Research Article | Open Access
Existence of Nontrivial Solutions for Perturbed -Laplacian Equation in with Critical Nonlinearity
We consider a perturbed -Laplacian equation with critical nonlinearity in . By using variational method, we show that it has at least one positive solution under the proper conditions.
1. Introduction and Main Results
In this paper, we are concerned with the existence of nontrivial solutions for the following nonlinear perturbed -Laplacian equation with critical nonlinearity: where is the -Laplacian operator with , , denotes the Sobolev critical exponent, is a nonnegative potential, is a bounded positive function, and is a superlinear but subcritical function.
For , (1) turns into the following Schrödinger equation of the form The equation (2) has been studied extensively under various hypotheses on the potential and nonlinearity by many authors including Ambrosetti and Rabinowitz , Bartsch and Wang , Brézis and Lieb , Brėzis and Nirenberg , and Del Pino and Felmer  in bounded domains. Meanwhile, we recall some works in unbounded domains which contain Cingolani and Lazzo , Clapp and Ding , Ding and Lin , Floer and Weinstein , Grossi , Jeanjean and Tanaka , Kang and Wei , Oh , Pistoia , Rabinowitz , and Tang .
For general , most of the work (see [17–19] and the reference therein) dealt with (1) with , and a certain sign potential . Liu and Zheng  considered the above mentioned problem with sign-changing potential and subcritical -superlinear nonlinearity. Cao et al.  also studied the similar problem. However, to our best knowledge, it seems that there is almost no work on the existence of semi-classical solutions to the equation in with critical nonlinearities. This paper will study the critical nonlinearity case in whole space.
Throughout the paper, we make the following assumption:(), and there exists such that the set has finite Lebesgue measure;(), ;() and uniformly in as ;()there are and such that for all ;()there exist , and such that and for all , where .
Our main result reads as follows.
Theorem 1. Assume that ()() hold. Then for any , there exists such that if , (1) has at least one positive solution of least energy which satisfied the following estimate: The main tool used in the proof of Theorem 1 is variational method which was mainly developed in . The main difficulty in the case is to overcome the loss of the compactness of the energy functional related to (1) because of unbounded domain and critical nonlinearity. Although the energy functional does not satisfy the condition, we can prove that it possesses condition at some energy level .
This outline of the paper is organized as follows. In Section 2, we give the variational settings and preliminary results. In Section 3, we show that the corresponding energy functional satisfies condition at the levels less than with some independent of . Furthermore, it possesses the mountain geometry structure. Section 4 is devoted to the proof of the main result.
Theorem 2. Assume that ()() is satisfied. Then for any , there exists such that if , (4) has at least one positive solution satisfying the following estimate:
Next, we introduce the space equipped with the norm Note that the norm is equivalent to the one for any . It follows from that continuously is embedded in . To prove Theorem 2, one considers the functional defined by where .
3. Necessary Lemmas
This section will show some lemmas which are important for the proof of the main result.
Lemma 3. Assume that ()–() is satisfied. For the sequence for , we get that and is bounded in the space .
Proof. By direct computation and the assumptions and , one has Together with and as , we easily get that the sequence is bounded in and the energy level .
By Lemma 3, there is such that in . Furthermore, passing to a subsequence, we have in for any and . in .
Lemma 4. For any , there is a subsequence such that, for any , there exists with where .
Proof. From in , we have Thus, there exists such that In particular, for , we have Note that there exists satisfying Then This completes the proof of Lemma 4.
Let be a smooth function satisfying , if and if . Define . It is clear that
Lemma 5. One has uniformly in with .
Proof. From (16) and the local compactness of Sobolev embedding, for any , we have uniformly in . For any , it follows from (14) that for all . By Lemma 4 and -, we obtain This shows that the desired conclusion holds.
Lemma 6. One has along a subsequence
Proof. By Lemma 2.1 of  and the arguments of , we have By (16) and the similar idea of proving the Brézis-Lieb Lemma , we easily get Together with the fact and , one has Next, we will check the fact in . For any , we have By the standard argument, it follows that uniformly in . Together with Lemma 5, we get the desired conclusion.
Set ; then, . From (16), it shows that in if and only if in .
Furthermore, we have where .
By the facts that and , one has Let , where is the positive constant in the assumption (). Since the set has finite measure and in , we get From ()–() and Young inequality, there exists such that Next, we consider the energy level of the functional below which the condition held.
Lemma 7. Assume that the assumptions of Theorem 2 are satisfied. There exists (independent of ) such that, for any sequence for with , either in or .
From Lemma 7, we will show that satisfies the following local condition.
Lemma 8. Assume that ()–() is satisfied. There exists a constant (independent of ) such that, if a sequence for satisfies , the sequence has a strongly convergent subsequence in .
Proof. By Lemma 7, we easily obtain the required conclusion.
Now, we consider . The following standard arguments show that the energy functional possesses the mountain-pass structure.
Lemma 9. Under the assumptions of Theorem 2, there exist such that
Proof. By (30) and , for any , there is such that Thus Observe that . Choosing , The fact implies the desired conclusion.
Lemma 10. For any finite dimensional subspace , we have
Proof. By the assumption (), one has Since all norms in a finite-dimensional space are equivalent and , this implies the desired conclusion.
Lemma 8 shows that satisfies condition for large enough and small sufficiently. In the following, we will find special finite-dimensional subspaces by which we establish sufficiently small minimax levels.
Define the functional It is apparent that and for all .
Note that For any , there is with and such that . Let , then . For any , we have where By direct computation, we easily get In connection with and , it shows that there exists such that for all , we have It follows from (47) that
Lemma 11. Assume that ()–() is satisfied. For any , there is such that ; there exists with ; we have and where is defined in Lemma 9.
Proof. This proof is similar to the one of Lemma 4.3 in , so we omit it.
4. Proof of Theorem 2
In the following, we will give the proof of Theorem 2.
Proof. By Lemma 11, for any with , there is such that for , we obtain
It follows from Lemma 8 that satisfies condition. Hence, by the mountain-pass theorem, there exists which satisfies and . Actually, is a weak solution of (4). Similar to the argument in , we also get that is a positive least energy solution.
In the end, we show that the solution satisfies the estimate (5). We easily get Note that and and it implies the required conclusion. The proof is complete.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to appreciate the referees for their precious comments and suggestions about the original manuscript. This research was supported by the Fundamental Research Funds for the Central Universities (2013XK03) and the National Training Programs of Innovation and Entrepreneurship for Undergraduates (201310290049).
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