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Article

Quasilinear Dirichlet Problems with Degenerated p-Laplacian and Convection Term

1
Department of Mathematics, University of Perpignan, 66860 Perpignan, France
2
Department of Mathematics and Computer Science, University of Palermo, 90123 Palermo, Italy
*
Author to whom correspondence should be addressed.
Mathematics 2021, 9(2), 139; https://doi.org/10.3390/math9020139
Submission received: 9 December 2020 / Revised: 29 December 2020 / Accepted: 6 January 2021 / Published: 11 January 2021
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)

Abstract

:
The paper develops a sub-supersolution approach for quasilinear elliptic equations driven by degenerated p-Laplacian and containing a convection term. The presence of the degenerated operator forces a substantial change to the functional setting of previous works. The existence and location of solutions through a sub-supersolution is established. The abstract result is applied to find nontrivial, nonnegative and bounded solutions.

1. Introduction

In this paper, we study the following quasilinear elliptic problem
div ( a ( x ) | u | p 2 u ) = f ( x , u , u ) in Ω u = 0 on Ω
on a bounded domain Ω R N with N 2 and p ( 1 , N ) . We assume that the boundary Ω of Ω is locally Lipschitzian, i.e., each point of Ω has a neighborhood whose intersection with Ω is the graph of a Lipschitz continuous function. Throughout the text we denote by | · | and · the standard Euclidean norm and scalar product on R N , respectively. A main feature of the present work is that the leading part of the equation in (P) is the differential operator in divergence form div ( a ( x ) | u | p 2 u ) known as the degenerated p-Laplacian with the weight a L loc 1 ( Ω ) . It is supposed that the function a be positive almost everywhere in Ω and that the following condition holds
a s L 1 ( Ω ) for some s N p , + 1 p 1 , + .
In the case where a ( x ) 1 we recover the ordinary p-Laplacian. Various examples of useful weights meeting the requirement (1) are given in [1]. For instance, it is obvious that defining a ( x ) = dist ( x , S ) for x Ω , with a nonempty closed subset S of Ω , one obtains a function a on Ω for which (1) holds true with any listed s.
The natural space associated with problem (P) is W 0 1 , p ( a , Ω ) that is the closure of C 0 ( Ω ) in the weighted Sobolev space W 1 , p ( a , Ω ) . In Section 2 we briefly survey the spaces W 1 , p ( a , Ω ) and W 0 1 , p ( a , Ω ) . The (negative) degenerated p-Laplacian with the weight a L loc 1 ( Ω ) under condition (1) is defined on W 0 1 , p ( a , Ω ) and takes values in the dual space ( W 0 1 , p ( a , Ω ) ) * .
Corresponding to the constant s in (1) we set
p s = p s s + 1
and the Sobolev critical exponent p s * = N p s N p s (we note that 1 p s < N ). There is a continuous embedding W 1 , p ( a , Ω ) L p s * ( Ω ) , so a continuous embedding L ( p s * ) ( Ω ) ( W 0 1 , p ( a , Ω ) ) * , where ( p s * ) stands for the Hölder conjugate of p s * , i.e., ( p s * ) = p s * p s * 1 . In order to handle problem (P) the idea is to arrange that the right-hand side f ( x , u , u ) become an element of L ( p s * ) ( Ω ) , which basically will be achieved through an adequate growth condition (see Hypothesis 1). We emphasize that the nonlinearity f ( x , u , u ) depends on the solution u and on its gradient u , which generally makes the variational methods be ineffective. Such a term f ( x , u , u ) is often called convection. It is expressed by means of a function f : Ω × R × R N R that is Carathéodory, i.e., f ( · , t , ξ ) is measurable for every ( t , ξ ) R × R N and f ( x , · , · ) is continuous for a.e. x Ω .
The goal of our work is to build a systematical approach to problem (P) via the method of sub-supersolution. It is for the first time when the method of sub-supersolution is implemented for problem (P) involving the degenerated p-Laplacian and related convection. In this respect, the functional setting is adapted to the novel situation of degenerated operators relying in an essential way on the associated exponent p s . For results on the method of sub-supersolution applied to problems exhibiting convection terms but not driven by degenerated differential operators we refer to [2,3,4,5,6].
By a (weak) solution to problem (P) we mean a function u W 0 1 , p ( a , Ω ) such that f ( x , u , u ) L ( p s * ) ( Ω ) and
Ω a ( x ) | u ( x ) | p 2 u ( x ) · v ( x ) d x = Ω f ( x , u ( x ) , u ( x ) ) v ( x ) d x , v W 0 1 , p ( a , Ω ) .
A function u ̲ W 1 , p ( a , Ω ) is called a subsolution for problem (P) if u ̲ 0 on Ω (in the sense of traces), f ( · , u ̲ ( · ) , u ̲ ( · ) ) L ( p s * ) ( Ω ) and
Ω a ( x ) | u ̲ ( x ) | p 2 u ̲ ( x ) · v ( x ) d x Ω f ( x , u ̲ ( x ) , u ̲ ( x ) ) v ( x ) d x
for all v W 0 1 , p ( a , Ω ) , v 0 a.e. in Ω . Symmetrically, a function u ¯ W 1 , p ( a , Ω ) is called a supersolution for problem (P) if u ¯ 0 on Ω (in the sense of traces), f ( · , u ¯ ( · ) , u ¯ ( · ) ) L ( p s * ) ( Ω ) and
Ω a ( x ) | u ¯ ( x ) | p 2 u ¯ ( x ) · v ( x ) d x Ω f ( x , u ¯ ( x ) , u ¯ ( x ) ) v ( x ) d x
for all v W 0 1 , p ( a , Ω ) , v 0 a.e. in Ω . Corresponding to a subsolution u ̲ and a supersolution u ¯ with u ̲ u ¯ a.e. in Ω we can consider the ordered interval
[ u ̲ , u ¯ ] = { w W 1 , p ( a , Ω ) : u ̲ w u ¯ } .
The following hypothesis for f ( x , s , ξ ) is adapted to an ordered sub-supersolution u ̲ u ¯ .
Hypothesis 1.
Given an ordered sub-supersolution u ̲ u ¯ for problem (P), the Carathéodory function f : Ω × R × R N R satisfies the growth condition
| f ( x , t , ξ ) | σ ( x ) + b | ξ | r for a . e . x Ω , for all t [ u ̲ ( x ) , u ¯ ( x ) ] , ξ R N ,
with a function σ L p s r ( Ω ) and constants b > 0 and r ( 0 , p s ( p s * ) ) .
According to Hypothesis 1 we have
f ( x , u , u ) L ( p s * ) ( Ω ) , u [ u ̲ , u ¯ ] ,
thus the integrals in the definitions above exist since
f ( x , u , u ) v L 1 ( Ω ) , u [ u ̲ , u ¯ ] , v W 0 1 , p ( a , Ω ) .
Under Hypothesis 1, our main result establishes the existence of a weak solution to problem (P) with the additional location property u [ u ̲ , u ¯ ] . We stress that this location property represents a significant qualitative information for the solution giving actually a priori estimates for it. As an application we prove the existence of a nontrivial nonnegative solution for a class of problems of type (P). The applicability of the stated result is demonstrated by an example.

