Abstract and Applied Analysis

Volume 2013 (2013), Article ID 834918, 7 pages

http://dx.doi.org/10.1155/2013/834918

Research Article

Existence and Multiplicity of Nonnegative Solutions for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions

Jincheng Huang1,2

¹Department of Mathematics and Physics, Hohai University, Changzhou Campus, Changzhou 213022, China

²College of Sciences, Hohai University, Nanjing 210098, China

Received 21 May 2013; Revised 27 October 2013; Accepted 27 October 2013

Academic Editor: Julio Rossi

Copyright © 2013 Jincheng Huang. 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.

Existence and multiplicity results are established for quasilinear elliptic problems with nonlinear boundary conditions in an exterior domain. The proofs combine variational methods with a fibering map, due to the competition between the different growths of the nonlinearity and nonlinear boundary term.

1. Introduction

Consider the following quasilinear elliptic problem:

where is a smooth exterior domain in , , and    is the unit vector of the outward normal on the boundary .

Equations of the type (1) arise in many and diverse contexts like differential geometry [1], nonlinear elasticity [2], non-Newtonian fluid mechanics [3], glaciology [4], and mathematical biology [5]. As a result, questions concerning the solvability of problem (1) have received great attention; see [6–10].

For with , , by using the fibering method, Kandilakis and Lyberopoulos [6] studied the existence of nonnegative solutions for problem (1) in unbounded domains with a noncompact boundary. When with , Lyberopoulos [7] studied the existence versus absence of nontrivial weak solutions for problem (1). Similar consideration can be found in Kandilakis and Magiropoulos [8]. In [9], Filippucci et al. established existence and nonexistence results for problem (1) via variational methods combined with the geometrical feature, where . Recently, Chen et al. [10] considered the existence and multiple of solutions for problem (1) by the variational principle and the mountain pass lemma.

Motivated by these findings, we consider the following quasilinear elliptic problem:

where is a smooth exterior domain in and   is the unit vector of the outward normal on the boundary . Since , problem (2) is essentially different from problem (1). Using the Nehari manifold and fibering map, Wu [11] considered problem (2) for ; Afrouzi and Rasouli [12] considered problem (2) for .

Throughout this paper, we make the following assumptions.

(), , and , where and   .() The function and .() The function and .() The function satisfies and with .() The function ,   in and with .

The purpose of this paper is to find existence and multiplicity of nonnegative solutions to problem (2). Our proofs are based on the variational method. The main difficulty is the lack of compactness of the Sobolev embeddings in unbounded domains. To overcome this difficulty, we impose the integrality conditions ()-() on and to establish compact Sobolev embedding theorems (see Lemmas 3 and 4).

The rest of the paper is organized as follows. In Section 2, we set up the variational framework of the problem and give some preliminaries. Section 3 is devoted to the existence results for problem (2). The multiplicity of nonnegative solutions for problem (2) is considered in the last section.

2. Variational Framework and Some Preliminaries

In this section, we set up the variational framework and give some preliminaries.

Define the weighted Sobolev space as the completion of under the norm

which is equivalent to the standard one under assumptions ()-(). Moreover, denote by and the weighted Lebesgue spaces equipped with the norm:

respectively. The definition of the weak solution of problem (2) reads as follows.

Definition 1. One says is a weak solution of problem (2) if

holds for all .

The energy functional corresponding to problem (2) is

where , . It is well known that the weak solutions of (2) are the critical points of the energy functional . If is a critical point of , then necessarily belongs to the Nehari manifold:

where

For all , we have , where

Moreover

The variational framework that we adopt is based on the so-called one-dimensional fibering method proposed by Pohozaev [13]. The central idea of this strategy consists in embedding the original variational problem into the “wider” space and then investigating the conditional solvability of the new problem in under an appropriately imposed constraint. To this end, we define the extended functional by setting for any

If is a critical point of , then necessarily ; that is,

In particular, if , then (12) is equivalent to

where

Now, suppose that solves (13) for all ; then . Furthermore, if exists and is unique for all , then (13) generates a bijection between and . Moreover, the following proposition holds.

Lemma 2 (see [13]). If is a conditional critical point of , under the constraint , then is a critical point of , where and is a nonnegative solution of (13).

In view of Lemma 2, the problem of finding solutions of (2) will be reduced to that of locating the critical point of under the constraint .

The following compact embedding theorems play an important role in the proof of our main results.

Lemma 3. Assume ()–(). Then the embedding is compact.

Proof. Let . Since , it follows that . Let be the standard Banach space endowed with the norm . By assumptions ()-(), . Similar to the proof of [10, Lemma 2] (see also the proof of [14, Theorem 7.9]), we can prove that is compact and so is . Let be the best trace embedding constant; that is,

By Hölder's inequality, we have

This shows that the embedding is continuous.

Assume is a bounded sequence in . Then by the compact embedding , there exist and a subsequence of (not relabelled) such that strongly in .

By Hölder's inequality again, we infer

This completes the proof.

Lemma 4. Assume ()-() and (). Then the embedding is compact.

