Abstract

The study of multiple solutions for quasilinear elliptic problems under Dirichlet or nonlinear Neumann type boundary conditions has received much attention over the last decades. The main goal of this paper is to present multiple solutions results for elliptic inclusions of Clarke's gradient type under Dirichlet boundary condition involving the -Laplacian which, in general, depend on two parameters. Assuming different structure and smoothness assumptions on the nonlinearities generating the multivalued term, we prove the existence of multiple constant-sign and sign-changing (nodal) solutions for parameters specified in terms of the Fučik spectrum of the -Laplacian. Our approach will be based on truncation techniques and comparison principles (sub-supersolution method) for elliptic inclusions combined with variational and topological arguments for, in general, nonsmooth functionals, such as, critical point theory, Mountain Pass Theorem, Second Deformation Lemma, and the variational characterization of the β€œbeginning”of the Fu\v cik spectrum of the -Laplacian. In particular, the existence of extremal constant-sign solutions and their variational characterization as global (resp., local) minima of the associated energy functional will play a key-role in the proof of sign-changing solutions.

1. Introduction

Let be a bounded domain with a -boundary , and let and , , denote the usual Sobolev spaces with their dual spaces and , respectively. We consider the following nonlinear multi-valued elliptic boundary value problem under Dirichlet boundary condition: find and parameters , such that where is the -Laplacian, and denotes Clarke's generalized gradient of some locally Lipschitz function which depends on and the parameters . For problem (1.1) reduces to which may be considered as a nonlinear and nonsmooth eigenvalue problem. We are going to study the existence of multiple solutions of (1.1) for two different classes of which are in some sense complementary. Our presentation is based on and extends the authors' recent results obtained in [1–3]. For the first class of we let and assume the following structure of : where is such that is a CarathΓ©odory function. Problem (1.1) reduces then to the following nonlinear eigenvalue problem: which will be considered in Section 2 when the parameter is small enough.

The second class of has the following structure: where and is the positive and negative part of , respectively, and is assumed to be the primitive of a measurable function that is merely bounded on bounded sets; that is, and is given by Problem (1.1) reduces then to the following parameter-dependent multi-valued elliptic problem: which will be studied in Section 3 for parameters and large enough. Note that stands for the generalized Clarke's gradient of the locally Lipschitz function . Obviously, if is a CarathΓ©odory function, that is, is measurable in for all and is continuous in for a.a. , then is single-valued, and thus problem (1.7) reduces to the following nonlinear elliptic problem depending on parameters and : find and constants , such that Multiple solution results for (1.8) were obtained by the authors in [4]. Furthermore, note that Therefore, if one assumes, in addition, , then (1.8) reduces to the nonlinear elliptic eigenvalue problem: find and a constant such that In a recent paper (see [5]) the authors considered the eigenvalue problem (1.10) for a CarathΓ©odory function . Combining the method of sub-supersolution with variational techniques and assuming certain growth conditions of at infinity and at zero the authors were able to prove the existence of at least three nontrivial solutions including one that changes sign. The results in [5] improve among others recent results obtained in [6]. For (1.7) reduces to the corresponding multivalued eigenvalue problem: find and a constant such that The existence of multiple solutions for (1.11) has been shown recently in [7] where techniques for single-valued problems developed in [5] and hemivariational methods applied in [8] have been used. Multiplicity results for (1.11) have been obtained also in [9].

The existence of multiple solutions for semilinear and quasilinear elliptic problems has been studied by a number of authors, for example, [10–24]. All these papers deal with nonlinearities that are sufficiently smooth.