2. Preliminary Material

The notation | Ω | stands for the Lebesgue measure of the bounded domain Ω in R N . In this section we discuss a few facts about the degenerated p-Laplacian entering problem (P). More details can be found in [1].
We note that (1) implies
a 1 p 1 L 1 ( Ω ) .
Indeed, it is seen that
Ω a ( x ) 1 p 1 d x = { a ( x ) < 1 } a ( x ) 1 p 1 d x + { a ( x ) 1 } a ( x ) 1 p 1 d x { a ( x ) < 1 } a ( x ) s d x + | Ω | <
since according to (1) one has s 1 p 1 and a s L 1 ( Ω ) .
The weighted Sobolev space W 1 , p ( a , Ω ) consists of all the functions u L p ( Ω ) for which a 1 p | u | L p ( Ω ) . It is endowed with the norm
u 1 , p , a = Ω | u | p d x + Ω a ( x ) | u | p d x 1 p
becoming a uniformly convex Banach space (due to the preceding property of the weight a ( x ) , see ([1], [Theorem 1.3])), thus reflexive, that contains C 0 ( Ω ) . The space W 0 1 , p ( a , Ω ) is the closure of C 0 ( Ω ) with respect to the norm · 1 , p , a .
There is an extensive literature devoted to the weighted Sobolev spaces including embeddings and traces related to different boundary value problems (see, e.g., [1,7,8]). The results depend strongly on what type of weight is used, generally attempting reduction to nonweighted spaces. As described below, under assumption (1), we can embed the space W 1 , p ( a , Ω ) into the ordinary Sobolev space W 1 , p s ( Ω ) , hence automatically having the trace (note the boundary Ω is Lipschitz). This fact is needed in the definition of the sub-supersolution.
From (1) it is known that s 1 p 1 , so one has p s 1 and the continuous embedding
W 1 , p ( a , Ω ) W 1 , p s ( Ω ) ,
which is relation (1.22) in [1]. More precisely, observing that p > p s , through Holder’s inequality and (1) we get
Ω | u | p s d x = Ω a p s p a p s p | u | p s d x Ω a s d x 1 s + 1 Ω a | u | p d x p s p
for all u W 1 , p ( a , Ω ) . As a consequence of the above inequality, we can endow W 0 1 , p ( a , Ω ) with an equivalent norm
u = Ω a ( x ) | u | p d x 1 p
for which it holds
u W 0 1 , p s ( Ω ) a s L 1 ( Ω ) 1 p s u .
The Sobolev embedding theorem ensures the continuous embedding W 0 1 , p s ( Ω ) L p s * ( Ω ) , with the critical exponent p s * = N p s N p s (note that 1 p s < N ). Hence there exists a constant T 0 > 0 such that
u L p s * ( Ω ) T 0 u W 0 1 , p s ( Ω ) , u W 0 1 , p s ( Ω ) .
The best embedding constant T 0 has been estimated by Talenti [9] as follows
T 0 π 1 2 N 1 p s p s 1 N p s 1 1 p s Γ 1 + N 2 Γ ( N ) Γ N p s Γ 1 + N N p s 1 N ,
where Γ is the Euler function
Γ ( t ) = 0 + z t 1 e z d z , t > 0 .
Moreover, by the Rellich–Kondrachov compact embedding theorem, if 1 r < p s * then the embedding W 0 1 , p s ( Ω ) L r ( Ω ) is compact.
By (7) and Hölder’s inequality we infer that
u L r ( Ω ) T 0 | Ω | p s * r p s * u W 0 1 , p s ( Ω )
for every u W 0 1 , p s ( Ω ) and r [ 1 , p s * ] . Combining (6) and (8) we arrive at
u L r ( Ω ) κ r u
for all u W 0 1 , p ( a , Ω ) and r [ 1 , p s * ] , with the constant
κ r = T 0 | Ω | p s * r p s * a s L 1 ( Ω ) 1 p s .
The (negative) degenerated p-Laplacian with the weight a L loc 1 ( Ω ) satisfying condition (1) is the operator A : W 0 1 , p ( a , Ω ) ( W 0 1 , p ( a , Ω ) ) * defined by
A ( u ) , v = Ω a ( x ) | u | p 2 u · v d x , u , v W 0 1 , p ( a , Ω ) .
