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International Journal of Differential Equations
Volume 2016, Article ID 3149482, 14 pages
Research Article

Multiplicity Results for the -Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition

College of Sciences at Dammam, University of Dammam, Dammam 31441, Saudi Arabia

Received 5 April 2016; Accepted 22 June 2016

Academic Editor: Julio D. Rossi

Copyright © 2016 K. Saoudi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We investigate the singular Neumann problem involving the -Laplace operator:   , in  , where is a bounded domain with boundary, is a positive parameter, and , and are assumed to satisfy assumptions (H0)(H5) in the Introduction. Using some variational techniques, we show the existence of a number such that problem has two solutions for one solution for , and no solutions for .

1. Introduction

The purpose of this paper is to study the existence of multiple solutions for the following inhomogeneous singular Neumann problem involving the -Laplace operator:

Here is a bounded domain with boundary; is a positive parameter. For any continuous and bounded function we define and . Associated with problem we have the singular functional given by where .

Definition 1. is called a generalized solution of the equationif for all

Obviously, every weak solution of problem is also a generalized solution of (3).

The operator is called -Laplace where is a continuous nonconstant function. This differential operator is a natural generalization of the -Laplace operator , where is a real constant. However, the -Laplace operator possesses more complicated nonlinearity than -Laplace operator, due to the fact that is not homogeneous. This fact implies some difficulties; for example, we cannot use the Lagrange Multiplier Theorem in many problems involving this operator.

The study of differential and partial differential equations involving variable exponent is a new and an interesting topic. The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, electrorheological fluids, image processing, flow in porous media, calculus of variations, nonlinear elasticity theory, heterogeneous porous media models (see Acerbi and Mingione [1], Diening [2]), and so forth. These physical problems were facilitated by the development of Lebesgue and Sobolev spaces with variable exponent.

At this point, we briefly recall related existence and multiplicity results for elliptic equations with Neumann boundary conditions. Neumann type problems are studied in [36] and references therein. The multiplicity result for Neumann problem with Sobolev critical nonlinearity has been studied in [5] where authors considered the problemHere , , and . They proved the existence of such that problem (5) admits at least two solutions for all , one solution when , and no solutions for The problem in two dimensions has been considered in [6] where the authors extended the results obtained by [5].

Results for -Laplacian problems Neumann boundary conditions are rare (see [7, 8]). In [7], Fan and Deng studied the Neumann problems with -Laplace operator and the nonlinear potential under appropriate assumptions. By using the subsupersolution method and variation method, the authors get the multiplicity of positive solutions. In [8], Sreenadh and Tiwari extend previous works on nonlinear parametric problems with the -Laplace operator to the case where the Neumann boundary condition is nonlinear. Precisely, under supplementary hypotheses on , the authors show that there exists a finite number such that the posed problem has two solutions for , one solution for , and no solutions for

Before stating our main results, we make the following assumptions throughout this paper:(H0), , and . As usual, .(H1) for some satisfying .(H2) satisfying .Let be a nondecreasing Carathéodory function satisfying the following: (H3) and for all .(H4)There exist such that, for , with and . (H5)There exist a constant and such thatNext we describe in a more precise way our main results.

Theorem 2. Assume that hold and in (6). Then there exists with the following properties:(1)Problem has a solution for every .(2)Problem has a solution if .(3)Problem does not have any solution if .

Theorem 3. Assume that hold and in (6). Then, problem has at least two distinct solutions and for every .

This paper is organized as follows. In Section 2, we will recall some basic facts about the variable exponent Lebesgue and Sobolev spaces which we will use later. Proofs of our results will be presented in Sections 3 and 5.

2. Generalized Lebesgue-Sobolev Spaces Setting

To deal with the -Laplacian problem, we need to introduce some functional spaces , , and properties of the -Laplacian which we will use later. Denote by the set of all measurable real-valued functions defined in . Note that two measurable functions are considered as the same element of when they are equal almost everywhere. Let with the norm The space becomes a Banach space. We call it variable exponent Lebesgue space. Moreover, this space is a separable, reflexive, and uniform convex Banach space; see [9, Theorems 1.6, 1.10, and 1.14].

