Sign-Changing Solutions to Equations of Elliptic TypeView this Special Issue
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Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters
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 “beginning”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.
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 Carathéodory 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 Carathéodory 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 . 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 ) the authors considered the eigenvalue problem (1.10) for a Carathéodory 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  improve among others recent results obtained in . 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  where techniques for single-valued problems developed in  and hemivariational methods applied in  have been used. Multiplicity results for (1.11) have been obtained also in .
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  (where and is Hölder continuous with respect to for every fixed ) and of Motreanu-Motreanu-Papageogiou . 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 Carathéodory (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  (see also ) 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 ) 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 ).
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 ) 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. ) 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 , 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 Vázquez ). 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  let us denote whenever , where Relation (2.48) justifies that . Then Proposition 1 of Anane  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  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 , 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., ). 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. : 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 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