1. Introduction

The aim of this work is to prove the existence of multiple solutions of constant sign and of nodal solutions (sign changing solutions) for nonlinear elliptic equations driven by the -Laplacian and having a nonsmooth potential (hemivariational inequalities). So let be a bounded domain with a -boundary The problem under consideration is the following:

Here is measurable function on , which in the variable is locally Lipschitz and stands for the generalized subdifferential of in the sense of Clarke [1]. Problem (1.1) is a hemivariational inequality. Hemivariational inequalities are a new type of variational expressions, which arise in applications if one considers more realistic mechanical laws of multivalued and nonmonotone nature. Then the corresponding energy (Euler) functional is nonsmooth and nonconvex. Various engineering applications of hemivariational inequalities can be found in the book of Naniewicz-Panagiotopoulos [2].

Multiple solutions of constant sign for problems monitored by the -Laplacian and with a -potential were obtained by Ambrosetti et al. [3], García Azorero-Peral Alonso [4], and García Azorero et al. [5]. In all these works, the authors consider nonlinear eigenvalue problems and prove the existence of positive and negative solutions for certain values of the parameter The question of existence of nodal solutions was first addressed within the framework of semilinear problems (i.e., ). We mention the works of Dancer-Du [6] and Zhang-Li [7], which contain two different approaches to the problem. In Dancer-Du [6], the authors use a combination of the variational method (critical point theory) with the method of upper and lower solutions. In contrast Zhang-Li [7] use invariance properties of the negative gradient flow of the corresponding equation in Recently these methods were extended to “smooth” problems driven by the -Laplacian differential operator. Carl-Perera [8] extended the work of Dancer-Du [6], by assuming the existence of upper and lower solutions for the problem. Zhang-Li [9] and Zhang et al. [10] extended the semilinear work of [7], by carefully constructing a pseudogradient vector field whose descent flow exhibits the necessary invariance properties. These works were extended recently by Filippakis-Papageorgiou [11]. Recently the approach based on the invariance properties of descent flow was used by Zhang-Perera [12] to produce nodal solutions for certain Kirchhoff type equations. Other recent works dealing with -Laplacian equations are those by Ahmad-Nieto [13] (monotone iterative technique), Kim et al. [14] (radial solutions), Lin et al. (singular odes) [15], and Väth [16] (degree theoretic approach).

In this paper using techniques from nonsmooth critical point theory in conjunction with the method of upper and lower solutions, we are able to extend the works of Dancer-Du [6] and Carl-Perera [8] to hemivariational inequalities. Helpful in this respect is the nonsmooth second deformation lemma of Corvellec [17]. Recently, sign-changing solutions for problems with discontinuous nonlinearities were obtained by Averna et al. [18], but in contrast to our work they deal with -superlinear problems.

2. Mathematical Background

In our analysis of problem (1.1), we use the nonsmooth critical point theory which is based on the subdifferential theory for locally Lipschitz functions and some basic facts about the spectrum of the negative -Laplacian with Dirichlet boundary conditions. For easy reference, we recall some definitions and results from these areas, which will be used in the sequel.

We start with the subdifferential theory for locally Lipschitz functions and the corresponding nonsmooth critical point theory. Details can be found in the books of Gasiński-Papageorgiou [19] and Motreanu-Panagiotopoulos [20]. So let be a Banach space and let be its topological dual. By we denote the duality brackets for the pair . Given a locally Lipschitz function the generalized directional derivative of at in the direction is defined as follows:

The function is sublinear continuous and so it is the support function of a nonempty, convex, and -compact set defined by

The multifunction is called the “generalized gradient" (or generalized subdifferential) of If is also convex, then coincides with the subdifferential in the sense of convex analysis defined by

Moreover if then is locally Lipschitz and

We say that is a critical point of the locally Lipschitz function if It is easy to see that if is a local extremum of (i.e., a local minimum or a local maximum of ), then is a critical point of .

