Applied Mathematics
Vol.4 No.3(2013), Article ID:28847,7 pages DOI:10.4236/am.2013.43067
Multiple Solutions for a Class of Concave-Convex Quasilinear Elliptic Systems with Nonlinear Boundary Condition
School of Basic Science, East China Jiaotong University, Nanchang, China
Email: wangli.423@163.com
Received June 7, 2012; revised February 6, 2013; accepted February 13, 2013
Keywords: Multiple Solutions; Quasilinear Elliptic Systems; Nehari Manifold; Fibering Method
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.
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.
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