International Journal of Differential Equations
Volume 2016 (2016), Article ID 3149482, 14 pages
http://dx.doi.org/10.1155/2016/3149482

Research Article

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

K. Saoudi, M. Kratou, and S. Alsadhan

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.

Abstract.

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 equation

if 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 [3–6] and references therein. The multiplicity result for Neumann problem with Sobolev critical nonlinearity has been studied in [5] where authors considered the problem

Here , , 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 that

 Next 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 inequality

holds 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 , and

with 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 that

Moreover, since , we have, for , small enough,

Set

Now, note that, for and , we have

Since , 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 and

Moreover, 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 by

where , , 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 equation

Now, using the strong maximum principle (see Theorem 8) and since is not a local minimizer of on By the definition of , and we have

Again, 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 get

Moreover, as , we have

It follows that

Now, from (7) there exists such that, for and for all ,

Moreover, using Theorem in [16], we get the existence of the constants , such that

where , and . On the other hand, we recall the following inequality due to Lieberman [17]. There exists a constant such that

Inserting (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 get

On the other hand, we have

Using assumption (H2) we have , and using (6) we have Therefore, from (41) and (42) we get

Now, since and , by the embedding of into and by the embedding of into we obtain

Substituting (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 get

Now since

it 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 equation

Therefore, 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 [22–24], 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 satisfying

be 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 is

Moreover, 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 that

where . To prove (58), we argue by contradiction: let

and 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 that

Letting 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 that

where , 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 get

and 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 for and using (52)

This yields a contradiction with the fact that is a global minimizer of on . In this case, using Lemma A.4 and from the Lagrange multiplier rule we have

We first show that We argue by contradiction. Suppose that ; then there exists such that

and then for small we have

This contradicts the fact that is a global minimizer of in .
We deal now with the two following cases.
Case 1 . In this case, we write (67) in its PDE form as

In this case, from (57), we have that Hence, we can apply Proposition 15 to conclude that for some constant independent of Therefore, using Theorem 16 we conclude that for some constant independent of and as

which contradicts the fact that is a local minimizer in .
Now, we deal with the second case.
Case 2 . From above, we can assume that for small enough. Furthermore, we can find a number independent of and , such that and are negative for all Then, from the weak comparison principle (see Lemma 7 and using ( as test function) we have that for small enough. Now, since is a local minimizer, is a weak solution to ; that is, satisfies ess over every compact set and

for all . Therefore, for every function , satisfies

Similarly,

Now, substracting the above relations with , with , as a test function in , integrate by parts and use the fact that is a monotone operator to obtain

Using the bounds of we get

where does not depend on and Now, using Hölder’s inequality and the bounds of combined with Lemma 4 we obtain

Therefore

Thus for any

Passing to the limit in (79) we get

Then, using (80) combined with Proposition 15, the uniform bounds for in as well as , we get that the right-hand side terms in are uniformly bounded in and in from which as in the first case we obtain that is bounded in independently of Finally, using Ascoli-Arzela Theorem we find a sequence such that

It follows that, for sufficiently small,

which contradicts the fact that is a local minimizer of for the topology. The proof of Lemma 14 is now completed.

5. Existence of a Second Weak Solution for

In this section, we fix and let , and be as in Section 3. Now, we are able to show the existence of a second solution using the generalized Mountain Pass Theorem. Since the functional is not , we use the cut-off functional defined in (86). Define the cut-off functions

by

by

by

and define now the corresponding functional given by

where , and . Firstly, we prove the following lemma on compactness of Palais-Smale sequences.

Lemma 17. The functional satisfies the Palais-Smale condition.

Proof. Let be a (PS) sequence; namely, is bounded and when Then,

Now, we estimate the boundary term from above as follows:

Hence, taking (88) in (87) and using (H2) combined with (7) we get

Now, using Lemma 4 and the fact that , we get

Hence, is bounded. Without loss of generality, we assume that there exists a subsequence of such that Therefore, using Theorems 5 and 6 we get

Observe that

We already know that

Using (91), we obtain

This together with the convergence of in implies that strongly in ; that is, satisfies the (PS) condition. The proof of Lemma 17 is now completed.

Proof of Theorem 3. Firstly, note that for any solution of (50). Hence, as in Section 3 we can conclude that is a local minimum for in By the strong comparison principle and Hopf lemma, we can conclude that any critical point of is also a critical point of and hence also solves It is easy to see that and are a subsolution and a supersolution to the problem associated with the functional energy Therefore, using the approach as in Theorem 2, we prove that this problem has a solution such that is a local minimizer of in the topology. Now, by the comparison principle we can see that and also solves If the conclusion of Theorem 3 holds. That is, we can assume and is a strict local minimum of in the topology. Then, from Lemma A.4, the functional and note that as Thus, we can apply Lemma 17 combined with the Mountain Pass Theorem to conclude that problem has a solution such that Therefore the proof of Theorem 3 is now completed.

Appendix

We start with an important technical tool which enables us to estimate the singularity in the Gâteaux derivative of the energy functional defined in (2).

Lemma A.1. Let . Then there exists a constant such that the inequality

holds true for all with .

An elementary proof of this lemma can be found in Takáč [25, Lemma  A.1, p. 233]. We continue by showing the Gâteaux differentiability of the energy functional at a point satisfying in with (for details see Theorem in [16]).

Lemma A.2. Let assumptions be satisfied. Assume that and satisfies in Then we have

Proof. We show the result only for the singular term ; the other two terms are treated in a standard way. So let

For we define

Consequently,

Notice that for almost every we have and

Moreover, the integral on the left-hand side (with nonnegative integrand) is dominated by

with constants independent of . Here, we have used the estimate (A.1) from Lemma A.1 above. Finally, we have , by and Hardy’s inequality. Hence, we are allowed to invoke the Lebesgue dominated convergence theorem in (A.5) from which the lemma follows by letting .

Corollary A.3. Let assumptions be satisfied. Then the energy functional is Gâteaux differentiable at every point that satisfies in . Its Gâteaux derivative at is given by

for .

We continue by proving the -differentiability of the cut-off energy functional defined below.

Lemma A.4. Let assumptions be satisfied, and such that in . Setting for

and for

we have that belongs to .

Proof. As in Lemma A.2, we concentrate on the singular term, the others being standard. Let

, and . Proceeding as in Lemma A.2, we obtain that, for all , has a Gâteaux derivative given by

Let , . Then

for all . Again, as in Lemma A.2, we use Hardy’s inequality to deduce that , so that by Lebesgue’s dominated convergence theorem we conclude that the Gâteaux derivative of is continuous which implies that .

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgments

This work is supported by the Research Center, Scientific Research Deanship, University of Dammam, KSA, under Award no. 2015078.

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