1. Introduction
In this paper, we consider a class of Dirichlet boundary value problems whose prototype is
where
is a bounded open subset of
,
,
,
is the so-called
p-Laplace operator, and
. We assume that
is a positive constant, and the coefficient
c and the datum
f are functions belonging to suitable Lorentz spaces.
When
, equations of this type are sometimes referred to as stationary viscous Hamilton–Jacobi equations, and appear in connection with stochastic equations subject to control processes (see, for example, [
1,
2] and the references therein).
The main feature of Problem (
1) is the presence of a first-order term, which grows like a
q-power of the gradient of
u with
, and the presence of a zero-order term with a coefficient
c.
Some comments on the restriction
are in order. We begin by dealing with the existence for Problem (
1). Existence results in the case where
and
are well known. Indeed, in this case, an a priori estimate for every solution
u of (
1) in
can be obtained by using
u as a test function, and existence follows by classical theory of pseudomonotone operators due to J. Leray and J.-L. Lions (see, for example, [
3]). The same occurs when
,
is small enough and
, since in this case, the operator
is coercive. If
is large, this is not the case. However, this problem has been solved in the linear case in [
4], and in the nonlinear case by various authors (see, for example, [
5] and the references therein); the approach in some sense allows one to reduce the problem to a finite sequence of problems with coefficient
having a norm in a suitable Lebesgue space small enough and to again obtain a priori estimates.
The limit case
has been studied by many authors, by proving a priori estimates as, for example, in [
6] and the references therein. In those papers, the authors prove a priori estimates and, therefore, the existence of solutions in
, which are not bounded in general, but satisfy a further regularity; the approach used in this case is based on a change of unknown function
, which allows one to cancel the term
.
In this paper, we focus our attention on the case
. When the study of the existence of solutions to Problem (
1) is faced, some necessary conditions are required on the data, as shown in [
7,
8] for
and
. These necessary conditions, when the datum is an element of a Lebesgue space, consist of the fact that the datum
f belongs to the
with
and the norm
is sufficiently small. Moreover, it is well-known that if
,
and
c is a constant, the existence and uniqueness of a weak solution are guaranteed if
, where
is the first eigenvalue of the Laplacian operator (see, for example, [
9], where the more general case with
is considered).
In order to explain the difficulties related to obtaining a priori estimates, let us use
u as test function in (
1) when
,
and
, where
. This leads to the following estimate:
where
is the best constant in the Sobolev embedding of
in
. Unfortunately, since
, this inequality does not provide any a priori bound for the gradient of solution
u, even if a size condition on
is required.
To overcome the difficulties in deriving an estimate from (
2), a method based on a continuity argument has been introduced in [
10], while a comparison result with symmetrization techniques has been used in [
11] when
. Sharp assumptions on the summability of the datum
f (which depend on
q) and on its smallness are also given in [
11].
In order to prove a priori estimates for weak solutions, the presence of a zero-order term of the type
does not allow us to use the continuity methods quoted above, nor to obtain a comparison result by using symmetrization techniques. A first attempt to study the existence of weak solutions to Problem (
1) has been made in [
12], when the datum
f is an element of the dual space
of the Sobolev space
, by proving the existence (without uniqueness) of a fixed point for a suitable operator.
As far as uniqueness of weak solutions concerns, a classical counterexample (recalled in
Section 4), with
, shows that uniqueness does not hold in general in the class of weak solutions and for values of
.
Nevertheless, various uniqueness results have been proven when the datum
f belongs to the dual space
in the case where the equation does not involve lower order terms of the type
and
, or in the case where the lower order term has natural growth with respect to the gradient, i.e., when
under further regularity assumptions on the weak solutions (see, for example, [
13]).
Uniqueness for the operator with a “superlinear” term of the type
are contained in [
14], where two different uniqueness results, according to the values of
p, namely
and
, are proven when
. The difference between the two cases
and
is due to the assumptions that are made on the operator in (
1); that is, the “strong monotonicity”. This means that uniqueness can be proven for operators like
when
, and both for operators
and for operators
when
. In both cases,
and
, the results give the uniqueness of the weak solution to Problem (
1) under the assumptions that
, where the value of
, is given by
The main difficulty which arises in proving uniqueness results are due to the lower order term
and to the fact that, in general,
is large so that the operator is not coercive. In this case, the idea by Bottaro and Marina [
4] was used, which consists of considering a coefficient
belonging to a suitable Lebesgue space
and reducing to the case where
is small enough.
In [
15], the uniqueness of solutions of (
1), with
and
or
c constant with
, is proven for a class of regular solutions that is consistent with the existence results proven in [
10]; that is, for weak solutions
u, such that
with
. Uniqueness is proven when
for the whole interval of values of
q, i.e.,
, but it is basic to use convexity of the function
or a linearization process, which assures a higher summability of the gradient of the solutions, for low values of
q given by
.
