Open Access

Least energy solutions for a quasilinear Schrödinger equation with potential well

Boundary Value Problems20132013:9

DOI: 10.1186/1687-2770-2013-9

Received: 23 October 2012

Accepted: 5 January 2013

Published: 21 January 2013

Abstract

In this paper, we consider the existence of least energy solutions for the following quasilinear Schrödinger equation:

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equa_HTML.gif

with a ( x ) 0 having a potential well, where N 3 and λ > 0 is a parameter. Under suitable hypotheses, we obtain the existence of a least energy solution u λ of ( E λ ) which localizes near the potential well int a 1 ( 0 ) for λ large enough by using the variational method and the concentration compactness method in an Orlicz space.

MSC:35J60, 35B33.

Keywords

quasilinear Schrödinger equation least energy solution Orlicz space concentration compactness method variational method

1 Introduction

Let us consider the following quasilinear Schrödinger equation:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equb_HTML.gif

for sufficiently large λ, where N 3 .

Our assumptions on a ( x ) are as follows:

( a 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq9_HTML.gif , the potential well Ω : = int a 1 ( 0 ) is a non-empty set and Ω ¯ = a 1 ( 0 ) ;

( a 2 ) There exists a constant M 0 > 0 such that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq14_HTML.gif , where μ denotes the Lebesgue measure on https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq15_HTML.gif .

Condition ( a 2 ) is very weak in dealing with the operator Δ + ( λ a ( x ) + 1 ) I on https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq15_HTML.gif , which was firstly used by Bartsch and Wang [1] in dealing with the semilinear Schrödinger equation.

Remark 1.1 Ω : = int a 1 ( 0 ) can be unbounded.

For f ( u ) , we assume that f is continuous and satisfies the following conditions:

( f 1 ) lim u 0 + f ( u ) u = 0 ;

( f 2 ) 0 f ( u ) C ( 1 + u p ) for u 0 , where C > 0 is a constant and 4 < p + 1 < 2 2 , where 2 = 2 N N 2 ;

( f 3 ) There is a number 4 < θ p + 1 such that for all u > 0 , we have f ( u ) u θ F ( u ) , where F ( u ) = 0 u f ( t ) d t .

Hypotheses ( a 1 ), ( a 2 ) and ( f 1 ), ( f 2 ), ( f 3 ) will be maintained throughout this paper.

Solutions of ( E λ ) are related to the existence of the standing wave solutions of the following quasilinear Schrödinger equation:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ1_HTML.gif
(1.1)

where V ( x ) is a given potential, k is a real constant and f, h are real functions. We would like to mention that (1.1) appears more naturally in mathematical physics and has been derived as models of several physical phenomena corresponding to various types of h. For instance, the case h ( s ) = s was used for the superfluid film equation in plasma physics by Kurihara [2] (see also [3]); in the case of h ( s ) = ( 1 + s ) 1 2 , (1.1) was used as a model of the self-changing of a high-power ultrashort laser in matter (see [47] and references therein).

In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ2_HTML.gif
(1.2)
For example, by using a constrained minimization argument, the existence of positive ground state solution was proved by Poppenberg, Schmitt and Wang [8]. Using a change of variables, Liu, Wang and Wang [9] used an Orlicz space to prove the existence of soliton solution of (1.2) via the mountain pass theorem. Colin and Jeanjean [10] also made use of a change of variables but worked in the Sobolev space https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq35_HTML.gif , they proved the existence of a positive solution for (1.2) from the classical results given by Berestycki and Lions [11]. By using the Nehari manifold method and the concentration compactness principle (see [12]) in the Orlicz space, Guo and Tang [13] considered the following equation:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ3_HTML.gif
(1.3)

with a ( x ) 0 having a potential well and 4 < p + 1 < 2 2 , where 2 = 2 N N 2 is the critical Sobolev exponent, and they proved the existence of a ground state solution of (1.3) which localizes near the potential well int a 1 ( 0 ) for λ large enough. In [14], Guo and Tang also considered ground state solutions of the corresponding quasilinear Schrödinger systems for (1.3) by the same methods and obtained similar results. For the stability and instability results for the special case of (1.2), one can also see the paper by Colin, Jeanjean and Squassina [15].

