##### Boundary Value Problems
Open Badges Research

# Multiple solutions to nonlinear Schrödinger equations with critical growth

Wulong Liu1* and Peihao Zhao2

Author Affiliations

1 Department of Mathematics, Jiangxi University of Science and Technology, Ganzhou, 341000, P.R. China

2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P.R. China

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Boundary Value Problems 2013, 2013:199  doi:10.1186/1687-2770-2013-199

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/199

 Received: 20 March 2013 Accepted: 24 May 2013 Published: 4 September 2013

© 2013 Liu and Zhao; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In 2000, Cingolani and Lazzo (J. Differ. Equ. 160:118-138, 2000) studied nonlinear Schrödinger equations with competing potential functions and considered only the subcritical growth. They related the number of solutions with the topology of the global minima set of a suitable ground energy function. In the present paper, we establish these results in the critical case. In particular, we remove the condition , which is a key condition in their paper. In the proofs we apply variational methods and Ljusternik-Schnirelmann theory.

MSC: 35J60, 35Q55.

##### Keywords:
nonlinear Schrödinger equations; critical growth; variational methods

### 1 Introduction and main result

We investigate the following nonlinear Schrödinger equation:

(1.1)

which arises in quantum mechanics and provides a description of the dynamics of the particle in a non-relativistic setting. ħ is the Planck’s constant, denotes the mass of the particle, is the electric potential, g is the nonlinear coupling, and ψ is the wave function representing the state of the particle. A standing wave solution of equation (1.1) is a solution of the form . It is clear that solves (1.1) if and only if solves the following stationary equation:

(1.2)

For simplicity and without loss of generality, we set , and , then equation (1.2) is equivalent to

(1.3)

A considerable amount of work has been devoted to investigating solutions of (1.3). The existence, multiplicity and qualitative property of such solutions have been extensively studied. For single interior spikes solutions in the whole space , please see [1-9]etc. For multiple interior spikes, please see [10,11]etc. For single boundary spike solutions with Neumann boundary condition, please see [6,12-15]etc. For multiple boundary spikes, please see [16-18]etc. In particular, Wang and Zeng [9] studied the existence and concentration behavior of solutions for NLS with competing potential functions. Cingolani and Lazzo in [19] obtained the multiple solutions for the similar equation. In those papers only the subcritical growth was considered. In the present paper, we complete these studies by considering a class of nonlinearities with the critical growth. In particular, we remove the condition , which is a key condition in [19].

In the sequel, we restrict ourselves to the critical case in which . More specifically, we study the following problem:

(1.4)

where if , and if . satisfies

(f1) for each ;

(f2) ;

(f3) there exists such that

(f4) there exists such that

where ;

(f5) the function is strictly increasing in for any .

Our main results are the following theorem.

Theorem 1.1Let. Suppose that f satisfies (f1)-(f5), Vis a continuous function inand satisfies. Then whenεis sufficiently small, the problem (1.4) has at leastdistinct nontrivial solutions.

Here denotes the Ljusternik-Schnirelmann category of Σ in . By definition (e.g., [20]), the category of A with respect to M, denoted by , is the least integer k such that , with () closed and contractible in M. We set and if there are no integers with the above property. We will use the notation for .

To prove Theorem 1.1, we mainly use the idea of [15,19,21]. More precisely, we can show that the -condition holds in the subset (see (4.6)). Hence the standard Ljusternik-Schnirelmann category theory can be applied in to yield the existence of at least critical points of . And then we construct two continuous mappings

(1.5)

and

(1.6)

where

(1.7)

Then a topological argument asserts that

We will also prove that if u is a critical point of satisfying , then u cannot change sign. Hence we obtain at least nontrivial critical points of .

The paper is organized as follows. In Section 2, we collect some notations and preliminaries. A compactness result is given in Section 3, which is a key step in our proof. Finally, in Section 4, we prove Theorem 1.1.

### 2 Notations and preliminaries

is the usual Sobolev space of real-valued functions defined by

with the normal

Let be the subspace of a Hilbert space with respect to the norm

We denote by S the Sobolev constant for the embedding , namely

(2.1)

where is the usual Sobolev space of real-valued functions defined by

We say that a function is a weak solution of the problem (1.4) if

In view of (f2) and (f3), we have that the associated functional given by

is well defined. Moreover, with the following derivative:

Hence, the weak solutions of (1.4) are exactly the critical points of .

