Open Access

Existence of multiple solutions for the Brezis-Nirenberg-type problem with singular coefficients

Boundary Value Problems20122012:137

DOI: 10.1186/1687-2770-2012-137

Received: 4 September 2012

Accepted: 19 September 2012

Published: 26 November 2012

Abstract

By energy estimates and by establishing a local (PS) condition, we obtain the multiplicity of solutions to a class of Brezis-Nirenberg-type problem with singular coefficients via minimax methods and the Krasnoselskii genus theory.

Keywords

Brezis-Nirenberg-type problem minimax method

1 Introduction and main results

This paper is concerned with multiple solutions for the semilinear Brezis-Nirenberg-type problem with singular coefficients
{ div ( D u | x | 2 a ) = λ | u | 2 2 u | x | 2 b + β | u | q 2 u | x | α , x Ω ; u = 0 , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ1_HTML.gif
(1)

where Ω R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq1_HTML.gif is a bounded smooth domain, and 0 Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq2_HTML.gif, < a < n 2 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq3_HTML.gif, a b < a + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq4_HTML.gif, 2 = 2 n n 2 d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq5_HTML.gif, d = a + 1 b ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq6_HTML.gif, 1 < q < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq7_HTML.gif, α < ( 1 + a ) q + n ( 1 q 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq8_HTML.gif. β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq9_HTML.gif, λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq10_HTML.gif are two real parameters.

The starting point of the variational approach to the problem is the Caffarelli-Kohn-Nirenberg inequality (see [1]): There is a constant C a , b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq11_HTML.gif such that
( R n | x | 2 b | u | 2 d x ) 2 / 2 C a , b R n | x | 2 a | D u | 2 d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ2_HTML.gif
(2)
for all u C 0 ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq12_HTML.gif, where
< a < n 2 2 , a b < a + 1 , 2 = 2 n n 2 d , d = a + 1 b . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equa_HTML.gif
Let D a 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq13_HTML.gif be the completion of C 0 ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq14_HTML.gif with respect to the weighted norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq15_HTML.gif defined by
u = ( Ω | x | 2 a | D u | 2 d x ) 1 / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equb_HTML.gif
From the boundedness of Ω and the standard approximation arguments, it is easy to see that (2) holds for any u D a 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq16_HTML.gif in the sense:
( Ω | x | α | u | r d x ) 2 / r C Ω | x | 2 a | D u | 2 d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ3_HTML.gif
(3)
for 1 r 2 = 2 n n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq17_HTML.gif, α r ( 1 + a ) + n ( 1 r 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq18_HTML.gif, that is, the embedding D a 1 , 2 ( Ω ) L r ( Ω , | x | α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq19_HTML.gif is continuous, where L r ( Ω , | x | α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq20_HTML.gif is the weighted L r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq21_HTML.gif space with the norm
u r , α : = u L r ( Ω , | x | α ) = ( Ω | x | α | u | r d x ) 1 / r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equc_HTML.gif
On D a 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq13_HTML.gif, we can define the energy functional
J ( u ) = 1 2 Ω | x | 2 a | D u | 2 d x λ 2 Ω | x | 2 b | u | 2 d x β q Ω | x | α | u | q d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ4_HTML.gif
(4)

From (4), J is well defined in D a 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq13_HTML.gif, and J C 1 ( D a 1 , 2 ( Ω ) , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq22_HTML.gif. Furthermore, the critical points of J are weak solutions of problem (1).

Breiz-Nirenberg-type problems have been generalized to many situations such as
{ div ( D u | x | 2 a ) μ u | x | 2 ( a + 1 ) = | u | 2 2 u | x | 2 b + λ u | x | 2 ( a + 1 ) c , x Ω , u = 0 , x Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ5_HTML.gif
(5)

Xuan et al.[2] derived the explicit formula for the extremal functions of the best embedding constant by applying the Bliss lemma [3]. They got a nontrivial solution for problem (5) including the resonant and nonresonant cases by variational methods. He and Zou [4] studied problem (5) and obtained the multiplicity of solutions with the aid of a pseudo-index theory. In [5], problem (5) has been extended to the p-Laplace case by Xuan.

The purpose of this paper is to study the multiplicity of solutions for the Breiz-Nirenberg-type problem (1) with the aid of a minimax method. We obtain multiple nontrivial solutions of (1) by proving the local (PS) condition and energy estimates.

Our main results are the following.

Theorem 1.1 Suppose 1 < q < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq23_HTML.gif, then
  1. (i)

    β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq24_HTML.gif, λ 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq25_HTML.gif such that if 0 < λ < λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq26_HTML.gif, problem (1) has a sequence of solutions { u m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq27_HTML.gif with J ( u m ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq28_HTML.gif and J ( u m ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq29_HTML.gif as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq30_HTML.gif.

     
  2. (ii)

    λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq31_HTML.gif, β 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq32_HTML.gif such that if 0 < β < β 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq33_HTML.gif, problem (1) has a sequence of solutions { u m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq27_HTML.gif with J ( u m ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq28_HTML.gif and J ( u m ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq29_HTML.gif as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq30_HTML.gif.

