Open Access

Ground state solution and multiple solutions to asymptotically linear Schrödinger equations

Boundary Value Problems20142014:216

https://doi.org/10.1186/s13661-014-0216-1

Received: 5 June 2014

Accepted: 11 September 2014

Published: 25 September 2014

Abstract

In this paper, we consider the Schrödinger equation Δ u + V ( x ) u = f ( x , u ) , x R N , where V and f are periodic in x 1 , , x N , asymptotically linear and satisfies a monotonicity condition. We use the generalized Nehari manifold methods to obtain a ground state solution and infinitely many geometrically distinct solutions when f is odd in u.

Keywords

Schrödinger equationground state solutionmultiplicity of solutionsasymptotically linear

1 Introduction

We consider the problem
Δ u + V ( x ) u = f ( x , u ) , u H 1 ( R N ) ,
(1.1)

where f and V are periodic in x 1 , , x N , asymptotically linear and satisfies a monotonicity condition. In the case that the nonlinear term is asymptotically linear at infinity, there are some results in the literature [1]–[12] and the references therein, where multiplicity results are considered in [1]–[3], [9], [10], [12]. As far as we know, there are only a few papers concerned with the existence of infinitely many solutions for the asymptotical linear case when f and V are also periodic in x 1 , , x N ; e.g. see [2]. Except for [5], there seem to be few results on the existence of a ground state solution in the asymptotically linear case. Motivated by [13], this paper is to present a different approach involving the critical point theory with the discreteness property of the Palais-Smale in search for a ground state solution and multiple solutions for the asymptotically linear Schrödinger equations. It should be pointed out that in [2], they cannot make sure the existence of a ground state solution. Our results can be regarded as complements or different attempts of the results in [2], [5].

Setting F ( x , u ) : = 0 u f ( x , s ) d s , we suppose that V and f satisfy the following assumptions:

(V): V is continuous, 1-periodic in x i , 1 i N , and there exists a constant a 0 > 0 such that V ( x ) a 0 for all x R N .

(f1): f is continuous, 1-periodic in x i , 1 i N .

(f2): f ( x , u ) = o ( u ) as u 0 , uniformly in x.

(f3): There is q ( x ) > V ( x ) , x R N , such that f ( x , u ) / u q ( x ) , as | u | , where q is continuous, 1-periodic in x i , 1 i N .

(f4): u f ( x , u ) / | u | is strictly increasing on ( , 0 ) and ( 0 , ) .

Let denote the action of Z N on H 1 ( R N ) given by
( k u ) ( x ) : = u ( x k ) , k Z N .
(1.2)
It follows from (V) and (f1) that if u 0 is a solution of (1.1), then so is k u 0 for all k Z N . Set
O ( u 0 ) : = { k u 0 : k Z N } .

O ( u 0 ) is called the orbit of u 0 with respect to the action of Z N , and it is called a critical orbit for a functional F if u 0 is a critical point of F and F is Z N -invariant, i.e., F ( k u ) = F ( u ) for all k Z N and all u (then of course all points of O ( u 0 ) are critical). Two solutions u 1 , u 2 of (1.1) are said to be geometrically distinct if O ( u 1 ) O ( u 2 ) .

Theorem 1.1

Suppose that (V), (f1)-(f4) are satisfied. Then (1.1) has a ground state solution. In addition, if f is odd in u, then (1.1) admits infinitely many pairs ±u of geometrically distinct solutions.

Notation

C , C 1 , C 2 , will denote different positive constants whose exact value is inessential. The usual norm in the Lebesgue space L p ( Ω ) is denoted by u p , Ω , and by u p if Ω = R N . E denotes the Sobolev space H 1 ( R N ) and S is the unit sphere in E. It follows from (V) that
u : = ( R N ( | u | 2 + V ( x ) u 2 ) ) 1 / 2
is an equivalent norm in E. It is more convenient for our purposes than the standard one and will be used henceforth. For a functional I, as in [14], we put
I d : = { u : I ( u ) d } , I c : = { u : I ( u ) c } , I c d : = { u : c I ( u ) d } .