2. Problem (1.4) for being Small

The aim of this section is to provide an existence result of multiple solutions for all values of the parameter in an interval , with , guaranteeing that for any such there exist at least three nontrivial solutions of problem (1.4), two of them having opposite constant sign and the third one being sign-changing (or nodal). More precisely, we demonstrate that under suitable assumptions there exist a smallest positive solution, a greatest negative solution, and a sign-changing solution between them, whereas the notions smallest and greatest refer to the underlying natural partial ordering of functions. This continues the works of Jin [25] (where and is HΓΆlder continuous with respect to for every fixed ) and of Motreanu-Motreanu-Papageogiou [26]. In these cited works one obtains three nontrivial solutions, two of which being of opposite constant sign, but without knowing that the third one changes sign. Here we derive the new information of having, in addition, a sign-changing solution by strengthening the unilateral condition for the right-hand side of the equation in (1.4) at zero. Furthermore, under additional hypotheses, we demonstrate that one can obtain two sign-changing solutions.

2.1. Hypotheses and Example

Let denote the positive cone of given by We impose the following hypotheses on the nonlinearity in problem (1.4)., with , is a function such that for a.a. , whenever , and one has the following. (i)For all , is CarathΓ©odory (i.e., is measurable for all and is continuous for almost all ). (ii)There are constants , , and functions () with as such that (iii)For all there exist constants , and a set with of Lebesgue measure zero such that uniformly with respect to .

In (iii), denotes the second eigenvalue of . As mentioned in the Introduction, the strengthening with respect to [26] (see also [25]) of the unilateral condition for the right-hand side in (1.4), which enables us to obtain, in addition, sign-changing solutions, consists in adding the part involving the limit superior in (iii).

Let us provide an example where all the assumptions formulated in are fulfilled.

Example 2.1. For the sake of simplicity we drop the dependence for the function in the right-hand side of (1.4). The function given by with and , satisfies hypotheses . Next we give an example of function verifying assumptions which is generally not odd with respect to : with , , , , , , .

2.2. Constant-Sign Solutions

The operator is maximal monotone and coercive; therefore there exists a unique solution of the Dirichlet problem With for , and using as a test function, we see that which implies that . From the nonlinear regularity theory (cf., e.g., [27, Theorem 1.5.6]) we have . Then from the nonlinear strong maximum principle (see [28]) we infer that . Here denotes the interior of the positive cone in the Banach space , given by where is the outer unit normal at .

Lemma 2.2. Let the data , , and be as in (ii). Then for every constant there is with the property that if , one can choose such that

Proof. On the contrary there would exist a constant and a sequence as such that Letting we get for all because we have as . Since , a contradiction is achieved as . Therefore (2.9) holds true.

We denote by the first eigenvalue of and by the eigenfunction of corresponding to satisfying

Lemma 2.3. Assume (i) and (ii) and the following weaker form of hypothesis (iii): for all there exist and with of Lebesgue measure zero such that uniformly with respect to
Fix a constant and consider the corresponding number obtained in Lemma 2.2. Then for any the function , with given by Lemma 2.2, is a supersolution for problem (1.4), and the function is a subsolution of problem (1.4) provided that the number is sufficiently small.

Proof. For a fixed , from (2.9) and (ii) we derive which says that is a supersolution for problem (1.4).
On the other hand, by hypothesis we can find and such that Choose . Then by (2.14) we have which ensures that is a subsolution of problem (1.4).

The following result which asserts the existence of two solutions of problem (1.4) having opposite constant sign and being extremal plays an important role in the proof of the existence of sign-changing solutions.

Theorem 2.4. Assume (i) and (ii) and the following weaker form of (iii): for all there exist constants , and a set with of Lebesgue measure zero such that uniformly with respect to . Then for all there exists a number with the property that if then there is a constant such that problem (1.4) has a least positive solution in the order interval and a greatest negative solution in the order interval .