The variable exponent Sobolev space can be equipped with the norm Note that is the closure of in under the norm . The spaces and are separable, reflexive, and uniform convex Banach spaces (see [9, Theorem ]). The inclusion between Lebesgue spaces also generalizes naturally: if and , are variable exponents so that almost everywhere in then there exists the continuous embedding .

We denote by the conjugate space of , where . For and , the Hölder type inequalityholds true.

An important role in manipulating the generalized Lebesgue spaces is played by the modular of the space, which is the mapping defined by If and , then the following relations hold true.

Lemma 4. Consider the following:

The following result generalizes the well-known Sobolev embedding theorem.

Theorem 5 (see [10, 11]). Let be an open bounded domain with Lipschitz boundary and assume that with for each . If and for all , then there exists a continuous embedding . Also, the embedding is compact almost everywhere in , where

Now, we recall the following boundary trace embedding theorem from [12].

Theorem 6. Let be an open bounded domain with Lipschitz boundary. If such that then .

Next we give a comparison principle as follows.

Lemma 7 (see [8, Lemma ]). Let be nonnegative functions satisfying Then almost everywhere in .

We recall the following strong maximum principle from [13].

Theorem 8. Let , for some , satisfy , andwith on , where are such that pointwise everywhere in . If where is the inward unit normal on , then, the following strong comparison principle holds:

3. Existence of a Solution

In this section, we show the existence of a local minimum for in a small neighborhood of the origin in . Firstly, let us define

Lemma 9. There exists such that admits a solution for

Proof. Using (14) and the embeddings in Theorem 5, we estimate for as follows:Hence, noting that and , we can choose small enough, and there exists such thatMoreover, since , we have, for , small enough, Set Now, note that, for and , we haveSince , this implies that as Thus, Now, let be a minimizing sequence for Then for Now by the Ekeland variational principle, there exists a sequence such that andMoreover, using the Brézis-Lieb lemma in [14] combined with the version of Theorem in Boccardo and Murat [15], it follows that for almost every ,Hence, from (28) and the compact boundary trace embedding in Theorem 5, we get Thus, it follows that Hence and it is a local minimizer of in

We prove now the existence of positive solution to for Precisely, we have the following result.

Lemma 10. Problem possesses a solution for

Proof. Fix . such that there exist solutions to for , say . Note that is a supersolution for It is clear that is not a local minimizer of on since and for Now, we show the existence of a local minimizer of the functional energy. For this, we use the cut-off argument. Define Also define the functional bywhere , , and From the dominated convergence and the compact boundary trace embedding in Theorem 5, it is easy to see that is bounded below and is weakly lower semicontinuous in Then, there exists such that achieves its global minimum in Moreover, since is from Lemma A.4, solves the equationNow, using the strong maximum principle (see Theorem 8) and since is not a local minimizer of on By the definition of , and we haveAgain, by the strong comparison principle (see Lemma 7), we conclude that in and on Hence is a solution to This completes the proof of Lemma 10.

Now, we show the following result.

Lemma 11. There exists at least one positive weak solution for to problem

Proof. Let , as , and be a solution of such that for all Now, taking as test function in , we getMoreover, as , we haveIt follows thatNow, from (7) there exists such that, for and for all , Moreover, using Theorem in [16], we get the existence of the constants , such thatwhere , and . On the other hand, we recall the following inequality due to Lieberman [17]. There exists a constant such thatInserting (36), (37), and (38) in (33), we get It follows that is bounded in since for all Without loss of generality, in and then by the Sobolev imbedding in and for a.e. By the -regularity results of [13], the boundedness of implies the boundedness of By the -regularity Theorem 16, the boundedness of implies the boundedness of , where is a constant. Thus, we have in For every , since is a solution of problem , we have that, for each ,Passing to the limit in (39) as yields which shows that is a solution of Obviously and Hence is a positive solution of in This completes the proof of Lemma 11.

Then we prove the following nonexistence result.

Lemma 12. Consider the following:

Proof. Let be a solution of Taking as a test function in the weak formulation of , we getOn the other hand, we haveUsing assumption (H2) we have , and using (6) we have Therefore, from (41) and (42) we getNow, since and , by the embedding of into and by the embedding of into we obtainSubstituting (44) in (43) we get Hence, is bounded by a constant independent of Now, taking as a test function in the weak formulation of and using (H2) we getNow sinceit follows that is finite. The proof of Lemma 12 is now completed.

