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Existence of solutions for nonlinear Robin problems with the p-Laplacian and hemivariational inequality
Boundary Value Problems volume 2014, Article number: 257 (2014)
Abstract
In this paper, we show the existence of at least three nontrivial solutions for a nonlinear elliptic equation driven by the p-Laplacian with a nonsmooth potential (hemivariational inequality) and Robin boundary condition. Two of these solutions are of constant sign (one is positive, the other negative). We mainly use a variational approach together with a sub-sup solution method.
1 Introduction
Consider the problem
where is a bounded domain with -boundary ∂Z, () is the p-Laplacian operator, , , , and on ∂Z. is a measurable potential function on , which is locally Lipschitz in the , stands for the generalized subdifferential of . Also denotes the outer normal derivative of x with respect to ∂Z. The aim of this paper is to prove the existence of two constant sign solutions and furthermore prove the existence of at least three nontrivial solutions for problem (1.1).
A multiplicity of solutions for problems driven by the p-Laplacian has been obtained by Ambrosetti et al.[1] and Garcia Azorero et al.[2]. In these works, the authors deal with a right-hand side nonlinearity of the form with being a real parameter, ( if ; otherwise) and prove the existence of positive and negative solutions. The question of the existence of a p-Laplacian Robin problem was also present in the work of Zhang et al.[3] for , the authors show that the Robin problem has at least four nontrivial solutions using a sub-sup solution method, the FucÃk spectrum, the mountain pass theorem, and the degree theorem together. In the work of Zhang et al.[4], [5] for , the authors show that the oscillating equations with the p-Laplacian Robin problem has infinitely many nontrivial solutions. In Anello [6] and Ricceri [7], the main tool is an abstract variational principle of Ricceri and its use is made possible by the hypothesis that ; by the fact that Sobolev space is compactly embedded in , the authors obtain infinitely many weak solutions for p-Laplacian Neumann problem.
In all the aforementioned works, the nonlinearity is a Carathéodory function, a.e. is continuously differentiable in the variable x. In Barletta and Papageorgiou [8], the authors consider a nonsmooth potential with an asymmetric behavior at +∞ and at −∞ to get two nontrivial solutions using degree methods. Also, in Dancer and Du [9], the authors use the critical point theory and a sub-sup solution method on smooth critical point theory.
In this paper, we use a combination of nonsmooth critical point theory with sub-sup solution methods. We also use the nonsmooth version of the second deformation theorem due to Corvellec [10]. Thus, we can extend the works of [9], [11]–[13] to a hemivariational inequality with the Robin boundary condition.
2 Preliminaries
Now we recall the subdifferential theory for locally Lipschitz functions and the corresponding nonsmooth critical point theory. Let X be a Banach space and let be its topological dual. We denote by the duality brackets for the pair . The generalized directional derivative of a locally Lipschitz function at along the direction is defined as follows:
It is well known that is sublinear continuous and it is the support function of a nonempty, convex, and -compact set defined by
The function is the ‘generalized subdifferential’ of φ. If , then φ is locally Lipschitz and . Moreover, if φ is also convex, then coincides with the subdifferential in the sense of convex analysis, , which is defined by
If , then we call critical point of φ. It is easy to see that if is a local minimum or a local maximum of φ, then is a critical point of φ.
A locally Lipschitz function φ satisfies the Palais-Smale condition at level , if every sequence satisfying and as has a strongly convergent subsequence. If φ satisfies the Palais-Smale condition at level for all , then we say that it satisfies the Palais-Smale condition. For the details, we refer to [14].
In the following study, denote , and we will use the following spaces:
Both are ordered Banach spaces, and we denote
It is well known that the principal eigenfunction , so .
Furthermore, define as the normalized principal eigenfunction of (see [15]). It is well known that , a.e. , from the nonlinear regularity (see Di Benedetto [16], [17], Chapter IX]), furthermore by virtue of the strong maximum principle of Vazquez [18].
We give the following minimax characterization (see [19]), suited for our purpose.
Proposition 2.1
Letand, where. Then the first eigenvalueofequals
Next we recall the definitions of sub-sup solutions for problem (1.1).