These results have been extended to more general classes of nonlinear elliptic equations (see, for example, [
16,
17,
18] and the references therein). In all these papers, uniqueness results have been proven for strongly monotone operators, and partial intervals of values of
p and of
q are considered. As far as we know, uniqueness results for operators having both first- and zero-order terms for general coefficient
c are not available in the literature.
Further questions concerning the existence, multiplicity or regularity of solutions related to Problem (
1) can be found in [
19,
20,
21,
22].
The main result of this paper is given by Theorem 1, stated in
Section 2. It states both the existence and uniqueness for any value of
p,
, for values of
q given by
, under smallness assumptions on the datum
f and the coefficient
c. As pointed out, the upper bound on the value
q is natural when we consider the example given in
Section 4, while the smallness assumption on
f is due to the fact that a “superlinear term”
with
appears. In
Section 3, we prove our existence and uniqueness result by applying the Schauder fixed point theorem. Once the operator
A is defined in such a way that its fixed points are weak solution to (
1), the main difficulty consists of proving that
A maps a ball
of
in itself for a suitable value of the radius
R; actually, this value is a positive zero of a suitable function
G given in the statement of the theorem. Actually, Theorem 1 improves the result proven in [
12], where just the existence was proven (without uniqueness) under stronger assumptions on the summability of
f and
c. Moreover, Theorem 1 gives the uniqueness of a weak solution in a suitable ball of the Sobolev space
for monotone operators
(and no more for strongly monotone operators), depending also on
u. In
Section 4, we explicitly show the effects of the different approaches in order to prove uniqueness for Problem (
1), and we highlight the novelty of the uniqueness result given by Theorem 1. Indeed, the “linearization process”, used to prove Theorem 2, can be just applied to operators satisfying further structural assumptions, such as strong monotonicity and, therefore, it gives uniqueness for a less general class of operators and for smaller interval of values of
p.
2. Preliminaries and Statement of Main Result
In this section, we firstly recall a few properties of Lorentz spaces, and then we state the main result of the paper.
Let us begin by recalling some properties of rearrangements. If
u is a measurable function defined in
, and
is its distribution function, then
is the decreasing rearrangement of
u, and
is the increasing rearrangement of
u.
Here, denotes the n-dimensional Lebesgue measure of any measurable set E.
If
is the measure of the unit ball of
, and
is the ball of
centered at the origin with the same measure as
,
denote the spherically decreasing and increasing rearrangements of
u, respectively.
For any
, the Lorentz space
is the collection of all measurable functions
u, such that
is finite, where we use the notation
if
;
if
.
These spaces give, in some sense, a refinement of the usual Lebesgue spaces. Indeed,
and
is the
-weak Marcinkiewicz space. The following embeddings hold true (see, for example, [
23]):
and
Moreover, the following inequalities hold:
and
where
For
,
, the generalized Hölder inequality holds true
for every
and
.
Here, for every , denotes the Hölder exponent .
More generally, if
and
with
then the following inequality holds true (cf., for example, [
23]):
As pointed out in the Introduction, the aim of this paper is to prove existence and uniqueness of weak solutions to the following more general class of Dirichlet boundary value problems:
where
is a bounded open subset of
,
,
,
are Carathéodory functions which satisfy the ellipticity condition
the monotonicity condition
and the growth conditions
with
, for every
and for every
. Moreover, the datum
f and the coefficient
c are measurable functions in suitable Lorentz spaces.
The main result of the paper is given by the following theorem, which state both existence and uniqueness for weak solutions satisfying a suitable a priori estimates in . Actually, such a uniqueness result is a consequence of the fact that the weak solution is obtained as a fixed point of a suitable map, and such a fixed point is unique by the Schauder fixed point theorem.
Theorem 1. Assume that (13)–(17) hold true withand If the norm of c in and the norm of f in are sufficiently small, that is,where S denotes the best constant in the embedding ,then there exists a unique weak solution to Problem (12), such thatwhere is the first positive zero of the functionfor any . Remark 1. Some comments on the smallness conditions (19) on the coefficient c and on the datum f are in order. Under Assumption (19), the function defined in (22) has a positive maximum , whereand Therefore, the smallness assumptions (19) guarantee that and are positive. Moreover, is the unique critical point with for and for . Since and , this allows us to assert that has two positive zeros. Remark 2. Let us explicitly remark that Theorem 1 gives an answer to the question of existence and uniqueness for Problem (1), since this model problem belongs to the class of operators considered in (12). 3. Proof of the Main Result
In this section, we prove our main existence and uniqueness result for a weak solution to Problem (
12), as stated in Theorem 1 above.
Proof. Consider the ball
defined by
where
is the first positive zero of the function
G. □
Consider the mapping
defined by
where
is the unique weak solution to problem
that is,
for any
.
