ un  w Jφ u0 and hence Jφ u0 = γ. □
Corollary 3.1.
If the Banach space X is reflexive, smooth and has the property (H), any duality map on X Jφ that is a-homogeneous is a-quasi-homogeneous related to Jφ.
Proof. 
Combine Propositions 3.1 and 3.4. □
We proceed to the basis proposition of this section. The conditions are slightly weakened. Previously:
Definition 3.3.
The map f : X → Y, where X and Y are normed spaces, is regularly surjective if it is surjective and ∀R > 0 ∃r > 0 e.g.,
f (x) ∥ ≤ R ⇒ ∥x∥ ≤ r.
Proposition 3.5.
Fredholm alternative. Let X and Y be real normed spaces, T: XY a(K, L, a) and a-homogeneous bijection, odd with a continuous inversei and S: X → Y an odd compact a-homogeneous operator. Theni for any λ ≠ 0, λTS is regularly surjective iff λ is not an eigenvalue for the couple (T, S).
Proof. 
Necessary. Let, ad absurdum, x0 ≠ 0 be from X such that
λT(x0) − S(x0) = 0.
Multiplying (3.5) by ta, we obtain
λT(tx0) − S(tx0) = 0
and as  lim t + tx0∥ = +∞, (3.6) imposes (ad absurdum!) the conclusion that λTS is not regularly surjective, which is a contradiction.
Sufficient. Firstly, we prove that:
ρ : = inf | | x | | = 1 λ T ( x ) S ( x )   >   0 .
Assume, ad absurdum,
ρ = 0.
With (3.8), we obtain a sequence (xn)nN, xnX,
xn∥ = 1
and
lim n [ λ T ( x n ) S ( x n ) ] = 0 .
The sequence (xn) being bounded, (S(xn))nN has a subsequence ( x k n ) convergent in Y, and let γ =  lim n S( x k n ). However, T is surjective and λ ≠ 0, so ∃ x0X such that λT(x0) = γ, and then, from (3.10),
lim n λ T ( x k n ) = λ T ( x 0 ) .
From (3.11), T having a continuous inverse, we obtain
lim n x k n = x 0 .
(3.12) imposes, on the one hand, ∥x0 = ( 3.9 ) 1 and, on the other hand,  lim n S( x k n ) = S(x0), which, combined with (3.10) and (3.11), gives λT(x0) − S(x0) = 0, which is a contradiction, and hence (3.7).
Thus, from (3.7),
λ T x | | x | | S x | | x | | ρ   x     X \ { 0 } ,
so
ρ∥xa ≤ ∥λT(x) − S(x) ∥ ∀xX\{0}.
From (3.13),
lim | | x   | | + λ T ( x ) S ( x ) = + ,
from which one concludes that λTS is surjective (see Proposition 2.1).
This surjectivity is regular. Indeed, assuming, ad absurdum, the contrary, we obtain R > 0 such that ∀nN x n X, ∥ x n ∥ > n and
λ T ( x n ) S ( x n )     R .
However, since  lim n x n ∥ = +∞, we have  lim n ∥λT( x n ) − S( x n )∥ = ( 3.14 ) +∞ and obtain a contradiction with (3.15). □
Proposition 3.6.
Let X be a real reflexive Banach space, smooth and having the property (H), which is compactly embedded in the real Banach space Z, and N: Z → Z* an a-homogeneous odd semicontinuous operator. Then the operator λJφ − N, with Jφ being the duality map on X with φ(t) = ta, λ ≠ 0, is regularly surjective iff λ is not an eigenvalue for the couple (Jφ, N) ([11]).
Explanation. In the expressions λJφN and (Jφ, N), N is actually i′ o N o i, i: XZ linear compact injection, i′: Z* → X* is its adjoint.
Proof 
([11]). We apply Proposition 3.5 with T : = Jφ, S : = i′ o N o i, correctly, as Jφ is (K, L, a) with K = L = 1, bijective with continuous inverse (Proposition 2.26), odd and S is odd, a-homogeneous and compact (see the proof of Proposition 2.30). □

3.2. Applications

3.2.1. Application for the p-Laplacian and p-Pseudo-Laplacian

In Proposition 3.6, we now take (X, · X) = ( W 0 1 , p (Ω), · 1, p), where p ∈ (1, +∞) and Ω is an open bounded set of C1 class from Rn, n ≥ 2 (hence Jφ = − ∆p, φ(t) = tp−1, Proposition 2.39), (Z, · Z) = ( L p (Ω), · 0,p), N L p (Ω) →  L p (Ω),  1 p + 1 p = 1, Nu = |u|p−2 u.
W 0 1 , p (Ω) is uniformly convex (Proposition 2.37) and hence also reflexive (Proposition 2.9) with the property (H), with its norm being Gâteaux differentiable (Corollary 2.6) and hence smooth (Proposition 2.3). It is compactly embedded in  L p (Ω). For the last assertion, we can mention the following:
Theorem 3.1.
Let Ω be a bounded set of the C1 class. Then
p <   n     W 1 , p   ( Ω )     L q ( Ω )   q   in   [ 1 ,   p ] ,   1 p   = 1 p 1 n , p = n   W 1 , p   ( Ω )     L q   q   in   [ 1 ,   + ) , p >   n     W 1 , p   ( Ω )     C   ( Ω ¯ ) ,
in all cases with compact injections (Rellich-Kondrashev).
Concerning N, it is the duality map on  L p (Ω) relative to the weight ttp−1 (see the following Proposition 3.8); consequently, N is a homeomorphism of  L p (Ω) on  L p (Ω) (Corollary 2.5), odd and (p − 1)-homogeneous.
We apply Proposition 3.6 in order to obtain the following statement [11,13].
Proposition 3.7.
Let p be from (1, + ∞) and λ ≠ 0. If
λ(−∆p u) = |u|p−2 u
does not have a nonzero solution in  W 0 1 , p (Ω), then, for any h from  W 1 , p (Ω), the equation
λ(−∆p u) = |u|p−2 u + h
has solution in  W 0 1 , p (Ω) in the sense of  W 1 , p (Ω).
Explanation. The term |u|p−2 u from (3.16) and (3.17) is actually considered to be its image through a compact embedding of  L p (Ω) in  W 1 , p (Ω) (use Schauder’s theorem).
Regarding the operator N, we can complete it with the following result.
Proposition 3.8.
The duality map on  L p (Ω), p ∈ (1, +∞), of weight φ(t) = tp−1 is
J φ   u = | u | p 1   sgn   u ,   u     L p ( Ω )
i.e.,
J φ   u ,   h = Ω | u | p 1 ( sgn u ) h dx   h     L p ( Ω )
Proof. 
Let Ψ: Ψ(u) =  1 p | | u | | 0 , p p . As Ψ(u) =  0 | | u | | 0 , p φ ( t ) dt, we have (Proposition 2.2, 3°)
Jφ (u) = ∂Ψ(u).
However, Ψ′(u)(h) =  Ω | u | p 1 ( sgn u ) h dxh from  L p (Ω) ([46]), and thus the conclusion. □
Proposition 3.9.
In the statement of Proposition 3.7, if p ∈ [2, +∞), then  p can be replaced by − Δ p s  ([11,13]).
Proof 
([11,13]). In Proposition 3.6, we take (X, · ) = ( W 0 1 , p (Ω), · 1, p) (see above), and (Z, · Z) = ( L p (Ω), · 0,p), N L p (Ω) →  L p (Ω),  1 p 1 p = 1, Nu = |u|p−2 u, and take into account that ( W 0 1 , p (Ω), · 1,p) is uniformly convex (see also Proposition 2.42 and Corollary 2.7 above). The compact embedding of  W 0 1 , p (Ω) in  L p (Ω) is given by the equivalence of the norms · 1,p and · 1,p since · 1,p is equivalent to the norm | · |1, p (see the p-pseudo-Laplacian in Section 2.3.2). □