It is worth pointing out that the existence of one-bump or multi-bump bound state solutions for the related semilinear Schrödinger equation (1.3) for k = 0 has been extensively studied. One can see Bartsch and Wang [1], Ambrosetti, Badiale and Cingolani [16], Ambrosetti, Malchiodi and Secchi [17], Byeon and Wang [18], Cingolani and Lazzo [19], Cingolani and Nolasco [20], Del Pino and Felmer [21, 22], Floer and Weinstein [23], Oh [24, 25] and the references therein.

In this paper, based on the idea from Liu, Wang and Wang [9], we consider the more general equation ( E λ ), the existence of least energy solutions for equation ( E λ ) with a potential well int a 1 ( 0 ) for λ large is proved under the conditions ( a 1 ), ( a 2 ) and ( f 1 ), ( f 2 ), ( f 3 ).

The paper is organized as follows. In Section 2, we describe our main result (Theorem 2.1). In Section 3, we give some preliminaries that will be used for the proof of the main result. Finally, Theorem 2.1 will be proved in Section 4.

Throughout this paper, we use the same C to denote different universal constants.

2 Main result

Let V λ ( x ) = λ a ( x ) + 1 . Formally, we define the following functional:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ4_HTML.gif
(2.1)

for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq39_HTML.gif . Note that under our assumptions, the functional J λ is not well defined on X.

We follow the idea of [9] and make the following change of variable.

Let v = h ( u ) = 1 2 u 1 + u 2 + 1 2 ln ( u + 1 + u 2 ) , then d v = 1 + u 2 d u . Moreover, h ( u ) satisfies
h ( u ) { u , | u | 1 , 1 2 u | u | , | u | 1 .
Since h ( u ) = 1 + u 2 > 0 , h ( u ) is strictly monotone and hence has an inverse function denoted by u = g ( v ) . Obviously,
g ( v ) { v , | v | 1 , 2 | v | v , | v | 1 , g ( v ) = 1 1 + g 2 ( v ) .
Let G ( v ) = g 2 ( v ) . Then it holds that
G ( v ) = g 2 ( v ) { v 2 , | v | 1 , 2 | v | , | v | 1
and G ( v ) is convex. Moreover, there exists C 0 > 0 such that G ( 2 v ) C 0 G ( v ) ,
G ( v ) = 2 g ( v ) 1 + g 2 ( v ) , G ( v ) = 2 ( 1 + g 2 ( v ) ) 2 > 0 .
Now we introduce the Orlicz space (see [26])
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equg_HTML.gif
equipped with the norm
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equh_HTML.gif

Then E G λ is a Banach space.

Let
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equi_HTML.gif
equipped with the norm
v λ = v L 2 + | v | G λ .
Using the change of variable, we define the functional Φ λ on H G λ by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ5_HTML.gif
(2.2)

where v + = max { v , 0 } is the positive part of v.

Let
N λ : = { v H G λ { 0 } | Φ λ ( v ) , v = 0 }
be the Nehari manifold and let
c λ : = inf v N λ Φ λ ( v )

be the infimum of Φ λ on the Nehari manifold N λ , where Φ λ ( v ) is the Gateaux derivative (see Proposition 3.3).

We say that u λ = g ( v λ ) is a least energy solution of ( E λ ) if v λ N λ such that c λ is achieved.

Note that under our assumptions, for λ large enough, the following Dirichlet problem is a kind of a ‘limit’ problem:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equm_HTML.gif

where Ω : = int a 1 ( 0 ) .

Similar to the definition of the least energy solution of ( E λ ), we can define the least energy solution of (D) which will be given in Section 4.

Our main result is as follows.

Theorem 2.1 Assume that ( a 1 ), ( a 2 ) and ( f 1 ), ( f 2 ), ( f 3 ) are satisfied. Then for λ large, c λ is achieved by a critical point v λ of Φ λ such that u λ = g ( v λ ) is a least energy solution of ( E λ ). Furthermore, for any sequence λ n , { v λ n } has a subsequence converging to v such that u = g ( v ) is a least energy solution of (D).

3 Preliminaries

In order to obtain the compactness of the functional Φ λ , we recall the following Lemmas 3.1 and 3.2 which can be found in [13].

Lemma 3.1 There exist two constants C 1 > 0 , C 2 > 0 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ6_HTML.gif
(3.1)

for any v H G λ .

