Existence and Multiplicity of Solutions for Semipositone Problems Involving p-Laplacian
We prove existence and multiplicity of positive solutions for semipositone problems involving -Laplacian in a bounded smooth domain of under the cases of sublinear and superlinear nonlinearities term.
In this paper, we shall study the following semipositone problem involving the -Laplacian: where is a smooth bounded domain, is a positive parameter, and is a continuous function satisfying the condition(F0).
Such problems are usually referred in the literature as semipositone problems. We refer the reader to , where Castro and Shivaji initially called them nonpositone problems, in contrast with the terminology positone problems, coined by Keller and Cohen in , when the nonlinearity was positive and monotone.
Under the case of , a novel variational approach is presented by Costa et al.  to the question of existence and multiplicity of positive solutions to problem (1.1), where they consider both the sublinear and superlinear cases. The aim of this paper is to extend their results to the case of -Laplacian. The main difficulty is in verifying -condition because of the operator is not self-adjoint linear.
Theorem 1.1. Assume (F0) and the following assumptions:(F1) (the sublinear case);(F2) for some , where .Then, there exist such that (1.3) has no nontrivial nonnegative solution for , and has at least two nontrivial nonnegative solutions , for all . Moreover, when is a ball , these two solutions are non-increasing, radially symmetric and, if , at least one of them is positive, hence a solution of (1.1).
Theorem 1.2. Assume (F0), (F2) and the following assumptions:(F3), (the sublinear, subcritical case);(F4)for all and some (the AR-condition).Then, (1.3) has at least one nonnegative solution for all . If then is nonincreasing, radially symmetric and one of the two alternatives occurs. (i)There exists such that, for all , is a positive solution of (1.1) having negative normal derivative on . (ii)For some sequence , problem (1.1) with has a positive solution with zero normal derivative on .
We start by recalling some basic results on variational methods for locally Lipschitz functionals. Let be a real Banach space and is its topological dual. A function is called locally Lipschitz if each point possesses a neighborhood such that for all , for a constant depending on . The generalized directional derivative of at the point in the direction is The generalized gradient of at is defined by which is a nonempty, convex, and -compact subset of , where is the duality pairing between and . We say that is a critical point of if . For further details, we refer the reader to Chang .
We list some fundamental properties of the generalized directional derivative and gradient that will be possibly used throughout the paper.
Proposition 2.1 (see [4, 5]).
(1) Let be a continuously differentiable function. Then , coincides with , and for all .
(2) for all .
(3) If is a convex functional, then coincides with the usual subdifferential of in the sense of convex analysis.
(4) If has a local minimum (or a local maximum) at , then .
(5) for all .
(6) for all .
(7) The function exists and is lower semicontinuous; that is, .
In the following we need the nonsmooth version of the Palais-Smale condition.
Definition 2.2. One says that nonsmooth satisfies the -condition if any sequence such that and .
In what follows we write the -condition as simply as the -condition if it holds for every level for the Palais-Smale condition at level .
We note that property (4) above says that a local minimum (or local maximum) of is a critical point of .Finally, we point out that many of the results of the classical critical point theory have been extended by Chang  to this setting of locally Lipschitz functionals. For example, one has the following celebrated theorem.
Theorem 2.3 (nonsmooth mountain pass theorem; see [4, 5]). If is a reflexive Banach space, is a locally Lipschitz function which satisfies the nonsmooth -condition, and for some and with , . Then has a nontrivial critical such that the critical value is characterized by the following minimax principle: where .
We would like to point out that we can obtain the same results for problem (1.3) through approximation of the discontinuous nonlinearity by a sequence of continuous functions. Variational methods were then applied to the corresponding sequence of problems and limits were taken. For the rest of this paper, we write with the norm by and denote by and the generic positive constants for simplicity.
3. Proof of the Main Results
Now, having listed some basic results on critical point theory for the Lipschitz functionals, let us consider the functional where and were defined in (1.2). Clearly is a locally Lipschitz continuous function and satisfies for . In view of [3, Theorems and 2.2], the above formula for defines a locally Lipschitz functional on whose critical points are solutions of the differential inclusion In our present case, it follows that for , for , and for .
We start with some preliminary lemmas.
Lemma 3.1. Assume (F0), (F1), and (F2). Then there exists such that problem (1.3) has no nontrivial solution for .
Proof . If is a solution of problem (1.3), then, multiplying the equation by and integrating over yields hence where we have chosen so that for (such a exists in view of (F0)). Now, since (F1) implies the existence of such that for all , we obtain from (3.4) that so that where this last constant is independent of both and . Therefore we must have
Lemma 3.2. Assume (F0) and either (F1) or (F3). Then is a strict local minimum of the functional .
Proof . Since (F1) or (F3) implies the existence of such that recall also that for . Then, with as in the proof of Lemma 3.1 and noticing that for all , we can write for an arbitrary , so that, using the Sobolev embedding theorem in the last inequality and with a constant independent of and , we obtain Therefore, for each , it follows that if . This shows that is a strict local minimum of .