We readily check that the operator A in (10) is well defined noticing by means of Hölder’s inequality that for all u , v W 0 1 , p ( a , Ω ) it holds
Ω a ( x ) | u | p 2 u · v d x Ω a ( x ) p 1 p | u | p 1 a ( x ) 1 p | v | d x Ω a ( x ) | u | p d x p 1 p Ω a ( x ) | v | p d x 1 p < .
Important properties of the operator A introduced in (10) are listed in the statement below.
Proposition 1.
Assume that the measurable function a : Ω R satisfies condition (1). Then the (negative) degenerated p-Laplacian A : W 0 1 , p ( a , Ω ) ( W 0 1 , p ( a , Ω ) ) * defined by (10) has the following properties:
(i) 
A is a bounded operator in the sense that it maps bounded sets to bounded sets;
(ii) 
A is a coercive operator, i.e.,
lim u A u , u u = + ;
(iii) 
A is a strictly monotone operator, i.e.,
A u A v , u v > 0 , u v ;
(iv) 
A has the S + property meaning that any sequence { u n } W 0 1 , p ( a , Ω ) that satisfies u n u in W 0 1 , p ( a , Ω ) and
lim sup n A ( u n ) , u n u 0
is strongly convergent.
Proof.
( i ) From (10) and (11) we infer that
| A u , v | = Ω a ( x ) | u | p 2 u · v d x u p 1 v , u , v W 0 1 , p ( a , Ω ) .
We obtain
A u ( W 0 1 , p ( a , Ω ) ) * = sup v W 0 1 , p ( a , Ω ) , v 1 | A u , v | u p 1 , u W 0 p ( a , Ω ) ,
whence A is bounded.
( i i ) By (10) we have that
A u , u = Ω a ( x ) | u | p d x = u p , u W 0 1 , p ( a , Ω ) .
Taking into account that p > 1 , it follows that the operator A is coercive.
( i i i ) In view of the strict monotonicity of the mapping ξ | ξ | p 2 ξ on R N , it turns out
A u A v , u v = Ω a ( x ) | u | p 2 u | v | p 2 v · ( u v ) d x > 0 , u v ,
so A is a strictly monotone operator.
( i v ) Let a sequence { u n } W 0 1 , p ( a , Ω ) satisfy u n u in W 0 1 , p ( a , Ω ) and (12). Using the monotonicity of the operator A and (12) we have
lim n A ( u n ) A ( u ) , u n u = 0 .
Through Hölder’s inequality we obtain
A ( u n ) A ( u ) , u n u = Ω a ( x ) | u n | p 2 u n | u | p 2 u · ( u n u ) d x = Ω a ( x ) | u n | p d x Ω a ( x ) | u n | p 2 u n · u d x Ω a ( x ) | u | p 2 u · u n d x + Ω a ( x ) | u | p d x Ω a ( x ) | u n | p d x Ω a ( x ) | u n | p d x p 1 p Ω a ( x ) | u | p d x 1 p Ω a ( x ) | u | p d x p 1 p Ω a ( x ) | u n | p d x 1 p + Ω a ( x ) | u | p d x = ( u n u ) ( u n p 1 u p 1 ) 0 ,
from which we find that lim n + u n = u . Due to the uniform convexity of W 0 1 , p ( a , Ω ) it follows that u n u in W 0 1 , p ( a , Ω ) , thus completing the proof. □
We also need the first eigenvalue λ 1 of the operator A : W 0 1 , p ( a , Ω ) ( W 0 1 , p ( a , Ω ) ) * in (10). Precisely, λ 1 > 0 is the least (positive) number for which the equation
div ( a ( x ) | u | p 2 u ) = λ 1 | u | p 2 u in Ω u = 0 on Ω
admits a nontrivial solution called eigenfunction corresponding to the first eigenvalue λ 1 . A solution to (13) is understood in the weak sense, i.e., u W 0 1 , p ( a , Ω ) satisfying
Ω a ( x ) | u ( x ) | p 2 u ( x ) · v ( x ) d x = λ 1 Ω | u ( x ) | p 2 u ( x ) v ( x ) d x , v W 0 1 , p ( a , Ω ) .
It is known that there exists an eigenfunction u 1 W 0 1 , p ( a , Ω ) corresponding to the first eigenvalue λ 1 such that u 1 ( x ) 0 for a.e. x Ω , u 1 ¬ 0 , and u 1 L ( Ω ) . For the proofs of these properties we refer to ([1], Chapter 3).

