Multiple Solutions for a Class of Concave-Convex Quasilinear Elliptic Systems with Nonlinear Boundary Condition
Li Wang
East China Jiaotong University, Nanchang, China.
DOI: 10.4236/am.2013.43067   PDF    HTML   XML   5,530 Downloads   11,682 Views   Citations

Abstract

In this paper, a quasilinear elliptic system is investigated, which involves concave-convex nonlinearities and nonlinear boundary condition. By Nehari manifold, fibering method and analytic techniques, the existence of multiple nontrivial nonnegative solutions to this equation is verified.

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Wang, L. (2013) Multiple Solutions for a Class of Concave-Convex Quasilinear Elliptic Systems with Nonlinear Boundary Condition. Applied Mathematics, 4, 449-455. doi: 10.4236/am.2013.43067.

1. Introduction

In this article, we are interested in the existence of two nontrivial nonnegative solutions of the following problem:

(1.1)

where is a bounded domain with smooth boundary, is the critical Sobolev exponent for the embedding.

is the outer normal derivative,

, the weight m(x) is a positive bounded function and are smooth functions which may change sign in Ω. By Nehari manifold, fibering method and analytic techniques, the existence of multiple positive solutions to this equation is verified.

In recent years, there have been many papers concerned with the existence and multiplicity of positive solutions for semilinear elliptic problems. Some interesting results can be found in Garcia-Azorero et al. [1], Wu [2-4] and the references therein. More recently, Hsu [5] has considered the following elliptic system:

(1.2)

By variational methods, he proved that problem (1.2) has at least two positive solutions if the pair of the parameters belongs to a certain subset of. However, as far as we know, there are few results of problem (1.1) in addition to concave-convex nonlinearities, i.e., , including nonlinear boundary condition. We focus on the existence of at least two nontrivial nonnegative solutions for problems (1.1) in the present paper.

Set

(1.3)

where satisfy

(1.4)

The main result of this paper is summarized in the following theorem.

Theorem 1.1. If the parameters satisfy

then problem (1.1) has at least two solutions and satisfy in and

It should be mentioned that the similar results about the existence of multiplicity of positive solutions for the Laplace problem with critical growth and sublinear perturbation have been discussed in the recent paper [6-8] and the reference therein.

This paper is organized as follows. Some preliminaries and properties of the Nehair manifold are established in Sections 2, and Theorems 1.1 is proved in Sections 3.

2. Preliminaries

Let denotes the usual Sobolev space. In the Banach space we introduce the norm which is equivalent to the standard one:

First, we give the definition of the weak solution of (1.1).

Definition 2.1. We say that is a weak solution to (1.1) if for all, we have

It is clear that problem (1.1) has a variational structure. Let be the corresponding energy functional of problem (1.1), and it is defined by

where

It is not difficult to verify that the functional I is not bounded neither from below nor from above. So it is convenient to consider I restricted to a natural constraint, the Nehari manifold, that contains all the critical points of I. First we introduce the following notation: for any functional we denote by the Gateaux derivative of F at in the direction of and

Define the Nehari manifold

. Note that N contains all solutions of (1.1) and if and only if

(2.1)

Lemma 2.1. is coercive and bounded below on N.

Proof. Suppose From (2.1), the Holder inequality and the Sobolev embedding theorem, it follows that

(2.2)

Thus is coercive and bounded below on since Define Then for all we have

(2.3)

Arguing as that in [9,10], we split into three parts:

Lemma 2.2. Supposeis a local minimizer of on and Then in

Proof. If is a local minimizer for I on N, then is a solution of the optimization problem minimize subject to

Hence, by the theory of Lagrange multipliers, there exists such that

in.

Here is the dual space of the Sobolev space. Thus,

But since Hence

Lemma 2.3. for all

Proof. We argue by contradiction. Suppose that for all

there is

then (2.3) and the Sobolev embedding theorem imply that

(2.4)

and

(2.5)

Thus from (2.4), (2.5) we have

(2.6)

and

Consequently,

which is a contradiction.

By Lemma 2.3, we can write for all

Define

.

Lemma 2.4. (i) for all

(ii) There exists a positive constant d0 depending on such that for all

Proof. (i) Suppose, then we have

for

Thus we get that

(ii) Suppose

and. Then (2.4) implies that

(2.7)

and (2.5) implies that

(2.8)

From (2.7) and (2.8) it follows that

which shows that

since

where is a positive constant.