Lemma 3.2 The map: v g ( v ) from H G λ into https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif is continuous for 2 q 2 2 .

Now we consider the functional Φ λ defined on H G λ by (2.2), the following Proposition 3.3 is due to [9].

Proposition 3.3
  1. (i)

    Φ λ is well defined on H G λ ;

     
  2. (ii)

    Φ λ is continuous in H G λ ;

     
  3. (iii)
    Φ λ is Gateaux differentiable, the Gateaux derivative Φ λ ( v ) for v H G λ is a linear functional and Φ λ ( v ) is continuous in v in the strong-weak topology, that is, if v n v strongly in H G λ , then Φ λ ( v n ) Φ λ ( v ) weakly. Moreover, the Gateaux derivative Φ λ ( v ) has the form
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ7_HTML.gif
    (3.2)
     

Recall that { v n } H G λ is called a Palais-Smale sequence ((PS) c sequence in short) for Φ λ if Φ λ ( v n ) c and Φ λ ( v n ) 0 in ( H G λ ) , the dual space of H G λ . We say that the functional Φ λ satisfies the (PS) c condition if any of (PS) c sequence (up to a subsequence, if necessary) { v n } converges strongly in H G λ .

Lemma 3.4 Any of (PS) c sequence { v n } for Φ λ is bounded.

Proof Suppose that { v n } is a (PS) c sequence of Φ λ . We have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq76_HTML.gif and Φ λ ( v n ) 0 in the space ( H G λ ) .

Taking w n = g ( v n ) g ( v n ) , then | w n | = ( 1 + g 2 ( v n ) 1 + g 2 ( v n ) ) | v n | 2 | v n | , we have w n λ C v n λ , thus
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ8_HTML.gif
(3.3)
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ9_HTML.gif
(3.4)
Taking (3.4) 1 θ (3.3) yields
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equn_HTML.gif
Note that
1 2 1 θ ( 1 + g 2 ( v n ) 1 + g 2 ( v n ) ) > 1 2 2 θ = θ 4 2 θ , 1 2 1 θ = θ 2 2 θ , f ( u ) u > θ F ( u ) , θ > 4 ,
we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equp_HTML.gif
It follows from Lemma 3.1 that
C 1 min { v λ , v λ 2 } 2 θ θ 4 ( c + o ( 1 ) + o ( v n λ ) ) ,
(3.5)

thus { v n } is bounded in H G λ .

Let K λ = { v H G λ | Φ λ ( v ) = 0 } be the critical set of Φ λ . Suppose v K λ , then it is easy to check that either v > 0 or v 0 in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq15_HTML.gif by the definition of Φ λ and the strong maximum principle. □

Lemma 3.5 There exists 0 < σ < 1 which is independent of λ such that v λ v 1 > σ for all v K λ { 0 } and λ 1 .

Proof Assume that v λ 1 for any v K λ { 0 } (otherwise, the conclusion is true). From ( f 1 ), ( f 2 ), we see that for any ε > 0 , there is a constant C ε > 0 such that f ( x , u ) ε u + C ε u p for u > 0 . We have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equq_HTML.gif

and we can easily deduce the desired result. □

Lemma 3.6 There exists a positive constant c 0 > 0 such that
lim sup n v n λ max { 2 θ ( θ 4 ) C 1 c , 2 θ ( θ 4 ) C 1 c }

and either c c 0 or c = 0 if { v n } is a (PS) c sequence for Φ λ , where C 1 is the constant in Lemma  3.1.

Proof Since { v n } is a (PS) c sequence, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equs_HTML.gif
It follows from (3.5) that
lim sup n v n λ max { 2 θ ( θ 4 ) C 1 c , 2 θ ( θ 4 ) C 1 c } .
On the other hand, for v n λ 1 , we have
o ( v n λ ) = Φ λ ( v n ) , g ( v n ) g ( v n ) C 1 min { v n λ , v n λ 2 } C 2 ( max { v n λ , v n λ 2 } ) p + 1 2 = C 1 v n λ 2 C 2 v n λ p + 1 2 .
(3.6)
Thus, there exists σ 1 > 0 ( σ 1 < 1 ) such that
Φ λ ( v n ) , g ( v n ) g ( v n ) 1 4 C 1 v n λ 2 for  v n λ σ 1 .
(3.7)
Taking c 0 = σ 1 max { 2 θ ( θ 4 ) C 1 , 2 θ ( θ 4 ) C 1 } , then we have
max { 2 θ ( θ 4 ) C 1 c , 2 θ ( θ 4 ) C 1 c } < σ 1 < 1
if c < c 0 . It follows from (3.6) and (3.7) that
1 4 v n λ 2 o ( v n λ ) ,

hence, v n λ 0 and c = 0 . Therefore, we have proved that there exists a constant c 0 such that either c c 0 or c = 0 . □