A locally Lipschitz function satisfies the Palais-Smale condition at level (-condition for short), if every sequence such that and as has a strongly convergent subsequence. We say that satisfies the -condition, if it satisfies the -condition for every

The following topological notion is crucial in the minimax characterization of the critical values of a locally Lipschitz functional Definition 2.1. Let be a Hausdorff topological space and and are nonempty closed subsets of with We say that the pair is linking with in if and only if
(a);(b)for any such that we have

Using this notion, we have the following general minimax principle for the critical values of a locally Lipschitz function Theorem 2.2. If is a reflexive Banach space, , , and are nonempty closed subsets of such that is linking with in is locally Lipschitz, and satisfies the -condition, then and is a critical value of Remark 2.3. From this general minimax principle, by appropriate choices of the linking sets, one can produce nonsmooth versions of the mountain pass theorem, of the saddle point theorem, and of the generalized mountain pass theorem.Definition 2.4. If is a subset of the Banach space a “deformation of ” is a continuous map such that . If , then we can say that is a “weak deformation retract of ”, if there exists a deformation such that and for all .

Given a locally Lipschitz function and we define

The next theorem is a partial extension to a nonsmooth setting of the so-called “second deformation theorem” (see, e.g., Gasiński-Papageorgiou [21, page 628]) and it is due to Corvellec [17]. In fact the result of Corvellec is formulated in the more general context of metric spaces, for continuous functions using the so-called weak slope. For our purposes, it suffices to use a particular form of the result which we state next.

Theorem 2.5. If is a Banach space, is locally Lipschitz and satisfies the -condition, has no critical points in and is discrete nonempty, then there exists a deformation such that
(a) for all (b) (c) for all and all

In particular the set is a weak deformation retract of .

Next let us recall some basic facts about the spectrum of the negative -Laplacian with Dirichlet boundary conditions. So let be a bounded domain with a -boundary and . We consider the following nonlinear weighted (with weight ) eigenvalue problem:

The least number for which problem (2.5) has a nontrivial solution is the first eigenvalue of and it is denoted by . The first eigenvalue is strictly positive (i.e., ); it is isolated and it is simple (i.e., the associated eigenspace is one dimensional). Moreover, using the Rayleigh quotient we have a variational characterization of , namely, (see also Cuccu et al. [22]).

The minimum in (2.6) is attained on the corresponding one-dimensional eigenspace. In what follows by we denote the normalized eigenfunction. Note that also realizes the minimum in (2.6). Hence we may assume that a.e. on Moreover, from nonlinear regularity theory (see, e.g., Gasiński-Papageorgiou [21, page 738]), we have The Banach space is an ordered Banach space with order cone given by

We know that and in fact

By virtue of the strong maximum principle of Vázquez [23], we have

Using the Lusternik-Schnirelmann theory, in addition to we obtain a whole strictly increasing sequence of eigenvalues of (2.5), such that as These are the so-called “variational eigenvalues” of ( ). When (linear case), then these are all the eigenvalues. For (nonlinear case), we do not know if this is true. Nevertheless exploiting the fact that is isolated, we can define

Because the set of eigenvalues of (2.5) is closed, we see that is an eigenvalue of ( In fact we have that is, the second eigenvalue and the second variational eigenvalue of coincide. Then for we have a variational expression provided by the Lusternik-Schnirelmann theory. The eigenvalues and exhibit some monotonicity properties with respect to the weight function More precisely, we have the following.

(a)If a.e. on with strict inequality on a set of positive measure, then (this is immediate from (2.6).(b) If a.e. on then (see Anane-Tsouli [24]).

If then we write and For there is an alternative variational characterization, due to Cuesta et al. [25]; namely, if , and then

Finally we recall the notions of upper and of lower solutions for problem (1.1).

(a)A function with is an “upper solution” for problem (1.1), if for all a.e. on and for some a.e. on for some .(b) A function with is a “lower solution” for problem (1.1), iffor all a.e. on and for some a.e. on for some .

3. Solutions of Constant Sign

In this section, we produce two nontrivial solutions of (1.1) which have constant sign. The first is positive and the second is negative. To do this, we will need the following hypotheses on the nonsmooth potential .