-
(1)
A function with is called a ‘sup solution’, if
for all , a.e. on Z and for some , a.e. on Z for some if , if .
-
(2)
A function with is called a ‘sub-solution’, if
for all , a.e. on Z and for some , a.e. on Z for some if , if .
Finally we recall the following topological notion which is crucial in critical point theory.
Definition 2.2
[20]
Let S, Q be closed subsets of a Banach space X, Q with relative boundary ∂Q. We say S and ∂Q link if
-
(1)
, and
-
(2)
for any map such that we have .
From the definition, we give the following general minimax principle for the critical values of a locally Lipschitz function φ.
Proposition 2.3
[20]
Suppose φ is locally Lipschitz and satisfies the-condition. Consider closed subsetsand Q with relative boundary ∂Q. Suppose
-
(1)
S and ∂Q link,
-
(2)
.
Let
Then the number
defines a critical valueof φ.
Remark 2.4
From the above general minimax principle, a nonsmooth version of the mountain pass theorem, the saddle point theorem, and the generalized mountain pass theorem are available by choosing the link sets appropriately (see [10], [14]).
The following result is the so-called ‘second deformation theorem’ for a nonsmooth setting. In fact, this result is due to Corvellec [10]. We give the following sets:
We know that K, , and are the critical set of φ, the critical set at level of φ, and the strict sublevel set of φ at c, respectively.
Proposition 2.5
Let X be a Banach space, be locally Lipschitz satisfying the Palais-Smale condition. with. Assume also thatandis a finite set containing only local minimizers of φ.
Then there exists a continuous deformation such that
-
(1)
for all , ,
-
(2)
,
-
(3)
for all , .
Definition 2.6
[21]
Let X be a topological space and A a subspace of X. A weak deformation retraction from X to A is a homology such that for all and , we have , , and .
In particular, the set is a weak deformation retract of .
We now recall another notion, which will be useful in the following. Suppose W is a Banach space and is a mapping, we say that A is a type if for every sequence such that and .
Considering the nonlinear mapping defined for all by
We have the following result (see [8], Proposition 4.1]).
Proposition 2.7
The mapping (2.1) is continuous and of the type.
Definition 2.8
[14]
Given a functional , is called a W-local minimizer of φ if there exists satisfying for all with , we have
Definition 2.9
[14]
is called a C-local minimizer of φ if there exists satisfying for all with , we have
As the study of problems like (1.1) is reduced to seeking the critical points of corresponding energy functional on or on , in this section we introduce the notations used along the paper together with the main abstract results that we will use later on for a C-local minimizer to be a W-local minimizer. Such a result for was first proved in [2]. Then it has been extended to the Neumann boundary condition and a nonsmooth potential by [8].
We denote for all
From Clarke [22], pp.32-34], we know that ψ is locally Lipschitz. By [12], we know that if we let be a C-local minimizer of ψ, then and it is a W-local minimizer of ψ.
3 Solutions of constant sign
In this section, by using a sub-sup solution method, we get two solutions of (1.1) with constant sign, one positive and the other negative.
Our general assumptions on the nonsmooth potential are the following:
:
-
(i)
is measurable for all ;
-
(ii)
is locally Lipschitz for a.e. ;
-
(iii)
for a.e. , all , , with and ;
-
(iv)
for a.e. , all , with satisfying a.e. in Z and in some set of positive measure;
-
(v)
for a.e. , all , with satisfying a.e. in Z and in some set of positive measure, is the first eigenvalue of with Robin boundary condition;
-
(vi)
for a.e. , all , .
Theorem 3.1
Assume that(i)-(vi) hold. Problem (1.1) has at least two solutionsand.
Example
The following potential function j satisfies assumptions (for the sake of simplicity we drop the z-dependence):
where , , and . Note that, if , then .
Note that
It is easy to see that j satisfies (i)-(iii), (vi). For all , we have
and
Then the potential function j satisfies assumptions .
Remark 3.2
In fact, problem (1.1) has the trivial solution for a.e. according to assumption (vi) and the upper semicontinuity of the subdifferential (see Clarke [22]). What we are interesting in is whether it has nontrivial solutions.
