Open Access

Multiple solutions to nonlinear Schrödinger equations with critical growth

Boundary Value Problems20132013:199

DOI: 10.1186/1687-2770-2013-199

Received: 20 March 2013

Accepted: 24 May 2013

Published: 4 September 2013

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 c 0 < c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq1_HTML.gif, 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:
i ħ ψ t = ħ 2 2 m Δ ψ + W ( x ) ψ g ( x , ψ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ1_HTML.gif
(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, m > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq2_HTML.gif denotes the mass of the particle, W : R N R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq3_HTML.gif 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 ψ ( x , t ) = u ( x ) e i E t h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq4_HTML.gif. It is clear that ψ ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq5_HTML.gif solves (1.1) if and only if u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq6_HTML.gif solves the following stationary equation:
ħ 2 2 m Δ u + ( W ( x ) E ) u = g ( x , u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ2_HTML.gif
(1.2)
For simplicity and without loss of generality, we set ε = ħ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq7_HTML.gif, V ( x ) = 2 m ( W ( x ) E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq8_HTML.gif and g ˜ = 2 m g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq9_HTML.gif, then equation (1.2) is equivalent to
ε 2 Δ u + V ( x ) u = g ˜ ( x , u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ3_HTML.gif
(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 R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq10_HTML.gif, please see [19]etc. For multiple interior spikes, please see [10, 11]etc. For single boundary spike solutions with Neumann boundary condition, please see [6, 1215]etc. For multiple boundary spikes, please see [1618]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 c 0 < c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq1_HTML.gif, which is a key condition in [19].

In the sequel, we restrict ourselves to the critical case in which g ˜ ( x , u ) = | u | 2 2 u + f ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq11_HTML.gif. More specifically, we study the following problem:
{ ε 2 Δ u + V ( x ) u = | u | 2 2 u + f ( x , u ) in  R N , u > 0 in  R N , lim | x | u ( x ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ4_HTML.gif
(1.4)

where 2 : = 2 N / ( N 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq12_HTML.gif if N 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq13_HTML.gif, and 2 : = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq14_HTML.gif if N = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq15_HTML.gif. f C ( R N × R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq16_HTML.gif satisfies

(f1) f ( x , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq17_HTML.gif for each t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq18_HTML.gif;

(f2) lim t 0 + f ( x , t ) t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq19_HTML.gif;

(f3) there exists q ( 2 , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq20_HTML.gif such that
lim sup t f ( x , t ) t q 1 < ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equa_HTML.gif
(f4) there exists 2 < θ < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq21_HTML.gif such that
0 < θ F ( x , t ) f ( x , t ) t , t > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equb_HTML.gif

where F ( t ) : = 0 t f ( τ ) d τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq22_HTML.gif;

(f5) the function f ( x , t ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq23_HTML.gif is strictly increasing in t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq24_HTML.gif for any x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq25_HTML.gif.

Our main results are the following theorem.

Theorem 1.1 Let N 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq26_HTML.gif. Suppose that f satisfies (f1)-(f5), V is a continuous function in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq10_HTML.gif and satisfies inf x R N V ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq27_HTML.gif. Then when ε is sufficiently small, the problem (1.4) has at least cat ( Σ , Σ δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq28_HTML.gif distinct nontrivial solutions.

Here cat ( Σ , Σ δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq28_HTML.gif denotes the Ljusternik-Schnirelmann category of Σ in Σ δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq29_HTML.gif. By definition (e.g., [20]), the category of A with respect to M, denoted by cat ( A , M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq30_HTML.gif, is the least integer k such that A A 1 A k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq31_HTML.gif, with A i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq32_HTML.gif ( i = 1 , , k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq33_HTML.gif) closed and contractible in M. We set cat ( , M ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq34_HTML.gif and cat ( A , M ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq35_HTML.gif if there are no integers with the above property. We will use the notation cat ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq36_HTML.gif for cat ( M , M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq37_HTML.gif.

To prove Theorem 1.1, we mainly use the idea of [15, 19, 21]. More precisely, we can show that the ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq38_HTML.gif-condition holds in the subset N ˜ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq39_HTML.gif (see (4.6)). Hence the standard Ljusternik-Schnirelmann category theory can be applied in N ˜ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq39_HTML.gif to yield the existence of at least cat ( N ˜ ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq40_HTML.gif critical points of I ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq41_HTML.gif. And then we construct two continuous mappings
ϕ ε : Σ N ˜ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ5_HTML.gif
(1.5)
and
β : N ˜ ε Σ δ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ6_HTML.gif
(1.6)
where
Σ δ = { x R N : dist ( x , Σ ) δ } , δ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ7_HTML.gif
(1.7)
Then a topological argument asserts that
cat ( N ˜ ε ) 2 cat ( Σ , Σ δ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equc_HTML.gif

We will also prove that if u is a critical point of I ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq41_HTML.gif satisfying I ε ( u ) ε N ( c 0 + h ( ε ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq42_HTML.gif, then u cannot change sign. Hence we obtain at least cat ( Σ , Σ δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq43_HTML.gif nontrivial critical points of I ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq41_HTML.gif.

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

H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif is the usual Sobolev space of real-valued functions defined by
H 1 ( R N ) : = { u : u L 2 ( R N )  and  u L 2 ( R N ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equd_HTML.gif
with the normal
u 2 : = R N ( | u | 2 + V ( a ) u 2 ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Eque_HTML.gif
Let H ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq45_HTML.gif be the subspace of a Hilbert space H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif with respect to the norm
u ε 2 : = R N ( ε 2 | u | 2 + V ( x ) u 2 ) d x < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equf_HTML.gif
We denote by S the Sobolev constant for the embedding D 1 , 2 ( R N ) L 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq46_HTML.gif, namely
S = inf 0 u D 1 , 2 R N | u | 2 d x ( R N | u | 2 d x ) 2 / 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ8_HTML.gif
(2.1)
where D 1 , 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq47_HTML.gif is the usual Sobolev space of real-valued functions defined by
D 1 , 2 ( R N ) : = { u : u L 2 ( R N )  and  u L 2 ( R N ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equg_HTML.gif
We say that a function u H ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq48_HTML.gif is a weak solution of the problem (1.4) if
R N ( ε 2 u v + V ( x ) u v | u | 2 2 u v f ( x , u ) v ) d x = 0 , v H ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equh_HTML.gif
In view of (f2) and (f3), we have that the associated functional I ε : H ε R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq49_HTML.gif given by
I ε ( u ) = 1 2 R N ( ε 2 | u | 2 + V ( x ) | u | 2 ) d x 1 2 R N | u | 2 d x R N F ( x , u ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equi_HTML.gif
is well defined. Moreover, I ε C 1 ( H ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq50_HTML.gif with the following derivative:
I ε ( u ) , v = R N ( ε 2 u v + V ( x ) u v | u | 2 2 u v f ( x , u ) v ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equj_HTML.gif

Hence, the weak solutions of (1.4) are exactly the critical points of I ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq41_HTML.gif.