3.2.2. Another Application for p-Laplacian

Here, in Proposition 3.6, we take (X, · X) = ( W 0 1 , p (Ω), · 1, p), where Ω is an open bounded set of C1 class in  R n , n ≥ 2, (Z, · Z) = ( L p (Ω), · 0,p), N L p (Ω) →  L p (Ω),  1 p + 1 p = 1, Nu = Nf u, Nf is the Nemytskii operator, with f : Ω × RR a Carathéodory function which verifies
| f (x, s)|c1 |s|p–1 + β(x) ∀sR, ∀x ∈ Ω\A, μ(A) = 0, where c1 ≥ 0, β ∈  L p (Ω);
f is odd and (p − 1)-homogeneous in the second variable.
Then, Nf is odd, (p − 1)-homogeneous and continuous (Proposition 2.46). We apply Proposition 3.6 (see also Section 3.2.1 above) and obtain the following:
Proposition 3.10.
Let p be from (1, +∞) and λ ≠ 0. If
λ(−∆p u) = Nf u
has no nonzero solution in  W 0 1 , p (Ω) in the sense of  W 1 , p (Ω), then, for any h from  W 1 , p (Ω), the equation
λ(−∆p u) = f (·, u(·)) + h, x ∈ Ω
has a solution in  W 0 1 , p (Ω) in the sense of  W 1 , p (Ω) ([11,13]).
Remark 3.1.
This statement can be compared with Proposition 2.48 above.
Remark 3.2.
Applications to real phenomena regarding the nonlinear elastic membrane with p-Laplacian and with p-pseudo-Laplacian will be developed in the second part of this article.

4. Surjectivity to Different Homogeneity Degrees

The propositions in this section originate from another assertion of the author [11,15], which generalizes a theorem of Fučik [42], but they are also based on other propositions obtained by the author. Applications to partial differential equations (weak solutions) are also given.

4.1. Theoretical Results

Proposition 4.1.
Let X and Y be real normed spaces, X complete and reflexive, T: X → Y (K, L, a) bijection odd with continuous inverse and S: X → Y odd compact operator b-strongly quasi-homogeneous relative to S0, b < a. For any λ ≠ 0, the operator λT − S is surjective.
Remark 4.1.
The author proposed this weakened version of the theorem from [42], i.e., with normed space instead of Banach space, and bijection with continuous inverse instead of homeomorphism.
Proof. 
According to Corollary 2.2, it is sufficient to prove:
lim ¯ | | x | | + | | S x | | | | x | | a = lim | | x | | + | | S x | | | | x | | a = 0 .
Supposing, ad absurdum, the contrary, we obtain a sequence (xn)nN, xn ∈ X\{0},  lim n ∥xn∥ = +∞, for which
| | S ( x n ) | | | | x n | | a     ε 0   n   in   N ,
where ε0 > 0. With X being complete and reflexive, the bounded sequence (yn)nN, yn : =  x n | | x n | | has a weakly convergent subsequence, one denotes this identically, yn  w y0. Then,
lim n S ( | | x n | | y n ) | | x n | | b = S 0   ( y 0 )
and as  lim n | | x n | | b | | x n | | a = 0, we obtain
lim n | | S ( x n ) | | | | x n | | a = 0 ,
in contradiction with (4.2), and hence (4.1). □
An immediate consequence:
Proposition 4.2.
Let X be a real reflexive Banach space and smooth with the property (H) which is compactly embedded in the real Banach space Z and N: Z→ Z* odd semicontinuous and b-homogeneous operator.
Then, for any λ ≠ 0,
λJφN,
Jφ the duality map on X with φ(t) = ta, a > b, is surjective ([11,15]).
Clarification. In the expression λJφN, N is actually (abbreviation!) the operator i′ o N o i, i′ is the adjoint of i.
Proof 
([11,15]). Applying Proposition 4.1, with T = Jφ (K = L = 1 is odd bijective with continuous inverse (Proposition 2.26)), S : = i′ o N o i is odd, compact and b-homogeneous. It remains only to prove that S is b-strongly quasi-homogeneous relative to S. Let tn ↓ 0 and un w u0, then  t n b S u n t n = S(un) and S(un) → S(u0): un w u0   i   compact i(un) → i(u0 N   semicontinuous N(i(un))  w N(i(u0)) i   compact S(un) → S(u0). □

4.2. Applications

4.2.1. First Application

We now take (X, · X) = ( W 0 1 , p (Ω), · 1, p) with p ∈ (1, +∞) and (X, · X) = ( W 0 1 , p (Ω), · 1,p) with p ∈ [2, +∞), respectively, and Ω is open bounded set of C1 class in  R n , n ≥ 2, φ(t) = tp−1; hence Jφ = −∆p (Proposition 2.39) and Jφ = −  Δ p s (Proposition 2.43), respectively, (Z, · Z) = ( L q (Ω), · 0,q) with q ∈ (1, p), N L q (Ω) →  L q (Ω), Nu = |u|q−2 u 1 q + 1 q = 1. N is odd, continuous (an even homeomorphism; see Proposition 3.8 and Corollary 2.5 above) and (q − 1)-homogeneous, q − 1 < p − 1. Applying Proposition 4.2, we obtain the following.
Proposition 4.3.
Under the above conditions, for any λ ≠ 0 and for any h from  W 1 , p (Ω), there exists u0 in  W 0 1 , p (Ω) such that
λ(−∆p)u0 = (i′ o N o i)u0 + h
and
λ ( Δ p s ) u 0   = ( i   o   N   o   i ) u 0 + h
respectively ([11,15]).