We introduce a useful extension of the notion of maximal monotonicity (see [14], p.83]).
Definition 3.3
Let X be a reflexive Banach space and an operator. We say that A is pseudomonotone if
-
(1)
A has nonempty, bounded and convex values;
-
(2)
A is upper semicontinuous for every finite dimensional subspace of X into ;
-
(3)
if in X, , and , then for every , there exists , such that
Definition 3.4
[14]
A is said to be demicontinuous on X if and together imply .
It is well known that (1.1) is the Euler-Lagrange equation of the functional ,
We introduce the truncation function by
then define the locally Lipschitz functional by
where for all , , which is locally Lipschitz.
We consider the nonlinear Robin problem for given and , :
Define the mapping for all by
It is well known that I is strictly monotone and demicontinuous, furthermore, maximal monotone (see [23]). We denote () and we have
which is bounded and continuous. Then the mapping is pseudomonotone from into , in fact, is compact embedding and is completely continuous.
Next, we will show that (3.1) has a solution .
Lemma 3.5
Letsatisfya.e. in Z andin some set of positive measure. Then (3.1) has a solutionforsmall enough.
Proof
First, we claim that there exists such that
In fact, from assumption , we know that , for all . Suppose the conclusion is false, we have , , . If we set , then (J is p-homogeneous). We may assume in , in by passing to a subsequence if necessary. Then
So, by passing to the limit of J, we have
This implies
Hence, we have for a.e. where . In fact, , if not, from the above inequality,
It produces a contradiction. On the other hand,
We have , together with in , so in , but , . So the assumption is false, we have the conclusion.
For all , from the above discussion, we get
So if small enough, we have is coercive. But a pseudomonotone coercive operator is surjective (see [23], Theorem 9.57]), for , we can find such that
That is,
It follows that is a solution of (3.1).
Next we show . Take for , then
So
But , we have , that is, . Since , from (3.2), we have and (nonlinear regularity theorem, see [24]), furthermore, on Z, so . □
Now we prove that the solution of (3.1) is a strict sup solution of (1.1) for small enough.
Lemma 3.6
Let assumptions(i)-(iv) hold. Then the solutionof (3.1) is a strict sup solution of (1.1) forsmall enough.
Proof
From (iv), for given , we can find , such that for all , , , we have
From (iii), we can find , , such that for all , , , we have
So for all , , , we have
From Lemma 3.5, we see that (3.1) has a solution , so when small enough, for all , , , we have
that is,
and from the definition of a sup solution, we know that is a sup solution of (1.1). □
Remark 3.7
We have found a sup solution of (1.1) and a.e. on Z, we also find is a sub-solution of (1.1). Define the set
Next, we will find a nontrivial solution of (1.1) in W.
Proof of Theorem 3.1
Step 1: Claim: We can find which is a local minimizer of and of φ.
From the discussion of Lemma 3.6, for a.e. , all , , we have
Furthermore, for a.e. , all , from assumptions (i), (ii),
then for a.e. , all , we have
So, for some , we have
Because of , , we see that is coercive, and together with weakly lower semicontinuous on W. Thus by the Weierstrass theorem, we can find , satisfying
We claim that . In fact, from assumption (v), we see that, for given , we can find some , for a.e. , all , ,
then for a.e. and all , we get
Furthermore, let be the first eigenfunction of Robin problem of (see [15]), then for , we can find , such that
Then , and
From assumption (v) and , we have
If we choose ε small enough, we can get for all small enough. So, we have
then we have , .
Step 2: The local minimizer of , is a nontrivial solution of (1.1).
Firstly, we claim that is also a local -minimizer of . In fact, the nonlinear regularity theory (see for example [24]) assures that . Hence, as the boundary relation is understood in a pointwise sense and we get , also, by , , and the nonlinear strong maximum principle of Vazquez, , . So we can find satisfying
Then
So, is also a local minimizer of on ; also from [24], is also a local -minimizer of and of φ too.