Let us recall some known facts about the limiting problem, namely the problem

(2.2)

here acts as a parameter instead of an independent variable. Solutions of (2.2) will be sought in the Sobolev space as critical points of the functional

The least positive critical value can be characterized as

where

(2.3)

An associated critical point w actually solves equation (2.2) and is called a ground state solution or the least energy solution, i.e., w satisfies

Moreover, there exist and such that

(2.4)

For more details, please see [22,23].

Set

For any , we denote . We need to estimate the super bound of . In order to do this, we estimate . We shall use a family of radial function defined by

It is known [20] that

Moreover, we have

Set , where is a cut-off function satisfying if , if and . After a detailed calculation, we have the following estimates:

(2.5)

(2.6)

(2.7)

Since , from (2.5)-(2.7), we conclude

(2.8)

then the maximum value of the right-hand side is achieved at

(2.9)

and

(2.10)

Hence we have

(2.11)

We denote the Nehari manifold of by

### 3 Compactness result

Proposition 3.1Letas. Assume thatsatisfiesas. Then uniformly in, there exist a subsequence of (still denoted by), andsuch that. Furthermore, converges strongly intow, the positive ground state solution of equation (2.2).

Proof Let be such that . Then, by a change of variable , we have

(3.1)

This implies that is bounded in . Noting that

(3.2)

hence

(3.3)

since . Now we prove that there exists a sequence and constants such that

(3.4)

Indeed, if this is not true, then the boundedness of in and a lemma due to Lions [[24], Lemma I.1] imply that in for all . Given , we can use (f2), (f3) and to get

Moreover,

as . Therefore

(3.5)

and consequently (3.2) yields

i.e.,

(3.6)

However, recall the definition of S in (2.1),

equivalent to , contradicting (3.6). Thus, (3.4) holds. Using the idea of [21,25], along a subsequence as , we may assume that

We now consider such that (see (2.3)). By a change of variable , it follows that

(3.7)

Hence , from which it follows that in .

Since and are bounded in and in , the sequence is bounded. Thus, up to a subsequence, . If , then , which does not occur. Hence , and therefore the sequence satisfies

(3.8)

For fixed , define

By the Hölder inequality,

Hence , the dual space of . Consequently, as , in implies , i.e.,

(3.9)

Since converges weakly to w in , is bounded in . Thus is bounded in . It then follows that there is a subsequence of , still denoted by , such that converges weakly to some in . Next we will show . Choose a sequence of open relatively compact subsets, with regular boundaries, of covering , i.e., . It is easy to see that, by compact embedding, in for any . Hence a.e. on . Hence a.e. on . By the Brezis and Lieb lemma [26], we conclude that strongly in . Thus a.e. on each , and then the diagonal rule implies a.e. on . Hence

(3.10)

Similarly, we have

(3.11)

By (f2) and (f3),

Hence when R is large enough, we get

Noting that in , . Therefore we have

Hence

(3.12)

By (3.9)-(3.12), we derive that

(3.13)

i.e., .

For any let us consider the measure sequence defined by

We assume

By the concentration-compactness lemma [24], there exists a subsequence of (denoted in the same way) satisfying one of the three following possibilities.

Compactness: There exists a sequence such that for any there is a radius with the property that

Vanishing: For all ,

Dichotomy: There exists a number , , such that for any there is a number and a sequence with the following property: Given there are non-negative measures , such that

(i) ,

(ii) , ,

(iii) .

We are going to rule out the last two possibilities so that compactness holds. Our first goal is to show that vanishing cannot occur. Otherwise,

Hence , contradicting .

Now for the harder part. Let η be a smooth nonincreasing cut-off function, defined in , such that if ; if ; and . Also, let . We define

a nondecreasing function on . Denote by . We show now that dichotomy does not occur. Otherwise there exists such that for some and the function splits into and with the following properties:

(3.14)

If we denote

(3.14) becomes

(3.15)

Denote by , then

Using Dichotomy (iii), we get

which implies

Hence

Now we observe that , therefore

(3.16)

(3.17)

(3.18)

and

(3.19)

where as .

Recall that (see (2.3)), which implies

(3.20)

Then using and in place of , respectively, we get

(3.21)

There exists such that , i.e.,

(3.22)

By (f2) and (f3), , we see cannot go zero, that is, . If , by (3.21), (3.22) and (f4), we get

(3.23)

since (f5). By (3.15),

(3.24)

a contradiction. Thus . Assume that , we will show . By (3.21) and (3.22), we have

Hence by the Lebesgue dominated convergence theorem, we get

By (f5), we have . Similarly, . Using this together with (3.16), (3.17), (3.18) and (3.19), we obtain

Contradiction! Thus dichotomy does not occur.