Proof. Since the proof of the existence of the greatest negative solution follows the same lines, we only provide the arguments for the existence of the least positive solution.
Applying Lemma 2.3 for we find as therein. Fix . Lemma 2.3 ensures that is a supersolution for problem (1.4), with given by Lemma 2.2, and is a subsolution for problem (1.4) if is small enough. Passing eventually to a smaller , we may assume that . Then by the method of sub-supersolution we know that in the order interval there is a least (i.e., smallest) solution of problem (1.4) (see [29]).
We thus obtain that for every positive integer sufficiently large there is a least solution of problem (1.4) in the order interval . Clearly, we have with some function satisfying . First we claim that Taking into account that solves (1.4), and the fact that belongs to the order interval , from (ii) we see that which implies the boundedness of the sequence in . Then due to (2.17) we have that as well as Since solves problem (1.4), one has Setting in (2.21) gives As already noticed that the sequence is uniformly bounded on , so (2.20) and (2.22) yield The -property of on implies The strong convergence in (2.24) and Lebesgue's dominated convergence theorem permit to pass to the limit in (2.21) that results in (2.18).
By (2.18) and the nonlinear regularity theory (cf., e.g., Theorem 1.5.6 in [27]) it turns out . The choice of guarantees that Thus, from (2.18), assumptions (ii) and (iii), and the boundedness of , we get with a constant . Applying the nonlinear strong maximum principle (cf. [28]) we conclude that either or .
We claim that Assume on the contrary that . Then (2.17) becomes Since we may consider Along a relabelled subsequence we may suppose for some . Moreover, one can find a function such that for almost all . Relation (2.21) reads Setting leads to By (iii) we know that there exist constants and such that while (ii) entails for a.a. and for all . Combining the two estimates gives with a constant . Since , and (2.35) holds, there exists a constant such that We see from (2.36) that Then, because the right-hand side of the above inequality is in , by means of (2.30) and (2.36) we can apply Lebesgue's dominated convergence theorem to get Consequently, from (2.32) we obtain The -property of on implies On the basis of (2.31) and (2.40) it follows Notice from (2.36) that for a.a. and for all . We are thus allowed to apply Fatou's lemma which in conjunction with (2.28), (2.30), and (2.16) ensures for all . Thanks to (2.41) we obtain Owing to (2.42) we may once again use Fatou's lemma; so according to (2.28), (2.30), and the last part of (2.16), we find for all . Then (2.41) ensures Combining (2.44) and (2.46) results in which guarantees to have (see [27, Theorem 1.5.5]). Since by (2.47) we know that , we are in a position to address Theorem 1.5.6 in [27], which provides with some . This regularity up to the boundary and the fact that a.e. in and (2.47) enable us to refer to the strong maximum principle (see Theorem 5 of VΓ‘zquez [28]). Recalling that does not vanish identically on (because ) we deduce that for all and for all which amounts to saying . Consequently, there exist constants and such that Following [30] let us denote whenever , where Relation (2.48) justifies that . Then Proposition 1 of Anane [30] implies . On the other hand a direct computation based on (2.48) and (2.47) shows This contradiction proves that the claim in (2.27) holds true.
In view of (2.18) it remains to establish that is the smallest positive solution of problem (1.4) in the interval . Let be a positive solution to (1.4) in . Since , then (1.4) and (ii) allow to deduce that . Using Theorem 1.5.6 of [27] leads to . Then, as is a solution to (1.4) and , with , by means of hypotheses (ii) and (iii), we are able to apply the strong maximum principle. So we get , hence for sufficiently large. The fact that is the least solution of (1.4) in ensures . Taking into account (2.17), we obtain . This completes the proof.

2.3. Sign-Changing Solution

The main result of this section is as follows.

Theorem 2.5. Under hypotheses , for all , there exists a number with the property that if then problem (1.4) has a (positive) solution , a (negative) solution and a nontrivial sign-changing solution satisfying , , .

