1. Introduction
This paper deals with the existence of two positive solutions for the following elliptic differential inclusion:
where
is a bounded open set in
,
, having a smooth boundary
,
is the classical Laplace operator and
is a set-valued mapping with convex compact values. Usually, problems of type (
1) are investigated assuming
G is either lower semicontinuous or upper semicontinuous. Indeed, in the first case, Michael’s selection theorem allows us to study differential inclusion through the equation with selection as its nonlinear term, while in the second case, fixed-point theorems for set-valued maps are applied (see, for instance [
1] (Introduction p. 93)). We cite here the seminal work of S.A. Marano, where, by applying the Ky Fan fixed-point theorem and assuming
G is upper semicontinuous, the existence of at least one solution for (
1) was obtained (see [
2] (Theorem 2.1)). As a consequence of this result, Marano obtained the existence of at least one solution for elliptic differential equations with discontinuous nonlinearities, where it is worth noticing that the set of discontinuity points may also be uncountable (see also [
3]). Elliptic differential Dirichlet problems have been widely investigated (see, for instance, the papers of Stuart and Toland, such as [
4]). We also cite the papers of Mi ([
5,
6]) for the research of positive solutions for a singular Dirichlet
p-Laplacian elliptic problem and singular elliptic systems, respectively, through the lower and upper solutions method. Additionally, we can refer to [
7,
8] for the study of positive solutions for fractional differential systems. In particular, we emphasize the fundamental contribution of the seminal work of Chang [
9] and the powerful paper of Ambrosetti and Badiale [
10], where such arguments have been studied by variational methods. We also mention the very interesting paper of Marano and Motreanu [
11], and we refer to Chapter 6.2, p. 321 in [
12] for a complete overview of this field.
The aim of this paper is to present an existence and multiplicity result for differential inclusion (
1). Specifically, we establish the existence of two positive solutions, assuming
G is an upper semicontinuous map with compact convex values and has subcritical growth. We observe that in [
2], in order to obtain a non-zero solution, it must be assumed in addition that
, which we do not require; in [
11], multiple solutions are obtained with a sublinear growth, while here, the growth may also be superlinear; in [
10,
13], the set of discontinuity points is assumed to be countable, while here, it may also be uncountable.
To give an idea of this, we now report a consequence of our main result.
Theorem 1. Let be an upper semicontinuous set-valued mapping with compact convex values. Assume that
there are and and two positive constants and , such that there exist two constants, and , such that
Then, there exists such that for each , denoting by the open ball of center 0 and radius R, problemadmits at least two positive solutions.
The meaning of a solution is that of a generalized weak solution defined in
Section 2 which returns the classical one, that is
such that
for a.a.
.
The main result of this paper is Theorem 3, which establishes, under essentially the same assumptions of
G as previously mentioned, the existence of a precise interval for the positive real parameter
such that the problem
admits at least two positive solutions. Moreover, a consequence is pointed out (see Theorem 4), and an application to Dirichlet problems with equations having discontinuous nonlinearities is established. Finally, two examples of applications are given (see Examples 1 and 2).
The reader can refer to the seminal books [
14,
15] for an overview of the non-smooth analysis theory and to [
1,
12] for basic notions of set-valued analysis. Finally, as references for boundary value problems with multivalued right-hand sides, we mention the papers [
16,
17,
18,
19].
The paper is organized as follows.
Section 2 is devoted to some preliminaries and auxiliary results of set-valued and non-smooth analysis, which are useful to define the variational framework for problem (
2). Additionally, we recall an abstract critical point theorem (see [
20] ([Theorem 2.10])), that is Theorem 2, which is the main tool in our investigation to obtain our main result, presented in
Section 3, together with its consequences and applications.
2. Basic Notions of Set-Valued and Non-Smooth Analysis
We recall here some notions from set-valued analysis, while for a general overview, we refer to the seminal book [
1]. Let
be two topological spaces. Given a set-valued mapping
, we say that
G is upper semicontinuous if, for any open set
, the set
is open in
X. We point out the following lemma (see, for instance, [
17] (Lemma 2.1)).
Lemma 1. Let be a set-valued mapping with compact convex values. The following propositions are equivalent:
G is upper semicontinuous;
the single-valued mappings are l.s.c. and u.s.c., respectively.
Our approach is variational and it is based on the non-smooth analysis theory as developed by Clarke (see [
14]). Let
X be a Banach space with norm
and let
be its topological dual. By
, we denote the duality brackets for the pair
. We recall that a functional
is said to be locally Lipschitz if for every
there exists a neighborhood
U of
x,
and
such that
Moreover, if
is a locally Lipschitz functional, the
generalized directional derivative in the sense of Clarke of
at the point
x in the direction
is defined by
The
(Clarke) generalized gradient of
at
is defined by
The next proposition lists some properties of the generalized directional derivative and generalized gradient that are useful in the subsequent sections.