2 Preliminary results

Consider the functional
I ( u ) : = 1 2 R N | u | 2 + 1 2 R N V ( x ) u 2 R N F ( x , u ) .
(2.1)
Then I is well defined on E and I C 1 ( E , R ) under the hypotheses (V), (f1)-(f3). Note also that (V), (f1) imply I is invariant with respect to the action of Z N given by (1.2). It is easy to see that
I ( u ) , v = R N u v + R N V ( x ) u v R N f ( x , u ) v
(2.2)

for all u , v E .

Let
M : = { u E { 0 } : I ( u ) , u = 0 } .
(2.3)

Recall that is called the Nehari manifold. We do not know whether is of class C 1 under our assumptions and therefore we cannot use minimax theory directly on . To overcome this difficulty, we employ the arguments developed in [13], [15], [16].

We assume that (V) and (f1)-(f4) are satisfied from now on. First, (f2) and (f3) imply that for each ε > 0 there is C ε > 0 such that
| f ( x , u ) | ε | u | + C ε | u | p 1 for all  u R ,
(2.4)

where 2 < p < 2 , 2 : = 2 N / ( N 2 ) if N 3 , 2 : = if N = 1  or  2 .

For t > 0 , let
h ( t ) : = I ( t u ) = t 2 2 R N | u | 2 + V ( x ) u 2 R N F ( x , t u ) .
Let
E : = { u E : R N | u | 2 + V ( x ) u 2 < R N q ( x ) u 2 } .

It follows from q ( x ) V ( x ) > 0 , x R N , that E .

Lemma 2.1

F ( x , u ) > 0 and 1 2 f ( x , u ) u > F ( x , u ) if u 0 .

This follows immediately from (f2) and (f4).

Lemma 2.2

  1. (1)

    For each u E there is a unique t u > 0 such that h ( t ) > 0 for 0 < t < t u and h ( t ) < 0 for t > t u . Moreover, t u M if and only if t = t u .

     
  2. (2)

    If u E , then t u M for any t > 0 .

     

Proof

  1. (1)
    For each u E , due to the Lebesgue dominated convergence theorem and (f2), (f3), we get
    lim t I ( t u ) t 2 = 1 2 R N | u | 2 + V ( x ) u 2 lim t u 0 F ( x , t u ) t 2 u 2 u 2 = 1 2 [ R N | u | 2 + V ( x ) u 2 R N q ( x ) u 2 ] < 0
     
and
lim t 0 I ( t u ) t 2 = 1 2 R N | u | 2 + V ( x ) u 2 lim t 0 u 0 F ( x , t u ) t 2 u 2 u 2 = 1 2 R N | u | 2 + V ( x ) u 2 > 0 .
Hence h has a positive maximum. The condition h ( t ) = 0 is equivalent to
u 2 = u 0 f ( x , t u ) t u u 2 .
By (f4), the first conclusion holds. The second conclusion follows from h ( t ) = t 1 I ( t u ) , t u .
  1. (2)
    If t u M for some t > 0 , then I ( t u ) , u = 0 and therefore using (f3) and (f4)
    u 2 = u 0 f ( x , t u ) t u u 2 < R N q ( x ) u 2 .
     

Hence u E . □

Lemma 2.3

  1. (1)

    There exists ρ > 0 such that c : = inf M I inf S ρ I > 0 .

     
  2. (2)

    u 2 2 c for all u M .

     

Proof

  1. (1)

    Using (2.4) and the Sobolev inequality we have inf S ρ I > 0 if ρ is small enough. The inequality inf M I inf S ρ I is a consequence of Lemma 2.2 since for every u M there is s > 0 such that s u S ρ (and I ( t u u ) I ( s u ) ).

     
  2. (2)
    For u M , by Lemma 2.1 we have
    c 1 2 u 2 R N F ( x , u ) 1 2 u 2 .
     

 □

We do not know whether I is coercive on . However, we can prove the following.

Lemma 2.4

All Palais-Smale sequences ( u n ) M are bounded.