: is a function such that a.e. on a.e. on , and(i)for every , is measurable;(ii)for almost all , is locally Lipschitz;(iii)for almost all all , and all we have(iv)there exists satisfying a.e. on with strict inequality on a set of positive measure, such that uniformly for almost all and all (v)there exist satisfying a.e. on where the first inequality is strict on a set of positive measure, such that uniformly for almost all and all (vi)for almost all all , and all we have (sign condition).Remark 3.1. Hypotheses (iv) and (v) are nonuniform nonresonance conditions at zero and at , respectively. Moreover, as we move from to the “slopes” cross the first eigenvalue So our framework incorporates the so-called asymptotically -linear equations. For , since the appearance of the pioneering work of Amann-Zehnder [26], these problems have attracted a lot of interest.

The next lemma is an easy consequence of the strict positivity of and of the hypotheses on (see (iv). We omit the proof.Lemma 3.2. If satisfies a.e. on with strict inequality on a set of positive measure, then there exists such that

Given and , we consider the following nonlinear Dirichlet problem:

In the next proposition, we establish the solvability of (3.5).Proposition 3.3. If satisfies a.e. on with strict inequality on a set of positive measure, then for all small problem (3.5) admits a solution Proof. In what follows by we denote the duality brackets for the pair ). We introduce the nonlinear operator defined by
It is straightforward to check that is strictly monotone and demicontinuous, hence maximal monotone too. Also let be the nonlinear, bounded, continuous map defined by
Because of the compact embedding of into , viewed as a map from into is completely continuous. Therefore is pseudomonotone from into Also for every we have
Therefore, if then by virtue of Poincare's inequality is coercive. But a pseudomonotone coercive operator is surjective. Hence we can find that such that
Thus is a solution of (3.5). We take duality brackets of (3.9) with the test function We obtain
But recall that So it follows that hence that is, Since from (3.10) it follows that and (nonlinear regularity theory). In addition, from (3.10), we see that

In fact the solution of (3.5) is an upper solution for problem (1.1).Proposition 3.4. If hypotheses (i)→(iv) hold and is small, then the solution of problem (3.5) obtained in Proposition 3.3 is a strict upper solution of problem (1.1) (strict means that is an upper solution of (1.1) which is not a solution).Proof. Because of hypotheses (iv), given we can find such that for almost all all , and all we have
Also due to hypothesis (iii), we can find such that for almost all all , and all we have
Therefore it follows that for almost all all , and all we have
So for and as above, we consider problem (3.5). From Proposition 3.3, we have a solution Then due to (3.15), for all with a.e. on we have

Since a.e. on is a lower solution for problem (1.1).

We introduce the set

and the truncation function defined by

Then we set and we consider the locally Lipschitz functional defined by

We will show that we can find a nontrivial solution of (1.1) in which is a local minimizer of and of To do this we will need the following simple result about ordered Banach spaces.Lemma 3.5. If is an ordered Banach space, is the order cone of , and then for every we can find such that Proof. Since we can find such that
Let (if then clearly the lemma holds for all ). We have the following:
So, if then

Using this lemma, we can prove the following result.Proposition 3.6. If hypotheses hold, then there exists which is a local minimizer of and of Proof. From (3.15), we know that given we can find such that
Because of hypotheses (i), (ii), for almost all is almost everywhere differentiable on (Rademacher's theorem) and at every point of differentiability we have
Integrating this inequality and since for almost all we obtain
Then for every we have
Choosing because from (3.25) and Poincare's inequality, we infer that is coercive. Also it is easy to see that is weakly lower semicontinuous on Hence by virtue of the theorem of Weierstrass, we can find such that
First we show that To this end, note that hypothesis (v) implies that given we can find such that
As before, integrating (3.27), we obtain
We know that (see Proposition 3.3). So using Lemma 3.5, we can find small such that
Then, because of (3.28), we have
Let Using the hypothesis on (see (v) and the fact that for all we see that So, if we choose we have
Note that for small, Hence
Given any we define Then is Lipschitz continuous, hence differentiable almost everywhere and for all From Chang [27, page 106], we know that we can find a.e. on such that
For any and we define
Clearly We use this