4.2.2. Second Application

This second application of Proposition 4.2 is made by replacing the operator N from Proposition 4.3 with Nf, the Nemytskii operator. More precisely, we take N L q (Ω) →  L q (Ω), N = Nf, with f : Ω × RR odd Carathéodory function and (q − 1)-homogeneous in the second variable, which verifies the growth condition
|f (x, s)|c1 |s|q−1 + β(x) ∀s in R, ∀x in Ω\A, μ(A) = 0,
where c1 ≥ 0, β ∈  L q (Ω).
Then, Nf is odd, (q − 1)-homogeneous and continuous (Proposition 2.46), and one can apply Proposition 4.2 to obtain:
Proposition 4.4.
Under the above conditions, for any λ ≠ 0 and for any h in  W 1 , p (Ω), there exists u0 in  W 0 1 , p (Ω) such that
λ(−∆p)u0 = (i′ o Nf o i)u0 + h
and
λ ( Δ p s ) u 0   = ( i   o   N f   o   i ) u 0 + h
respectively ([11,15]).
Remark 4.2.
Applications to models of real phenomena involving a nonlinear elastic membrane and a nonlinear elastic membrane with p-Laplacian and the p-pseudo-Laplacian will be provided in the second part of this work.

5. Weak Solutions Starting from Ekeland Variational Principle

5.1. Critical Points and Weak Solutions for Elliptic Type Equations

The theoretical results in the following two subsections were obtained by the author in [17].

5.1.1. Theoretical Support

In order to introduce the first result, theoretical support will be given, starting with:
Ekeland Principle.
Let (X, d) be a complete metric space and φ: X → (−∞, +∞] bounded from below, lower semicontinuous and proper. For any ε > 0 and u of X with
φ(u) ≤ inf φ(X) + ε
and for any λ > 0, there exists vε in X such that
φ ( v ε )   <   φ ( w ) + ε λ d ( v ε ,   w )   w     X \ { v ε }
and
φ(vε) ≤ φ(u), d(vε, u) ≤ λ
([22,63,64]).
We continue with the following:
Definition 5.1.
Let X be a real normed space, β a bornology (Definition 2.3) on X, and let φ: XR. Let c be in R and F a nonempty subset of X. φ verifies the Palais-Smale condition on level c around F (or relative to F), (PS)c,F, with respect to β, when ∀(un)n≥1 a sequence of points in X for which
lim n φ ( u n ) = c , lim n | | β φ ( u n ) | |   = 0   and   lim n dist ( u n ,   F ) = 0 ,
this sequence has a convergent subsequence.
To clarify the above notation, see Definition 2.3 regarding the β-derivative.
Let us introduce the definition of the metric gradient in order to provide other observations related to this central notion for the following statement.
Definition 5.2.
In a real normed space X, consider the Gâteaux-differentiable functional f : X R. The metric gradient of f is the multiple-valued mapping:
f   :   X   P   ( X ) ,     f ( x ) = i 1 J f w ( x ) ,
where  J : X* → P (X**) is the duality mapping on X* corresponding to the identity, and i is the canonical injection of X into X**: i(x) = x**, 〈x**, x*=x*, x〉, ∀x* ∈ X*.
Consequently, for any xX: ∇f (x) = {yX: i(y) ∈  J f w (x)} = {yX: 〈i(y),  f w (x)〉 = 〈 f w (x), y〉 = f w (x)2, i(y) = y = f w (x)}. If X is reflexive, for any xX, ∇f (x) is nonempty. X** being strictly convex,  J is single-valued. So, if X is reflexive and strictly convex, then ∇f : XX, ∇f (x) = i−1 J f w (x), and the following equalities hold:
f w ( x ) ,   f   ( x ) = | | f w ( x ) | | 2 ,   | | f   ( x ) | | = | | f w ( x ) | | .
Through the minimization of a functional on F (minimization with constraints), its global critical points may be obtained.
As a preliminary, we generalize some results from [65] by introducing Banach space instead of Hilbert space and Gâteaux differentiability instead of C1-class Fréchet.
Proposition 5.1.
Let X be real reflexive strictly convex Banach space, let φ: X R be lower semicontinuous and Gâteaux differentiable and let F be a closed subset of X such that for every u from F with the metric gradient ∇φ (u) ≠ 0, for sufficiently small r > 0,
u δ φ ( u ) | | φ ( u ) | | F , δ 0 , r .
Then, if φ is lower bounded on F, for every (vn)n≥1 a minimizing sequence for φ on F, there exists a sequence (un)n≥1 in F such that
φ u n   ε n ,
φ ( u n ) φ ( v n )   n
lim n u n v n   = 0 ,
where  ε n > 0 and  ε n 0 .
Remark 5.1.
This result is reported in [65] as Lemma 9 in the frame of Hilbert spaces having the function φ of the Fréchet C1 class, but the condition (5.3) is more complicated due to another condition imposed on the set F.
Proof. 
Denote c : = inf φ(F) and let n be from N. For  ε n :   = φ v n c + 1 n , hence  ε n > 0, we have φ( v n ) < c ε n . We apply the enounced Ekeland principle with  λ = ε n , ∃ u n in F, with known properties. Thus, we obtain the sequence  ( u n ) n 1 satisfying (5.4), (5.5) ( | | u n v n | | ≤  ε n ε n → 0) and
φ ( v ) φ ( u n ) ε n | | v u n | |     v F .
Next, we verify (5.3). It is sufficient to work under the assumption that  | | φ w ( u n ) | | > 0 n. Thus, we apply the hypothesis made in the statement with respect to F with  u = u n and, denoting, for  δ 0 , r ,   v δ : = u n δ φ ( u n ) | | φ ( u n ) | | (∈ F), replace vδ in (5.6) and find
ε n | | v δ u n | | φ ( u n ) φ ( v δ ) ,
multiply this inequality by  1 δ , δ > 0, and take the limit for δ → 0+ in order to keep the sense of the inequality. We remark that  lim δ 0 v δ = u n lim δ 0 | | v δ u n | | δ lim δ 0 δ | | φ ( u n ) | | | | φ ( u n ) | | δ = 1. Consider that the existence of the limit for δ → 0 implies the existence of the limit for δ → 0±, together with their equality,  lim δ 0 + φ ( u n ) φ ( v δ ) δ lim δ 0 + φ u n δ φ ( u n ) | | φ ( u n ) | | φ ( u n ) δ = φw′(un) φ ( u n ) | | φ ( u n ) | | 1 | | φ ( u n ) | | 〈φw′(un), ∇φ(un)〉 =  1 | | φ ( u n ) | | || φ w ( u n ) ||2 = || φ w ( u n ) ||; taking into account the definition of the Gâteaux derivative and the above considerations on the metric gradient, (5.3) is also fulfilled. □
Remark 5.2.
The Gâteaux derivative from the above statement can be replaced by any β-derivative, and the result remains the same. In the case of the Fréchet derivative, the condition “φ lower semicontinuous” must be removed from the statement.
Notation. φ: XR is β-differentiable, cR
K c ( φ ) : = { x X : φ ( x ) = c ,   β φ ( x ) = 0 } .
Proposition 5.2.
Let X be a real reflexive strictly convex Banach space and φ: X R lower semicontinuous and Gâteaux differentiable and let F be a nonempty convex closed subset such that (I −∇φ)(F) ⊂ F, where I is the identity map. If φ is lower bounded on F, then for every (vn)n≥1, a minimizing sequence for φ on F, there is a sequence (un)n≥1 in F such that
φ ( u n )     φ ( v n )   n ,   lim n u n v n   = 0 ,   lim n φ w ( u n )   = 0 .
Moreover, if φ satisfies (PS)c,F, where c = inf φ(F), then
F K c ( φ ) .
Proof. 
Applying Proposition 5.1, (5.2) is satisfied; indeed, if uF and  φ w ( u ) ≠ 0, then, F being convex,
u δ φ ( u ) | | φ ( u ) | | = 1 δ | | φ w ( u ) | |   u + δ | | φ w ( u ) | | ( I φ )   ( u ) F .
Let (un)n≥1 be the sequence given by the statement. c ≤ φ(un) ≤ φ(vn) ∀n, hence φ(un) → c | | φ w ( u n ) | | ( 5.3 ) ε n , hence  | | φ w ( u n ) | |   0 , clearly dist (un, F) = 0, and consequently, (un)n≥1 has a convergent subsequence  ( u k n ) n 1 u k n u 0 F . This implies  | | φ w ( u k n ) | |  →  | | φ w ( u 0 ) | | = 0 and thus u0 is a global critical point of φ contained in F. □