Proof. Let . Since , it follows that . Hence the embedding is compact (see [15, 16]). Let be the best trace embedding constant; that is,

By Hölder's inequality, we have

This shows that the embedding is continuous.

Assume is a bounded sequence in . Then by the compact embedding , there exist and a subsequence of (not relabelled) such that strongly in .

By Hölder's inequality again, we infer

This completes the proof.

We also need the following mountain pass lemma (see [17, 18]).

Lemma 5. Let be a real Banach space and with . Suppose

() there are such that for ;() there is , such that .

Define

Then

is finite and possess a sequence at level . Furthermore, if satisfies the (PS) condition, then is a critical value of .

To get multiplicity results, we need the following fountain theorem due to Bartsch [19] and a critical point theorem in [20, 21].

Let be a reflexive and separable Banach space. It is well known that there exist and () such that

(1), where for and for ,(2) and .

For convenience, we write

Lemma 6 (fountain theorem [19]). Assume is an even functional that satisfies the condition. If for every there exist such that

() as ,(),

then J has a sequence of critical points with .

Lemma 7 (see [20, 21]). Let , where is a Banach space. Assume that satisfies the condition and is even and bounded from below, and . If for any there exists a -dimensional subspace and such that

then has a sequence of critical values satisfying as .

3. Existence of Nonnegative Solutions

In this section, the existence results are established for problem (2). The proofs combine variational methods with a fibering map. Since , we may suppose that the solution to problem (2) is nonnegative throughout this paper.

Theorem 8. Let ()–() hold with either or . Then problem (2) admits a nonnegative nontrivial weak solution which is also a ground state.

Proof. Suppose . Rewriting (13) as

we immediately see that for every (where is defined by (9)) there exists a unique satisfying (25). Moreover, it can be easily checked that

Consider now the variational problem

Let be a minimizing sequence in with . Then there exists such that in . By Lemmas 3 and 4, we have and .

We first assert that . Suppose the contrary; then . In view of (25),

Letting , it follows that . Thus

which contradicts .

Next, we prove . If not, then . So, there exists such that . From (25), we have

This and (26) yield

On the other hand, it follows from (29) that is bounded and so there exists a subsequence (not relabelled) such that . Thus by (25), we have

Hence . Notice that

is strictly decreasing for all ; we have

which is a contradiction. So, and is a critical point of . By Lemma 2, is a nontrivial solution of problem (2). Since is a unique solution of (25), then (25) generates a bijection between and and so the obtained solution is actually a ground state.

The case can be treated in a similar way.

Remark 9. Afrouzi and Rasouli [12] consider the following problem:

where is a bounded domain in and . The functions and are continuous functions which change sign in . Using the Nehari manifold and fibering map, they proved that problem (36) has at least two nontrivial nonnegative solutions if is sufficiently small. In fact, by slight modification, we can prove that the result they established is still true if is a smooth exterior domain or the parameters satisfy . But for , we can prove that problem (36) has at least one nontrivial nonnegative solution for is sufficiently small via the method used in [12]. Notice that our result (Theorem 8) does not need to be small.

Theorem 10. Let ()–() hold with . Then problem (2) has a nonnegative nontrivial weak solution.

Proof. From Lemma 3, we have

Thus

So is coercive by .

By Lemmas 3 and 4, it is easy to verify that is weakly lower semicontinuous. So has a minimum point in and is a weak solution of (2).

In the following, we prove . Let . Then

Thus for is sufficiently small. Notice that ; we obtain

Thus the minimum point of is nontrivial.

Theorem 11. Let ()–() hold with . Then problem (2) has a nonnegative nontrivial weak solution in .

Proof. Let . Then

Thus for small and for large.

Let satisfy in and in . Then

This shows that is bounded in . Up to a subsequence, we obtain in . Thus

It follows from Lemmas 3 and 4 that

Hence

Then (45) give that . Notice that ; we have

where

Using the standard inequality in given by

we have from (46) that in . Thus satisfies (PS) condition. Then the assertion of this theorem follows from Lemma 5.

Next, we seek for a solution in with . In this case, we find it necessary to strengthen our hypothesis by assuming that the function is positive. That is, () will be replaced by

() the function and with .

Theorem 12. Let ()–() and () hold with . Suppose also

where . Then problem (2) has a nonnegative nontrivial weak solution in .

Proof. Define

Fix with . In view of assumption (), we have . For all , there exist and with such that . Moreover, . Thus

as . Hence is coercive in . By Lemmas 3 and 4, it is easy to verify that is weakly lower semicontinuous. So has a minimum point in and is a weak solution of (2).

In the following, we prove that . Notice that as , as , and ; we infer that attain its maximum at , where

If , then (13) has exactly two solutions and with . Let . We have from (11) and (12) that

Let . It follows that

which ensures . Thus . This implies that the weak solution of (2) is nontrivial.

Remark 13. Condition (49) may be viewed as grading the “strength” of interaction induced by and . Hence, qualitatively speaking, one may rephrase Theorem 12 as saying that if , then problem (2) admits a nontrivial weak solution provided that “prevails” over .