Proof . We shall follow some of the arguments in . As mentioned earlier, if is a critical point of , then it is shown in  that is a solution of the differential inclusion
Since is only discontinuous at , the above differential inclusion reduces to an equality, except possibly on the subset where . Since is subcritical, using the regularity results for quasilinear elliptic equations with -growth condition (see, for example, ), we have . And, in this latter case, takes on values in the bounded interval . Therefore, by the Michael selection theorem (see Theorem 1.2.5 of ), we see that admits a continuous selection. Using the regularity results for quasilinear elliptic equations with -growth condition again, it follows that , .
Next, using a Morrey-Stampacchia theorem [7, Theorem 3.2.2, page 69], we have that a.e. in . Therefore, since we defined , it follows that Replacing the inclusion (3.2) on , we conclude that is a solution of (1.3). Finally, recalling that for , it is clear that . The proof of Lemma 3.3 is complete.
Lemma 3.4. Assume either (F1) or (F3), (F4). Then satisfies the nonsmooth -condition at every .
Proof. Let be a sequence such that for all and as . In the superlinear and subcritical case, from (F4), we have
where . Hence is bounded.
Thus by passing to a subsequence if necessary, we may assume that in as . We have with , where and . From (F3) and Chang  we know that (). Since is embedded compactly in , we have that as in . So using the Hölder inequality, we have
Therefore, we obtain that . But we know that is a mapping of type (). Thus we have Using the similar method, we can more easily get the nonsmooth -condition in the case of sublinear.
Remark 3.5. Note that we cannot directly apply the results of  because the operator is not self-adjoint linear.
Lemma 3.6. Under assumptions (F0), (F1), and (F2), let with , and let be a radially symmetric, nonincreasing function such that and is a minimizer of with . Then, does not vanish in ; that is, for all .
Proof. Since is discontinuous at zero, we note that the conclusion does not follow directly from uniqueness of solution for the Cauchy problem with data at (in fact, writing , , we may have and ).
Now, since by assumption, satisfies . If there is nothing to prove in view of the fact that is non-increasing. On the other hand, if then and for . It is not hard to prove that this contradicts that is a minimizer of . Indeed, if then A simple calculation shows that the rescaled function satisfies where is less than 1. Therefore, since we are assuming , we would reach the contradiction .
Remark 3.7. Note that the condition (F2) is necessary because of which guarantee can have positive values.
Proof of Theorem 1.1. We observe that the functional is weakly lower semicontinuous on . Moreover, the sublinearity assumption (F1) on implies that is coercive. Therefore, the infimum of is attained at some : And, in view of Lemma 3.3, is a nonnegative solution of (1.3). We now claim that is nonzero for all large.
Claim 1. There exists such that for all .
In order to prove the claim it suffices to exhibit an element such that for all large. This is quite standard considering that by (F2). Indeed, letting for small, define so that for and for . Then where we note that the expression in the above parenthesis is positive if we choose sufficiently small. Therefore, there exists such that for all , which proves the claim.
On the other hand, when , let denote the Schwarz symmetrization of , namely, is the unique radially symmetric, nonincreasing, nonnegative function in which is equi-measurable with . As is well known, and (the Faber-Krahn inequality; see ), so that . Therefore, we necessarily have and may assume that . Moreover, in by Lemma 3.6. Therefore, is a positive solution of both problems (1.1) and (1.3).
Next, we recall that is a strict local minimum of by Lemma 3.2. Therefore, since satisfies the nonsmooth Palais-Smale condition by Lemma 3.4, we can use the minima and of to apply the nonsmooth mountain pass theorem and conclude that there exists a second nontrivial critical point with . Again, is a nonnegative solution of problem (1.3) in view of Lemma 3.3. In addition, when , arguments similar to above passage show that we may assume . The proof of Theorem 1.1 is complete.
Proof of Theorem 1.2. As is well-known, the AR condition (F4) readily implies the existence of an element such that . On the other hand, Lemma 3.2 says that is a (strict) local minimum of and Lemma 3.4 says that satisfies nonsmooth for every . Therefore, an application of the nonsmooth mountain pass theorem stated in section 2 yields the existence of a critical point such that
In particular, , and it follows that is a nonnegative solution of problem (1.3) by Lemma 3.3. As in the proof of Theorem 1.1, we may assume that in the case of a ball .
Finally, still in the case of a ball , we claim that there exists such that, for all , is a positive solution of problem (1.3) (hence of problem (1.1)) having negative normal derivative on . Indeed, if that is not the case, then, for any given , we can find such that the nonnegative solution of problem (1.1) with obtained a bove satisfies for some . Therefore, the rescaled function is a positive solution of (1.1) with (again in the ball ), with . This shows that we can always construct a decreasing sequence satisfying alternative (ii) of Theorem 1.2, in case alternative (i) does not hold.
The author is very grateful to an anonymous referee for his or her valuable suggestions. Research supported by NSFC (no. 11061030, no. 10971087).
L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, vol. 8, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005.
C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Springer, New York, NY, USA, 1966.