3. Main Results

Our main abstract result provides the existence of a solution to problem (P) and its location within the ordered interval determined by a sub-supersolution.
Theorem 1.
Let the weight a L loc 1 ( Ω ) fulfill the requirement (1) and assume that Hypothesis 1 for a subsolution u ̲ and a supersolution u ¯ with u ̲ u ¯ a.e. is satisfied. Then problem (P) possesses at least a solution u W 0 1 , p ( a , Ω ) with the location property u ̲ u u ¯ for a.e. x Ω .
Proof. 
By means of the given sub-supersolution u ̲ u ¯ for problem (P), we introduce some related mappings. The cut-off function for problem (P), we introduce some related mappings. The cut-off function π : Ω × R R is defined by
π ( x , t ) = ( u ̲ ( x ) t ) r p s r if t < u ̲ ( x ) 0 if u ̲ ( x ) t u ¯ ( x ) ( t u ¯ ( x ) ) r p s r if t > u ¯ ( x ) ,
where s and r are the constants given in (1) and Hypothesis 1. Using (14) in conjunction with u ̲ , u ¯ L p s * ( Ω ) enables us to find that
| π ( x , t ) | c | t | r p s r + ϱ ( x ) for a . e . x Ω , all t R ,
with a constant c > 0 and a function ϱ L p s * ( p s r ) r ( Ω ) . Moreover, proceeding as in [4], we can establish that
Ω π ( x , u ( x ) ) u ( x ) d x b 1 u L p s p s r ( Ω ) p s p s r b 2 for all u W 0 1 , p ( a , Ω ) ,
with positive constants b 1 and b 2 .
In view of (15), the Nemytskij operator u π ( · , u ( · ) ) generated by π maps continuously L p s * ( Ω ) to L p s * ( p s r ) r ( Ω ) . Therefore, the mapping Π : W 0 1 , p ( a , Ω ) ( W 0 1 , p ( a , Ω ) ) * defined by
Π ( u ) , v = Ω π ( x , u ) v d x , u , v W 0 1 , p ( a , Ω )
is completely continuous. This is true because the inclusion L p s * ( p s r ) r ( Ω ) ( W 0 1 , p ( a , Ω ) ) * is compact being the adjoint of the compact inclusion W 0 1 , p ( a , Ω ) L p s * ( p s r ) p s * ( p s r ) r ( Ω ) (note that p s * ( p s r ) p s * ( p s r ) r < p s * owing to the assumption r ( 0 , p s ( p s * ) ) in Hypothesis 1).
Hypothesis 1 and (5) imply that the Nemytskij operator u f ( · , u ( · ) , u ( · ) ) maps continuously [ u ̲ , u ¯ ] W 1 , p ( a , Ω ) to L p s r ( Ω ) with r ( 0 , p s ( p s * ) ) . Composing the preceding Nemytskij operator with the inclusion L p s r ( Ω ) ( W 0 1 , p ( a , Ω ) ) * , which is compact because it is the adjoint operator of the compact inclusion W 0 1 , p ( a , Ω ) L p s p s r ( Ω ) (note that p s p s r < p s * since r ( 0 , p s ( p s * ) ) in Hypothesis 1), we obtain a completely continuous mapping N f : [ u ̲ , u ¯ ] ( W 0 1 , p ( a , Ω ) ) * given by
N f ( u ) , v = Ω f ( x , u ( x ) , u ( x ) ) v ( x ) d x
for all u [ u ̲ , u ¯ ] and v W 0 1 , p ( a , Ω ) .
We also make use of the truncation operator T : W 0 1 , p ( a , Ω ) W 1 , p ( a , Ω ) given by
( T u ) ( x ) = u ̲ ( x ) if u ( x ) < u ̲ ( x ) u ( x ) if u ̲ ( x ) u ( x ) u ¯ ( x ) u ¯ ( x ) if u ( x ) > u ¯ ( x )
for all u W 0 1 , p ( a , Ω ) and a.e. x Ω . It is a continuous and bounded mapping (in the sense that it maps bounded sets to bounded sets). Notice that its range lies in [ u ̲ , u ¯ ] , so T can be composed with the operator N f .
Now we consider for every λ > 0 the operator A λ : W 0 1 , p ( a , Ω ) ( W 0 1 , p ( a , Ω ) ) * defined by