Proposition 3.7 Let M > 0 be a constant. Then for any ε > 0 , there exist Λ ε > 0 , R ε > 0 such that
lim sup n B R ε c ( 1 2 f ( g ( ( v n ) + ) ) g ( ( v n ) + ) F ( g ( ( v n ) + ) ) ) d x ε

if { v n } is a (PS) c sequence of Φ λ with λ > Λ ε , c M , where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq109_HTML.gif .

Proof For all R > 0 , let
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equx_HTML.gif
We have
A ( R ) g 2 ( v n ) d x 1 λ M 0 + 1 A ( R ) ( λ a ( x ) + 1 ) g 2 ( v n ) d x 1 λ M 0 + 1 A ( R ) ( | v n | 2 + ( λ a ( x ) + 1 ) g 2 ( v n ) ) d x 1 λ M 0 + 1 ( 2 θ θ 4 c + o ( v n λ ) ) = 1 λ M 0 + 1 ( 2 θ θ 4 c + o ( 1 ) ) 0 ( λ ) .
(3.8)
On the other hand, by the Hölder inequality and interpolation inequality, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ14_HTML.gif
(3.9)
By using the Gagliardo-Nirenberg inequality, we obtain
B R ε c ( 1 2 f ( g ( ( v n ) + ) ) g ( ( v n ) + ) F ( g ( ( v n ) + ) ) ) d x B R ε c ( 1 2 f ( g ( ( v n ) + ) ) g ( ( v n ) + ) C ( g ( ( v n ) + ) ) θ ) d x ( by  ( f 3 ) ) C B R c | g ( v n ) | p + 1 d x C ( B R c | g ( v n ) | 2 d x ) ( p + 1 ) β 2 ( B R c g 2 ( v n ) d x ) ( p + 1 ) ( 1 β ) 2 ( β = N ( p 1 ) 2 ( p + 1 ) ) C v n λ ( p + 1 ) β ( A ( R ) g 2 ( v n ) d x + B ( R ) g 2 ( v n ) d x ) ( p + 1 ) ( 1 β ) 2 C ( A ( R ) g 2 ( v n ) d x + B ( R ) g 2 ( v n ) d x ) ( p + 1 ) ( 1 β ) 2 ( v n λ  is bounded ) .

Let λ and R be large enough, from (3.8) and (3.9), we get the desired result. □

Lemma 3.8 c λ = inf N λ Φ λ ( v ) is achieved by some v > 0 .

Proof By the definition of c λ and the Ekeland variational principle, there exists a (PS) c sequence { v n } , by Lemma 3.4, we know that { v n } is bounded. Hence (up to a subsequence) we have v n v in L 2 , v n v in L 2 , v n v a.e. in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq15_HTML.gif , g ( v n ) g ( v ) in L q for 2 q 2 2 .

It is sufficient to prove that v 0 and v N λ . In fact,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ15_HTML.gif
(3.10)
it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equz_HTML.gif
Let ε 0 = 1 4 c , since g ( v n ) g ( v ) strongly in L p + 1 ( B R ) for R > 0 , by Proposition 3.7, there exist Λ 0 > 0 , R 0 > 0 such that for λ Λ 0 , R R 0 ,
B R c | ( 1 2 f ( g ( ( v n ) + ) ) g ( ( v n ) + ) F ( g ( ( v n ) + ) ) ) ( 1 2 f ( g ( v + ) ) g ( v + ) F ( g ( v + ) ) ) | d x < ε 0 ,
thus
B R ( 1 2 f ( g ( v + ) ) g ( v + ) F ( g ( v + ) ) ) d x 1 2 c > 0 .

Hence v 0 .