Let us recall some known facts about the limiting problem, namely the problem
Δ u + V ( a ) u = | u | 2 2 u + f ( a , u ) in  R N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ9_HTML.gif
(2.2)
here a R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq51_HTML.gif acts as a parameter instead of an independent variable. Solutions of (2.2) will be sought in the Sobolev space H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif as critical points of the functional
J a ( u ) = 1 2 R N ( | u | 2 + V ( a ) | u | 2 ) d x 1 2 R N | u | 2 d x R N F ( a , u ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equk_HTML.gif
The least positive critical value G ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq52_HTML.gif can be characterized as
G ( a ) : = inf u M a J a ( u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equl_HTML.gif
where
M a : = { v H 1 ( R N ) { 0 } : J a ( u ) , u = 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ10_HTML.gif
(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
J a ( w ) = inf u M a J a ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equm_HTML.gif
Moreover, there exist C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq53_HTML.gif and δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq54_HTML.gif such that
w ( x ) e δ | x | , w ( x ) e δ | x | for all  x R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ11_HTML.gif
(2.4)

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

Set
c 0 : = inf a R N G ( a ) , Σ : = { a R N : G ( a ) = c 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equn_HTML.gif
For any δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq54_HTML.gif, we denote Σ δ = { x R N : dist ( x , Σ ) δ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq55_HTML.gif. We need to estimate the super bound of c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq56_HTML.gif. In order to do this, we estimate G ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq52_HTML.gif. We shall use a family of radial function defined by
U ε ( x ) = ( N ( N 2 ) ) N 2 4 ( ε ε 2 + | x | 2 ) N 2 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equo_HTML.gif
It is known [20] that
Δ U ε = U ε ( N + 2 ) / ( N 2 ) in  R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equp_HTML.gif
Moreover, we have
R N | U ε | 2 d x = R N | U ε | 2 d x = S N / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equq_HTML.gif
Set u ε ( x ) = ϕ ( x ) U ε ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq57_HTML.gif, where ϕ C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq58_HTML.gif is a cut-off function satisfying ϕ ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq59_HTML.gif if | x | δ / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq60_HTML.gif, ϕ ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq61_HTML.gif if | x | δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq62_HTML.gif and 0 ϕ ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq63_HTML.gif. After a detailed calculation, we have the following estimates:
R N | ( ϕ U ε ) | 2 d x = S N / 2 + O ( ε N 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ12_HTML.gif
(2.5)
R N | ϕ U ε | 2 d x = S N / 2 + O ( ε N ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ13_HTML.gif
(2.6)
R N | ϕ U ε | 2 d x = α ( ε ) : = { C ε 2 | log ε | + O ( ε 2 ) if  N = 4 , C ε 2 + O ( ε N 2 ) if  N 5 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ14_HTML.gif
(2.7)
Since F ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq64_HTML.gif, from (2.5)-(2.7), we conclude
G ( a ) max t > 0 J a ( t u ε ) max t > 0 ( t 2 2 R N ( | u ε | 2 + V ( a ) | u ε | 2 ) d x t 2 2 R N | u ε | 2 d x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ15_HTML.gif
(2.8)
then the maximum value of the right-hand side is achieved at
τ = ( R N ( | u ε | 2 + V ( a ) | u ε | 2 ) d x R N | u ε | 2 d x ) N 2 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ16_HTML.gif
(2.9)
and
max t > 0 J a ( t u ε ) 1 N ( R N ( | u ε | 2 + V ( a ) | u ε | 2 ) d x ) N / 2 ( R N | u ε | 2 d x ) ( N 2 ) / 2 = 1 N ( R N ( | u ε | 2 + V ( a ) | u ε | 2 ) d x ( R N | u ε | 2 d x ) 2 / 2 ) N / 2 1 N ( S N / 2 + V ( a ) α ( ε ) + O ( ε N 2 ) S ( N 2 ) / 2 + O ( ε N 2 ) ) N / 2 < 1 N S N / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ17_HTML.gif
(2.10)
Hence we have
c 0 < 1 N S N / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ18_HTML.gif
(2.11)
We denote the Nehari manifold of I ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq41_HTML.gif by
N ε = { u H ε , A { 0 } : I ε ( u ) , u = 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equr_HTML.gif

3 Compactness result

Proposition 3.1 Let ε n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq65_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif. Assume that ( u n ) N ε n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq67_HTML.gif satisfies ε n N I ε n ( u n ) c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq68_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq69_HTML.gif. Then uniformly in a Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq70_HTML.gif, there exist a subsequence of v n ( y ) : = u n ( ε n y + a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq71_HTML.gif (still denoted by v n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq72_HTML.gif), and t n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq73_HTML.gif such that w n : = t n v n M a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq74_HTML.gif. Furthermore, w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq75_HTML.gif converges strongly in H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif to w, the positive ground state solution of equation (2.2).

