Multiple Solutions for a Class of Concave-Convex Quasilinear Elliptic Systems with Nonlinear Boundary Condition ()
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. Suppose
is 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 functional
has a minimizer 
and it satisfies
(i) 
(ii) 
is a nontrivial solution of (1.1).
Proof. Let
be a minimizing sequence such that
(3.1)
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
. So
is 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.