Proof. Let . Consider the positive number given by Theorem 2.4 and fix . Let and be the two extremal solutions determined in Theorem 2.4. We introduce on the truncation functions and then define the following associated functionals: It is clear that .
We observe that if is a critical point of , then which implies Similarly, it follows that . This leads to
Since the function is coercive and weakly lower semicontinuous, there exists a global minimizer of it. Using (2.14), it is seen that and so . Relation (2.55) shows that is a nontrivial solution of problem (1.4) belonging to the order interval . Via assumptions (ii) and (iii) and the boundedness of , we may apply the strong maximum principle which ensures on . In view of the minimality property of as stated in Theorem 2.4, it follows that . In fact, is the unique global minimizer of .
Since , there exists a neighborhood of in the space such that . Therefore on , which guarantees that is a local minimizer of on . It results that is also a local minimizer of on the space (see [27], pages 655-656 ). Employing the functional and proceeding as in the case of , we establish that is a local minimizer of on .
As in the case of (2.55), we verify that every critical point of belongs to the set , which implies that every critical point of is a solution to problem (1.4). The functional is coercive, weakly lower semicontinuous, with . Thus has a global minimizer with . The above properties ensure that is a nontrivial solution of problem (1.4) belonging to the order interval . Assume and . We claim that changes sign. Indeed, if not, would have constant sign, for instance a.e. on . Using assumptions (ii) and (iii) and the boundedness of , we may apply the strong maximum principle which leads to on . This is impossible because it contradicts the minimality property of the solution as given by Theorem 2.4. According to the claim, we obtain the conclusion of the theorem setting .
Thus, the proof reduces to consider the cases or . To make a choice, suppose . We may also admit that is a strict local minimizer of This is true since on the contrary we would find (infinitely many) critical points of belonging to the order interval which are different from , , , and if does not change sign, taking into account the strong maximum principle, the extremality properties of the solutions , given in Theorem 2.4 will be contradicted. A straightforward argument allows then to find such that where . Relation (2.57) in conjunction with the Palais-Smale condition (which holds for due to its coercivity) enables us to apply the mountain pass theorem to the functional (see, e.g., [31]). In this way we get satisfying and where We infer from (2.57) and (2.58) that and .
The next step in the proof is to show that By the equality in (2.58), it suffices to produce a path such that Let , where , and be endowed with the topologies induced by and , respectively. We set Making use of the first inequality in assumption (iii), we fix numbers and such that (2.14) holds, and then let . We recall the following variational expression for given by Cuesta et al. [32]: where By (2.63) there exists such that Choose some number with . The density of in implies that is dense in so there is satisfying Then the choice of establishes The boundedness of the set in ensures the existence of some such that Since (see Theorem 2.4), for every and any bounded neighborhood of in there exist positive numbers and such that whenever , , and This fact and the compactness of in allow to determine a number for which one has We now focus on the continuous path in joining and with a fixed constant satisfying . By (2.70), (2.67), (2.68), (2.14) with and taking into account the choice of as well as we obtain
At this point we apply the second deformation lemma (see, e.g., [27, page 366]) to the functional . Towards this let us denote It was already shown that is the unique global minimizer of and so we have Taking into account (2.55), has no critical values in the interval (for, otherwise, the minimality of the positive solution of (1.4) would be contradicted). Using also that the functional satisfies the Palais-Smale condition (because it is coercive), the second deformation lemma can be applied to yielding a continuous mapping such that and for all as well as whenever and . Introducing by for all , it is seen that is a continuous path in joining and . (Note the mapping is continuous from into itself.) The properties of the deformation imply for all . Similarly, applying the second deformation lemma to the functional , we construct a continuous path joining and such that The union of the curves , and gives rise to a path . We see from (2.75), (2.71), and (2.74) that (2.61) is satisfied. Hence (2.60) holds, and so . Recalling that the critical points of are in the order interval we derive that is a nontrivial solution of (1.4) distinct from and with . By the nonlinear regularity theory we have that . The extremality properties of the constant sign solutions and as described in Theorem 2.4 force to be sign-changing. This completes the proof.

2.4. Two Sign-Changing Solutions

The goal of this section is to show that under hypotheses stronger than those in Theorem 2.5, problem (1.4) possesses at least two sign-changing solutions.

The new hypotheses on the nonlinearity in problem (1.4) are the following., with , is a function such that for a.a. , whenever . (i)For all , .(ii)There are constants , , and functions () with as such that (iii)For all there exist constants , and a set with of Lebesgue measure zero such that uniformly with respect to .(iv)There exist constants such that for all we have (v)For every , there exist and such that

We notice that hypotheses are stronger than . In particular, for every , we added the Ambrosetti-Rabinowitz condition for (see hypothesis (v)).