Now we prove v N λ . Indeed, since { v n } is a (PS) c sequence, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ16_HTML.gif
(3.11)

where o ( v n λ ) 0 as n .

Let l n : = g ( v n ) 1 + g 2 ( v n ) , then { l n } is bounded in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif for 2 q 2 2 , by the continuity of g, we have, up to a subsequence, l n l = g ( v ) 1 + g 2 ( v ) in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif .

Similarly, we have s n : = f ( g ( v n ) ) 1 + g 2 ( v n ) f ( g ( v n ) ) is bounded in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq135_HTML.gif . Again, by the continuity of g, we have s n s = f ( g ( v ) ) 1 + g 2 ( v ) in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq135_HTML.gif . Passing to the limits in (3.11), we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equac_HTML.gif

which is equivalent to Φ λ ( v ) , v = 0 , that is, v N λ . □

4 Proof of the main result

Consider the following quasilinear Schrödinger equation in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq138_HTML.gif ( N 3 ):
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equad_HTML.gif
We have the same change of variables and the same notation as in the previous sections. Define the corresponding Orlicz space E G ( Ω ) by
E G ( Ω ) = { v | Ω g 2 ( v ) d x < + }
with the norm
| v | G ( Ω ) : = inf ξ > 0 ξ ( 1 + Ω G ( ξ 1 v ) d x ) .
The space H G ( Ω ) is defined by
H G ( Ω ) : = { v | Ω | v | 2 d x < + , Ω g 2 ( v ) d x < + }
with the norm
v Ω = v L 2 + | v | G ( Ω ) .

The following Lemma 4.1 is a counterpart of Lemma 3.1.

Lemma 4.1 There exist two constants C 1 > 0 , C 2 > 0 such that
C 1 min { v Ω , v Ω 2 } Ω | v | 2 d x + Ω g 2 ( v ) d x C 2 max { v Ω , v Ω 2 }

for any v H G ( Ω ) .

We denote by H G 0 ( Ω ) the closure of C 0 ( Ω ) in H G ( Ω ) . We define the functional Φ Ω on H G 0 ( Ω ) by
Φ Ω ( v ) = 1 2 Ω ( | v | 2 + g 2 ( v ) ) d x Ω F ( g ( v + ) ) d x ,
(4.1)
and we define the Nehari manifold N Ω by
N Ω : = { v H G ( Ω ) { 0 } | Φ Ω ( v ) , v = 0 } .
Let
c ( Ω ) = inf N Ω Φ Ω ( v ) .

We recall that u = g ( v ) is a least energy solution of (D) if v N Ω such that c ( Ω ) is achieved.

Lemma 4.2 Suppose c λ = inf N λ Φ λ . Then lim λ + c λ = c ( Ω ) .