For all such that, set

Lemma 2.5. Suppose that

and is a function satisfying

(i) If, then there exists a unique

such that and .

(ii) If, then there exist and such that

Furthermore,

Proof. Fix with For all, let

then it is obvious that Ψ(0) = 0, Ψ(t) → −∞ as t → +∞, as small enough. So we can deduce that Ψ′(t) = 0 at for, for Then Ψ(t) that achieves its maximum at is increasing for and decreasing for Moreover,

(i) If, then there exists a unique such that Note that

thus we get

From

we have. For all it follows that

So we get that

(ii) If for

then there exist and such that

and

By the similar argument in (i), we get and

for

for

Then it follows that

The proof of this Lemma is completed.

For each with, we write

(2.9)

Lemma 2.6. Suppose that

and is a function satisfying.

(i) If then there exists a unique such that and

(ii) If, then there exist and such that and. Furthermore,

Proof. Fix with For all let

(2.10)

then it is obvious that. So we can deduce that at

for.

Then that achieves its maximum at is increasing for and decreasing for

Using the similar argument in Lemma 2.5, we can obtain the result of Lemma 2.6.

3. Proof of Theorem 1.1

Lemma 3.1. Suppose that

then the functionalhas a minimizer and it satisfies

(i)

(ii) is a nontrivial solution of (1.1).

Proof. Let be a minimizing sequence such that

(3.1)

Since I is coercive on N, we get that is bounded on. Passing to a subsequence (still denoted by), there exists such that

weakly in,

a.e. in,          (3.2)

strongly in and in. This implies

Since, we get

By Lemma 2.4 (i) we get and then. Now we prove that

strongly in Suppose otherwise, then either

(3.3)

Fix with. Let

, where is as in (2.10).

Clearly, as, and

as Since

by an argument similar to the one in the proof of Lemma 2.6, we have that the function achieves its maximum at, is increasing for and decreasing for where is as in (2.9). Since by Lemma 2.6, there is unique such that

Then

(3.4)

By (3.3) and (3.4), we obtain for n sufficiently large for the sequence Since

we have Moreover,

and is increasing for. This implies for all and sufficiently large. We obtain. But

and this implies

which is a contradiction. Hence strongly in W. This implies as. Thus is a minimizer for on Since

and, by Lemma 2.2 we may assume that is a nontrivial nonnegative solution of Equation (1.1).

Next we prove Arguing by contradiction, without loss of generality, we may assume that v ≡ 0. Then as u is a nonzero solution of

(3.5)

we have

(3.6)

Choose such that

(3.7)

then

By Lemma 2.6, there is a unique such that. Moreover, from (3.6) and (3.7), it follows that

and

This implies

which contradict with that (u,0) is the minimizer and hence. Sois a nontrivial nonnegative solution of Equation (1.1).

Lemma 3.2. Suppose that

Then the functional has a minimizer and it satisfies

(i)

(ii) is a nontrivial solution of (1.1).

Proof. Let be a minimizing sequence such that

(3.8)

Since I is coercive on N, we get that is bounded on. Passing to a subsequence (still denoted by), there exists such that

weakly in,

a.e. in,        (3.9)

strongly in and in

This implies

Moreover, by (2.3) we obtain

then Now we prove that

strongly in W. Suppose otherwise, then either

(3.10)

By Lemma 2.6, there is unique to such that

Since

for all, we have

and this is a contradiction. Hence

strongly in W. This implies

as. Thus is a minimizer for I on.

Since and, by Lemma 2.4 and the similar argument as that in Lemma 3.1 we can get is also a nontrivial nonnegative solution of Equation (1.1).

Proof of Theorem 1.1. From Lemma 3.1 and Lemma 3.2, we obtain that Equation (1.1) has two nontrivial nonnegative solutions and satisfy

and. It remains to show that the solutions found in Lemma 3.1 and Lemma 3.2 are distinct. Since this implies that and are distinct. This concludes the proof.

4. Acknowledgements

The author is indebted to the referees for carefully reading this paper and making valuable comments and suggestions.

Conflicts of Interest

The authors declare no conflicts of interest.

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