Proof

Arguing by contradiction, suppose there exists a sequence ( u n ) M such that u n and I ( u n ) d for some d [ c , ) . Let v n : = u n / u n . Then v n v and v n ( x ) v ( x ) a.e. in R N after passing to a subsequence. Choose y n R N so that
B 1 ( y n ) v n 2 = max y R N B 1 ( y ) v n 2 .
(2.5)
Since I and are invariant with respect to the action of Z N given by (1.2), we may assume that ( y n ) is bounded in R N . If
B 1 ( y n ) v n 2 0 as  n ,
(2.6)
then it follows that v n 0 in L r ( R N ) for 2 < r < 2 by Lions’ lemma (cf.[17], Lemma 1.21), and therefore (2.4) implies that R N F ( x , s v n ) 0 for every s R . Lemma 2.2 implies that
d I ( u n ) I ( s v n ) = s 2 2 R N F ( x , s v n ) s 2 2 .

Taking a sufficiently large s, we get a contradiction. Hence (2.6) cannot hold and, since v n v in L loc 2 ( R N ) , v 0 . Hence | u n ( x ) | if v ( x ) 0 .

Let φ C 0 ( R N ) . Then I ( u n ) , φ 0 and hence
R N v n φ + V ( x ) v n φ R N f ( x , u n ) u n v n φ 0 .
By the Lebesgue dominated convergence theorem we therefore have
R N v φ + V ( x ) v φ = R N q ( x ) v φ .

So v 0 and Δ v + V ( x ) v = q ( x ) v . This is impossible because Δ + V q has only an absolutely continuous spectrum. The proof is complete. □

Lemma 2.5

If V is a compact subset of , then there exists R > 0 such that I 0 on ( R + V ) B R ( 0 ) .

Proof

We may assume without loss of generality that V S . Arguing by contradiction, suppose there exist u n V and w n = t n u n , where u n u , t n and I ( w n ) 0 . We have
0 I ( t n u n ) t n 2 = 1 2 R N | u n | 2 + V ( x ) u n 2 u n 0 F ( x , t n u n ) t n 2 u n 2 u n 2 1 2 R N | u | 2 + V ( x ) u 2 1 2 R N q ( x ) u 2 < 0 .

 □

Let U : = E S and define the mapping m : U M by setting
m ( w ) : = t w w ,

where t w is as in Lemma 2.2.

Lemma 2.6

U is an open subset of S.

Proof

Obvious because is open in E. □

Lemma 2.7

Assume u n U , u n u 0 U , and t n u n M , then I ( t n u n ) .

Proof

Since u 0 U , R N | u 0 | 2 + V ( x ) u 0 2 = R N q ( x ) u 0 2 . Using this, we have
I ( t u 0 ) = 1 2 t 2 R N | u 0 | 2 + V ( x ) u 0 2 t 2 R N F ( x , t u 0 ) t 2 u 0 2 u 0 2 = 1 2 t 2 R N ( q ( x ) 2 F ( x , t u 0 ) t 2 u 0 2 ) u 0 2 = 1 2 t 2 R N ( q ( x ) f ( x , t u 0 ) t u 0 ) u 0 2 + R N 1 2 f ( x , t u 0 ) t u 0 F ( x , t u 0 ) .
Note that by (f4), we have for large enough s, there is δ > 0 such that
1 2 f ( x , s ) s F ( x , s ) δ
(see [4], Remark 1.5). So I ( t u 0 ) , as t (we have used Fatou’s lemma). Given C > 0 , choose t > 0 such that I ( t u 0 ) C . Since u n u 0 ,
lim n I ( t n u n ) lim n I ( t u n ) = I ( t u 0 ) C

and hence I ( t n u n ) . □

The following lemmas are taken from [13], [15].

Below we shall use the notations
K : = { w S : Ψ ( w ) = 0 } , K d : = { w K : Ψ ( w ) = d } .
Since f is odd in u, we can choose a subset of K such that F = F and each orbit O ( w ) K has a unique representative in . We must show that the set is infinite. Arguing indirectly, assume
F  is a finite set .
(2.7)

Lemma 2.8

The mapping m is a homeomorphism between U and , and the inverse of m is given by m 1 ( u ) = u u .

