Existence of Nontrivial Solutions and High Energy Solutions for a Class of Quasilinear Schrödinger Equations via the Dual-Perturbation Method
We study the quasilinear Schrödinger equation of the form , . Under appropriate assumptions on and , existence results of nontrivial solutions and high energy solutions are obtained by the dual-perturbation method.
1. Introduction and Preliminaries
In this paper we consider the quasilinear Schrödinger equation of the form where and . Solutions of (1) are standing waves of the following quasilinear Schrödinger equation: where is a given potential, is a real constant, and and are real functions. The quasilinear Schrödinger equations (2) are derived as models of several physical phenomena; for example, see [1–5]. Several methods can be used to solve (1). For instance, the existence of a positive ground state solution has been proved in [6, 7] by using a constrained minimization argument; the problem is transformed to a semilinear one in [8–11] by a change of variables (dual approach); Nehari method is used to get the existence results of ground state solutions in [12, 13].
Recently, some new methods have been applied to these equations. In , the authors prove that the critical points are functions by the Moser’s iteration; then the existence of multibump type solutions is constructed for this class of quasilinear Schrödinger equations. In , by analysing the behavior of the solutions for subcritical case, the authors pass to the limit as the exponent approaches to the critical exponent in order to establish the existence of both one-sign and nodal ground state solutions. Another new method which works for these equations is perturbations. In  4-Laplacian perturbations are led into these equations; then high energy solutions are obtained on bounded smooth domain.
In this paper, the perturbation, combined with dual approach, is applied to search the existence of nontrivial solution and sequence of high energy solutions of (1) on the whole space . For simplicity we call this method the dual-perturbation method.
We need the following several notations. Let be the collection of smooth functions with compact support. Let with the inner product and the norm Let the following assumption hold: satisfies and .
Set with the inner product and the norm Then both and are Hilbert spaces.
By the continuity of the for we know that, for each , there exists constant such that where denotes the -norm. In the following, we use or to denote various positive constants. Moreover, we need the following assumptions: there if and if such that uniformly in , there exist and such that for all and , where .
By Lemma 3.4 in  we know that, under the assumption , the embedding is compact for each .
Equation (1) is the Euler-Lagrange equation of the energy functional where . Due to the presence of the term , is not well defined in . To overcome this difficulty, a dual approach is used in [9, 10]. Following the idea from these papers, let be defined by on , and on . Then has the following properties: is uniquely defined function and invertible; for all ; for all ;;, ; for all and for all ; for all ; the function is strictly convex; there exists a positive such that there exist positive constants and such that for all ; for all ; for each , there exists such that .
Indeed, by , , and , there exist and such that, for , and for , Since there exists a such that (see ), we can assume that . For , we have , and hence for , one has , and hence and for , there exist and such that and . Then we have Hence , where .
After the change of variable, can be reduced to From [8, 9, 11] we know that if is a critical point of , that is, for all , then is a weak solution of (1). Particularly, if is a critical point of , then is a classical solution of (1).
A sequence is called a Cerami sequence of if is bounded and in . We say that satisfies the Cerami condition if every Cerami sequence possesses a convergent subsequence.
2. Some Lemmas
Consider the following perturbation functional defined by where . We have the following lemmas.
Lemma 1. If assumptions , , and hold, then the functional is well defined on and .
Proof. By conditions and , the properties , , , and imply that there exists such that
for all . By (26) and the continuity of the embedding (),
Hence is well defined in .
Now, we prove that . It suffices to prove that
For any and , by the mean value theorem, (25) and -, we have The Hölder inequality implies that Hence, by the Lebesgue theorem, we have for all . Now, we show that , , are continuous. Indeed, if in , then in for all .
On the space , we define the norm Then Moreover, on the space , we define the norm By (25), we have where and . Then Theorem A.4 in  implies as . If with and , one has Hence and hence as . Therefore, .
Define with the norm . On the space , we define the norm On the space , we define the norm From in , one has and as . Since , by the following Lemma 2, we have If with and , one has Hence and hence as . Therefore, . This completes the proof.
Lemma 2. Assume that , and Then, for every , , and the operator is continuous.
Proof. Let be a smooth cut-off function such that for and for . Define We can assume that . Hence for all . Assume in . Then in and in . As in the proof of Lemma A.1 in , there exists a subsequence of and such that and for a.e. . Hence, from (51), one has a.e. on . It follows from the Lebesgue theorem that in . Consequently, in . Similarly, we can prove in . Since it follows that in . This completes the proof.
Lemma 3. Let , , and hold. Then every bounded sequence with possesses a convergent subsequence.
Proof. Since is bounded, then, by the compactness of the embedding (), passing to a subsequence, one has in , in for all , and for a.e. . By (25) as . Similarly, as . Hence, by the property of , we have where as . This shows that as . This completes the proof.
Lemma 4. If a.e. in and , then as .
3. Main Results
Theorem 5. Assume conditions , hold. Let be such that . Let be a critical point of with for some constant independent of . Then, up to subsequence, one has in , and is a critical point of .
Proof. By , for , there exists such that
By , for ( is the constant appearing in condition ), we have
where is the constant appearing in condition . Hence
Since , there exists such that
for all . Hence
Since is a critical point of ,
for all . Consequently, taking , by and we have
for some constant independent of . By the boundedness of , there exists such that
for all . Hence, by the Sobolev embedding theorem, one has
Next, we prove that and , where the positive constant is independent of . Setting , , define , where is a smooth function satisfying for , ; for , and is decreasing in .
This means that , for ; , for ; , for , where .
Let ; then . By (61) . Hence where For , . By the properties of and , the mean value theorem implies Hence Consequently, Combining (67) and (68), we have For any , by and , there exists such that Combining (66), (72), and (73), one has By the Hölder inequality and (65), Moreover, HenceSince , . Set . Then Take such that . Since , . Hence, from (65), we have Since as , taking in (78) with , we have Set . Then Inductively, we have