A λ = A + λ Π N f T .
Explicitly, it reads as
A λ ( u ) , v = Ω a ( x ) | u | p 2 u · v d x + λ Ω π ( x , u ) v d x Ω f ( x , T u , ( T u ) ) v d x for all u , v W 0 1 , p ( a , Ω ) .
From Proposition 1 ( i ) it is known that the operator A : W 0 1 , p ( a , Ω ) ( W 0 1 , p ( a , Ω ) ) * is bounded, while the above comments demonstrate that the operators Π , N f and T are all of them bounded. Therefore from (18) we infer that the operator A λ : W 0 1 , p ( a , Ω ) ( W 0 1 , p ( a , Ω ) ) * is bounded.
We claim that A λ : W 0 1 , p ( a , Ω ) ( W 0 1 , p ( a , Ω ) ) * is a pseudomonotone operator. In this respect, let a sequence { u n } W 0 1 , p ( a , Ω ) satisfy u n u in W 0 1 , p ( a , Ω ) and
lim sup n A λ ( u n ) , u n u 0 .
The sequence { Π ( u n ) } is bounded in L p s * ( p s r ) r ( Ω ) , while u n u in L p s * ( p s r ) p s * ( p s r ) r ( Ω ) by the compact embedding W 0 1 , p ( a , Ω ) L p s * ( p s r ) p s * ( p s r ) r ( Ω ) , thus
lim n Π ( u n ) , u n u = 0 .
The sequence { N f T ( u n ) } is bounded in L p s r ( Ω ) , while u n u in L p s p s r ( Ω ) by the compact embedding W 0 1 , p ( a , Ω ) L p s p s r ( Ω ) , producing
lim n N f T ( u n ) , u n u = 0 .
Consequently, complying with (18), we see that (20) reduces to (12). This, in conjunction with the weak convergence u n u , enables us to apply Proposition 1 ( i v ) ensuring that the strong convergence u n u in W 0 1 , p ( a , Ω ) holds.
From the strong convergence a ( · ) 1 p u n ( · ) a ( · ) 1 p u ( · ) in ( L p ( Ω ) ) N it follows the strong convergence a ( · ) p 1 p | u n ( · ) | p 2 u n ( · ) a ( · ) p 1 p | u ( · ) | p 2 u ( · ) in ( L p p 1 ( Ω ) ) N . This amounts to saying that A u n A u in ( W 0 1 , p ( a , Ω ) ) * since
A u n , v = Ω a ( x ) | u n | p 2 u n · v d x Ω a ( x ) | u | p 2 u · v d x = A u , v , v W 0 1 , p ( a , Ω ) .
Again, from the strong convergence a ( · ) 1 p u n ( · ) a ( · ) 1 p u ( · ) in ( L p ( Ω ) ) N we infer that
A u n , u n = Ω a ( x ) | u n | p d x Ω a ( x ) | u | p d x = A u , u
as n . Taking into account the continuity of the mappings Π and N f T , we have
A λ u n , v A λ u , v , v W 0 1 , p ( a , Ω ) ,
and
A λ u n , u n A λ u , u
as n , for every λ > 0 . We can conclude that A λ : W 0 1 , p ( a , Ω ) ( W 0 1 , p ( a , Ω ) ) * is a pseudomonotone operator (see, e.g., ([2], Definition 2.97)).
The next step in the proof is to show that the operator A λ : W 0 1 , p ( a , Ω ) ( W 0 1 , p ( a , Ω ) ) * is coercive provided λ > 0 is large enough. Taking advantage of the fact that T u [ u ̲ , u ¯ ] whenever u W 0 1 , p ( a , Ω ) , let us note by (16), (19) and Hypothesis 1 that
A λ ( u ) , u = A ( u ) , u + λ Ω π ( x , u ) u d x Ω f ( x , T u , ( T u ) ) u d x u p + λ ( b 1 u L p s p s r ( Ω ) p s p s r b 2 ) σ L p s r ( Ω ) u L p s p s r ( Ω ) b Ω | ( T u ) | r | u | d x
for all u W 0 1 , p ( a , Ω ) . Now we estimate the last term in (21) based on the fact that by (5) we know that u ( L p s ( Ω ) ) N , and so ( T u ) ( L p s ( Ω ) ) N . Using the definition of T u in (17), Hölder’s inequality and the continuous embedding in (9) it turns out that