Proof of Theorem 2. Theorem 2 follows from Lemmas 10, 11, and 12.

Now, we prove that the solution of problem obtained in Lemma 10 is a local minimum for the functional energy associated to problem Precisely, we have the following result.

Lemma 13. Let be the weak solution of problem obtained in Lemma 10. Then, is a local minimum for .

Proof. Fix and let be solutions to for and , respectively, such that . By Lemma 7,   on Define the following cut-off functions: Then the corresponding functional given by where , , and Firstly, from Lemma A.4   is Then, it is simple to see that is coercive and bounded below. Let denote the global minimum of on which satisfies the equationTherefore, from the regularity results Theorem 16, we conclude that is in for some Now, using (H2) and since is nondecreasing, by the definition of , , and we have Again, by the strong comparison principle (see Lemma 7), we conclude that in Let We claim that If not, then there exists such that and But this contradicts the boundary data as Therefore, if then in since on the set Hence is a local minimum for This completes the proof of Lemma 13.

4. versus Local Minimizers of the Energy

The following lemma is crucial in showing multiplicity of solutions. It has been shown in the case in [18] for the case of critical growth functionals : , , and later for critical growth functionals , , , in [19]. A key feature of these latter works is the uniform estimate they obtain for equations like but involving two -Laplace operators. Using constraints based on -norms rather than Sobolev norms as in [19], the equations for which uniform estimates are required can be simplified to a standard type involving only one -Laplace operator. This approach was followed in [20] in the subcritical case, in [21] in the critical case, in [2224], and also in this work to deal with the boundary value problem involving the nonlinear -Laplacian case. More precisely, we have the following result.

Lemma 14. Suppose that conditions are satisfied. Let satisfyingbe a local minimizer of in topology. Then, is a local minimum of in also.

For proving Lemma 14, we will need the following uniform -estimates for a family of solutions to

Proposition 15. Let be a family of solutions to , where satisfies (52) and solves Let be such that Then,

The proof of Proposition 15 is a consequence of the results proved in Appendix  A in [13]. Hence, the regularity results of Saoudi and Ghanmi [13] give the following regularity result for weak solutions to problem .

Theorem 16. Let be a solution to problem Then, there exists such that any weak solution to problem belongs to for some .

Proof of Lemma 14. Assume that the conclusion of Lemma 14 is not true. We define the following constraint for each :We consider the following constraint minimization problem: Firstly, clearly Moreover, we note that is a convex set. Using the trace embeddings we see that is also a closed set in which implies that is weakly closed in ; with the fact that is weakly low semicontinuous in , it follows that for is achieved on some ; that isMoreover, since and , we may assume that
We now consider the following two cases.
(1) Let . Then is also a local minimizer of in We now show that admits Gâteaux derivatives on to derive that satisfies the Euler-Lagrange equation associated with . For this, according to Lemma A.2, in the Appendix, we need to prove that such thatwhere . To prove (58), we argue by contradiction: letand suppose that has a nonzero measure.
Let and for set . Then, there exists satisfying such that   . Then, from Lemma A.4   is differentiable for and . Thus, From (H1)(H3), we see that for small enough.
Now, since is nonincreasing for small enough uniformly to (by (H1)(H3)) and from the monotonicity of the operator , we have that for small enough Therefore, from Taylor’s expansion and since , there exists such thatLetting we have We obtain a contradiction with (62) and then for some (which depends a priori on ). Since is a local minimizer of and is Gâteaux differentiable in , we get that is defined and Recalling that is the solution to the pure singular problem given by Theorem in [16] and from the weak comparison principle, there exist , such thatwhere , for some (independent of ). Since and from the fact that satisfies , we get that is uniformly bounded in . Now, using Proposition 15 and Theorem 16, we getand as which contradicts the fact that is a local minimizer in .
Now, we deal with the second case.
(2) : we again show that in for some . Taking ,  , we obtain as above that for and small enough.
Then is decreasing. This implies that