The Existence and Uniqueness Result for a Relativistic Nonlinear Schrödinger Equation

Abstract

We study the existence and uniqueness of positive solutions for a class of quasilinear elliptic equations. This model has been proposed in the self-channeling of a high-power ultrashort laser in matter.

1. Introduction

In this paper, we consider the following quasilinear Schrödinger equation: where , , , and . Solutions of (1) are related to standing waves for the following quasilinear Schrödinger equation: where , is a given potential, is real constant, and and are real functions. Quasilinear equations such as (2) have been accepted as models of several physical phenomena corresponding to various types of ; see [1–5] for physical backgrounds.

The superfluid film equation in plasma physics has this structure for (see [6]). Putting , where and is a real function, (2) turns into the following equation: where is the new potential function and is the new nonlinearity. In this case, the first existence results are due to [7]. In [7], the main existence results are obtained through a constrained minimization argument. Subsequently, a general existence result was derived in [8]. The idea in [8] is to make a change of variables and reduce the quasilinear problem to semilinear one and Orlicz space framework was used to prove the existence of positive solutions via the Mountain pass theorem. The same method of changing of variables was also used in [9] but the usual Sobolev space framework was used as the working space. Precisely, since the energy functional associated (3) is not well defined in , they first make the changing of unknown variables , where is defined by ODE as follows: and , . Then, after the changing of variable, to find the solutions of (2), it suffices to study the existence of solutions for the following semilinear equation: where By using the classical results given by [10], they proved the existence of a spherically symmetric solution. In [11], the authors give a sufficient condition for uniqueness of the ground state solutions by using the same change of variables as [9].

In the case , (2) models the self-channeling of a high-power ultrashort laser in matter (see [12]). In this case, few results are known. In [13], the authors proved global existence and uniqueness of small solutions in transverse space dimensions 2 and 3 and local existence without any smallness condition in transverse space dimension 1. But they did not study the existence of standing waves. But we have to point out that the method of change of variables as (4) cannot be generalized to treat the case . In [14], the authors made the changing of known variable (see also [15]) and proved the existence of nontrivial solution with and . In this paper, for and , we will show the existence and uniqueness result for (1) by using a change of variables due to [14, 15]. One main difficulty in dealing with this problem seems to be that of obtaining the boundedness of a (PS) sequence for the corresponding functional. We overcome this difficulty by using Jeanjean’s result [16].

Our main result is the following.

Theorem 1. Assume that , , , and . There exists such that if , then the positive solution of (1) is unique.

In this paper, C denotes positive (possibly different) constant, denotes the usual Lebesgue space with norm , , and denotes the Sobolev space with norm .

2. Preliminaries

We note that the solutions of (1) are the critical points of the following functional: Since the functional may not be well defined in the usual Sobolev spaces , we make a change of variables as where . Since is monotonous with , the inverse function of exists. Then after the change of variables, can be written as By Lemma 2 listed below, we have and , so is well defined in and .

If is a nontrivial solution of (1), then for all it should satisfy

We show that (11) is equivalent to Indeed, if we choose in (11), then we get (12). On the other hand, since , if we let in (12), we get (11). Therefore, in order to find the nontrivial solutions of (1), it suffices to study the existence of the nontrivial solutions of the following equation:

Before we close this section, we give some properties of the change of variables.

Lemma 2. for all ;
for all ;
;
;
for all ;
for all .

Proof . Since and , so , for ; that is, , for , which proves (1).
Since and is increasing, so properties (2) and (3) are obvious.
For (4), the result is obvious since is an increasing bounded function.
Since which proves (5).
For (6), since is a increasing function, then , which implies that . On the other hand, by (1) and , we get .

3. Existence

At first, we give two Lemmas.

Lemma 3. There exist such that for all .

Proof. Let Then, by Lemma 2 and , we have Thus, for sufficiently small, there exists a constant such that Then, we have Thus, by choosing small, we get the result when .

Lemma 4. There exists such that .

Proof. Given with , we will prove that as , which will prove the result if we take with large enough. By Lemma 2, we have as , so as . Thus, we get the result.

We will use the following Theorem which is due to Jeanjean [16].

Theorem 5. Let be a Banach space equipped with the norm and let be an interval. One considers a family of -functionals on of the form where , for all , and such that either or as . One assumes that there are two points in such that setting there hold, for all , Then, for almost every , there is a subsequence such that (i) is bounded;(ii);(iii) in the dual of .

We consider the functional where .

Let . We find that for all . On the other hand, if , then either , which implies , or ; in this case, to verify that , we start splitting since and by Lemma 2 (6), we have , so so .

For defined above with , using Lemma 4, we get a such that . Also from Lemma 2 we know that as . Thus setting we have, for all , Therefore, using Theorem 5, for almost all , there exists a subsequence such that (i) is bounded in ;(ii); (iii) in .