Proof Let ( u n ) N ε n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq76_HTML.gif be such that ε n N I ε n ( u n ) c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq77_HTML.gif. Then, by a change of variable x = ε n y + a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq78_HTML.gif, we have
c 0 + 1 ε n N ( I ε n ( u n ) 1 θ I ε n ( u n ) , u n ) ε n N ( 1 2 1 θ ) u n ε n 2 = ( 1 2 1 θ ) R N ( | u n ( ε n y + a ) | 2 + V ( ε n y + a ) | u n ( ε n y + a ) | 2 ) d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ19_HTML.gif
(3.1)
This implies that ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq79_HTML.gif is bounded in H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif. Noting that
c 0 + o ( 1 ) = 1 2 R N ( | u n + V ( ε n y + a ) | u n | 2 ) d y 1 2 R N | u n | 2 d y R N F ( x , u n ) d y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ20_HTML.gif
(3.2)
hence
m : = lim n R N ( | u n + V ( ε n y + a ) | u n | 2 ) d y > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ21_HTML.gif
(3.3)
since c 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq80_HTML.gif. Now we prove that there exists a sequence ( z n ) R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq81_HTML.gif and constants R , γ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq82_HTML.gif such that
lim inf n B R ( z n ) | u n | 2 d y γ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ22_HTML.gif
(3.4)
Indeed, if this is not true, then the boundedness of ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq79_HTML.gif in H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif and a lemma due to Lions [[24], Lemma I.1] imply that u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq83_HTML.gif in L s ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq84_HTML.gif for all 2 < s < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq85_HTML.gif. Given δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq54_HTML.gif, we can use (f2), (f3) and u n N ε n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq86_HTML.gif to get
R N f ( x , u n ) u n d y δ R N | u n | 2 d y + C δ R N | u n | q d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equs_HTML.gif
Moreover,
R N ( | u n + V ( ε n y + a ) | u n | 2 ) d y = R N | u n | 2 d y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equt_HTML.gif
as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif. Therefore
m = lim n R N | u n | 2 d y > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ23_HTML.gif
(3.5)
and consequently (3.2) yields
c 0 = m 2 m 2 = m N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equu_HTML.gif
i.e.,
m = N c 0 < S N / 2 (see (2.11)) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ24_HTML.gif
(3.6)
However, recall the definition of S in (2.1),
m = lim n R N ( | u n + V ( ε n y + a ) | u n | 2 ) d y S lim n ( R N | u n | 2 d y ) 2 / 2 = S m 2 / 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equv_HTML.gif
equivalent to m S N / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq87_HTML.gif, contradicting (3.6). Thus, (3.4) holds. Using the idea of [21, 25], along a subsequence as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif, we may assume that
v n ( y ) : = u n ( ε n y + a ) v 0 weakly in  H 1 ( R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equw_HTML.gif
We now consider t n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq73_HTML.gif such that w n : = t n v n M a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq74_HTML.gif (see (2.3)). By a change of variable x = ε n y + a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq78_HTML.gif, it follows that
c 0 J a ( w n ) = J a ( t n v n ) sup t > 0 J a ( t v n ) sup t > 0 ε n N I ε n ( t u n ) + o ( 1 ) = ε n N I ε n ( u n ) + o ( 1 ) = c 0 + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ25_HTML.gif
(3.7)

Hence J a ( w n ) c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq88_HTML.gif, from which it follows that w n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq89_HTML.gif in H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif.

Since ( v n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq90_HTML.gif and ( w n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq91_HTML.gif are bounded in H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif and v n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq92_HTML.gif in H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif, the sequence ( t n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq93_HTML.gif is bounded. Thus, up to a subsequence, t n t 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq94_HTML.gif. If t 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq95_HTML.gif, then w n H 1 ( R N ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq96_HTML.gif, which does not occur. Hence t 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq97_HTML.gif, and therefore the sequence ( w n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq91_HTML.gif satisfies
J a ( w n ) c 0 , w n w : = t 0 v 0 weakly in  H 1 ( R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ26_HTML.gif
(3.8)
For fixed v H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq98_HTML.gif, define
b ( w ) = R N w v d y for all  w H 1 ( R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equx_HTML.gif
By the Hölder inequality,
| b ( w ) | | w | 2 | v | 2 w H 1 ( R N ) v H 1 ( R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equy_HTML.gif
Hence b H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq99_HTML.gif, the dual space of H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif. Consequently, as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif, w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq100_HTML.gif in H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif implies b ( w n ) b ( w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq101_HTML.gif, i.e.,
R N w n v d x = R N w v d x + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ27_HTML.gif
(3.9)
Since w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq75_HTML.gif converges weakly to w in H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif, w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq75_HTML.gif is bounded in L 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq102_HTML.gif. Thus | w n | 2 2 w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq103_HTML.gif is bounded in L ( 2 ) ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq104_HTML.gif. It then follows that there is a subsequence of ( w n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq91_HTML.gif, still denoted by ( w n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq91_HTML.gif, such that | w n | 2 2 w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq103_HTML.gif converges weakly to some w ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq105_HTML.gif in L ( 2 ) ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq104_HTML.gif. Next we will show w ˜ = | w | 2 2 w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq106_HTML.gif. Choose a sequence ( K m ) m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq107_HTML.gif of open relatively compact subsets, with regular boundaries, of R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq10_HTML.gif covering R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq10_HTML.gif, i.e., R N = m 1 K m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq108_HTML.gif. It is easy to see that, by compact embedding, w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq109_HTML.gif in L q ( K m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq110_HTML.gif for any q < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq111_HTML.gif. Hence w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq109_HTML.gif a.e. on K m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq112_HTML.gif. Hence | w n | 2 2 w n | w | 2 2 w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq113_HTML.gif a.e. on K m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq112_HTML.gif. By the Brezis and Lieb lemma [26], we conclude that | w n | 2 2 w n | w | 2 2 w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq114_HTML.gif strongly in L ( 2 ) ( K m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq115_HTML.gif. Thus w ˜ = | w | 2 2 w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq106_HTML.gif a.e. on each K m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq112_HTML.gif, and then the diagonal rule implies a.e. on R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq10_HTML.gif. Hence
R N | w n | 2 2 w n v d x = R N | w | 2 2 w v d x + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ28_HTML.gif
(3.10)
Similarly, we have
R N V ( a ) w n v d x = R N V ( a ) w v d x + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ29_HTML.gif
(3.11)
By (f2) and (f3),
R N | f ( x , w n ) v | d x δ R N | w n | | v | d x + C δ R N | w n | q 1 | v | d x δ | w n | 2 | v | 2 + C δ | w n | q q 1 | v | q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equz_HTML.gif
Hence when R is large enough, we get
B R c ( z n ) R N | f ( a , w n ) v | d x = o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equaa_HTML.gif
Noting that w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq109_HTML.gif in L q ( B R ( z n ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq116_HTML.gif, 2 q < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq117_HTML.gif. Therefore we have
B R ( z n ) R N f ( a , w n ) v d x = B R ( z n ) R N f ( a , w ) v d x + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equab_HTML.gif
Hence
R N f ( a , w n ) v d x = R N f ( a , w ) v d x + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ30_HTML.gif
(3.12)
By (3.9)-(3.12), we derive that
Δ w + V ( a ) w = | w | 2 2 w + f ( a , w ) in  R N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ31_HTML.gif
(3.13)

i.e., J a ( w ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq118_HTML.gif.