5.1.2. Weak Solutions

Open set of C1 class in RN. We use the following notations (the norm is that Euclidean from RN−1): R + N = {x = (x′, xN): xN > 0},  Q = x = ( x ,   x N ) : x   < 1 ,   x N < 1 ,   Q + = Q  ∩   R + N , Q0 = {x = (x′, xN): x < 1, xN = 0}. Let Ω be an open nonempty set in RN, Ω ≠ RN and ∂Ω its boundary. By definition, Ω is of C1 class if ∀x from ∂Ω ∃U is a neighborhood of x in RN and f : QU is bijective such that fC1( Q ¯ ), f−1C1( U ¯ ), f ( Q + ) = U ∩ Ω, and
f (Q0) = U ∩ ∂Ω.
Weak solution. Let Ω be an open bounded nonempty set in RN, N > 1, f : Ω × RNR, and  u 0 W 0 1 , p ( Ω ) . Consider the problems:
( )   Δ p u = f ( x , u ) ,   x Ω u = 0   on   Ω ,
and
( ) Δ p s u = f ( x , u ) ,   x Ω u = 0   on   Ω .
The equality u = 0 on ∂Ω for both problems is in the sense of the trace (Definition 2.8).  u ¯ from X =  W 0 1 , p (Ω) is, by definition, a weak solution for (∗) and (∗∗) if  u ¯ = 0 on ∂Ω in the sense of the trace and
Ω | u ¯ | p 2 u ¯ v d x Ω f ( x , u ¯ ( x ) ) v d x = 0 v   W 0 1 , p
and
i = 1 n Ω u ¯ x i p 2 u ¯ x i v x i d x Ω f ( x , u ¯ ( x ) ) v d x = 0 v   W 0 1 , p
respectively.
Remark 5.3.
Here, ∇w is the weak gradient (see Section 2.3.1 here and what follows). X : =  W 0 1 , p (Ω) is endowed in the first case (∗) with the norm ∥ · ∥1, p that was defined in Section 2.3.1, which is equivalent to the norm u →  | | u | | L p ( Ω ) p + i = 1 n u x i L p ( Ω ) p 1 p  (also highlighted there). For the second case (∗∗), equip the same vector space with the norm uu1,p i = 1 n u x i L p ( Ω ) p 1 p , which is equivalent to u|u|1, p i = 1 n u x i L p ( Ω )  (Section 2.3.2).
For the Nemytskii operator, see also Section 2.3.1 and suppose now that μ(Ω) < +∞. Then,
u n ( x ) x Ω μ u 0   ( x )     N f u n ( x ) x Ω μ N f u 0 ( x ) .
Assume that f satisfies the growth condition:
| f ( x ,   s ) |     c   | s | p 1 + b ( x ) ,   x     Ω \ A   with   μ ( A ) = 0 , x     R ,
where c ≥ 0, p > 1 and b ∈  L q (Ω), q ∈ [1, +∞]. Then Nf ( L q ( p 1 ) (Ω)) ⊂  L q (Ω); Nf is continuous (q < +∞) and bounded on  L ( p 1 ) q (Ω) (Proposition 2.46). If Ω is bounded and  1 p + 1 q = 1 , then Nf( L p (Ω)) ⊂  L q (Ω) with Nf continuous; moreover, NF( L p (Ω)) ⊂ L1(Ω), with NF continuous (ibidem), where  F ( x , s ) = 0 s f ( x , t ) d t , and Φ:  L p (Ω) → R, Φ(u) = Ω F ( x , u ( x ) ) d x is of Fréchet C1 class and Φ′ = Nf [65], so it is also Gâteaux differentiable.
Theorem 5.1.
Let Ω be an open bounded nonempty set in RN and f : Ω × R → R a Carathéodory function with the growth condition:
|f (x, s)|c|s|p−1 + b(x),
where  c > 0 ,   2 p 2 N N 2  when N ≥ 3 and 2 ≤ p < +∞ when N = 1, 2, and where b ∈  L q (Ω),  1 p + 1 q = 1 .
Then, the energy functional  φ : W 0 1 , p (Ω) → R, and
φ ( u ) = 1 p | u | 1 , p p Ω F x , u x d x ,   f o r   t h e   p r o b l e m   ( )
and
φ ( u ) = 1 p u 1 , p p Ω F ( x , u ( x ) ) d x ,   f o r   t h e   p r o b l e m   ( ) ,
where  F ( x , s ) = 0   s f ( x , t ) d t  is Gâteaux differentiable on  W 0 1 , p (Ω)\{0} and, respectively,
φ w u v = Ω | u | p 2 u v d x Ω f x , u x v d x , u , v W 0 1 , p ( Ω )
and
φ w u v = i = 1 N Ω u x i p 2 u x i v x i d x Ω f x , u x v d x , u , v W 0 1 , p Ω .
Proof. 
One may consider φ, in both cases, to be the sum of two other functions. The second of these functions being Gâteaux differentiable (see the above considerations), it is sufficient to remark that the maps u →  1 p | | u | | 1 , p p and u →  1 p u 1 , p p are also Gâteaux differentiable on  W 0 1 , p ( Ω ) \{0} (Propositions 2.39 and 2.43) and then φ is Gâteaux differentiable on  W 0 1 , p Ω \{0}. □
Corollary 5.1.
Let Ω and f be as in Theorem 5.1 above. Then the weak solutions of (∗) and (∗∗) are precisely the critical points of the functional  φ : W 0 1 , p Ω R, respectively:
φ ( u ) = 1 p | | u | | 1 , p p Ω F x , u x d x ,   F x , s : = 0 s f ( x , t ) d t
and
φ ( u ) = 1 p u 1 , p p Ω F x , u x d x ,   F x , s : = 0 s f ( x , t ) d t .
Proof. 
Indeed, if  u ¯ is a weak solution of (∗) and (∗∗), then  φ w ( u ¯ ) ( v ) = 0   v ∈  W 0 1 , p ( Ω ) ((5.7) and (5.8) respectively (Theorem 5.1)), hence,  φ w ( u ¯ ) = 0 . The inverse assertion is obvious. □
Weak subsolutions and weak supersolutions of (∗) and (∗∗). Let Ω be an open bounded set of C1 class in RN, N ≥ 3, f : Ω × RR a Carathéodory function and let  u ¯ ∈  W 0 1 , p ( Ω ) u ¯ is a weak subsolution and a weak supersolution, respectively, of (∗) or (∗∗) if
u ¯     0   on   Ω   and   u ¯ 0   on Ω ,   respectively ,   and
Ω | u ¯ | p 2 u ¯ v d x Ω f ( x , u ¯ ( x ) ) v d x   v W 0 1 , p ( Ω ) ,   v 0 respectively Ω | u ¯ | p 2 u ¯ v d x Ω f ( x , u ¯ ( x ) ) v d x   v W 0 1 , p ( Ω ) ,   v 0 .
or
i = 1 n Ω u ¯ x i p 2 u ¯ x i v x i d x Ω f ( x , u ¯ ( x ) ) v d x   v W 0 1 , p ( Ω ) ,   v 0 respectively i = 1 n Ω u ¯ x i p 2 u ¯ x i v x i d x Ω f ( x , u ¯ ( x ) ) v d x   v W 0 1 , p ( Ω ) ,   v 0 .
Proposition 5.3.