We state now the main result of this section, which produces two sign-changing solutions for problem (1.4).

Theorem 2.6. Assume that hypotheses are fulfilled. Then there exists a number with the property that if then problem (1.4) has a minimal (positive) solution , a maximal (negative) solution and two nontrivial sign-changing solutions satisfying , , a.e. in (so ) and , where .

Proof. Since hypotheses are stronger than , we can apply Theorem 2.5 with , which ensures the existence of a number such that for every , problem (1.4) possesses a positive solution , a negative solution and a sign-changing solution with . The proof of Theorem 2.5 shows that and can be chosen to be the minimal positive solution and the maximal negative solution, respectively.
On the other hand, hypotheses enable us to apply Theorem 1.1 of Bartsch et al. [33]. It follows that there exists a sign-changing solution (by the nonlinear regularity theory) with and . Therefore, we have , which shows that the sign-changing solutions and are different. This completes the proof.

Remark 2.7. In fact, under hypotheses , for , problem (1.4) admits at least six nontrivial solutions: two positive solutions, two negative solutions, and two sign-changing solutions, as seen in Theorem 5 in [34].

3. Problem (1.7) for Parameters and being Large

The main goal of this section is to provide a detailed multiplicity analysis of the nonsmooth elliptic problem (1.7) in dependence of the two parameters and . Conditions in terms of the Fučik spectrum are formulated that ensure the existence of sign-changing solutions. As for the precise formulation of this result we recall the Fučik spectrum, see, for example, [13].

The set of those points for which the problem has a nontrivial solution is called the Fučik spectrum of the negative -Laplacian on . Hence, clearly contains the two lines and with being the first Dirichlet eigenvalue of . In addition, the spectrum of the negative -Laplacian has an unbounded sequence of variational eigenvalues , , satisfying a standard min-max characterization, and contains the corresponding sequence of points , . A first nontrivial curve in through asymptotic to and at infinity was constructed and variationally characterized by a mountain-pass procedure by Cuesta et al. [32] (see Figure 1), which implies the existence of a continuous path in joining and provided is above the curve . Here the functional on is given by The hypothesis on the parameters and that will finally ensure the existence of sign-changing solutions is as follows. (H)Let be above the curve of the Fučik spectrum constructed in [32]; see Figure 1.


Figure 1 
Fučik Spectrum.
3.1. Hypotheses, Definitions, and Preliminaries

We impose the following hypotheses on the nonlinearity whose primitive is of problem (1.7) is measurable in each variable separately. There exists , and such that for a.a. and for all , where denotes the critical Sobolev exponent which is if , and if . One has One has

In view of assumptions (g1) and (g2) the function is locally Lipschitz and the functional defined by is well defined and locally Lipschitz continuous as well. The generalized gradients and can be characterized as follows: Define for every , Proposition 1.7 in [35] ensures that while Theorem 4.5.19 of [36] implies with . The next result is an immediate consequence of [37, Proposition 2.1.5].

Lemma 3.1. Suppose in , in , and for all . Then .

Definition 3.2. A function is called a solution of (1.7) if there is an such that

Remark 3.3. Due to assumption (g3) we have for almost all . Hence, in view of (3.8), problem (1.7) always possesses the trivial solution.

Definition 3.4. A function is called a subsolution of (1.7) if , and if there is an such that

Similarly, we define a supersolution as follows.

Definition 3.5. A function is called a supersolution of (1.7) if , and if there is an such that

Lemma 3.6. Let be the uniquely defined solution of (2.6). If , then there exists a constant such that for any the function is a positive supersolution of problem (1.7).

Proof. Let . By (g4) there is such that and by (g2) we get which implies and thus in view of the definition of we obtain Let , where is a positive constant to be specified. Then we get which shows that for the function is in fact a supersolution of (1.7) with .

In a similar way the following lemma on the existence of a negative subsolution can be proved.

Lemma 3.7. Let be the uniquely defined solution of (2.6). If , then there exists a constant such that for any the function is a negative subsolution of problem (1.7).

