Advances in Mathematical Physics
Volume 2016 (2016), Article ID 3042493, 10 pages
http://dx.doi.org/10.1155/2016/3042493

Research Article

A Nonlinear Schrödinger Equation Resonating at an Essential Spectrum

Shaowei Chen and Haijun Zhou

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

Received 12 December 2015; Accepted 14 February 2016

Academic Editor: Jacopo Bellazzini

Copyright © 2016 Shaowei Chen and Haijun Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider the nonlinear Schrödinger equation . The potential function satisfies that the essential spectrum of the Schrödinger operator is and this Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. The nonlinearity satisfies the resonance type condition . Under some additional conditions on and , we prove that this equation has infinitely many solutions.

1. Introduction and Statement of Results

In this paper, the following nonlinear Schrödinger problem in () is considered:

The nonlinearity satisfies the resonance type condition:

Here, denotes the essential spectrum (see [1, Chapter 7] or [2, Chapter 7.4]) of the Schrödinger operator defined as follows:

Equation (1) arises in quantum mechanics and is related to the study of the nonlinear Schrödinger equation for a particle in an electromagnetic field and it has attracted considerable attention from researchers in recent decades. One can see [3–6] and the references therein. However, there are very few results on (1) with the resonance type condition (2).

The following nonlinear elliptic problem with resonance type conditions in bounded domain has been studied by many authors and numerous existence and multiplicity results have been obtained in the past forty years:

Here, is a bounded domain in , and satisfies the resonance type condition:

is the th eigenvalue of with -Dirichlet boundary condition on . We refer to [7–10] and references therein for more detailed discussions of some historical results. This problem has deeply inspired developments in critical point theory in the last forty years, such as the Landesman-Lazer type conditions (e.g., [8, 10]), Morse theory (e.g., [9]), and the variational reduction method (e.g., [11]). The difficulty of this problem lies in the proof of the boundedness of the Palais-Smale sequence or the sequence (see Definition 5 in Section 2) of its corresponding functional.

The resonant Schrödinger problem (1) is much more difficult than the bounded domain case and it has fewer studies. Unlike the case of bounded domains for which the linear operators in bounded domain are compact, there are continuous spectra of the linear operator . Moreover, the proofs of boundedness and compactness of the sequence of the corresponding functional of (4) are greatly different from and more difficult than the case of bounded domain (see [4, 5, 12]).

More precisely, this paper considers the nonlinear Schrödinger equation:

Here, and is the Sobolev space:

For , the following is assumed:

(V ₁)Consider , in , , with if and , for some if , and there exist , , and such that, for ,

(V ₂)For any , and there exists such that

For , the following is assumed:

(f ₁) is a continuous function:

(f ₂)There exists such that, for any ,

 Here,

 For , let

(f ₃)For any , and there exists such that if , then

(f ₄)For any , .

The main result of this paper is as follows.

Theorem 1. Suppose that and are satisfied; then, problem (6) has infinitely many solutions.

Remark 2. (i) Under the assumptions and , the essential spectrum of defined by (3) is (see Lemma 3 in Section 2). Therefore, condition (11) indicates that satisfies a resonance type condition at the essential spectrum of .
(ii) A typical example for which satisfies is

Here, , , , , and in .
(iii) There are many functions satisfying . Let . Using

we can construct a function satisfying from a given . For example, let

where

Then, by (19), for ,

and, for , It follows that

and for any . Because the even function is strictly increasing in , as , it can be deduced that satisfies , . Moreover, it is easy to verify that as , as , as , and as . Because the even function is strictly increasing in , as , and as , it can be deduced that, for sufficiently small , if , then Therefore, satisfies .

Let

The inner product and the norm in are

respectively. Here, is the constant in (10).

Let

Then, for any , the Gateaux derivative of is as follows:

It follows that is a solution of (6) if and only if is a critical point of . In Section 2, it is shown that satisfies the Cerami condition. In Section 3, an abstract critical point theorem from [13] is used to show that has infinitely many critical points.

Notation. Let be Banach space. We denote the dual space of by and denote strong and weak convergence in by and , respectively. For , we denote the Fréchet derivative of at by . The Gateaux derivative of is denoted by . denotes the standard space . We use and to mean and , respectively.

2. The Cerami Condition for

Lemma 3. If satisfies the assumptions , then the essential spectrum of the Schrödinger operator defined by (3) is and has infinitely many negative eigenvalues accumulating at zero.

Proof. Because satisfies , by [14, Theorem XIII.15(b)] or [2, Theorem 10.29(b)], the essential spectrum of the Schrödinger operator defined by (3) is . The second result of this lemma comes from [2, Theorem 10.31] or [14, Theorem XIII.6(a)].

Lemma 4. If satisfies the assumptions and , then has no eigenvalue in .

Proof. Because satisfies the assumptions and , by [15, Corollary ], to prove this lemma, it suffices to prove that satisfies the conditions I and II in [15, Corollary ]. For this, it suffices to prove that, for , the functions

are bounded functions in . Because and , there exists such that, for ,

Because , for , the following is true:

Because , the following holds:

Combining (29) and (30) yields that is a bounded function in . From , we have , . It follows that is also a bounded function in .

Definition 5 (see [16]). Let be a Banach space. Let and . One can call the fact that satisfies the Cerami condition at , denoted by condition, if for any satisfying

there exists such that, up to a subsequence, as . If satisfies the condition for every , then is said to satisfy the Cerami condition. A sequence satisfying (32) is called a sequence of .

It was shown in [13] that the Cerami condition actually suffices to get a deformation theorem (see [8]) and then, by standard minimax arguments, it allows rather general minimax results.

Lemma 6. Suppose that satisfies that and . If in , then

Proof. Because in , is bounded. It follows that

From , we deduce that, for any , there exists such that

It follows that

For and , let

Because and in , by Lemma of [6], we have

Combining (36) and (38) yields (33).