For any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq119_HTML.gif let us consider the measure sequence μ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq120_HTML.gif defined by
R N μ n ( d y ) = R N ( | w n | 2 + | w n | 2 + | w n | q ) d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equac_HTML.gif
We assume
R N μ n ( d y ) l . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equad_HTML.gif

By the concentration-compactness lemma [24], there exists a subsequence of ( μ n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq121_HTML.gif (denoted in the same way) satisfying one of the three following possibilities.

Compactness: There exists a sequence z n R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq122_HTML.gif such that for any δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq54_HTML.gif there is a radius R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq123_HTML.gif with the property that
lim n B R ( z n ) μ n ( d y ) l δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equae_HTML.gif
Vanishing: For all R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq123_HTML.gif,
lim n ( sup z R N B R ( z ) μ n ( d y ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equaf_HTML.gif
Dichotomy: There exists a number a ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq124_HTML.gif, 0 < a ˜ < l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq125_HTML.gif, such that for any δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq54_HTML.gif there is a number R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq123_HTML.gif and a sequence ( z n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq126_HTML.gif with the following property: Given R > R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq127_HTML.gif there are non-negative measures μ n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq128_HTML.gif, μ n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq129_HTML.gif such that
  1. (i)

    0 μ n 1 + μ n 2 μ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq130_HTML.gif,

     
  2. (ii)

    supp ( μ n 1 ) B R ( z n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq131_HTML.gif, supp ( μ n 2 ) R N B R ( z n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq132_HTML.gif,

     
  3. (iii)

    lim sup n ( | a ˜ R N μ n 1 ( d y ) | + | ( l a ˜ ) R N μ n 2 ( d y ) | ) δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq133_HTML.gif.

     
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,
w n H 1 ( R N ) = R N | w n | 2 d y + R N f ( x , w n ) w n d y R N | w n | 2 d y + δ R N | w n | 2 d y + C δ R N | w n | q d y 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equag_HTML.gif

Hence J ( w n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq134_HTML.gif, contradicting c 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq80_HTML.gif.

Now for the harder part. Let η be a smooth nonincreasing cut-off function, defined in [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq135_HTML.gif, such that η = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq136_HTML.gif if 0 t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq137_HTML.gif; η = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq138_HTML.gif if t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq139_HTML.gif; 0 η 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq140_HTML.gif and | η ( t ) | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq141_HTML.gif. Also, let η r ( ) = η ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq142_HTML.gif. We define
ξ ( t ) = 1 η ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equah_HTML.gif
a nondecreasing function on [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq135_HTML.gif. Denote by ξ r ( ) = ξ ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq143_HTML.gif. We show now that dichotomy does not occur. Otherwise there exists a ˜ ( 0 , l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq144_HTML.gif such that for some R > R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq145_HTML.gif and z n R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq122_HTML.gif the function μ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq120_HTML.gif splits into μ n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq128_HTML.gif and μ n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq129_HTML.gif with the following properties:
R N μ n 1 ( d y ) a ˜ , R N μ n 2 ( d y ) l a ˜ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ32_HTML.gif
(3.14)
If we denote
w n 1 = η R ( x z n ) w n ( x ) , w n 2 = ξ R ( x z n ) w n ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equai_HTML.gif
(3.14) becomes
R N ( | w n 1 | 2 + | w n 1 | 2 + | w n 1 | q ) d y a ˜ , R N ( | w n 2 | 2 + | w n 2 | 2 + | w n 2 | q ) d y l a ˜ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ33_HTML.gif
(3.15)
Denote by Ω : = B R ( z n ) B R ( z n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq146_HTML.gif, then
0 = J a ( w n ) , χ Ω w n = Ω ( | w n | 2 + V ( a ) | w n | 2 ) d y Ω | w n | 2 d y Ω f ( a , w n ) w n d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equaj_HTML.gif
Using Dichotomy (iii), we get
Ω μ n ( d y ) = R N μ n ( d y ) B R ( z n ) μ n ( d y ) B R c ( z n ) μ n ( d y ) R N μ n ( d y ) B R ( z n ) μ n 1 ( d y ) B R c ( z n ) μ n 2 ( d y ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equak_HTML.gif
which implies
Ω | w n | 2 d y 0 , Ω | w n | 2 d y 0 , Ω | w n | q d y 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equal_HTML.gif
Hence
Ω ( | w n | 2 + V ( a ) | w n | 2 ) d y = Ω | w n | 2 d y + Ω f ( a , w n ) w n d y Ω | w n | 2 d y + δ Ω | w n | 2 d y + C δ Ω | w n | q d y 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equam_HTML.gif
Now we observe that supp  w n 1 supp  w n 2 = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq147_HTML.gif, therefore
R N ( | w n | 2 + V ( a ) | w n | 2 ) d y = R N ( | w n 1 | 2 + V ( a ) | w n 1 | 2 ) d y + R N ( | w n 2 | 2 + V ( a ) | w n 2 | 2 ) d y + o ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ34_HTML.gif
(3.16)
R N | w n | 2 d y = R N | w n 1 | 2 d y + R N | w n 2 | 2 d y + o ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ35_HTML.gif
(3.17)
R N F ( a , w n ) d x = R N F ( a , w n 1 ) d y + R N F ( a , w n 2 ) d y + o ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ36_HTML.gif
(3.18)
and
R N f ( a , w n ) w n d y = R N f ( a , w n 1 ) w n 1 d y + R N f ( a , w n 2 ) w n 2 d y + o ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ37_HTML.gif
(3.19)

where o ( 1 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq148_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif.