In the next lemma we demonstrate that small constant multiples of may be sub- and supersolutions of (1.7). More precisely we have the following result.

Lemma 3.8. Let be the normalized positive eigenfunction corresponding to the first eigenvalue of . If , then for sufficiently small and any the function is a positive subsolution of problem (1.7). If , then for sufficiently small and any the function is a negative supersolution of problem (1.7).

Proof. By (g3) there is a constant such that which implies Define with . Applying (3.19) and the definition of we get provided . The latter can be satisfied by choosing sufficiently small such that , where stands for the supremum-norm of . This proves that is a subsolution if . In a similar way one can show that for sufficiently small the function is a negative supersolution.

Applying a recently obtained comparison result that holds for even more general elliptic inclusions (see [38, Theorem 4.1, Corollary 4.1] we immediately obtain the following theorem.

Theorem 3.9. Let hypotheses (g1)-(g2) be satisfied and assume the existence of a subsolution and supersolution of (1.7) such that . Then there exist extremal solutions of (1.7) within .

3.2. Extremal Constant-Sign Solutions and Their Variational Characterization

Combining the results of Lemmas 3.6, 3.7, and 3.8 and Theorem 3.9 we immediately deduce the existence of nontrivial positive solutions of problem (1.7) provided the parameter satisfies that and the existence of negative solutions of problem (1.7) provided that the parameter satisfies . Our main goal of this section is to show that problem (1.7) has a smallest positive solution and a greatest negative solution . More precisely the following result will be shown.

Theorem 3.10. Let hypotheses (g1)–(g4) be fulfilled. For every and there exists a smallest positive solution of (1.7) within the order interval with the constant as in Lemma 3.6. For every and there exists a greatest negative solution of (1.7) within the order interval with the constant as in Lemma 3.7.

Proof. Let . Lemmas 3.6 and 3.8 ensure that is a supersolution of problem (1.7) and is a subsolution of problem (1.7) provided that is sufficiently small. We may choose such that, in addition, . Thus by Theorem 3.9 there exists a smallest and a greatest solution of (1.7) within the ordered interval . Let us denote the smallest solution by . Moreover, the nonlinear regularity theory for the -Laplacian (cf., e.g., [27, Theorem 1.5.6]) and VΓ‘zquez's strong maximum principle [28] ensure that . Thus for every positive integer sufficiently large there is a smallest solution of problem (1.7) within . In this way we inductively construct a sequence of smallest solutions which is monotone decreasing; that is, we have with some function satisfying .Claim 1. is a solution of problem (1.7).As and are solutions of (1.7) we have where and for almost all . Since , the last equation together with (g2) implies that the sequence is bounded in . Taking into account (3.21) we obtain that and The solution of (1.7) satisfies which yields with in (3.24) the equation Using the convergence properties (3.23) of and (g2) as well as the uniform boundedness of the sequence , we get by applying Lebesgue's dominated convergence theorem which by the -property of on implies Since are uniformly bounded, from (g2) we see that there exists a constant such that and thus we get (for some subsequence if necessary) in . By the strong convergence (3.27), Lemma 3.1 can be applied to show that for almost every . Passing to the limit in (3.24) (for some subsequence if necessary) proves Claim 1.
As belongs, in particular, to , Claim 1 and Assumption (g2) implies . The nonlinear regularity theory (cf., e.g., Theorem 1.5.6 in [27]) ensures that for some , so . In view of (g2) (g3) a constant can be found such that which yields in conjunction with Claim 1 that We now apply VΓ‘zquez's strong maximum principle [28] where in its statement the function is chosen as for all , which is possible because . This result guarantees that if , then in and on which means that .Claim 2. .Suppose that Claim 2 does not hold. Then by VΓ‘zquez's strong maximum principle we must have , and thus the sequence satisfies Setting we may suppose that along a relabelled subsequence one has with some , and there is a function such that Since are positive solutions of (1.7), we get for the following variational equation: With the special test function in (3.35) we obtain From (3.29) and (3.34) we get the estimate As the right-hand side of the last inequality is in , we may apply Lebesgue's dominated convergence theorem, which in conjunction with (3.33) yields From (3.33) and (3.36) we conclude which in view of the -property of on results in and therefore, in particular, . Taking into account (g3), (3.31), and (3.40), we may pass to the limit in (3.35) which results in As , relation (3.41) expresses the fact that is an eigenfunction of corresponding to the eigenvalue . As this is impossible according to Anane [30], because must change sign. This contradiction proves that Claim 2 holds true. Note that unlike in the proof of Theorem 2.4, here the contradiction is achieved by the sign-changing property of eigenfunctions belonging to eigenvalues bigger than .Claim 3. is the smallest positive solution of (1.7) in .We already know that . Assume that is any positive solution of (1.7) belonging to . Since , then by (1.7) and (g3) we deduce . Using [27, Theorem 1.56] we derive , and applying VΓ‘zquez's strong maximum principle [28] we infer , which yields for sufficiently large. This in conjunction with the fact that is the least solution of (1.7) in ensures if is large enough. In view of (3.21), we obtain , which proves Claim 3.
The proof of the existence of the greatest negative solution of (1.7) within the ordered interval can be done in a similar way. This completes the proof of the theorem.