Proposition 1 (See [
14])
. Let be two locally Lipschitz functions. Then, for every , the following conditions hold:;
or, equivalently,
;
the function is upper semicontinuous;
if X is finite-dimensional, is upper semicontinuous in X;
.
Clearly, the generalized directional derivative extends the classical directional derivative. To be precise, one has
Lemma 2 ([
14])
. Let be a continuously Gâteaux differentiable functional. Then, Φ is locally Lipschitz and
Finally, we say that
is a critical point of
when
, namely
for all
(see [
9]).
For a full treatment of these topics, we refer to [
15].
Now, consider the problem
where
,
, is a bounded open set, with a smooth boundary
,
is a positive real parameter and
is an upper semicontinuous set-valued mapping with compact convex values. We say that a function
is a generalized weak solution of (
2) if there is a function
such that
- (i)
- (ii)
u is a weak solution of the problem
that is, we have
Clearly, when
G is a single-valued mapping, the previous definition coincides with the usual definition of a weak solution. We also observe that, in a general case, since
is smooth enough (see, for instance, [
21] (Theorem 9.25), [
22] (Theorem 9.15)), we have
and
for a.a.
.
We assume that
there are and such that
where for all . In particular, we observe that from Lemma 1, is a measurable function. Moreover, put , the function defined by . Clearly, is a locally Lipschitz function. Moreover, we define and .
Remark 1. Thanks to Lemma 1, and are measurable selections and any convex combination of these functions is still a measurable selection. However, we choose as a particular selection the minimum of G since its growth behaviour does not impose restrictions on the growth of the set-valued mapping G, which can also have supercritical growth. Therefore, our choice is motivated by the aim of considering a more general case that is possible.
Now, we introduce the variational setting for our problem, which involves Dirichlet boundary conditions, but we observe that it can be adapted to other boundary conditions, such as Neumann, mixed, or periodic conditions.
We take as Banach space
X the Sobolev space
, endowed with the norm
Next, we define the functionals
by
for all
. Notice that
and
Moreover, standard computations show that the functional
is well-defined and sequentially weakly continuous on
X. Further, it is a locally Lipschitz functional on
X (see [
9]).
We point out the following proposition, which is based on a result of Chang (see [
9] (Corollary p. 111)).
Proposition 2. If is a critical point of functional defined by (6), then u is a generalized weak solution for problem (2). Proof. Let
be a critical point of
. We have
Therefore, we have
Hence,
, that is
Now, ref. [
9] (Corollary p. 111) ensures that
is locally Lipschitz on X and we have
where the latest must be read as follows: for all
, there is
such that
for all
and
a.e. in
. Therefore, from (7), there is
such that
for all
and
a.e. in
.
In other words,
is a weak solution of
Finally, in order to obtain the conclusion, we claim
Indeed, for all
, we have
for which, taking into account that
is u.s.c., we have
. In a similar way, it follows
, and hence, our claim is proven.
At this point, we have the conclusion, since for a.a. . □
Finally, we recall that a functional
satisfies the Palais–Smale condition (briefly the
-condition) if, for any sequence
, such that
is bounded and
where
, has a convergent subsequence.
We conclude this section by pointing out that, in order to assure the existence of multiple solutions to problem (
2), our main tool will be the following abstract critical point theorem for locally Lipschitz continuous functions (see Theorem 2.10 in [
20]). We recall that this result is based on the Ambrosetti–Rabinowitz theorem ([
23]).
Theorem 2. Let X be a real Banach space and let be two locally Lipschitz continuous functions such that . Suppose that there exist and with such thatand for each the functional fulfills the -condition and it is unbounded from below. Then, for each , the functional admits at least two nontrivial critical points such that .
3. Elliptic Differential Inclusions
In this section, is an upper semicontinuous set-valued mapping with compact convex values such that the following condition holds:
there are and and two positive constants and such that
where
is defined as in
Section 2. Clearly, condition
implies the condition
given in
Section 2, so that Proposition 2 holds.
Denoted by
, the critical Sobolev exponent, it is well known that we have the continuous embedding
where the constant
C, given by
is the best constant, see Talenti [
24], and
stands for the Gamma function. By Hölder’s inequality and (10) we obtain
for all
and for all
, where
denotes the Lebesgue measure on
. As in
Section 2, put
for all
and
for all
and for
. In addition, we set
for all
and
for which there exists
such that
. Furthermore, put
and
for two positive constants
and
, respectively.
The main result in this section reads as follows.