Recall that ( w n ) M a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq149_HTML.gif (see (2.3)), which implies
J a ( w n ) , w n = R N ( | w n | 2 + V ( a ) | w n | 2 ) d y R N | w n | 2 d y R N f ( a , w n ) w n d y = o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ38_HTML.gif
(3.20)
Then using w n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq150_HTML.gif and w n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq151_HTML.gif in place of w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq75_HTML.gif, respectively, we get
J a ( w n ) , w n 1 = R N ( | w n 1 | 2 + V ( a ) | w n 1 | 2 ) d y R N | w n 1 | 2 d y R N f ( a , w n 1 ) w n 1 d y J ( w n ) , w n 1 = o ( 1 ) , J a ( w n ) , w n 2 = R N ( | w n 2 | 2 + V ( a ) | w n 2 | 2 ) d y R N | w n 2 | 2 d y R N f ( a , w n 2 ) w n 2 d y J ( w n ) , w n 2 = o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ39_HTML.gif
(3.21)
There exists t n 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq152_HTML.gif such that t n 1 w n 1 M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq153_HTML.gif, i.e.,
( t n 1 ) 2 R N ( | w n 1 | 2 + V ( a ) | w n 1 | 2 ) d y R N | t n 1 w n 1 | 2 d y R N f ( x , t n 1 w n 1 ) t n 1 w n 1 d y = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ40_HTML.gif
(3.22)
By (f2) and (f3), f ( a , w n ) w n δ | w n | 2 + C δ | w n | q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq154_HTML.gif, we see t n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq155_HTML.gif cannot go zero, that is, t n 1 t 0 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq156_HTML.gif. If t n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq157_HTML.gif, by (3.21), (3.22) and (f4), we get
R N | w n 1 | 2 d y + R N f ( x , w n 1 ) w n 1 d y = R N ( | w n 1 | 2 + V ( a ) | w n 1 | 2 ) d y = R N | t n 1 w n 1 | 2 + f ( x , t n 1 w n 1 ) t n 1 w n 1 ( t n 1 ) 2 d y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ41_HTML.gif
(3.23)
since (f5). By (3.15),
R N | w n 1 | 2 d y + R N f ( x , w n 1 ) w n 1 d y R N | w n 1 | 2 d y + δ R N | w n 1 | 2 d y + C δ R N | w n 1 | q d y C R N ( | w n 1 | 2 + | w n 1 | 2 + | w n 1 | q ) d y C ( a ˜ + o ( 1 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ42_HTML.gif
(3.24)
a contradiction. Thus 0 < t 0 1 t n 1 C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq158_HTML.gif. Assume that t n 1 t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq159_HTML.gif, we will show t 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq160_HTML.gif. By (3.21) and (3.22), we have
o ( 1 ) = ( t n 1 ) 2 2 R N | w n 1 | 2 d y + R N f ( x , t n 1 w n 1 ) w n 1 t n 1 d y R N | w n 1 | 2 d y R N f ( x , w n 1 ) w n 1 d y = ( ( t n 1 ) 2 2 1 ) R N | w n 1 | 2 d y + R N ( f ( x , t n 1 w n 1 ) t n 1 f ( x , w n 1 ) ) w n 1 d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equan_HTML.gif
Hence by the Lebesgue dominated convergence theorem, we get
( ( t 1 ) 2 2 1 ) R N | w 1 | 2 d y + R N ( f ( x , t 1 w 1 ) t 1 f ( x , w 1 ) ) w 1 d y = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equao_HTML.gif
By (f5), we have t 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq160_HTML.gif. Similarly, t n 2 t 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq161_HTML.gif. Using this together with (3.16), (3.17), (3.18) and (3.19), we obtain
c 0 + o ( 1 ) = J a ( w n ) = J a ( w n 1 ) + J a ( w n 2 ) + o ( 1 ) = J a ( t n 1 w n 1 ) + J a ( t n 2 w n 2 ) + o ( 1 ) 2 c 0 + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equap_HTML.gif

Contradiction! Thus dichotomy does not occur.

With vanishing and dichotomy ruled out, we obtain the compactness of a sequence μ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq120_HTML.gif, i.e., there exist z n R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq122_HTML.gif and for each δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq54_HTML.gif, there exists R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq123_HTML.gif such that
B R c ( z n ) ( | w n | 2 + | w n | 2 + | w n | q ) d y δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ43_HTML.gif
(3.25)
Then ( z n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq126_HTML.gif must be bounded, for otherwise (3.25) would imply, in the limit n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif,
R N ( | w | 2 + | w | 2 + | w | q ) d y C 1 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ44_HTML.gif
(3.26)

for some positive constants C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq162_HTML.gif, independent of δ, which implies w 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq163_HTML.gif, contrary to (3.8).