Under hypotheses (g1)–(g4), Theorem 3.10 ensures the existence of extremal positive and negative solutions of (1.7) for all and denoted by and , respectively. In what follows we are going to characterize these extremal solutions as global (local) minimizers of certain locally Lipschitz functionals that are generated by truncation procedures. To this end let us introduce truncation functions related to the extremal solutions and as follows: The truncations are continuous, uniformly bounded, and Lipschitzian with respect to . The extremal positive and negative solutions and of (1.7), respectively, ensured by Theorem 3.10 satisfy where and for a.a. . By means of we introduce the following truncations of the nonlinearity : and define functionals by Due to (g2) the functionals are locally Lipschitz continuous. Moreover, in view of the truncations involved these functionals are bounded below, coercive, and weakly lower semicontinuous such that their global minimizers exist. The following lemma provides a characterization of the critical points of these functionals.

Lemma 3.11. Let and be the extremal constant-sign solutions of (1.7). Then the following holds. (i)A critical point of is a (nonnegative) solution of (1.7) satisfying .(ii)A critical point of is a (nonpositive) solution of (1.7) satisfying .(iii)A critical point of is a solution of (1.7) satisfying .

Proof. To prove (i) let be a critical point of , that is, , which results in for some and such that almost everywhere in , with Let us show that holds. As is a positive solution of (1.7), it satisfies the first equation in (3.43), and by subtracting that equation from (3.47) and applying the special test function we get By the definition of the truncations introduced above we have and for a.a. , and thus the right-hand side of (3.49) is zero which leads to and hence it follows . Because this implies which proves . To prove we test (3.47) with and get which results in , and thus , that is, . Thus the critical point of which is a solution of the Dirichlet problem (3.47) satisfies , and therefore . Because it follows , and therefore must be a solution of (1.7). This proves (i). In just the same way one can prove also (ii) and (iii) which is omitted.

The following lemma provides a variational characterization of the extremal constant-sign solutions and .

Lemma 3.12. Let and . Then the extremal positive solution of (1.7) is the unique global minimizer of the functional , and the extremal negative solution of (1.7) is the unique global minimizer of the functional . Both and are local minimizers of .

Proof. The functional is bounded below, coercive, and weakly lower semicontinuous. Thus there exists a global minimizer of , that is, As is a critical point of , so by Lemma 3.11 it is a nonnegative solution of (1.7) satisfying . Since there is a such that . By (g3) we infer the existence of a such that and thus for sufficiently small such that we obtain (note ) Hence it follows that , and thus . Applying nonlinear regularity theory for the -Laplacian (cf., e.g., [27, Theorem 1.5.6]) and VΓ‘zquez's strong maximum principle, we see that . As is the smallest positive solution of (1.7) in and , it follows , which shows that the global minimizer must be unique and equal to . By similar arguments one can show that the global minimizer of must be unique and . It remains to prove that and are local minimizers of . Let us show this last assertion for only. By definition we have Since is a global minimizer of and , it follows that is a local minimizer of with respect to the topology. Due to a result by Motreanu and Papageorgiou in [39, Proposition 4], we conclude that is also a local minimizer of with respect to the topology. This completes the proof of the lemma.