4. Multiplicity of Nonnegative Solutions

In this section, we establish multiplicity results for the cases and by Lemmas 6 and 7, respectively. To this purpose, the assumption () will be replaced by the following:

() the function satisfies in and with .

Theorem 14. Let ()–(), (), and () hold with . Then problem (2) has a sequence of solutions in E with as .

Proof. We will prove this theorem by fountain theorem. The proof is divided into three steps.

(1) Let and be defined by (23) and . Then it follows that (see [22]). Therefore, we have

Choosing , we obtain that if ,, then

Thus in Lemma 6 is proved.

(2) Since in the finite dimensional space all norms are equivalent, there exist such that hold for all . Thus by (19),

Therefore in Lemma 6 is satisfied for every large enough.

(3) Let be a sequence of . Then we have

Therefore is bounded in . Similar to the proof of Theorem 11, we can verify that satisfies (PS) condition.

Obviously, is an even functional and . Thus the assertion of Theorem 14 follows from Lemma 6.

Theorem 15. Let ()–(), (), and () hold with . Then problem (2) has a sequence of solutions in E with and as .

Proof. Since

and ,   is coercive and bounded from below. As before, we can verify that (PS) condition holds.

Let be defined by (23) and . Since is a finite dimensional space and , we can choose small enough such that

We obtain a sequence of solutions by Lemma 7.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (2013B09914).

References

J. L. Kazdan, Prescribing the Curvature of a Riemannian Manifold, vol. 57 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1985.

P. G. Ciarlet, Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity, vol. 20 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1988.

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries. Vol. I. Elliptic Equations, vol. 106 of Research Notes in Mathematics, Pitman, Boston, Mass, USA, 1985.

M.-C. Pélissier and L. Reynaud, “Étude d'un modèle mathématique d'écoulement de glacier,” Comptes Rendus de l'Académie des Sciences A, vol. 279, pp. 531–534, 1974.

D. G. Aronson and H. F. Weinberger, “Multidimensional nonlinear diffusion arising in population genetics,” Advances in Mathematics, vol. 30, no. 1, pp. 33–76, 1978.

D. A. Kandilakis and A. N. Lyberopoulos, “Indefinite quasilinear elliptic problems with subcritical and supercritical nonlinearities on unbounded domains,” Journal of Differential Equations, vol. 230, no. 1, pp. 337–361, 2006.

A. N. Lyberopoulos, “Existence and Liouville-type theorems for some indefinite quasilinear elliptic problems with potentials vanishing at infinity,” Journal of Functional Analysis, vol. 257, no. 11, pp. 3593–3616, 2009.

D. A. Kandilakis and M. Magiropoulos, “Existence results for a p-Laplacian problem with competing nonlinearities and nonlinear boundary conditions,” Electronic Journal of Differential Equations, no. 95, pp. 1–6, 2011.

R. Filippucci, P. Pucci, and V. Rădulescu, “Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions,” Communications in Partial Differential Equations, vol. 33, no. 4–6, pp. 706–717, 2008.

C. Chen, S. Liu, and H. Yao, “Existence of solutions for quasilinear elliptic exterior problem with the concave-convex nonlinearities and the nonlinear boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 383, no. 1, pp. 111–119, 2011.

T.-F. Wu, “A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential,” Electronic Journal of Differential Equations, no. 131, pp. 1–15, 2006.

G. A. Afrouzi and S. H. Rasouli, “A variational approach to a quasilinear elliptic problem involving the p-Laplacian and nonlinear boundary condition,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2447–2455, 2009.

S. I. Pohozaev, “On the fibering method for the solution of nonlinear boundary value problems,” Proceedings of the Steklov Institute of Mathematics, vol. 192, pp. 157–173, 1992.

I. Kuzin and S. Pohozaev, Entire Solutions of Semilinear Elliptic Equations, Progress in Nonlinear Differential Equations and their Applications, 33, Birkhäuser, Basel, Switzerland, 1997.

L. F. O. Faria, O. H. Miyagaki, and F. R. Pereira, “Existence results for quasilinear elliptic exterior problems involving convection term and nonlinear Robin boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 368, no. 2, pp. 578–586, 2010.

R. A. Adams, Sobolev Spaces, Academic Press, New York, NY, USA, 1975.

A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” Journal of Functional Analysis, vol. 14, pp. 349–381, 1973.

M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, vol. 34, Springer, New York, NY, USA, 3rd edition, 2000.

T. Bartsch, “Infinitely many solutions of a symmetric Dirichlet problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 20, no. 10, pp. 1205–1216, 1993.

Z.-Q. Wang, “Nonlinear boundary value problems with concave nonlinearities near the origin,” Nonlinear Differential Equations and Applications, vol. 8, no. 1, pp. 15–33, 2001.

H.-P. Heinz, “Free Ljusternik-Schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems,” Journal of Differential Equations, vol. 66, no. 2, pp. 263–300, 1987.

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser, Boston, Mass, USA, 1996.