From the foregoing, it follows that there exist bounded nonnegative measures μ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq164_HTML.gif, ν ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq165_HTML.gif on R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq10_HTML.gif such that | w n | 2 μ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq166_HTML.gif weakly and | w n | 2 ν ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq167_HTML.gif tightly as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif. Lemma I.1 in [27] declares that there exist sequences ( x j ) R + N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq168_HTML.gif, ( μ ˜ j ) , ( ν ˜ j ) ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq169_HTML.gif such that
( 1 ) μ ˜ | w | 2 + j J ˜ μ ˜ j δ x j , ( 2 ) ν ˜ = | w | 2 + j J ˜ ν ˜ j δ x j , ( 3 ) μ ˜ j S ν ˜ j 2 / 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ45_HTML.gif
(3.27)
where δ x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq170_HTML.gif denotes a Dirac measure, j J ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq171_HTML.gif. Take x j R + N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq172_HTML.gif in the support of the singular part of μ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq164_HTML.gif, ν ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq165_HTML.gif. We consider ϕ C c ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq173_HTML.gif such that
ϕ = 1 on  B ( x j , ε ) , ϕ = 0 on  B ( x j , 2 ε ) c , | ϕ | 2 / ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ46_HTML.gif
(3.28)
Choosing the test function ϕ w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq174_HTML.gif, from I ε ( w n ) , ϕ w n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq175_HTML.gif, we have
R N ϕ μ ˜ ( d x ) + R N ϕ V ( a ) | w | 2 d x R N ϕ ν ˜ ( d x ) R N ϕ f ( x , w ) w d x = lim n R N w n w n ϕ d x C 1 lim n ( B ( x j , 2 ε ) | w n | 2 d x ) 1 / 2 ε 0 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ47_HTML.gif
(3.29)
This reduces to
μ ˜ j = ν ˜ j , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equaq_HTML.gif
hence (3.27)(3) states
S ν ˜ j 2 / 2 ν ˜ j , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equar_HTML.gif
i.e.,
ν ˜ j = 0 or ν ˜ j S N / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equas_HTML.gif
Consequently,
ν ˜ | w | 2 + S N / 2 j J δ x j , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equat_HTML.gif
and hence
R N | w n | 2 d x R N ν ˜ ( d x ) R N | w | 2 d x + S N / 2 Card J , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ48_HTML.gif
(3.30)
which implies that the set J is at most finite. Here CardJ is the cardinal numbers of set J. Hence
J a ( w n ) 1 2 J a ( w n ) , w n = ( 1 2 1 2 ) R N | w n | 2 d y + R N ( f ( a , w n ) w n F ( a , w n ) ) d y 1 N R N | w n | 2 d y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ49_HTML.gif
(3.31)
since
f ( x , t ) t θ F ( t ) F ( t ) > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equau_HTML.gif
When n is large enough, recall c 0 < 1 N S N / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq176_HTML.gif (see (2.11)), together with (3.30) and (3.31), we obtain
1 N S N / 2 > c 0 1 N Ω | w | 2 d x + 1 N j J ν j 1 N S N / 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equav_HTML.gif
a contradiction. Therefore J is empty, that is, | w n | 2 2 | w | 2 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq177_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif. By the Brezis and Lieb lemma [26] again, we get
| w n w | 2 2 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ50_HTML.gif
(3.32)
Equation (3.25) and compact embedding theorem imply
R N f ( a , w n ) w n d x = R N f ( a , w ) w d x + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ51_HTML.gif
(3.33)
This together with (3.13), (3.20) and (3.32) allows us to deduce easily
w n H 1 ( R N ) = w H 1 ( R N ) + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equaw_HTML.gif
Since H 1 ( R + N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq178_HTML.gif is a uniformly convex Banach space, hence
w n w H 1 ( R N ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ52_HTML.gif
(3.34)
From (3.32), (3.33) and (3.34), we can obtain
c 0 = lim n J a ( w n ) = J a ( w ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ53_HTML.gif
(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 4.1 Suppose f satisfies (f2)-(f4). Then I ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq41_HTML.gif satisfies the ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq38_HTML.gif-condition for all c < ε N S N / 2 / N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq179_HTML.gif, that is, every sequence ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq79_HTML.gif in H ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq45_HTML.gif such that I ε ( u n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq180_HTML.gif, I ε ( u n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq181_HTML.gif, as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif, possesses a convergent subsequence.

Proof Suppose that ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq79_HTML.gif is a sequence in H ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq45_HTML.gif such that I ε ( u n ) c < ε N S N / 2 / N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq182_HTML.gif, I ε ( u n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq181_HTML.gif, as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif. Using (f4), by a change of variable x = ε y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq183_HTML.gif, we obtain that
c + o ( 1 ) u n ε I ε ( u n ) 1 θ I ε ( u n ) , u n = ( 1 2 1 θ ) R N ( ε 2 | u n | 2 + V ( x ) | u n | 2 ) d x + ( 1 θ 1 2 ) R N | u n | 2 d x + 1 θ R N ( f ( x , u n ) u n θ F ( x , u n ) ) d x ( 1 2 1 θ ) u n ε 2 = ( 1 2 1 θ ) u n 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ54_HTML.gif
(4.1)
This implies that ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq79_HTML.gif is bounded in H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq184_HTML.gif. Therefore we may assume u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq185_HTML.gif in H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq184_HTML.gif and u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq186_HTML.gif a.e. Let u n = v n + u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq187_HTML.gif. Then
R N ( | u n | 2 + V ( ε y ) | u n | 2 ) d y = R N ( | v n | 2 + V ( ε y ) | v n | 2 ) d y + R N ( | u | 2 + V ( ε y ) | u | 2 ) d y + o ( 1 ) , R N f ( x , u n ) u n d y = R N f ( x , v n ) v n d y + R N f ( x , u ) u d y + o ( 1 ) , R N F ( x , u n ) d y = R N F ( x , v n ) d y + R N F ( x , u ) d y + o ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equax_HTML.gif
and by the Brezis-Lieb lemma [26],
R N | u n | 2 d y = R N | v n | 2 d y + R N | u | 2 d y + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equay_HTML.gif
For convenience, we denote by
I ( u ) = 1 2 R N ( | u | 2 + V ( ε y ) | u | 2 ) d y 1 2 R N | u | 2 d y R N F ( x , u ) d y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ55_HTML.gif
(4.2)
and
I ( u ) , v = R N ( u v + V ( ε y ) u v | u | 2 2 u v f ( x , u ) v ) d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ56_HTML.gif
(4.3)
It is clear that
I ε ( u ) = ε N I ( u ) , I ε ( u ) , v = ε N I ( u ) , v . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ57_HTML.gif
(4.4)
It is easy to verify that I ( u ) , u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq188_HTML.gif. Hence we have
o ( 1 ) = I ( u n ) , u n = I ( v n ) , v n + I ( u ) , u + o ( 1 ) = I ( v n ) , v n + o ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equaz_HTML.gif
and thus
lim n R N ( | v n | 2 + V ( x ) | v n | 2 ) d y = lim n R N | v n | 2 d y = : , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equba_HTML.gif
since R N f ( | v n | 2 ) | v n | 2 d y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq189_HTML.gif by (f2) and (f3). If = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq190_HTML.gif, then v n 2 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq191_HTML.gif, hence v n ε 2 = ε N v n 2 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq192_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif, and we can obtain the desired conclusion. Hence it remains to show that = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq193_HTML.gif. By a change of variable, from
I ( v n ) = I ( v n ) 1 2 I ( v n ) , v n 1 N R N | v n | 2 d y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbb_HTML.gif
and
c = I ( u n ) + o ( 1 ) = I ( v n ) + I ( u ) + o ( 1 ) I ( v n ) + o ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbc_HTML.gif
we get
ε N N c < ε N S N / 2 N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbd_HTML.gif
i.e.,
< S N / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ58_HTML.gif
(4.5)
By the Sobolev inequalities,
( R N | v n | 2 d x ) 2 / 2 S 1 R N ( | v n | 2 + V ( y ) | v n | 2 ) d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Eqube_HTML.gif

Letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif, we get 2 / 2 S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq194_HTML.gif, so either S N / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq195_HTML.gif which contradicts (4.5) or = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq190_HTML.gif. □

Let ι > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq196_HTML.gif be fixed. Let η be a smooth nonincreasing cut-off function, defined in [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq135_HTML.gif, such that η = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq136_HTML.gif if 0 t ι https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq197_HTML.gif; η = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq138_HTML.gif if t 2 ι https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq198_HTML.gif; 0 η 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq140_HTML.gif and | η ( t ) | C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq199_HTML.gif for some C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq53_HTML.gif. For any a Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq70_HTML.gif, let
ψ ε ( a ) ( x ) = η ( | x a | ) w ( x a ε ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbf_HTML.gif
where w is the positive ground state of (2.2). We may assume that t ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq200_HTML.gif is the unique positive number such that
max t 0 I ε ( t ψ ε ( ξ ) ( x ) ) = I ε ( t ε ψ ε ( ξ ) ( x ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbg_HTML.gif
Let h ( ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq201_HTML.gif be any positive function tending to 0 as ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq202_HTML.gif, we define the sublevel
N ˜ ε : = { u N ε : I ε ( u ) ε N ( c 0 + h ( ε ) ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ59_HTML.gif
(4.6)
By Lemma 4.2 below, N ˜ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq39_HTML.gif is not empty for ε sufficiently small. By noticing that t ε ψ ε ( ξ ) N ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq203_HTML.gif, we can define ϕ ε : Ω N ˜ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq204_HTML.gif as
ϕ ε : = t ε ψ ε ( ξ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbh_HTML.gif
Lemma 4.2 Uniformly in a Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq70_HTML.gif, we have
lim ε 0 ε N I ε ( ϕ ε ( a ) ) = c 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ60_HTML.gif
(4.7)
Proof Let a Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq70_HTML.gif. Computing directly, we have
ψ ε ( a ) ( x ) ε 2 = R N ( ε 2 | ψ ε ( a ) ( x ) | 2 + V ( x ) | ψ ε ( a ) ( x ) | 2 ) d x = R N ( ε 2 | w ( x a ε ) η ( | x a | ) + 1 ε η ( | x a | ) w ( x a ε ) | 2 + V ( x ) | η ( | x a | ) w ( x a ε ) | 2 ) d x = R N ( η 2 ( | x a | ) ( | w ( x a ε ) | 2 + V ( x ) | w ( x a ε ) | 2 ) ) d x + R N ( ε 2 | η ( | x a | ) | 2 | w ( x a ε ) | 2 + 2 ε η ( | x a | ) w ( x a ε ) η ( | x a | ) w ( x a ε ) ) d x = : I 1 + I 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ61_HTML.gif
(4.8)
By a change of variable y = x a ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq205_HTML.gif, we obtain
I 1 = ε N R N η 2 ( ε y ρ ) ( | w ( y ) | 2 + V ( ε y + a ) | w ( y ) | 2 ) d y = ε N ( R N ( | w | 2 + V ( a ) w 2 ) d y + o ( 1 ) ) as  ε 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ62_HTML.gif
(4.9)
uniformly for a Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq70_HTML.gif.
I 2 = ε N R N ( ε 2 | η ( | ε y | ρ ) | 2 ω 2 ( y ) + 2 ε η ( | ε y | ρ ) ω ( y ) η ( | ε y | ρ ) ω ( y ) ) d y ε N + 2 { y : ρ / ε | y | 2 ρ / ε } C | ω ( y ) | 2 d y + 2 ε N + 1 { y : ρ / ε | y | 2 ρ / ε } C 1 | ω ( y ) | | ω ( y ) | d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ63_HTML.gif
(4.10)
By the exponential decay of ω, we get
I 2 = ε N ( o ( 1 ) ) as  ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ64_HTML.gif
(4.11)
uniformly for a Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq70_HTML.gif. Therefore, in the limit that ε is very small, thanks to (4.8) (4.9) and (4.11), we find
ψ ε ( a ) ( x ) ε 2 = ε N ( R N ( | w | 2 + V ( a ) w 2 ) d y + o ( 1 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ65_HTML.gif
(4.12)
On the other hand, following the idea of [21, 25], from I ε ( t ε ψ ε ( ξ ) ( x ) ) , t ε ψ ε ( ξ ) ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq206_HTML.gif, by the change of variables y : = ( x ξ ) / ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq207_HTML.gif, we get
t ε ψ ε ( ξ ) ( x ) ε 2 = R N | t ε ψ ε ( ξ ) ( x ) | 2 d x + R N f ( x , t ε ψ ε ( ξ ) ( x ) ) t ε ψ ε ( ξ ) ( x ) d x = ε N ( R N | t ε η ρ ( ε y ) w ( y ) | 2 d y + R N f ( x , t ε η ρ ( ε y ) w ( y ) ) t ε η ρ ( ε y ) w ( y ) d y ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ66_HTML.gif
(4.13)
ε N ψ ε ( ξ ) ( x ) ε 2 = | t ε | 2 2 R N | η ρ ( ε y ) w ( y ) | 2 d y + R N f ( x , t ε η ρ ( ε y ) w ( y ) ) t ε η ρ ( ε y ) w ( y ) d y as  t ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ67_HTML.gif
(4.14)

which contradicts (4.12). Thus, up to a subsequence, t ε t 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq208_HTML.gif.