Lemma 3.13. The functional has a global minimizer which is a nontrivial solution of (1.7) satisfying .

Proof. One easily verifies that is coercive and weakly lower semicontinuous, and thus a global minimizer exists which is a critical point of . Apply Lemma 3.11(iii) and note that, for example, , which shows that .

3.3. Sign-Changing Solutions

Theorem 3.10 ensures the existence of a smallest positive solution in and a greatest negative solution of (1.7) in . The idea to show the existence of sign-changing solutions is to prove the existence of nontrivial solutions of (1.7) satisfying with and , which then must be sign-changing, because and are the extremal constant-sign solutions.

Theorem 3.14. Let hypotheses (g1)–(g4) and (H) be satisfied. Then problem (1.7) has a smallest positive solution in , a greatest negative solution in , and a nontrivial sign-changing solution with .

Proof. Clearly the existence of the extremal positive and negative solution and follows from Theorem 3.10, because (H), in particular, implies that and . As for the existence of a sign-changing solution we first note that by Lemma 3.13 it follows that the global minimizer of is a nontrivial solution of (1.7) satisfying . Therefore, if and then must be a sign-changing solution as asserted, because is the greatest negative and is the smallest positive solution of (1.7). Thus, we still need to prove the existence of sign-changing solutions in case that either or .
Let us consider the case only, since the case can be treated quite similarly. By Lemma 3.12, is a local minimizer of . Without loss of generality we may even assume that is a strict local minimizer of , because on the contrary we would find (infinitely many) critical points of that are sign-changing solutions thanks to and the extremality of the solutions , obtained in Theorem 3.10 which proves the assertion.
Therefore, it remains to prove the existence of sign-changing solutions under the assumptions that the global minimizer of is equal to , and is a strict local minimizer of . This implies the existence of such that where . The functional satisfies the Palais-Smale condition, because it is bounded below, locally Lipschitz continuous, and coercive; see, for example, [40, Corollary 2.4]. Thus in view of (3.57) we may apply the nonsmooth version of Ambrosetti-Rabinowitz's Mountain-Pass Theorem (see, e.g., [41, Theorem 3.4]) which ensures the existence of a critical point satisfying and where It is clear from (3.57) and (3.58) that and and thus is a sign-changing solution provided . To prove the latter we claim for which it suffices to construct a path such that The construction of such a path can be done by adopting an approach due to the authors in [3] and applying the Second Deformation Lemma for locally Lipschitz functionals as it can be found in [42, Theorem 2.10]. This completes the proof.

Remark 3.15. The multiplicity results and the existence of sign-changing solutions obtained in this section generalize very recent results due to the authors obtained in [3, 4, 7]. Moreover, if the function is continuous on and , then , and problem (1.7) reduces to However, even in this setting the results obtained here are more general than obtained in [6, Theorem 3.9], because we do not assume that for all .

Remark 3.16. Theorem 3.14 improves also Corollary 3.2 of [8]. In fact, let , let be a solution of (1.7) in case and , with satisfying . By definition of Clarke's gradient we have, for any , As is a solution, the following holds: and (), which yields That is, turns out to be a solution of the hemivariational inequality studied in [8]. Since the hypotheses of [8, Corollary 3.2] imply (g1)–(g4), the assertion follows.

Remark 3.17. Multiplicity results for a nonsmooth version of problem (1.4) in form of (1.2) can be established under structure conditions for Clarke's gradient similar to H(f).
Multiplicity and sign-changing solutions results have been obtained recently in [43] for the following nonlinear Neumann-type boundary value problem: find and parameters such that For problem (3.66) conditions on the parameters have been given in terms of the β€œSteklov-Fučik” spectrum to ensure multiplicity results.