Proof It is easy to see that c λ c ( Ω ) for λ 1 . We claim that c λ is monotone increasing with respect to λ. In fact, for λ 1 < λ 2 , we assume that c λ 1 , c λ 2 are achieved for v λ 1 > 0 , v λ 2 > 0 . Obviously,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ18_HTML.gif
(4.2)
We first prove that there exists 0 < α < 1 such that α v λ 2 N λ 1 . This is sufficient to prove that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equal_HTML.gif
That is,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equam_HTML.gif
Let
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equan_HTML.gif
Then by ( f 1 ), we can obtain lim α 0 I ( α ) 0 and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equao_HTML.gif
Hence, there exists 0 < α 0 < 1 such that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq160_HTML.gif , i.e., α 0 v λ 2 N λ 1 . Thus
c λ 1 Φ λ 1 ( α 0 v λ 2 ) .
In the following, we will prove that
Φ λ 1 ( α 0 v λ 2 ) Φ λ 2 ( v λ 2 ) = c λ 2 .
In fact, we consider the function ρ ( α ) defined by
ρ ( α ) = ( f ( g ( α v λ 2 ) ) ( λ 2 a + 1 ) g ( α v λ 2 ) ) v λ 2 α 1 + g 2 ( α v λ 2 ) .
By g ( t ) 1 + g 2 ( t ) 2 t for t 0 , we have g ( α v λ 2 ) g ( α v λ 2 ) = g ( α v λ 2 ) 1 + g 2 ( α v λ 2 ) 2 α v λ 2 ( θ 2 ) α v λ 2 . It follows that
α v λ 2 g ( α v λ 2 ) g ( α v λ 2 ) / g ( α v λ 2 ) + α v λ 2 α v λ 2 g ( α v λ 2 ) ( θ 2 ) α v λ 2 + α v λ 2 .
Obviously,
f ( g ( α v λ 2 ) ) g ( α v λ 2 ) v λ 2 2 α g ( α v λ 2 ) f ( g ( α v λ 2 ) ) v λ 2 ( g ( α v λ 2 ) + α g ( α v λ 2 ) v λ 2 ) 0
and hence it is easy to check that
d d α ( f ( g ( α v λ 2 ) ) v λ 2 α 1 + g 2 ( α v λ 2 ) ) = d d α ( g ( α v λ 2 ) 1 + g 2 ( α v λ 2 ) ) f ( g ( α v λ 2 ) ) v λ 2 α g ( α v λ 2 ) + d d α ( f ( g ( α v λ 2 ) ) v λ 2 α g ( α v λ 2 ) ) g ( α v λ 2 ) 1 + g 2 ( α v λ 2 ) = v λ 2 2 ( 1 + g 2 ( α v λ 2 ) ) 2 f ( g ( α v λ 2 ) ) v λ 2 α g ( α v λ 2 ) + f ( g ( α v λ 2 ) ) g ( α v λ 2 ) v λ 2 2 α g ( α v λ 2 ) f ( g ( α v λ 2 ) ) v λ 2 ( g ( α v λ 2 ) + α g ( α v λ 2 ) v λ 2 ) α 2 g 2 ( α v λ 2 ) × g ( α v λ 2 ) 1 + g 2 ( α v λ 2 ) f ( g ( α v λ 2 ) ) g ( α v λ 2 ) v λ 2 2 α g ( α v λ 2 ) f ( g ( α v λ 2 ) ) v λ 2 ( g ( α v λ 2 ) + α g ( α v λ 2 ) v λ 2 ) α 2 g ( α v λ 2 ) 1 + g 2 ( α v λ 2 ) 0 .
On the other hand,
d d α ( ( λ 2 a + 1 ) g ( α v λ 2 ) v λ 2 α 1 + g 2 ( α v λ 2 ) ) = ( λ 2 a + 1 ) α v λ 2 2 g ( α v λ 2 ) 1 + g 2 ( α v λ 2 ) ( 1 + g 2 ( α v λ 2 ) ) α 2 ( 1 + g 2 ( α v λ 2 ) ) 2 v λ 2 ,
by v = 1 2 g ( v ) 1 + g 2 ( v ) + 1 2 ln ( g ( v ) + 1 + g 2 ( v ) ) , it is easy to check that for any t 0 ,
t g ( t ) 1 + g 2 ( t ) ( 1 + g 2 ( t ) ) 0 ,
which implies
d d α ( ( λ 2 a + 1 ) g ( α v λ 2 ) v λ 2 α 1 + g 2 ( α v λ 2 ) ) 0

for any α > 0 , thus we have proved that ρ ( α ) is monotone increasing for α > 0 .

Now we consider the function γ ( α ) defined by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equay_HTML.gif
Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equaz_HTML.gif
for 0 < α < 1 . Therefore, γ ( α ) is monotone increasing with respect to α < 1 . Thus, we deduce that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equba_HTML.gif

Assume that lim λ c λ = k c ( Ω ) . If k < c ( Ω ) , then for any sequence { λ n } ( λ n + ), we have c λ n k < c ( Ω ) .

We assume that v n is such that c λ n is achieved, by Lemma 3.4, { v n } is bounded in H G λ n . Since v n H G 0 v n H G λ n , { v n } is bounded in H G 0 , as a result, we have v n v in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq181_HTML.gif , g ( v n ) g ( v ) in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq182_HTML.gif for 2 q 2 2 , v n v in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif for 2 q 2 2 , v n v a.e. in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq15_HTML.gif .

We claim that v | Ω c = 0 , where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq184_HTML.gif . Indeed, it is sufficient to prove g ( v ) | Ω c = 0 . If not, then there exists a compact subset Σ Ω c with dist { Σ , Ω } > 0 such that g ( v ) | Σ 0 and
Σ g 2 ( v n ) d x Σ g 2 ( v ) d x > 0 .