Since f has subcritical growth and t ε ψ ε ( ξ ) N ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq203_HTML.gif, it follows that t 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq97_HTML.gif. Thus, we can take the limit in (4.13) to obtain
R N | ( t 0 w ) | 2 + | t 0 w | 2 d y = R N | t 0 w | 2 d y + R N f ( a , t 0 w ) t 0 w d y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ68_HTML.gif
(4.15)
from which it follows that t 0 w M a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq209_HTML.gif. Since w also belongs to M a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq210_HTML.gif, we conclude that t 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq211_HTML.gif. This and Lebesgue’s theorem imply that
R N | t ε ψ ε ( ξ ) | 2 d x = ε N ( R N | w | 2 d y + o ( 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ69_HTML.gif
(4.16)
and
Ω F ( x , t ε ψ ε ( a ) ) d x = ε N ( R N F ( a , w ) d y + o ( 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equ70_HTML.gif
(4.17)
uniformly for a Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq70_HTML.gif. Noting t ε 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq212_HTML.gif, from (4.12), (4.16) and (4.17), we have
ε N I ε ( ϕ ε ( ξ ) ) = 1 2 R N ( | w | 2 + V ( a ) w 2 ) d y + 1 2 R N | w | 2 d y + R N F ( a , w ) d y + o ( 1 ) = c 0 + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbi_HTML.gif

Thus (4.7) is proved. □

Let β ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq213_HTML.gif be the center of mass of u N ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq214_HTML.gif in terms of the L 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq215_HTML.gif norm:
β ( u ) : = R N x | u | 2 d x R N | u | 2 d x , u N ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbj_HTML.gif
Lemma 4.3 Let ε n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq65_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif. Then for u n N ˜ ε n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq216_HTML.gif,
β ( u n ) a , as  n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbk_HTML.gif

uniformly for a Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq70_HTML.gif.

Proof By change of variable x = ε n y + a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq78_HTML.gif, we have
β ( u n ) = R N x | u n | 2 d x R N | u n | 2 d x = R N ( ε n y + a ) | v n | 2 d y R N | v n | 2 d y = a + ε n R N y | v n | 2 d y R N | v n | 2 d y = a + ε n R N y | t n v n | 2 d y R N | t n v n | 2 d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbl_HTML.gif

By Proposition 3.1, w n = t n v n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq217_HTML.gif converges strongly in H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq44_HTML.gif 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 β ( u n ) a Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq218_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq66_HTML.gif. This completes the proof of Lemma 4.3. □

Proof of Theorem 1.1 By Proposition 4.1, I ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq41_HTML.gif satisfies the ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq38_HTML.gif-condition for all c < ε N S N / 2 / N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq219_HTML.gif. Now let us choose a function h ( ε ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq220_HTML.gif such that h ( ε ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq221_HTML.gif as ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq202_HTML.gif and such that ε N ( c 0 + h ( ε ) ) < ε N S N / 2 / N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq222_HTML.gif is not a critical level for I ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq41_HTML.gif. For such h ( ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq201_HTML.gif, let us introduce the set N ˜ ε N ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq223_HTML.gif as in (4.6). Then the standard Ljusternik-Schnirelmann theory implies that I ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq41_HTML.gif has at least cat ( N ˜ ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq40_HTML.gif critical points on N ˜ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq39_HTML.gif (also see [21]).

By Lemma 4.3, we can assume that for any δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq54_HTML.gif, there exists ε δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq224_HTML.gif such that β ( N ˜ ε ) Σ δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq225_HTML.gif for any ε < ε δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq226_HTML.gif. For such ε, by Lemma 4.2, we have I ε ( ϕ ε ( a ) ) ε N ( c 0 + h ( ε ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq227_HTML.gif uniformly for a Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq70_HTML.gif, thus ϕ ε ( Σ ) N ˜ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq228_HTML.gif. Recall t ε 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq212_HTML.gif, calculating directly, we get
β ( ϕ ε ( a ) ) = R N x | t ε ψ ε ( a ) ( x ) | 2 d x R N | t ε ψ ε ( a ) ( x ) | 2 d x = R N x | η ( | x a | ) w ( x a ε ) | 2 d x R N | η ( | x a | ) w ( x a ε ) | 2 d x + o ( 1 ) = R N ( ε y + a ) | η ( | ε y | ) w ( y ) | 2 d x R N | η ( | ε y | ) w ( y ) | 2 d x + o ( 1 ) = a + R N ε y | η ( | ε y | ) w ( y ) | 2 d x R N | η ( | ε y | ) w ( y ) | 2 d x = a + o ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbm_HTML.gif
as ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq202_HTML.gif uniformly for a Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq70_HTML.gif. Hence the map β ϕ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq229_HTML.gif is homotopical equivalence to the inclusion i : Σ Σ δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq230_HTML.gif for ε small enough. We denote N ˜ ε + = N ˜ ε { u N ε : u 0  in  R N } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq231_HTML.gif. It is easy to verify that cat ( N ˜ ε + , N ˜ ε ) cat ( Σ , Σ δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq232_HTML.gif and cat ( N ˜ ε , N ˜ ε ) cat ( Σ , Σ δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq233_HTML.gif (cf. [[19], Lemma 2.2]). Hence we have
cat ( N ˜ ε ) = cat ( N ˜ ε + , N ˜ ε ) + cat ( N ˜ ε , N ˜ ε ) 2 cat ( Σ , Σ δ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbn_HTML.gif
Next we show that if u is a critical point of I ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq41_HTML.gif satisfying I ε ( u ) ε N ( c 0 + h ( ε ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq42_HTML.gif, then u cannot change sign. Indeed, if u = u + + u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq234_HTML.gif with u + 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq235_HTML.gif and u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq236_HTML.gif, then from I ε ( u ) , u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq237_HTML.gif, we have
I ε ( u + ) , u + = 0 , I ε ( u ) , u = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbo_HTML.gif
By change of variable x = ε n y + a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq78_HTML.gif, we get
J a ( u + ) , u + = 0 , J a ( u ) , u = 0 , as  ε n 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbp_HTML.gif
i.e.,
u + , u M a . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbq_HTML.gif
Also, noting
I ε ( u ) = ε N J a ( u ) , as  ε n 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbr_HTML.gif
Hence
c 0 + h ( ε ) J a ( u ) = J a ( u + ) + J a ( u ) 2 G ( a ) 2 c 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_Equbs_HTML.gif

which is a contradiction. Therefore there exist at least cat ( Σ , Σ δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq43_HTML.gif nonzero critical points of I ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq41_HTML.gif and thus cat ( Σ , Σ δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-199/MediaObjects/13661_2013_Article_452_IEq43_HTML.gif solutions of equation (1.4). □

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Jiangxi University of Science and Technology
(2)
School of Mathematics and Statistics, Lanzhou University

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