With vanishing and dichotomy ruled out, we obtain the compactness of a sequence , i.e., there exist and for each , there exists such that

(3.25)

Then must be bounded, for otherwise (3.25) would imply, in the limit ,

(3.26)

for some positive constants , independent of δ, which implies , contrary to (3.8).

From the foregoing, it follows that there exist bounded nonnegative measures , on such that weakly and tightly as . Lemma I.1 in [27] declares that there exist sequences , such that

(3.27)

where denotes a Dirac measure, . Take in the support of the singular part of , . We consider such that

(3.28)

Choosing the test function , from , we have

(3.29)

This reduces to

hence (3.27)(3) states

i.e.,

Consequently,

and hence

(3.30)

which implies that the set J is at most finite. Here CardJ is the cardinal numbers of set J. Hence

(3.31)

since

When n is large enough, recall (see (2.11)), together with (3.30) and (3.31), we obtain

a contradiction. Therefore J is empty, that is, as . By the Brezis and Lieb lemma [26] again, we get

(3.32)

Equation (3.25) and compact embedding theorem imply

(3.33)

This together with (3.13), (3.20) and (3.32) allows us to deduce easily

Since is a uniformly convex Banach space, hence

(3.34)

From (3.32), (3.33) and (3.34), we can obtain

(3.35)

i.e., w is the ground state solution of (2.2) in view of (3.13). The proof of Proposition 3.1 is complete. □

### 4 Proof of Theorem 1.1

Proposition 4.1Supposefsatisfies (f2)-(f4). Thensatisfies the-condition for all, that is, every sequenceinsuch that, , as, possesses a convergent subsequence.

Proof Suppose that is a sequence in such that , , as . Using (f4), by a change of variable , we obtain that

(4.1)

This implies that is bounded in . Therefore we may assume in and a.e. Let . Then

and by the Brezis-Lieb lemma [26],

For convenience, we denote by

(4.2)

and

(4.3)

It is clear that

(4.4)

It is easy to verify that . Hence we have

and thus

since by (f2) and (f3). If , then , hence as , and we can obtain the desired conclusion. Hence it remains to show that . By a change of variable, from

and

we get

i.e.,

(4.5)

By the Sobolev inequalities,

Letting , we get , so either which contradicts (4.5) or . □

Let be fixed. Let η be a smooth nonincreasing cut-off function, defined in , such that if ; if ; and for some . For any , let

where w is the positive ground state of (2.2). We may assume that is the unique positive number such that

Let be any positive function tending to 0 as , we define the sublevel

(4.6)

By Lemma 4.2 below, is not empty for ε sufficiently small. By noticing that , we can define as

Lemma 4.2Uniformly in, we have

(4.7)

Proof Let . Computing directly, we have

(4.8)

By a change of variable , we obtain

(4.9)

uniformly for .

(4.10)

By the exponential decay of ω, we get

(4.11)

uniformly for . Therefore, in the limit that ε is very small, thanks to (4.8) (4.9) and (4.11), we find

(4.12)

On the other hand, following the idea of [21,25], from , by the change of variables , we get

(4.13)

(4.14)

which contradicts (4.12). Thus, up to a subsequence, .

Since f has subcritical growth and , it follows that . Thus, we can take the limit in (4.13) to obtain

(4.15)

from which it follows that . Since w also belongs to , we conclude that . This and Lebesgue’s theorem imply that

(4.16)

and

(4.17)

uniformly for . Noting , from (4.12), (4.16) and (4.17), we have

Thus (4.7) is proved. □

Let be the center of mass of in terms of the norm:

Lemma 4.3Letas. Then for,

uniformly for.

Proof By change of variable , we have

By Proposition 3.1, converges strongly in to w, which is a positive ground state solution of equation (2.2). Thanks to the exponential decay of w (see (2.4)), we obtain as . This completes the proof of Lemma 4.3. □

Proof of Theorem 1.1 By Proposition 4.1, satisfies the -condition for all . Now let us choose a function such that as and such that is not a critical level for . For such , let us introduce the set as in (4.6). Then the standard Ljusternik-Schnirelmann theory implies that has at least critical points on (also see [21]).