Lemma 6. Assume that is a bounded Palais-Smale sequence of the functional for . Then there exists a nontrivial critical point of .

Proof. We first note that satisfies and, for any , Since is a bounded Palais-Smale sequence, there exists such that in and in for . By the Lebesgue Dominated Theorem, we have Hence, is a weak solution of (1). If , then we get the result.
Otherwise, if , we claim that for all , cannot occur. Suppose by contradiction that (33) occurs, that is, vanish; then, by the Lions compactness Lemma (see [17, 18]), in for any . Since , then by the proof of Lemma 2, we get which implies that Since and , then On the other hand, note by Lemma 2 (5) that Combing Lemma 2, we have . In fact, we only need to show that ; let ; then by Lemma 2 (5), we have so for and for , which implies that . Thus, since is dense in , by choosing in (31), we deduce that So Combing (36) and (34), we have so we get a contradiction since . Thus, does not vanish and there exist , and such that Define and . Since is a Palais-Smale sequence for , is also a Palais-Smale sequence for with if in . Since does not vanish, we have that is a nontrivial solution of (1).

From Lemma 6, we see that, for almost all , there exists a solution to the following Schrödinger equation: where

Therefore, we can choose such that . Setting , we have . We can deduce that is a solution to (13) if we show that . To prove this, in view of Lemma 6, we first check that is bounded in .

Notice that the Pohozaev identity implies that the solutions of (45) satisfy

Lemma 7. The sequence is bounded.

Proof . Since is a solution to (45) with , by (47), we have which implies that is bounded. On the other hand, together with (41), we have Since is bounded, so is bounded. To verify that is bounded in , we start splitting since so is bounded.

Lemma 8. Assume that , , and . Then (1) has a nontrivial solution.

Proof. The boundedness of in follows from Lemma 7; we have that is bounded in for . Then for any , we have since so as ; thus we have as . By knowing that since so , and we distinguish two cases. Either or . In the first case, we get and the result follows from Lemma 6.
In the second case, we define the sequence by with satisfying (if for a , defined by (56) is not unique, we choose the smaller possible value). By construction is bounded. Moreover by the definition of (56), we have so . Then following the proof above, we have and . On the other hand, by the proof of Lemmas 3 and 2 (6), there exists a constant such that as , uniformly in . Thus, since , there is such that , . Similarly, following the proof of Lemma 3, we have with as . Then recording that , we obtain from (56) that Using Lemma 6 again, we complete the proof of Lemma 8 which implies that is a solution for (1).

Remark 9. In [14], the authors considered the existence of solutions for the following quasilinear Schrödinger equation: where the nonlinearity is Hölder continuous and satisfies the following conditions:   if ;  as ;  there exists such that ;  there exists such that for any , there holds .
If we take , , and , (59) turns into (1) with . We point out that the existence result in [14] does not cover our result.

Now, we show that is not satisfied for if . In fact, if and only if By Lemma 2 (5), we have Thus, we only need to show that is under the hypothesis . Then, by (63), we have

Remark 10. In [14], is used to prove the boundedness of (PS) sequence. In this paper, since does not satisfy our condition, we obtain the boundedness of (PS) sequence by using Jeanjean's result [16].

4. Uniqueness

In this section, we study the uniqueness of the positive radial solution of (13). We put We apply the following uniqueness result due to Serrin and Tang [19].

Theorem 11. Suppose that there exists such that (1) is continuous on , on , and for ;(2) and on .
Then the semilinear problem has at most one positive radial solution.

Now we can see that defined in (65) is of the class . Moreover, by the proof of Lemma 2, we have that is increasing and ; then . So ; then there exists a unique such that , and So (1) of Theorem 11 holds. From , we can also observe that . Since is increasing and , this implies that

Lemma 12. Suppose and . Then there exists such that if , then satisfies (2) of Theorem 11.

Proof. We observe that Thus we have only to show that , for . Since so Then by complicated computations, we have where
For , it follows that . Thus it suffices to show that , for , in order to prove that .
By (4) of Lemma 2 and , we have Thus, for sufficiently large , we obtain if and only if for .
Next, we investigate the sign of . Firstly, we express in terms of and , and since , so and Thus we obtain We note that so Moreover, we have , , and as . Then, from , we have so for if is sufficiently large; that is, for . From (68), there exists such that if , then we obtain for .

By Lemma 8, we can apply Theorem 11, Hence we obtain the uniqueness of positive radial solutions of (13).

5. Conclusion

By the discussion of Section 3, we have a nontrivial solution of (1). Then using the result of Gidas et al. [20], we know that the nontrivial solution is a positive radial solution with as and . Combined with the discussion of Section 4, we complete the proof of Theorem 1. That is, if , , , and , there exists such that if , then the positive solution of (1) is unique.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper is supported by NSFC (nos. 11201154 and 11371146) and the Fundamental Research Funds for the Central Universities (nos. 2013ZM0112 and 2013ZM0113).