Let Ω be an open bounded set of C1 class in RN, N ≥ 3, and let f : Ω × RR be a Carathéodory function and u1 and u2 from  W 0 1 , p ( Ω ) bounded weak subsolution and weak supersolution of (∗), respectively, with u1 (x) ≤ u2 (x) a.e. on Ω. Suppose that f verifies (5.9) and there is ρ > 0 such that the function g: g(x, s) = f (x, s) + ρs is strictly increasing in s on [inf u1(Ω), supu2(Ω)]. Then there is a weak solution  u ¯ of (∗) in  W 0 1 , p ( Ω ) with the property
u 1 ( x ) u ¯ ( x ) u 2 ( x )   a.e.   on   Ω .
Proof. 
Taking the equivalent norm on X =  W 0 1 , p ( Ω ) , we obtain
| | u | | = ρ | | u | | L p ( Ω ) p + i = 1 n u x i L p ( Ω ) p 1 p .
Considering the functional φ:  W 0 1 , p ( Ω ) R,
φ ( u ) = 1 p | | u | | p Ω G ( x , u ( x ) ) d x , G ( x , s ) : = 0   s g ( x , t ) d t .
where φ is Gâteaux differentiable, and its critical points are the weak solutions of (∗) (see Corollary 5.1 above). φ is also lower-bounded, with the norm on  L p (Ω) actually being of Fréchet C1 class (see, for instance, [43], Volume 2). Use Proposition 5.2, (X, · ) being a reflexive strictly convex Banach space (see Section 2). Let
F   :   = { u     W 0 1 , p ( Ω ) :   u 1 ( x ) u ( x ) u 2 ( x )   a.e.   on   Ω } .
F is closed convex. We also obtain
(I − ∇φ)F ⊂ F.
Here, ∇φ denotes the metric gradient of φ. Since (X, · ) is reflexive and strictly convex (see in Section 2), ∇φ is thus uni-valued, and it has the above-described properties. Indeed, let u be in F and  v : = ( I φ ) ( u ) . We should prove that v ∈ F. v = u − ∇φ(u) ∈  W 0 1 , p (Ω) and u1(x) ≤ v(x) ≤ u2(x). Since the definition relation of the subsolution for u1 actually means φw′(u1)(w) ≤ 0 ∀w in  W 0 1 , p (Ω) with w(x) ≥ 0 almost everywhere (a.e.) on Ω, and that of the supersolution for u2 is φw′(u2)(w) ≥ 0 ∀w in  W 0 1 , p (Ω), verifying w(x) ≥ 0 a.e. on Ω, we will prove that v(x) − u1(x) ≥ 0 a.e. on Ω and u2(x) − v(x) ≥ 0 a.e. on Ω using the Gâteaux derivatives of φ in u1 and u2, respectively. φw′(u1)(v − u1) = φw′(u1)(u1 − u) − φw′(u1)(∇φ(u)) ≤ −φw′(u1)(∇φ(u)) ≤ −φw′(u)(∇φ(u)) = −φw′(u)2 ≤ 0 (take into account that u1u, φw′(u1) is a linear map and some properties of the metric gradient). Also φw′(u2)(u2 − v) = φw′(u2)(u2 − v) + φw′(u2)(∇φ(u)) ≥ φw′(u2)(∇φ(u)) ≥ φw′(u)(∇φ(u)) = ∥φw′(u)∥2 ≥ 0. φ is lower bounded on F, φ being continuous, actually (for this assertion, see Section 2). Until now, applying Proposition 5.2, for every (vn)n≥1, a minimizing sequence for φ on F, there is a sequence (un)n≥1 in F such that φ(un) ≤ φ(vn) ∀n lim n | | u n v n | |   = 0 ,   lim n | | φ w ( u n ) | |   = 0 . So  lim n φ(un) = c and since c : = inf φ(F), we have  lim n | | φ w ( u n ) | |   = 0 already, and the last property from the (PS)c,F condition is verified. To finish the proof, we once again apply Proposition 5.2. □
Example 1.
Consider the problemis an open bounded set of C1 class in RN, N ≥ 3)
Δ p u = α ( x ) u | u | p 2 on   Ω , u = 0 on   Ω ,
where p 2 N N 2 and α is continuous with 1 ≤ α(x) ≤ a < +∞ on Ω. Then u1 : = 1 is a weak subsolution, u2 : = M, M > 1 sufficiently big, is a weak supersolution, |f (x, s)| ≤ a|s|p−1 (condition (5.9)), and s →  α ( x ) s | s | p 2 +   s is increasing in s on [1, M]; consequently, according to Proposition 5.3, (5.17) has a weak solution  u with 1 ≤  u ¯ ( x ) M a.e. on Ω.
Proposition 5.4.
Let Ω be an open bounded set of C1 class in RN, N ≥ 3, and f : Ω × RR a Carathéodory function and u1, u2 from  W 0 1 , p (Ω) bounded weak subsolution and weak supersolution of (∗∗), respectively, with u1 (x) ≤ u2 (x) a.e. on Ω. Suppose that f verifies (5.9) and there is ρ > 0 such that the function g: g(x, s) = f (x, s) + ρs is strictly increasing in s on [inf u1(Ω), sup u2(Ω)]. Then there is a weak solution  u ¯ of (∗∗) in  W 0 1 , p (Ω) with the property
u 1 ( x ) u ¯ ( x ) u 2 ( x )   a.e.   on   Ω .
Proof. 
We follow, step by step, the above proof for Proposition 5.3 considering the real reflexive strictly convex Banach space X W 0 1 , p (Ω) endowed with the norm uu1, p i = 1 N u x i L p ( Ω ) p 1 p or the equivalent norm uu1,p i = 1 N u x i L p ( Ω ) , which are both also equivalent to the other two norms used in Remark 5.3. The function φ is from (5.11), having the weak derivative given in Theorem 5.1. Using similar calculus, we obtain a similar conclusion. □
Example 2.
Consider the problemis an open bounded set of the C1 class in RN, N ≥ 3)
Δ p s u = α ( x ) u | u | p 2 on   Ω , u = 0 on     Ω   ,
where p =   2 N N 2 and α is continuous with 1 ≤ α(x) ≤ a < +∞ on Ω. Then, u1 : = 1 is a weak subsolution, u2 : = M, M > 1 being sufficiently big, is a weak supersolution, |f (x, s)|a|s|p−1 (condition (5.9)), and s → α(x)s|s|p−2 + s is increasing in s on [1, M]; consequently, according to Proposition 5.8, (5.18) has a weak solution  u ¯ with 1 ≤  u ¯ ( x ) M a.e. on Ω.
Remark 5.4.
The results from Section 5.1.1, Section 5.1.2 and Section 5.2.1 have been reported by the author in [17].
Remark 5.5.
Applications to real phenomena, as well as an application in glaciology, a nonlinear elastic membrane, the pseudo-torsion problem and a nonlinear elastic membrane with the p-pseudo-Laplacian, will be presented in the second part of this article.