By Lemma 4.3, we can assume that for any , there exists such that for any . For such ε, by Lemma 4.2, we have uniformly for , thus . Recall , calculating directly, we get

as uniformly for . Hence the map is homotopical equivalence to the inclusion for ε small enough. We denote . It is easy to verify that and (cf. [[19], Lemma 2.2]). Hence we have

Next we show that if u is a critical point of satisfying , then u cannot change sign. Indeed, if with and , then from , we have

By change of variable , we get

i.e.,

Also, noting

Hence

which is a contradiction. Therefore there exist at least nonzero critical points of and thus solutions of equation (1.4). □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

WL carried out the genetic studies, participated in the sequence alignment and drafted the manuscript. PZ checked the references.

### Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (11261052). The authors are grateful to Prof. Guowei Dai for pointing out several mistakes and valuable comments.

### References

1. Floer, A, Weinstein, A: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal.. 69, 397–408 (1986). Publisher Full Text

2. Oh, Y-G: Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class . Commun. Partial Differ. Equ.. 13, 1499–1519 (1988). Publisher Full Text

3. Oh, Y-G: Corrections to ‘Existence of semi-classical bound state of nonlinear Schrödinger equations with potentials of the class ’. Commun. Partial Differ. Equ.. 14, 833–834 (1989)

4. Ambrosetti, A, Badiale, M, Cingolani, S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal.. 140, 285–300 (1997). Publisher Full Text

5. Del Pino, M, Felmer, P: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var.. 4, 121–137 (1996). Publisher Full Text

6. Del Pino, M, Felmer, P: Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting. Indiana Univ. Math. J.. 48(3), 883–898 (1999)

7. Rabinowitz, P: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys.. 43, 270–291 (1992). Publisher Full Text

8. Wang, X: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys.. 153, 229–244 (1993). Publisher Full Text

9. Wang, X, Zeng, B: On concentration of positive bound states of nonlinear Schrödinger equation with competing potential functions. SIAM J. Math. Anal.. 28, 633–655 (1997). Publisher Full Text

10. Del Pino, M, Felmer, P: Multi-peak bound states of nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 15(2), 127–149 (1998). Publisher Full Text

11. Gui, C: Existence of multi-bump solutions for nonlinear Schrödinger equations. Commun. Partial Differ. Equ.. 21, 787–820 (1996). Publisher Full Text

12. Lin, C, Ni, W-M, Takagi, I: Large amplitude stationary solutions to a chemotaxis systems. J. Differ. Equ.. 72, 1–27 (1988). Publisher Full Text

13. Ni, WM, Takagi, I: On the shape of least-energy solutions to a semilinear Neumann problem. Commun. Pure Appl. Math.. 45, 819–851 (1990)

14. Ni, WM, Takagi, I: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J.. 70, 247–281 (1993). Publisher Full Text

15. Wang, ZQ: On the existence of multiple, single peaked solutions for a semilinear Neumann problem. Arch. Ration. Mech. Anal.. 120, 375–399 (1992). Publisher Full Text

16. Gui, C: Multi-peak solutions for a semilinear Neumann problem. Duke Math. J.. 84, 739–769 (1996). Publisher Full Text

17. Li, YY: On a singularly perturbed equation with Neumann boundary condition. Commun. Partial Differ. Equ.. 23, 487–545 (1998)

18. Wei, J, Winter, M: Multiple boundary spike solutions for a wide class of singular perturbation problems. J. Lond. Math. Soc.. 59(2), 585–606 (1999). Publisher Full Text

19. Cingolani, S, Lazzo, M: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ.. 160, 118–138 (2000). Publisher Full Text

20. Ambrosetti, A, Malchiodi, A: Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, Cambridge (2007)

21. Liu, W, Zhao, P: Critical semilinear Neumann problem with magnetic fields. Preprint

22. Berestycki, H, Lions, PL: Nonlinear scalar field equations I. Existence of a ground state. Arch. Ration. Mech. Anal.. 82, 313–345 (1983)

23. Zhang, J, Zou, W: A Berestycki-Lions theorem revisited. Commun. Contemp. Math.. 14(5), (2012) Article ID 1250033

24. Lions, PL: The concentration-compactness principle in the calculus of variation. The locally compact case. II. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 1, 223–283 (1984)

25. Alves, CO, Figueiredo, GM, Furtado, MF: Multiple solutions for a nonlinear Schrödinger equation with magnetic fields. Commun. Partial Differ. Equ.. 36, 1565–1586 (2011). Publisher Full Text

26. Brezis, H, Lieb, E: A relation between pointwise convergence of functions and convergence of functional. Proc. Am. Math. Soc.. 88, 486–490 (1983)

27. Lions, PL: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoam.. 1, 145–201 (1985)