Existence of positive solutions for a critical nonlinear Schrödinger equation with vanishing or coercive potentials
© Chen; licensee Springer 2013
Received: 2 May 2013
Accepted: 22 August 2013
Published: 8 September 2013
In this paper we investigate the existence of positive solutions for the following nonlinear Schrödinger equation:
where and as with , , , and .
1 Introduction and statement of results
By this definition, .
With respect to the functions V and K, we assume that
(A1) for every , and .
When , the potentials are vanishing at infinity and when , the potentials are coercive.
Equation (1.1) arises in various applications, such as chemotaxis, population genetics, chemical reactor theory and the study of standing wave solutions of certain nonlinear Schrödinger equations. Therefore, they have received growing attention in recent years (one can see, e.g., [1–6] and [7–10] for reference).
is well defined in E. And it is easy to check that Φ is a functional and the critical points of Φ are solutions of (1.1) in E.
In a recent paper , Alves and Souto proved that the space E can be embedded compactly into if and and Φ satisfies the Palais-Smale condition consequently. Then, by using the mountain pass theorem, they obtained a nontrivial solution for Eq. (1.1). Unfortunately, when , the embedding of E into is not compact and Φ no longer satisfies the Palais-Smale condition. Therefore, the ‘standard’ variational methods fail in this case. From this point of view, should be seen as a kind of critical exponent for Eq. (1.1). If the potentials V and K are restricted to the class of radially symmetric functions, ‘compactness’ of such a kind is regained and ‘standard’ variational approaches work (see  and ). However, this method does not seem to apply to the more general equation (1.1) where K and V are non-radially symmetric functions.
It is not easy to deal with Eq. (1.1) directly because there are no known approaches that can be used directly to overcome the difficulty brought by the loss of compactness. However, in this paper, through an interesting transformation, we find an equivalent equation for Eq. (1.1) (see Eq. (2.9) in Section 2). This equation has the advantages that its Palais-Smale sequence can be characterized precisely through the concentration-compactness principle (see Theorem 5.1), and it possesses partial compactness (see Corollary 5.8). By means of these advantages, a positive solution for this equivalent equation and then a corresponding positive solution for Eq. (1.1) are obtained.
Before stating our main result, we need to give some definitions.
We denote this infimum by .
Our main result reads as follows.
then Eq. (1.1) has a positive solution .
This implies that (1.10) is satisfied if ϵ is chosen such that .
We use to mean as .
2 An equivalent equation for Eq. (1.1)
This completes the proof. □
This theorem implies that the problem of looking for solutions of (1.1) can be reduced to a problem of looking for solutions of (2.9).
3 The variational functional for Eq. (2.9)
The following inequality is a variant Hardy inequality.
Then the desired result of this lemma follows from (3.4), (3.8) and (3.9) immediately. □
and the critical points of J are nonnegative solutions of (2.9).
4 Some minimizing problems
is independent of with .
Proof In this proof, we always view a vector in as a matrix, and we use to denote the conjugate matrix of a matrix A.
By (4.5), (4.7) and (4.8), we get that . This together with (4.5) leads to the result of this lemma. □
Since the infimum (4.4) is independent of with , we denote it by S.
Lemma 4.2 Let be the infimum in (1.9). Then .
and the critical points of this functional are solutions of (4.9).
This together with (4.14) yields the result of this lemma. □
5 The Palais-Smale condition for the functional J
Recall that J is the functional defined by (3.16). By a sequence of J, we mean a sequence such that and in as , where denotes the dual space of . J is called satisfying the condition if every sequence of J contains a convergent subsequence in .
Our main result in this section reads as follows.
, , , ,
This theorem gives a precise representation of the sequence for the functional J. Through it, partial compactness for J can be regained (see Corollary 5.8).
To prove this theorem, we need some lemmas. Our proof of this theorem is inspired by the proof of [, Theorem 8.4].
It follows that . Now let .
Let . Then we get the desired result of this lemma. □
then in .
Therefore, . Now let . □
One can follow the proof of [, Lemma 8.1] step by step and use Lemma 5.2 to give the proof of this lemma.
is bounded in ,
a.e. on , then
The left proof is the same as the proof of [, Lemma 1.32]. □
- (2)Lemma 5.5 implies(5.5)
- (3)Since in and , it is easy to verify that . For ,(5.6)
Combining (5.6) and (5.7) leads to . Then, by in and , we obtain that in . □
- (1)Since in , it is clear that
- (2)For any ,(5.8)
- (3)From the definition of ,