5.2. Critical Points for Nondifferentiable Functionals

5.2.1. Theoretical Results

The meaning of the title is actually “not compulsory differentiable”. We start this section with the following:
Definition 5.3.
x0 is a critical point (in the sense of the Clarke subderivative) for the real function f if 0 ∈ ∂ f (x0). In this case, f (x0) is a critical value (in the sense of the Clarke subderivative) for f. To clarify this notion, the Clarke derivative should be introduced. Let X be a real normed space, EX, f : ER x 0 E o and v ∈ X. We set
f 0 ( x 0 ;   v ) : = lim x x 0 t 0 + ¯ f ( x + t v ) f ( x ) t .
The upper limit obviously exists. f0 (x0; v) is by definition the Clarke derivative (or the generalized directional derivative) of the function f at x0 in the direction v. The functional ξ from X* is by definition Clarke subderivative (or generalized gradient) of f in x0 if f0 (x0; v) ≥ ξ(v) ∀vX. The set of these generalized gradients is designated as ∂ f (x0).
Here it is a generalization at p-Laplacian and p-pseudo-Laplacian of an application of this concept from [66].
Let Ω be a bounded domain of RN with the smooth boundary ∂Ω (topological boundary). Consider the nonlinear boundary value problems (∗) and (∗∗) from Section 5.1.2 above, where f : Ω × RR is a measurable function with subcritical growth, i.e.,
(I) |f (x, s)|a + b|s|σsR, x ∈ Ω a.e.,
where a, b > 0, 0 ≤ σ <  N + 2 N 2 for N > 2 and σ ∈ [0, +∞) for N = 1 or N = 2.
Set [67]:
f ¯ ( x ,   t )   = lim ¯ s t f   ( x ,   s ) , f ¯ ( x ,   t )   = lim ¯ s t f   ( x ,   s ) .
Suppose
( II )   f ¯   and   f ¯ :   Ω   ×   R     R   are   measurable   with   respect   to   x .
We emphasize that (II) is verified in the following two cases:
f is independent of x;
f is Baire measurable and sf (x, s) is decreasing ∀x ∈ Ω, in which case we have:
f ¯ ( x ,   t ) = max {   f   ( x ,   t + ) ,   f   ( x ,   t ) } , f ¯ ( x ,   t ) = min   {   f   ( x ,   t + ) ,   f   ( x ,   t ) } .
Definition 5.4.
u from  W 0 1 , p (Ω), p > 1, is solution of (∗) and (∗∗) if u = 0 on ∂ Ω in the sense of the trace (see in Section 2.3.2 above) and
Δ p   u ( x )     [ f ¯ ( x ,   u ( x ) ) , f ¯ ( x ,   u ( x ) ) ]   in   Ω   a . e .
and
Δ p s u ( x )     [ f ¯ ( x ,   u ( x ) ) , f ¯ ( x ,   u ( x ) ) ]   in   Ω   a . e .
respectively.
Let X : =  W 0 1 , p (Ω), but in the first case (∗), the norm endowing X is ∥ · ∥1, p i.e., ∥u1, p = notation | | u | | W 0 1 , p ( Ω ) = | | u | | L p ( Ω ) + i = 1 N u x i L p ( Ω ) , which is equivalent to the norm u →  | | u | | L p ( Ω ) p + i = 1 N u x i L p ( Ω ) p 1 p . For the second case (∗∗), equip (also as previously) the same set X with the norm uu1,p i = 1 N u x i L p ( Ω ) p 1 p , which is equivalent to u|u|1, p i = 1 N u x i L p ( Ω ) .
Associate with (∗) the locally Lipschitz functional Φ : XR,
Φ ( u ) = 1 p u 1 , p p Ω F ( x , u ) dx ,   u     X ,
and associate with (∗∗)
Φ ( u ) = 1 p u 1 , p p Ω F ( x , u ) dx ,   u     X ,
where F(x, s) =  0   s f ( x , t ) dx. Set
Q ( u ) :   = 1 p u 1 , p p ,   u     X ,   Ψ 1   ( u ) :   = Ω F ( x , u ) dx ,   u     X ,
and
Q ( u ) :   = 1 p u 1 , p p ,   u     X ,   Ψ 1   ( u ) :   = Ω F ( x , u ) dx ,   u     X ,  
respectively, where F, a map defined on Ω × R, taking values in R, is locally Lipschitz (use (I)). The functional Ψ: Lσ+1(Ω) → R, Ψ(u) =  Ω F ( x , u ) dx, is also locally Lipschitz (again (I)). Using the Sobolev embedding XLσ+1(Ω), we find that Ψ1: = Ψ|X is locally Lipschitz on X, which implies that Φ is locally Lipschitz on X, and consequently, according to a local extremum result for Lipschitz functions (if x0 is a point of local extremum for f, then 0 ∈ ∂ f (x0)), the critical points of Φ for Clarke subderivative can be taken into account. One may state:
Proposition 5.5.
Suppose (I) and (II) are satisfied. Then Ψ is locally Lipschitz on Lσ+1 (Ω) and
(i) ∂ Ψ (u) ⊂ [ f _ (x, u (x)),  f ¯ (x, u (x))] in Ω a.e.
(ii) If Ψ1 = Ψ|X, where X W 0 1 , p (Ω) endowed with the norm ∥ · ∥1, p for the problem (∗) and · 1, p for the problem (∗∗), respectively, then
∂ Ψ1 (u) ⊂ ∂ Ψ (u) ∀uX.
Proof. 
The proof for (i) can be found in [67], Theorem 2.1, which remains the same here, while the problem was solved for the Laplacian with X H 0 1 ( Ω ) only. In order to prove (ii), we use 2.2 from [67], observing for both cases (X is endowed with each one from those two norms) that X is reflexive and dense in Lσ+1(Ω), as can be seen, for instance, in Section 2, where they are summarized. □
Proposition 5.6.
If (I) and (II) are verified, every critical point of Φ is a solution for (∗) and (∗∗), respectively.
Proof. 
Problem (∗). Let u0 be a critical point for Φ. We have
0 ∈ ∂ Φ(u0) ⊂ ∂ Q(u0) + ∂ (−Ψ1)(u0)
since Φ = ( 5.21 ) Q − Ψ1, and we apply some rules of subdifferential calculus concerning finite sums. ∂ Q(u0) = {Q′(u0)}, where Q′(u0)(v) =  Ω | u 0 | p 2 v dx = 〈−∆p u0 , v〉 (Section 2).
Using (5.25) and a specific property of a function f Lipschitz around x0 (f0 (x0; v) =  sup ξ f ( x 0 ) ξ(v), ∀vX, f0 the Clarke derivative of f), we find
0     Ω | u 0 | p 2   v dx + ( Ψ 1 ) 0   ( u 0 ;   v ) .
However, (−Ψ1)0 (u0; v) =  Ψ 1 0 (u0; −v) (a property of the Clarke derivative; see [22]), and thus,
Ω | u 0 | p 2   ( v ) dx     Ψ 0 ( u 0 ;   v )   v     X ;
that is,
μ 0   ( v ) :   =   Ω | u 0 | p 2 v dx     Ψ 1 0 ( u 0 ;   v )   v     X ,
μ0 = −Δp u0 ∈ ∂ Ψ1 (u0) and, using Proposition 5.5, −Δp u0 ∈ ∂ Ψ(u0). Since ∂ Ψ(u0) ⊂ (Lσ+1(Ω))* = L(σ+1)/σ(Ω), we obtain u0W2, (σ+1)/σ(Ω) and (5.19):
Δ p   u 0   ( x )     [ f ¯ ( x ,   u 0 ( x ) ) , f ¯ ( x ,   u 0 ( x ) ) ]   in   Ω   a . e .
Problem (∗∗). Let u0 be a critical point for Φ. We have
0 ∈ ∂ Φ(u0) ⊂ ∂ Q(u0) + ∂(−Ψ1)(u0)
since Φ = ( 5.22 ) Q − Ψ1, and we apply some rules of subdifferential calculus concerning finite sums (Section 2).
  Q ( u 0 ) = { Q ( u 0 ) } ,   where   Q ( u 0 ) ( v ) = i = 1 N Ω u 0 x i p 2 u 0 x i v x i dx = Δ p s u 0 ,   v .
Using (5.26) and a mentioned property of a function f Lipschitz around x0, we find
0   i = 1 N Ω u 0 x i p 2 u 0 x i v x i dx + ( Ψ 1 ) 0   ( u 0 ;   v ) .
However, (−Ψ1)0 (u0; v) =  Ψ 1 0 (u0; −v) (a property of the Clarke derivative, see [22]), and thus,
i = 1 N Ω u 0 x i p 2 u 0 x i ( v ) x i dx     Ψ 0   ( u 0 ; v )   v     X ;
that is
μ 0   ( v ) :   = i = 1 N Ω u 0 x i p 2 u 0 x i v x i dx   Ψ 1 0 ( u 0 ,   v )   v     X ,
μ0 = − Δ p s u0 ∈ ∂ Ψ1(u0) and, using Proposition 5.5, − Δ p s u0 ∈ ∂ Ψ(u0). Since ∂ Ψ(u0) ⊂ (Lσ+1(Ω))* = L(σ+1)/σ(Ω), we obtain u0W2, (σ+1)/σ(Ω) and (5.20),
Δ p s u 0   ( x )     [ f ¯ ( x ,   u 0 ( x ) ) , f ¯ ( x ,   u 0 ( x ) ) ]   in   Ω   a . e .
Remark 5.6.
Some applications to characterize the solution of the modeling given in [68] and solutions using this kind of definition for Dirichlet problems derived from the previously presented problems of the movement of glacier, nonlinear elastic membrane, pseudo-torsion problem or a nonlinear elastic membrane with the p-pseudo-Laplacian will be developed in Part two of this article.