Ω | ( T u ) | r | u | d x = { u ̲ u u ¯ } | u | r | u | d x + { u < u ̲ } | u ̲ | r | u | d x + { u > u ¯ } | u ¯ | r | u | d x Ω | u | r | u | d x + c 1 u , u W 0 1 , p ( a , Ω ) ,
with a constant c 1 > 0 . We can insert the preceding inequality in (21) to derive
A λ ( u ) , u u p + λ ( b 1 u L p s p s r ( Ω ) p s p s r b 2 ) c 2 u b Ω | u | r | u | d x ,
with a constant c 2 > 0 . The Hölder’s and Young’s inequalities in conjunction with embedding (5) imply
Ω | u | r | u | d x u L p s ( Ω ) r u L p s p s r ( Ω ) c 3 u p s + c 4 u L p s p s r ( Ω ) p s p s r ,
with constants c 3 > 0 and c 4 > 0 . Then (22) entails
A λ ( u ) , u u p + λ ( b 1 u L p s p s r ( Ω ) p s p s r b 2 ) c 2 u b ( c 3 u p s + c 4 u L p s p s r ( Ω ) p s p s r )
for all u W 0 1 , p ( a , Ω ) . Recalling from (16) that b 1 > 0 , we can choose λ > 0 so large to have λ b 1 > b c 4 . Hence due to p > p s 1 (see (1)), (23) yields the coercivity of A λ , i.e.,
lim u + A λ ( u ) , u u = + .
We have shown that the nonlinear operator A λ : W 0 1 , p ( a , Ω ) ( W 0 1 , p ( a , Ω ) ) * is bounded, pseudomonotone and coercive provided λ > 0 is sufficiently large. Therefore, for such an A λ we can apply the main theorem of pseudomonotone operators (see, e.g., ([2], Theorem 2.99)) ensuring that there exists a solution u W 0 1 , p ( a , Ω ) to the equation
A λ ( u ) = 0 .
Fix an admissible λ > 0 as pointed out above. We are going to prove that u W 0 1 , p ( a , Ω ) resolving (24) is a weak solution of the original problem (P), which means that (2) is satisfied. To this end, notice that (19) and (24) yield
Ω a ( x ) | u ( x ) | p 2 u ( x ) · v ( x ) d x + λ Ω π ( x , u ) v d x = Ω f ( x , T u , ( T u ) ) v d x for all v W 0 1 , p ( a , Ω ) .
We proceed by comparing u with the subsolution u ̲ and supersolution u ¯ postulated in Hypothesis 1. We claim that u u ¯ a.e. in Ω . Towards this, it can be readily checked that ( u u ¯ ) + = max { u u ¯ , 0 } W 0 1 , p ( a , Ω ) , where the condition u ¯ 0 on Ω in the sense of traces is essentially used. Thus, we can insert v = ( u u ¯ ) + in (25) and (4) which gives
Ω a ( x ) | u ( x ) | p 2 u ( x ) · ( u u ¯ ) + ( x ) d x + λ Ω π ( x , u ( x ) ) ( u u ¯ ) + ( x ) d x = Ω f ( x , T u ( x ) , ( T u ) ( x ) ) ( u u ¯ ) + ( x ) d x
and
Ω a ( x ) | u ¯ ( x ) | p 2 u ¯ ( x ) · ( u u ¯ ) + ( x ) d x Ω f ( x , u ¯ ( x ) , u ¯ ( x ) ) ( u u ¯ ) + ( x ) d x .
From (26) and (27), by subtraction we are led to
Ω a ( x ) | u ( x ) | p 2 u ( x ) | u ¯ ( x ) | p 2 u ¯ ( x ) · ( u u ¯ ) + ( x ) d x + λ Ω π ( x , u ( x ) ) ( u u ¯ ) + ( x ) d x Ω f ( x , T u ( x ) , ( T u ) ( x ) ) f ( x , u ¯ ( x ) , u ¯ ( x ) ) ( u u ¯ ) + ( x ) d x .
By (14), (17), and the preceding inequality we get
{ u > u ¯ } a ( x ) | u ( x ) | p 2 u ( x ) | u ¯ ( x ) | p 2 u ¯ ( x ) · ( u u ¯ ) d x + λ { u > u ¯ } ( u ( x ) u ¯ ( x ) ) p s p s r d x { u > u ¯ } f ( x , T u , ( T u ) ) f ( x , u ¯ , u ¯ ) ( u u ¯ ) d x = 0 .
Since the function a ( x ) is positive almost everywhere in Ω and the mapping ξ | ξ | p 2 ξ on R N is monotone, we arrive at
{ u > u ¯ } ( u ( x ) u ¯ ( x ) ) p s p s r d x 0 .
Therefore, the Lebesgue measure of the set { u > u ¯ } is zero, i.e., u u ¯ a.e. in Ω .
