The nonlinear elliptic eigenvalue problem
,
where and are studied. The key ingredient is a special constrained minimization method.
1. Introduction
In this paper, we are interested in finding nontrivial weak solutions for the nonlinear eigenvalue problem
where and , , , , satisfy the following conditions:
(A) and uniformly in ,(B) uniformly for in bounded subsets of .Remark 1.1. We can see if , , then
where and .
Problem (1.1) comes, for example, from a general reaction-diffusion system:
where . This system has a wide range of applications in physics and related sciences such as biophysics, plasma physics, and chemical reaction design. In such applications, the function describes a concentration, the first term on the right-hand side of (1.3) corresponds to the diffusion with a diffusion coefficient ; whereas the second one is the reaction and relates to source and loss processes. Typically, in chemical and biological applications, the reaction term is a polynomial of with variable coefficients.
When , problem (1.1) is a normal Schrodinger equation which has been extensively studied, for example, [1–8]. The authors used many different methods to study the equation. In [8], the authors established some embedding results of weighted Sobolev spaces of radially symmetric functions which are used to obtain ground state solutions. In [6], the authors studied the equation depending upon the local behavior of near its global minimum. In [3], the authors used spectral properties of the Schrodinger operator to study nonlinear Schrodinger equations with steep potential well. In [9], the author imposed on functions and conditions ensuring that this problem can be written in a variational form. We know that is not a Hilbert space for , except for . The space with does not satisfy the Lieb lemma (e.g., see [9]). And results in the loss of compactness. So there are many difficulties to study equation (1.1) of by the usual methods. There seems to be little work on the case for problem (1.1), to the best of our knowledge. In this paper, we overcome these difficulties and study (1.1) of .
Recently, when , and then the problem is the following eigenvalue problem has been studied by many authors:
where . We can see [10–13]. In [13], Szulkin and Willem generalized several earlier results concerning the existence of an infinite sequence of eigenvalues.
When and is constant then the problem is the following quasilinear elliptic equation:
where , , is a parameter, is an unbounded domain in . There are many results about it we can see [14–18]. Because of the unboundedness of the domain, the Sobolev compact embedding does not hold. There are many methods to overcome the difficulty. In [15], the authors used the concentration-compactness principle posed by P. L. Lions and the mountain pass lemma to solve problem (1.5). In [17, 18], the authors studied the problem in symmetric Sobolev spaces which possess Sobolev compact embedding. By the result and a min-max procedure formulated by Bahri and Li [16], they considered the existence of positive solutions of
where satisfies some conditions. We can see if is function, then it cannot easily be proved by the above methods.
When is positive constant, He and Li used the mountain pass theorem and concentration-compactness principle to study the following elliptic problem in [19]:
where , and , tends to a positive constant as . The authors prove in this paper that the problem possesses a nontrivial solution even if the nonlinearity does not satisfy the Ambrosetti-Rabinowitz condition.
In [20], Li and Liang used the mountain pass theorem to study the following elliptic problem:
where . They generalized a similar result for -Laplacian type equation in [15].
It is our purpose in this paper to study the existence of ground state to the problem (1.1) in . We call any minimizer a ground state for (1.1). We inspired by [9, 16, 21] try to use constrained minimization method to study problem (1.1). Let us point out that although the idea was used before for other problems, the adaptation to the procedure to our problem is not trivial at all. But since both - and -Laplacian operators are involved, careful analysis is needed. A typical difficulty for problem (1.1) in is the lack of compactness of the Sobolev imbedding due to the invariance of under the translations and rotations. However, our method has essential difference with the methods used in [19, 20]. In order to obtain the results, we have to overcome two main difficulties; one is that results in the loss of compactness; the other is that is not a Hilbert space for and it does not satisfy the Lieb lemma, except for .
The paper is organized as follows. In Section 2, we state some condition and many lemmas which we need in the proof of the main theorem. In Section 3, we give the proof of the main result of the paper.
2. Preliminaries
Let
and we define variational functionals and by
Solutions to problem (1.1) will be found as minimizers of the variational problem
To find a solution of problem () we introduce the (limit) variational problem
Lemma 2.1. Let a bounded sequence and . Going if necessary to a subsequence, one may assume that in , almost everywhere, where is an open subset.
Then,
Proof. When from Brezis-Lieb lemma (see [21, Lemma ]) we have
when , using the lower semicontinuity of the -norm with respect to the weak convergence and in , we deduce
Then,
From in
So
So we have
Then,
when , there exists a that . Then, we only need to prove the following inequality:
The proof of it is similar to the above, so we omit it here. So, the lemma is proved.
Lemma 2.2. Let be a bounded sequence in such that
for some . Then in for , where .
Proof. We consider the case . Let and . Holder and Sobolev inequalities imply that
where . Choosing , we obtain
Now, covering by balls of radius , in such a way that each point of is contained in at most balls, we find
Under the assumption of the lemma, in , . The proof is complete.
Corollary 2.3. Let be a sequence in satisfying and such that in . Then there exist a sequence and a function such that up to a subsequence in .
Lemma 2.4. Let and suppose that
uniformly in and
for all and . If in and a.e. on , then
where .
Proof. Let . Applying the mean value theorem we have
where depends on and and satisfies . We now write
For each fixed
Applying (2.20) and the Holder inequality we get that
Since is bounded in we see that
The result follows from (2.21) and (2.23).
Lemma 2.5. Functions and are continuous on and minimizing sequences for problems () and () are bounded in .
Proof. From condition , we observe that for each there exists such that
where and .
By the Holder and Sobolev inequalities we have
where .
Similarly we have
Consequently by the Young inequality we have
for , where is a constant.
Because so we can by Sobolev embedding and letting , we derive the following estimates for and :
Choosing and so that
we see that and are finite and moreover minimizing sequences for problems () and () are bounded. It is easy to check that and are continuous on .
We observe that for all . Indeed, let and
then for each we have