5.3. Other Solutions

The results from this section have been obtained by the author in [18].

5.3.1. Basic Results

Let us now consider the two problems (∗) and (∗∗) from the above two subsections, but the boundary condition is now by Bu = 0 instead of u = 0. Once again, we take the function f as in Section 5.2.1 with the corresponding  f _ and  f ¯ , as was performed there.
Definition 5.5.
u from  W 2 , p (Ω), p > 1, is solution of (∗) and (∗∗) from this section if Bu = 0 on ∂ Ω in the sense of the trace (whose meaning is introduced above) and
Δ p   u ( x )     [ f ¯ ( x ,   u ( x ) ) , f ¯ ( x ,   u ( x ) ) ]   in   Ω   a . e .
and
Δ p s u ( x )     [ f ¯ ( x ,   u ( x ) ) , f ¯ ( x ,   u ( x ) ) ]   in   Ω   a . e .
respectively.
We now continue with some necessary results on Lipschitz functions and Palais-Smale type conditions. First, we provide some comments related to the Clarke derivative. From Definition 5.3, the Clarke derivative is:
f 0 ( x 0 ; v ) = inf V V ( x 0 ) r ( 0 , + ) sup x V t ( 0 , r ) f ( x + t v ) f ( x ) t .
Proposition 5.7.
Let f be Lipschitz around x0 with the constant L. Then
the function vf0 (x0; v) has values in R, is positive homogeneous and subadditive on X and
f 0 ( x 0 ;   v ) L v   v     X ;
f0 (x0; − v) = (− f)0 (x0; v) ∀vX, λ ≥ 0 ⇒ (λ f)0 (x0; v) = λ f0 (x0; v) ∀vX;
vf0 (x0; v) is Lipschitz on X with the constant L [69].
Proof. 
f0 (x0; v) ∈ R. For x near x0 and with t strictly positive near 0, we have
f ( x + t v ) f ( x ) t 1 t L t v = L v .
From (5.30), we obtain
f 0 ( x 0 ; v ) L   v     and   so   f 0   ( x 0 ;   v )     R .
Indeed, suppose ad absurdum that f0 (x0; v) > Lv∥, for instance. Then, with ∀V from V (x0) and ∀r from (0, +∞), we have
sup x V t ( 0 , r ) f ( x + t v ) f ( x ) t >   L v ,
in contradiction with (5.30).
vf0(x; v) is positive homogeneous. Since f0 (x0; 0) = 0, let λ > 0. Then, f0 (x0; λv) =  λ lim x x 0 λ t 0 + ¯ f ( x + t λ v ) f ( x ) λ t , etc.
vf0(x; v) is subadditive.
f ( x 0 ;   v 1 + v 2 ) = lim x x 0 t 0 + ¯ f ( x + t v 1 ) + t v 2 f ( x + t v 1 ) t + f ( x + t v 1 ) f ( x ) t   lim x x 0 t 0 + ¯ f ( x + t v 1 ) + t v 2 f ( x + t v 1 ) t + lim x x 0 t 0 + ¯ f ( x + t v 1 ) f ( x ) t .
As  lim x x 0 t 0 + ( x + t v 1 ) = x , in the third member of (5.32), the first term is equal to f0(x0; v2), and the second is equal to f0(x0; v1).
2°  f 0 ( x 0 ; v ) = lim x x 0 t 0 + ¯ f ( x t v ) f ( x ) t = u = x tv lim u x 0 t 0 + ¯ ( f ) ( u + t v ) ( f ) ( u ) t = (−f)0 (x0; v), since  lim x x 0 t 0 + ( x t v ) = x 0 . For the second statement, we use Formula (5.29).
3° Let v and w be arbitrary in X. For x near x0 and with t strictly positive near 0, we have
f ( x 0 + t v ) f ( x 0 ) f ( x 0 + t w ) f ( x 0 ) + L t v w .
Dividing by t and taking the upper limit for xx0 and t → 0+, one finds:
f 0 ( x 0 ; v ) f 0 ( x 0 ; w ) + L v w .
Exchanging v with w, we come to the desired conclusion. □
Proposition 5.8.
Let f be locally Lipschitz on X. The function Φ : X × X → R,
Φ(x; v) = f0 (x0; v)