Similarly, we can prove that u ̲ u a.e. in Ω . Specifically, relying on the condition u ̲ 0 on Ω (in the sense of traces), it holds ( u ̲ u ) + = max { u ̲ u , 0 } W 0 1 , p ( a , Ω ) , which allows us to test (25) and (3) with v = ( u ̲ u ) + W 0 1 , p ( a , Ω ) . This results in
Ω a ( x ) | u ( x ) | p 2 u ( x ) · ( u ̲ u ) + ( x ) d x + λ Ω π ( x , u ( x ) ) ( u ̲ u ) + ( x ) d x = Ω f ( x , T u ( x ) , ( T u ) ( x ) ) ( u ̲ u ) + ( x ) d x
and
Ω a ( x ) | u ̲ ( x ) | p 2 u ̲ ( x ) · ( u ̲ u ) + ( x ) d x Ω f ( x , u ̲ ( x ) , u ̲ ( x ) ) ( u ̲ u ) + ( x ) d x .
Arguing as before, we deduce from (28), (29), (14), and (17) the following estimate
{ u ̲ > u } a ( x ) | u ̲ ( x ) | p 2 u ̲ ( x ) | u ( x ) | p 2 u ( x ) · ( u ̲ u ) d x + λ { u ̲ > u } ( u ̲ ( x ) u ( x ) ) p s p s r d x Ω ( f ( x , u ̲ , u ̲ ) f ( x , T u , ( T u ) ) ( u ̲ u ) + d x = { u ̲ > u } ( f ( x , u ̲ , u ̲ ) f ( x , T u , ( T u ) ) ) ( u ̲ u ) + d x = 0 .
At this point, the positivity of the function a ( x ) on Ω and the monotonicity of the mapping ξ | ξ | p 2 ξ on R N confirm that
{ u ̲ > u } ( u ̲ ( x ) u ( x ) ) p s p s r d x 0 ,
from which we can readily derive that u ̲ u a.e in Ω .
Based on the enclosure property u ̲ u u ¯ a.e. in Ω , it follows through (17) that T ( u ) = u and through (14) that Π ( u ) = 0 . As a result, (25) takes the form of (2), thus the proof is complete. □
Now we present an application of Theorem 1 describing how the existence of a nontrivial nonnegative solution can be established by effectively determining a sub-supersolution. In the sequel, by λ 1 we denote the first eigenvalue of problem (13) (see Section 2).
Theorem 2.
Let the weight a L loc 1 ( Ω ) fulfill the requirement (1). Assume that the Carathéodory function f : Ω × R × R N R satisfies the conditions:
(j) 
there is a constant μ > 0 such that
λ 1 t p 1 f ( x , t , ξ ) for a . e . x Ω , all t [ 0 , μ ] , ξ R N ;
(jj) 
there is a constant C > 0 such that
f ( x , C , 0 ) 0 for a . e . x Ω ;
(jjj) 
there are a function σ L p s r ( Ω ) and constants b > 0 and r ( 0 , p s ( p s * ) ) such that
| f ( x , t , ξ ) | σ ( x ) + b | ξ | r for a . e . x Ω , all t [ 0 , C ] , ξ R N .
Then problem (P) has a nondegenerate, nonnegative and bounded weak solution u W 0 1 , p ( a , Ω ) satisfying the estimate u C .
Proof. 
Our goal is to apply Theorem 1 by constructing an appropriate sub-supersolution. In order to determine a subsolution, we use an eigenfunction u 1 W 0 1 , p ( a , Ω ) corresponding to the first eigenvalue λ 1 of problem (13) with the properties u 1 ( x ) 0 for a.e. x Ω , u 1 ¬ 0 , and u 1 L ( Ω ) as mentioned in Section 2. Then we choose an ε > 0 sufficiently small to verify
ε u 1 ( x ) μ for a . e . x Ω ,
where μ is the positive constant postulated in assumption ( j ) . Then assumption ( j ) implies
λ 1 ( ε u 1 ) p 1 f ( x , ε u 1 , ( ε u 1 ) ) for a . e . x Ω .
For a possibly smaller ε > 0 we can suppose
ε u 1 ( x ) C for a . e . x Ω ,
with C > 0 in assumption ( j j ) .
Let us fix an ε > 0 for which (30) and (32) are fulfilled. We claim that u ̲ = ε u 1 is a subsolution to problem (P). Indeed, by (13) with u 1 in place of u and (31) we note that