Theorem 3. Let be an upper semicontinuous set-valued mapping with compact convex values satisfying condition . Assume that there exist two positive constants , with , such thatand suppose that there exist two constants and such thatThen, for each , problem (2) admits at least two distinct positive generalized weak solutions. Proof. Let
be fixed. From (15) and (16), we easily see that the interval
is not empty. We want to apply Theorem 2. First, we mention that the Ambrosetti–Rabinowitz condition stated in (17) implies that the functional
is unbounded from below and satisfies the Palais–Smale condition (see, for example, [
25]). So, we only need to show that inequality (9) is satisfied. To this end, put
and note that the growth condition in
implies
Taking into account (12), (18) and (19), we have
where
This implies that
In order to prove the other inequality, let
We easily see that
. Moreover, one has
and
Combining these estimates yields
From (20) and (22), we obtain
We only need to show that
. Setting
we get from (21) that
Recall that
and we are going to show that
.
First, we observe that
Now, we apply (16) in combination with the growth condition in (19) to obtain
Arguing by contradiction, we assume that
. Then, this fact together with
and (24) gives
which is a contradiction to (25). Hence,
and this implies
.
Now, taking (23) also into account, we are in the position to apply Theorem 2, which says that
admits two distinct non-zero critical points
. Therefore, Proposition 2 ensures that they are solutions of problem (
2). So, in particular, there are
,
, such that
are weak solutions of
where
a.e. in
,
, owing to
. Hence, the strong maximum principle guarantees that
,
, are positive in
, and the conclusion is achieved. □
Now put
We point out the following consequences of Theorem 3.
Theorem 4. Let be an upper semicontinuous set-valued mapping with compact convex values satisfying conditions and (17). Assume thatThen, for each , problem (2) admits at least two distinct positive generalized weak solutions. Proof. Let
be fixed. We easily see that
, and so there is
such that
. On the other side, since
there exists
such that
Hence,
, and so condition (16) is fulfilled. Therefore, the statement of the corollary follows from Theorem 3. □
Remark 2. Theorem 1 in the Introduction is a consequence of Theorem 4. It is enough to pickand observe that if and only if . Now, we give an application of previous results to Dirichlet problems with discontinuous nonlinearities.
Theorem 5. Let be almost every continuous function satisfying the following:
there are and and two positive constants and such that
Put , , and assume that there exist two constants and such thatandWe denote byand we requireThen, for each , problemadmits at least two distinct positive weak solutions. Proof. First, we observe that the assumption “
f is a.e. continuous in
” signifies
. Moreover, assumption
implies that
f is a locally bounded function. So,
F is well defined, and it is a locally Lipschitz function. Moreover, we can define
,
as in
Section 2. Consider
,
. Moreover, put
. Since
and
are homeomorphic sets, from Propositions 2.1.5 (d) and 2.1.2 (a) in [
14],
L is an upper semicontinuous set-valued mapping with convex and compact values. Moreover, from Example 1 in [
9] we have
a.e. in
. So, in particular, one has
for a.a.
since
. Moreover, taking into account that
for a.a.
, simple computations show that
L satisfies the assumptions
, (17) and (27) of Theorem 4. Hence, problem
admits two positive generalized weak solutions. Let
u be one of them. There is a function
such that
- (j)
- (jj)
Since
is smooth enough, from usual regularity results, it follows that
, and from
, one has
for a.a.
. Now, we claim that
. Arguing by a contradiction, assume
. From one side, Lemma 1 by De Giorgi, Buttazzo, and Dal Maso [
26] ensures that
for a.a.
. On the other side,
ensures
for a.a.
, for which
for a.a.
. Therefore, from (30), we obtain
for a.a.
. Hence, we have
for a.a.
and this is absurd, so our claim is proven.
Hence, we have
for a.a.
, and from
follows
This completes the proof. □
We now give two examples of applications of Theorem 5. The first has a nonlinearity with only one discontinuity point, while the nonlinearity on the second one has an uncountable set of discontinuity points.
Example 1. The following problem has been introduced in [13]. Putand consider the problemwith and . Theorem 5 ensures that there issuch that for each and for each , problem (35) admits two positive weak solutions. Indeed, by choosing and , all assumptions of Theorem 5 are verified. In particular, we havefor all . Hence, Theorem 5 ensures that the problemadmits two positive weak solutions for each whereso that if and only if . Example 2. Putwhere C is the Cantor set, and consider the problemwith and . Theorem 5 ensures that there issuch that for each , problem (37) admits two positive weak solutions. Indeed, by choosing and , all assumptions of Theorem 5 are verified. Hence, Theorem 5 ensures that the problemadmits two positive weak solutions for each , whereso that if and only if . Remark 3. Consider the following problem involving a hemivariational inequalitywhere stands for Clarke’s generalized directional derivative of a locally Lipschitz function given by , , and is a locally essentially bounded function. If we assume , (27) and (28), and then for each , problem (38) admits at least two positive solutions. Indeed, arguing as in the first part of the proof of Theorem 5, condition and are obtained. Hence, taking into account that if and only if , one haswhich is the conclusion.