Lemma 7. Let be a sequence of . Then, up to a subsequence, is bounded.

Proof. By Definition 5, we have

Arguing indirectly, assume that and set .
Case 1. is nonvanishing; that is, there exist and such that

Set . Then, by (40), in
By (39) and (27), for any , we have, as ,

It follows that, as ,

where . If , sending in (42), by (11), and the fact that, for , , we get that

This is impossible, because . If has a subsequence bounded in , up to a subsequence, it can be assumed that , and sending in (42), we get that

It follows that

This implies that is a nonzero eigenfunction of corresponding to eigenvalue . It contradicts Lemma 4.
Case 2. is vanishing; that is,

The Lions lemma (see, e.g., [6, Lemma ]) shows that, for any ,

Here, denotes norm. Equations (39) and (27) show that, as ,

where is defined in (15). It follows that

By Lemma 6 and in , we have

Together with (49), this implies

By (39),

Let

By , (17), and (52), the following is true:

Here, denotes the Lebesgue measure of . It follows that

Let

Because (see ), we have

Under the assumption , there exists such that

It follows that for some constant . Then, by (55) and (47), for ,

Combining (57) and (59) yields

This contradicts (51).
From the two cases given above, it can be deduced that is bounded in .

Lemma 8. The functional satisfies the condition for every .

Proof. Let be a sequence of . By Lemma 7, is bounded. It follows that there exists such that, up to a subsequence, in . Because the embedding is compact, we get that, up to a subsequence,

As the proof of Lemma of [6], we have . It follows that

We are going to prove that, up to a subsequence, as .
By (39), as ,

By Lemma 6,

Combining (62)–(64) yields

Because in , by [17, Theorem ], we have

Because for any (see (12)), by (61) and the Fatou lemma, we have

Using (65)–(67), it can be deduced that, up to a subsequence,

By (13), is a sequence of nonnegative functions. Because

by the Fatou lemma and (69), the following is true:

It follows that

By (12), we deduce that in . Then, by

and the Fatou lemma, the following holds:

Combining (72) and (74) leads to

By (10) and (11), it can be deduced that there exists such that

Then, by the mean value theorem, the Hölder inequality, and (76), we get that, for any and for any sufficiently small , there exists such that

Because (61), (75), and (77) hold and is bounded, Theorem of [18] shows that

It follows that, for every sufficiently small ,

Here,

Equations (10) and (12) show that

Combining (81) and (82) yields that, for any sufficiently small , there exists such that

By (79) and (83), we get that, for every sufficiently small ,

From

we get that

By (86), the following holds:

where and

Combining (84) and (87) leads to

Together with (68), this implies in . This completes the proof.

3. Proof of Theorem 1

For the proof of Theorem 1, the following abstract theorem, which is a corollary of [13, Theorem ], is used.

Theorem 9. Let be a real Hilbert space with norm . Suppose that is a -functional defined in and satisfies the following conditions:

(A ₁) satisfies the condition for every , and .(A ₂) is an even functional; that is, for every .(A ₃)There exist and a closed subspace of with finite codimension such that there exists such that

 Here, (A ₄)For any , there exists a subspace with dimension larger than or equal to such that

Then, has infinitely many pairs of critical points.

Proof. Arguing indirectly, assume that has only a finite number of critical points, say . The codimension of is here denoted by By , there exists a subspace with dimension larger than or equal to such that

This indicates that there exists such that

Then, by [13, Theorem ], has at least critical points. This contradicts the assumption that has only critical points.

Let

be the eigenvalues of below the essential spectrum of . Each has been repeated in the sequence according to its finite multiplicity. Then, by Lemma 3, the following is true:

Let

Here, is the constant in (10). From this definition,

Let denote the spectral family of the operator defined by (3).

Let

Here, is the identity map and

It is easy to see that the codimension of equals .

Lemma 10. There exists such that

Proof. From (96) and (98), the following holds:

Here,

It follows that

If , then there exists satisfying , , and

Up to a subsequence, it can be assumed that in and a.e. in . Since is a closed subspace of , by in , we have . Because in , by [17, Theorem ], we have

Because a.e. in , by the Fatou lemma, the following holds:

Moreover, by Lemma 6,

Combining (104)–(107) yields

Because , by (101) and (108), we have . Then, by (104) and (107),

This contradicts , . Therefore, .

Proof of Theorem 1. We are going to prove that the functional satisfies the conditions in Theorem 9.
By Lemma 8, the functional satisfies the condition . By , satisfies the condition .
By (10) and (11), it can be deduced that, for any , there exists such that

Here,

Let be the space defined by (98). It has a finite codimension. By Lemma 10 and (110), we have, for any ,

where the positive constant comes from the Sobolev inequality , . Choosing and , by (112), the following holds:

It follows that satisfies the assumption .
Let such that and . Then, for the function , and . Then, by (8), it follows that, for ,

where is a positive constant independent of . Because , it follows that there exists such that

Let , . These functions have mutually disjoint compact supports. Let

Then, is an -dimensional subspace of . By (10) and (11), it can be deduced that, for any , there exist and such that

Because have mutually disjoint compact supports, by (115), we have, for ,

where

Let

Then, is a bounded set in , since every has a compact support. By (117),

where

Choosing

by (118) and (121), we get that

Since , for every , (124) implies

Therefore, satisfies the condition of Theorem 9.
Because satisfies the conditions of Theorem 9, by Theorem 9, it has infinitely many critical points. It follows that (6) has infinitely many solutions.

Conflict of Interests

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

Acknowledgments

Shaowei Chen was supported by Science Foundation of Huaqiao University (13BS208) and Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University (ZQN-PY119).

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