We consider the functional Ψ : U R given by
Ψ ( w ) : = I ( m ( w ) ) .

Lemma 2.9

  1. (1)
    Ψ C 1 ( U , R ) and
    Ψ ( w ) , z = m ( w ) I ( m ( w ) ) , z for all  z T w ( U ) .
     
  1. (2)

    If ( w n ) is a Palais-Smale sequence for Ψ, then ( m ( w n ) ) is a Palais-Smale sequence for I. If ( u n ) M is a bounded Palais-Smale sequence for I, then ( m 1 ( u n ) ) is a Palais-Smale sequence for Ψ.

     
  2. (3)

    w is a critical point of Ψ if and only if m ( w ) is a nontrivial critical point of I. Moreover, the corresponding values of Ψ and I coincide and inf U Ψ = inf M I .

     
  3. (4)

    Ψ is even (because I is).

     

By (2.4), the following lemma also holds.

Lemma 2.10

Let d c . If ( v n 1 ) , ( v n 2 ) Ψ d are two Palais-Smale sequences for Ψ, then either v n 1 v n 2 0 as n or lim sup n v n 1 v n 2 ρ ( d ) > 0 , where ρ ( d ) depends on d but not on the particular choice of Palais-Smale sequences.

It is well known that Ψ admits a pseudo-gradient vector field H : U K T U (see e.g.[18], p.86). Moreover, since Ψ is even, we may assume H is odd. Let η : G U K be the flow defined by
{ d d t η ( t , w ) = H ( η ( t , w ) ) , η ( 0 , w ) = w ,
(2.8)
where
G : = { ( t , w ) : w U K , T ( w ) < t < T + ( w ) }

and ( T ( w ) , T + ( w ) ) is the maximal existence time for the trajectory t η ( t , w ) . Note that η is odd in w because H is and t Ψ ( η ( t , w ) ) is strictly decreasing by the properties of a pseudogradient.

Let P U , δ > 0 and define U δ ( P ) : = { w U : dist ( w , P ) < δ } .

Lemma 2.11

Let d c . Then for every δ > 0 there exists ε = ε ( δ ) > 0 such that
  1. (a)

    Ψ d ε d + ε K = K d and

     
  2. (b)

    lim t T + ( w ) Ψ ( η ( t , w ) ) < d ε for w Ψ d + ε U δ ( K d ) .

     

Part (a) is an immediate consequence of (2.7) and (b) has been proved in [15]; see Lemmas 2.15 and 2.16 there. The argument is exactly the same except that S should be replaced by U. We point out that an important role in the proof of Lemma 2.11 is played by the discreteness property of the Palais-Smale sequences expressed in Lemma 2.10.

3 Proof of Theorem 1.1

Proof of Theorem 1.1

Taking a similar argument as in the proof of Theorem 1.1 in [15], it is easy to get a ground state solution. Noting that by Lemma 2.7 and Ekeland’s variational principle, it can make sure the existence of a ( PS ) c sequence belonging to U.

For the multiplicity the argument is the same as in Theorem 1.2 (cf.[15]). However, there are details which need to be clarified.

Let η be the flow given by (2.8). If T + ( w ) < , then lim t T + ( w ) η ( t , w ) exists (cf.[15], Lemma 2.15, Case 1) but unlike the situation in [15], this limit may be a point w 0 U . This possibility is ruled out by Lemma 2.7.

Finally, we need to show that U contains sets of arbitrarily large genus. Since the spectrum of Δ + V q in L 2 ( R N ) is absolutely continuous, E { 0 } contains an infinite-dimensional subspace E 0 . Hence E 0 S U and γ ( E 0 S ) = . □

Remark 3.1

There is a small gap in the proof of Theorem 1.2 in [13]. Lemma 4.6 as stated there does not exclude the possibility of η ( t , w ) approaching the boundary as t T + ( w ) (because we only know that η ( t , w ) goes to infinity). But it is easy to prove that I ( η ( t , w ) ) goes to infinity as well in [13]. In Lemma 2.7 of this paper we make some proper modifications which also apply to [13] and were proposed by Andrzej Szulkin.