Moreover, there exists ε 0 > 0 such that a ( x ) ε 0 for any x Σ .

By the choice of v n , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbc_HTML.gif
hence,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbd_HTML.gif

This contradiction shows that g ( v ) | Ω c = 0 and so does v.

Now we show that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ19_HTML.gif
(4.3)
Suppose that (4.3) is not true, then by the concentration compactness principle of Lions (see [12]), there exist δ > 0 , ϱ > 0 and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq194_HTML.gif with | x n | + such that
lim sup n B ϱ ( x n ) | g ( v n ) g ( v ) | 2 δ > 0 .
On the other hand, by the choice of { v n } , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbf_HTML.gif

which shows that g ( v n ) g ( v ) in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif for 2 < q < 2 . In the above proof, we have used the fact that μ { B ϱ ( x n ) { x | a ( x ) M 0 } } 0 as n and the L 2 bounded property of g ( v n ) .

Now, since { v n } is bounded, by the Fatou lemma, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbg_HTML.gif
But, by the choice of v n , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbh_HTML.gif
hence,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ20_HTML.gif
(4.4)
In the following, we will prove that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbi_HTML.gif
Indeed,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbj_HTML.gif
Since f ( g ( v + ) ) g ( ( v n ) + ) v n f ( g ( v + ) ) g ( v + ) v , one can easily see that I 2 0 as n , and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbk_HTML.gif
by using g ( v n ) g ( v ) in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif for 2 < q < 2 . It follows from (4.4) that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbl_HTML.gif
thus, there exists 0 < α 0 1 such that α 0 v N Ω and
Φ Ω ( α 0 v ) Φ Ω ( v ) ,

hence c ( Ω ) Φ Ω ( α 0 v ) < Φ Ω ( v ) lim n Φ λ n = k < c ( Ω ) . A contradiction. Thus we have proved that lim λ c λ c ( Ω ) as λ + . □

Proof of Theorem 2.1 Suppose that { v n } is a sequence such that v n N λ , Φ λ n ( v n ) = c λ n , by the proof of Lemma 3.2, we have v n v in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq181_HTML.gif , g ( v n ) g ( v ) in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif for 2 < q < 2 and v | Ω c = 0 . Moreover, Φ Ω ( v ) lim n Φ λ n ( v n ) c ( Ω ) , and if v N Ω , then Φ Ω ( v ) = c ( Ω ) . Hence, in the following, we need only to prove that v N Ω . To do this, it is sufficient to prove that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbn_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbo_HTML.gif
In fact, if one of the above three limits does not hold, by the Fatou lemma, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbp_HTML.gif

Similar to above, there exists α 0 ( 0 , 1 ) such that α 0 v N Ω and c ( Ω ) Φ Ω ( α 0 v ) < Φ Ω ( v ) lim n Φ λ n ( v n ) c ( Ω ) . A contradiction, and thus we complete the proof of Theorem 2.1. □

Declarations

Acknowledgements

The author would like to thank the referee for some valuable comments and helpful suggestions. This study was supported by the National Natural Science Foundation of China (11161041, 31260098) and the Fundamental Research Funds for the Central Universities (zyz2012080, zyz2012074).

Authors’ Affiliations

(1)
College of Mathematics and Computer Science, Northwest University for Nationalities
(2)
College of Mathematics and Statistics, Northwest Normal University