Ω a ( x ) | u ̲ ( x ) | p 2 u ̲ ( x ) · v ( x ) d x = ε p 1 λ 1 Ω u 1 ( x ) p 1 v ( x ) d x Ω f ( x , ε u 1 ( x ) , ( ε u 1 ) ( x ) ) v ( x ) d x = Ω f ( x , u ̲ ( x ) , u ̲ ( x ) ) v ( x ) d x
for all v W 0 1 , p ( a , Ω ) , v 0 a.e. in Ω , thereby proving the claim.
Next we claim that the constant function u ¯ = C , with C > 0 in assumption ( j j ) , is a supersolution to problem (P). Accordingly, from assumption ( j j ) we find that
Ω a ( x ) | u ¯ ( x ) | p 2 u ¯ ( x ) · v ( x ) d x = 0 Ω f ( x , C , 0 ) v ( x ) d x = Ω f ( x , u ¯ ( x ) , u ¯ ( x ) ) v ( x ) d x
for all v W 0 1 , p ( a , Ω ) , v 0 a.e. in Ω , which proves the claim.
It is clear from (32) that u ̲ ( x ) u ¯ ( x ) for a.e. in Ω . Assumption ( j j j ) ensures that the growth condition required in Hypothesis 1 of Theorem 1 holds true. Therefore, all the hypotheses of Theorem 1 are verified, which permits the conclusion that there exists a solution u W 0 1 , p ( a , Ω ) of problem (P) within the ordered interval [ u ̲ , u ¯ ] . Since the function u ̲ = ε u 1 is nontrivial and nonnegative, and u u ̲ , we have that u is nontrivial and nonnegative, whereas u [ u ̲ , u ¯ ] renders the boundedness of u and the a priori estimate u C . The proof is complete. □
We end the paper with a simple example for which Theorem 2 applies.
Example 1.
Fix a positive weight a L loc 1 ( Ω ) with the property (1). Let the function f : Ω × R × R N R be defined by
f ( x , t , ξ ) = 0 if t < 0 t p 1 ( ρ ( x ) + | ξ | r ) if 0 t 1 ( 2 t ) ( ρ ( x ) + | ξ | r ) if t > 1 ,
with some r [ 1 , p s ( p s * ) ) and ρ L ( Ω ) satisfying ρ ( x ) λ 1 for a.e. x Ω . It follows that f is a Carathéodory function for which conditions ( j ) ( j j j ) in Theorem 2 are verified. Precisely, condition ( j ) holds with μ = 1 because ρ ( x ) λ 1 , condition ( j j ) holds with C = 2 , and condition ( j j j ) is fulfilled with the given r. Hence Theorem 2 applies to problem (P) whose equation has the right-hand side expressed with the function f ( x , t , ξ ) given above.

Author Contributions

All authors (D.M. and E.T.) contributed equally to this paper. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Acknowledgments

The last author is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM). The paper is partially supported by PRIN 2017—Progetti di Ricerca di rilevante Interesse Nazionale, Nonlinear Differential Problems via Variational, Topological and Set-valued Methods.

Conflicts of Interest

The authors declare no conflict of interest.

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Motreanu, D.; Tornatore, E. Quasilinear Dirichlet Problems with Degenerated p-Laplacian and Convection Term. Mathematics 2021, 9, 139. https://doi.org/10.3390/math9020139

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Motreanu D, Tornatore E. Quasilinear Dirichlet Problems with Degenerated p-Laplacian and Convection Term. Mathematics. 2021; 9(2):139. https://doi.org/10.3390/math9020139

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Motreanu, Dumitru, and Elisabetta Tornatore. 2021. "Quasilinear Dirichlet Problems with Degenerated p-Laplacian and Convection Term" Mathematics 9, no. 2: 139. https://doi.org/10.3390/math9020139

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