Declarations

Acknowledgements

The authors thank the referee for providing some of the references and suggestions. The authors thank Professor Szulkin for his encouragement. This work was supported by NSFC 11171047. The first author was supported the Fundamental Research Funds for the Central Universities.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Dalian University of Technology, Dalian, P.R. China
(2)
State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian, P.R. China

References

  1. Costa DG, Tehrani H:On a class of asymptotically linear elliptic problem in R N . J. Differ. Equ. 2001, 173: 470-494. 10.1006/jdeq.2000.3944MathSciNetView ArticleGoogle Scholar
  2. Ding YH, Lee C: Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. J. Differ. Equ. 2006, 222: 137-163. 10.1016/j.jde.2005.03.011MathSciNetView ArticleGoogle Scholar
  3. Ding YH, Luan SX: Multiple solutions for a class of nonlinear Schrödinger equations. J. Differ. Equ. 2004, 207: 423-457. 10.1016/j.jde.2004.07.030MathSciNetView ArticleGoogle Scholar
  4. Jeanjean L:On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on R N . Proc. R. Soc. Edinb., Sect. A 1999, 129: 787-809. 10.1017/S0308210500013147MathSciNetView ArticleGoogle Scholar
  5. Liu CY, Wang ZP, Zhou HS: Asymptotically linear Schrödinger equation with potential vanishing at infinity. J. Differ. Equ. 2008, 245: 201-222. 10.1016/j.jde.2008.01.006MathSciNetView ArticleGoogle Scholar
  6. Liu ZL, Wang ZQ:Existence of a positive solution of an elliptic equation on R N . Proc. R. Soc. Edinb., Sect. A 2004, 134(1):191-200. 10.1017/S0308210500003152View ArticleGoogle Scholar
  7. Polidoro S, Ragusa MA: Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term. Rev. Mat. Iberoam. 2008, 24: 1011-1046. 10.4171/RMI/565MathSciNetView ArticleGoogle Scholar
  8. Stuart CA, Zhou HS:Applying the mountain pass theorem to an asymptotically linear elliptic equation on R N . Commun. Partial Differ. Equ. 1999, 24(9-10):1731-1758. 10.1080/03605309908821481MathSciNetView ArticleGoogle Scholar
  9. van Heerden FA, Francois A: Homoclinic solutions for a semilinear elliptic equation with an asymptotically linear nonlinearity. Calc. Var. Partial Differ. Equ. 2004, 20: 431-455. 10.1007/s00526-003-0242-9View ArticleGoogle Scholar
  10. van Heerden FA, Wang ZQ: Schrödinger type equations with asymptotically linear nonlinearities. Differ. Integral Equ. 2003, 16: 257-280.MathSciNetGoogle Scholar
  11. Zhou HS: Positive solution for a semilinear elliptic equation which is almost linear at infinity. Z. Angew. Math. Phys. 1998, 49: 896-906. 10.1007/s000330050128MathSciNetView ArticleGoogle Scholar
  12. Zhao FK, Zhao LG, Ding YH: Existence and multiplicity of solutions for a non-periodic Schrödinger equation. Nonlinear Anal. 2008, 69: 3671-3678. 10.1016/j.na.2007.10.024MathSciNetView ArticleGoogle Scholar
  13. Fang XD, Szulkin A: Multiple solutions for a quasilinear Schrödinger equation. J. Differ. Equ. 2013, 254: 2015-2032. 10.1016/j.jde.2012.11.017MathSciNetView ArticleGoogle Scholar
  14. Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.Google Scholar
  15. Szulkin A, Weth T: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 2009, 257: 3802-3822. 10.1016/j.jfa.2009.09.013MathSciNetView ArticleGoogle Scholar
  16. Szulkin A, Weth T: The method of Nehari manifold. In Handbook of Nonconvex Analysis and Applications. Edited by: Gao DY, Motreanu D. International Press, Boston; 2010:597-632.Google Scholar
  17. Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.View ArticleGoogle Scholar
  18. Struwe M: Variational Methods. Springer, Berlin; 1996.View ArticleGoogle Scholar

Copyright

© Fang and Han; licensee Springer 2014

This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.