Also, from [25], there exists , satisfying
Using
We have for all , , then we have
From the monotonicity of I, we have
From the definition of a sup solution of (1.1), we have
From (vi), we have . Furthermore,
Also,
and
So, we have
As , for all , we obtain
That is,
Then is a solution of (1.1).
Step 3: In a similar way, we introduce another truncation function by
then define the locally Lipschitz functional by
where for all , which is locally Lipschitz. Then we have another nontrivial solution which is a local minimum of and of φ too. □
4 Existence of the third nontrivial solution
In this section, we prove the existence of the third solution. Then we give the new assumptions which differ slightly from (v):
:
-
(i)
is measurable for all ;
-
(ii)
is locally Lipschitz for a.e. ;
-
(iii)
for a.e. , all , , with and ;
-
(iv)
for a.e. , all , with satisfying a.e. in Z and in some set of positive measure;
-
(v)
for a.e. , all , with satisfying a.e. in Z and in some set of positive measure, is the first eigenvalue of with the Robin boundary condition;
-
(vi)
for a.e. , all , .
Theorem 4.1
Let assumptions(i)-(vi) hold. Then we can find three nontrivial solutions, , andof (1.1).
Proof
From Theorem 3.1, we have two constant sign solutions and which are the local minimizers of and of , also of φ. We may assume that is the only nontrivial critical point of and is the only nontrivial critical point of . In fact, if there exists another nontrivial critical point of , . Then, by a similar discussion, and it solves (1.1). Thus we have a third nontrivial solution, a.e. .
Moreover, as for φ, we see that φ is coercive and so we can easily prove the Palais-Smale condition. In fact, as in the proof of Theorem 3.1, for a.e. , all , we have
where ω satisfies (iii), , .
Then, using Lemma 3.5, we have
It follows that φ is coercive.
We set , , with relative boundary . If we choose . Then S and ∂Q link. In fact, , and for any map such that , we can choose some satisfying
so , S and ∂Q link.
When we choose δ, we can also assume δ satisfy and (, are local minimizers of φ), we may assume that . Therefore, we can apply Proposition 2.3; let , produce , a critical point of φ, such that
From the above inequality, we have , .
From , we know that
and from the regularity theory (see [24]), we have , hence (4.1) holds in all , we get .
Finally, we prove that . It is equivalent to proving that there is a path such that for all ,
From Proposition 2.1, recall that , endowed with the -topology. Furthermore, set equipped with the -topology. Then we can find by virtue of the density of in S in the -topology, so is dense in , and
From assumption (v), we can find , such that for a.e. , all , , we get
So for a.e. , all ,
Since , for the , we can find , such that for a.e. , , we have
Then let be such that , from (4.2), (4.3), (4.4), and , we have
We consider the continuous path , then for all ,
Next recall that φ is coercive and satisfies the Palais-Smale condition. From the discussion, we set , , φ has no critical points in , . Then with the help of Proposition 2.5, there exists a deformation such that
In fact, the continuous path Γ can be seen as . Then we define by
Then it is a continuous path, so from (4.5), we have
Thus, we construct a continuous path joining and such that
Similarly, we construct a continuous path joining and such that
Then we join , , , and we construct a continuous path such that
It follows that and so .
Therefore, we find the third nontrivial solution of (1.1). □
5 Open related questions
Consider the problem
where is a bounded domain with -boundary ∂Z, () is the p-Laplacian operator, , , and on ∂Z. is a measurable potential function on , which is locally Lipschitz in the , stands for the generalized subdifferential of . Also denotes the outer normal derivative of x with respect to ∂Z.
Whether problem (5.1) has more solutions and whether it has oscillating solutions, we will discuss in the future.
Author’s contributions
The author wrote, read, and approved the final manuscript.
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The author was supported by Tianyuan Foundation for Mathematics under Grant No. 11326098 of National Natural Science Foundation and Doctor Scientific Research Startup Foundation of Harbin Normal University No. XKB 201311.
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Zhang, J. Existence of solutions for nonlinear Robin problems with the p-Laplacian and hemivariational inequality. Bound Value Probl 2014, 257 (2014). https://doi.org/10.1186/s13661-014-0257-5
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DOI: https://doi.org/10.1186/s13661-014-0257-5