References

  1. Bartsch T, Wang Z: Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys. 2000, 51: 366-384.MathSciNetView ArticleGoogle Scholar
  2. Kurihura S: Large-amplitude quasi-solitons in superfluid. J. Phys. Soc. Jpn. 1981, 50: 3262-3267. 10.1143/JPSJ.50.3262View ArticleGoogle Scholar
  3. Laedke EW, Spatschek KH, Stenflo L: Evolution theorem for a class of perturbed envelop soliton solutions. J. Math. Phys. 1983, 24: 2764-2769. 10.1063/1.525675MathSciNetView ArticleGoogle Scholar
  4. Brandi HS, Manus C, Mainfray G, Lehner T, Bonnaud G: Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Phys. Fluids B 1993, 5: 3539-3550. 10.1063/1.860828View ArticleGoogle Scholar
  5. Chen XL, Sudan RN: Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse. Phys. Rev. Lett. 1993, 70: 2082-2085. 10.1103/PhysRevLett.70.2082View ArticleGoogle Scholar
  6. De Bouard A, Hayashi N, Saut JC: Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Commun. Math. Phys. 1997, 189: 73-105. 10.1007/s002200050191MathSciNetView ArticleGoogle Scholar
  7. Ritchie B: Relativistic self-focusing channel formation in laser-plasma interactions. Phys. Rev. E 1994, 50: 687-689. 10.1103/PhysRevE.50.R687View ArticleGoogle Scholar
  8. Poppenberg M, Schmitt K, Wang Z: On the existence of solutions to quasilinear Schrödinger equation. Calc. Var. Partial Differ. Equ. 2002, 14: 329-344. 10.1007/s005260100105MathSciNetView ArticleGoogle Scholar
  9. Liu J, Wang Y, Wang Z: Soliton solutions for quasilinear Schrödinger equation II. J. Differ. Equ. 2003, 187: 473-493. 10.1016/S0022-0396(02)00064-5View ArticleGoogle Scholar
  10. Colin M, Jeanjean L: Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal. 2004, 56: 213-226. 10.1016/j.na.2003.09.008MathSciNetView ArticleGoogle Scholar
  11. Berestycki H, Lions PL: Nonlinear scalar field equations I. Arch. Ration. Mech. Anal. 1983, 82: 313-346.MathSciNetGoogle Scholar
  12. Lions PL: The concentration-compactness principle in the calculus of variations. The locally compact case. Part I. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1984, 1: 109-145.Google Scholar
  13. Guo Y, Tang Z: Ground state solutions for the quasilinear Schrödinger equation. Nonlinear Anal. 2012, 75: 3235-3248. 10.1016/j.na.2011.12.024MathSciNetView ArticleGoogle Scholar
  14. Guo Y, Tang Z: Ground state solutions for the quasilinear Schrödinger systems. J. Math. Anal. Appl. 2012, 389: 322-339. 10.1016/j.jmaa.2011.11.064MathSciNetView ArticleGoogle Scholar
  15. Colin M, Jeanjean L, Squassina M: Stability and instability results for standing waves of quasi-linear Schrödinger equations. Nonlinearity 2010, 23: 1353-1385. 10.1088/0951-7715/23/6/006MathSciNetView ArticleGoogle Scholar
  16. Ambrosetti A, Badiale M, Cingolani S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 1997, 140: 285-300. 10.1007/s002050050067MathSciNetView ArticleGoogle Scholar
  17. Ambrosetti A, Malchiodi A, Secchi S: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 2001, 159: 253-271. 10.1007/s002050100152MathSciNetView ArticleGoogle Scholar
  18. Byeon J, Wang Z: Standing waves with a critical frequency for nonlinear Schrödinger equations II. Calc. Var. Partial Differ. Equ. 2003, 18: 207-219. 10.1007/s00526-002-0191-8MathSciNetView ArticleGoogle Scholar
  19. Cingolani S, Lazzo M: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ. 2000, 160: 118-138. 10.1006/jdeq.1999.3662MathSciNetView ArticleGoogle Scholar
  20. Cingolani S, Nolasco M: Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations. Proc. R. Soc. Edinb. 1998, 128: 1249-1260. 10.1017/S030821050002730XMathSciNetView ArticleGoogle Scholar
  21. Del Pino M, Felmer P: Semi-classical states for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré 1998, 15: 127-149.MathSciNetView ArticleGoogle Scholar
  22. Del Pino M, Felmer P: Multi-peak bound states for nonlinear Schrödinger equations. J. Funct. Anal. 1997, 149: 245-265. 10.1006/jfan.1996.3085MathSciNetView ArticleGoogle Scholar
  23. Floer A, Weinstein A: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 1986, 69: 397-408. 10.1016/0022-1236(86)90096-0MathSciNetView ArticleGoogle Scholar
  24. Oh YG: On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 1990, 131: 223-253. 10.1007/BF02161413View ArticleGoogle Scholar
  25. Oh YG:Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class ( V ) a . Commun. Partial Differ. Equ. 1988, 13: 1499-1519. 10.1080/03605308808820585View ArticleGoogle Scholar
  26. Rao M, Ren Z: Theory of Orlicz Space. Dekker, New York; 1991.Google Scholar

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© Jiao; licensee Springer. 2013

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