The existence and uniqueness of such a solution
u is followed by classical results (see, for example, [
3]). Indeed, since
,
and
, then
. Moreover, since
, also
. Hence, the right-hand side of the equation in (
25) belongs to
and by Sobolev embedding of
in
, it is an element of the dual space
. Therefore, by classical results, the solution
u exists, and is unique.
Now, the proof proceeds by showing that the Schauder fixed point theorem can be applied. Therefore, in Step 1 in the following, we prove that A maps into itself; in Step 2, we prove that A is continuous; and, finally, in Step 3, we prove that A is a compact operator.
We proceed by dividing the proof into steps.
- Step 1:
In this step, we prove that A maps into itself.
Let
, and let
be the unique weak solution to Problem (
25). Let us choose
u as test function in (
26). Then, we obtain
By the ellipticity condition (
13) and the growth conditions (
16) and (
17), we have
By using generalized Hölder inequality (
10) and inequality (
7), since
, we obtain
By sharp Sobolev embedding
, denoting with
S the best constant in such embedding, we deduce
By smallness assumptions (
19), the function
defined in (
22) has two positive zeros
. Therefore,
R satisfies the following equality:
and the assertion follows.
- Step 2:
In this step, we prove that A is continuous with respect to the strong convergencein .
Assume that
, such that
By definition of
A,
is the unique solution to Problem (
25) with
; that is,
for all
. Therefore, in order to show (
33), we prove that
and
By Step 1,
for any
. Hence, we can extract a subsequence, still denoted by
, such that
- Step 2a:
Proof of strong convergence in (35) of
.
In this step, we prove that
Indeed, by (
38) and Lemma 5 in [
24] (p. 190), we can deduce
Here, we just recall that Lemma 5 in [
24] is a subtle generalization of the classical result, which asserts that the strong convergence of the sequence is a consequence of the weak convergence of the sequence and the convergence of the norms of its elements.
In order to prove (
38), we use
as test function in (
34), and we obtain
We prove that each term on the right-hand side of (
40) goes to zero as
n goes to
. Indeed, since
strongly in
by Lebesgue dominate convergence, and since
weakly in
, then
Indeed, by (
37) and (
32), since for a subsequence, denoted again with
,
a.e., and
is a Carathéodory function, we have
Moreover, such a sequence is equintegrable. Indeed, by (
32),
strongly in
and for a subsequence, denoted again as
, there exists a function
such that
a.e. Therefore, since the sequence
is bounded, and the sequence
is bounded when
, i.e., when
, by Hölder inequality, we obtain
for any measurable subset
.
Here,
C denotes a positive constant, which depends only on the data, does not depend on
n, and can vary from line to line. Then, by equi-integrability and Vitali theorem, (
42) is proven.
Finally, let us consider the last term on the right-hand side of (
40). Since
converges almost everywhere to
w, by (
37), we have
Moreover, such a sequences is equintegrable. Indeed, by (
32),
strongly in
, and for a subsequence denoted
, there exists a function
such that
a.e. in
. Therefore, by Hölder inequality, since the sequence
is bounded, we obtain
for every measurable set
. Then, by equi-integrability and Vitali theorem, we obtain
Combining (
40)–(
43), we conclude the proof of (
38).
- Step 2b:
In this step, we prove that .
Let us consider (
34) and pass to the limit for
n, which goes to
. By (
39), since
is a Carathéodory function bounded in
, we obtain
This implies
as
n, which goes to
.
Moreover, by (
32), we can extract a subsequence, still denoted by
, such that
Therefore, since
and
are Carathéodory functions, by using the Vitali theorem, we obtain
as
n, which goes to
.
Now, we pass to the limit in (
34). By combining (
44)–(
46), we obtain
for all
. This yields that
u is a weak solution to (
34). Since the solution to (
34) is unique, one has
.
By Step 2a and Step 2b, the conclusion that A is continuous follows.
- Step 3:
In this step, we prove that A is a compact operator.
To this aim, we prove that
is precompact in
. Let
, and let
be the unique weak solution to Problem (
25) with
. In order to prove that
is precompact in
, we prove that there exists a subsequence, still denoted with
, which strongly converges to
. Since
, they are bounded in
, and we can extract subsequences still denoted by
and
, respectively, such that
and
Now, we proceed as in Step 2a to prove that strongly in , and as in Step 2b to prove that .
This proves that is precompact in , and yields the conclusion.
- Step 4:
The conclusion. By the Schauder fixed point theorem, since
A maps
into itself and is a continuous compact operator, there exists a unique fixed point of
A. By definition of
A, the unique fixed point of
A is the unique weak solution to Problem (
12).
Remark 3. As pointed out in the Introduction, an existence result has been proven in [12] under the assumptions We explicitly remark that Theorem 1 completes the existence result given in [12], since it also gives a uniqueness result, which cannot be deduced by the analogous result in [12].