     

2 Preliminary results

Lemma 2.1[5]

Suppose that Ω R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq1_HTML.gifis an open bounded domain with C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq34_HTML.gifboundary and 0 Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq35_HTML.gif, < a < ( n 2 ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq36_HTML.gif. The embedding D a 1 , 2 ( Ω ) L r ( Ω , | x | α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq37_HTML.gifis compact if 1 r < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq38_HTML.gif, α < ( 1 + a ) r + n ( 1 r 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq39_HTML.gif.

Lemma 2.2 (Concentration compactness principle [5])

Let < a < ( n 2 ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq36_HTML.gif, a b a + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq40_HTML.gif, 2 = 2 n / ( n 2 d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq41_HTML.gif, d = 1 + a b [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq42_HTML.gif, and M ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq43_HTML.gifbe the space of bounded measures on R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq44_HTML.gif. Suppose that { u m } D a 1 , 2 ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq45_HTML.gifis a sequence such that
u m u in  D a 1 , 2 ( R n ) , μ m : = | | x | a D u m | 2 d x μ in  M ( R n ) , ν m : = | | x | b u m | 2 d x ν in  M ( R n ) , u m u a.e. on  R n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equd_HTML.gif
Then there are the following statements:
  1. (1)
    There exists some at most countable set I, a family { x ( i ) : i I } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq46_HTML.gif of distinct points in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq44_HTML.gif, and a family { ν ( i ) : i I } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq47_HTML.gif of positive numbers such that
    ν = | | x | b u | 2 d x + i I ν ( i ) δ x ( i ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ6_HTML.gif
    (6)
     
where δ x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq48_HTML.gifis the Dirac-mass of mass 1 concentrated at x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq49_HTML.gif.
  1. (2)
    The following inequality holds
    μ | | x | a D u | 2 d x + i I μ ( i ) δ x ( i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ7_HTML.gif
    (7)
     
for some family { μ ( i ) > 0 : i I } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq50_HTML.gif satisfying
S ( ν ( i ) ) 2 / 2 μ ( i ) for all  i I , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ8_HTML.gif
(8)
where S : = inf u D a 1 , 2 ( R n ) { 0 } E a , b ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq51_HTML.gifto be the best embedding constants, and
E a , b ( u ) = R n | x | 2 a | D u | 2 d x ( R n | x | 2 b | u | 2 d x ) 2 / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Eque_HTML.gif

In particular, i I ( ν ( i ) ) 2 / 2 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq52_HTML.gif.

Lemma 2.3 Assume { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gifis a (PS) c sequence with c < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq54_HTML.gif, 1 < q < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq23_HTML.gif, then
  1. (1)

    λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq31_HTML.gif, there exists β 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq55_HTML.gif such that for any 0 < β < β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq56_HTML.gif, { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif has a convergent subsequence in D a 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq57_HTML.gif.

     
  2. (2)

    β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq24_HTML.gif, there exists λ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq58_HTML.gif such that for any 0 < λ < λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq59_HTML.gif, { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif has a convergent subsequence in D a 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq57_HTML.gif.

     

Proof (1) The boundedness of (PS) c sequence.

For { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif is a (PS) c sequence, then
J ( u n ) = 1 2 Ω | x | 2 a | D u n | 2 d x λ 2 Ω | x | 2 b | u n | 2 d x β q Ω | x | α | u n | q d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ9_HTML.gif
(9)
J ( u n ) , u n = Ω | x | 2 a | D u n | 2 d x λ Ω | x | 2 b | u n | 2 d x β Ω | x | α | u n | q d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ10_HTML.gif
(10)
So, we get
o ( 1 ) ( 1 + u n ) + | c | J ( u n ) 1 2 J ( u n ) , u n = d n u n 2 ( 1 q 1 2 ) β Ω | u n | q | x | α d x d n u n 2 ( 1 q 1 2 ) β C α u n q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equf_HTML.gif
We have the boundedness of { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif for 1 < q < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq23_HTML.gif, then there exists a subsequence, we still denote it by { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif, such that
u n u in  D a 1 , 2 ( Ω ) , u n u in  L 2 ( Ω , | x | 2 b ) , u n u in  L r ( Ω , | x | α ) , 1 r < 2 n n 2 , α < ( 1 + a ) r + n ( 1 r 2 ) , u n u a.e. on  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equg_HTML.gif
From the concentration compactness principle, there exist nonnegative measures μ, ν and a countable family { x i } Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq60_HTML.gif such that
| x | 2 b | u n | 2 d x ν = | x | 2 b | u | 2 d x + i I ν ( i ) δ x ( i ) , | x | 2 a | D u n | 2 d x μ | x | 2 a | D u | 2 d x + S i I ( ν ( i ) ) 2 / 2 δ x ( i ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equh_HTML.gif
  1. (2)

    Up to a subsequence, u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq61_HTML.gif in L 2 ( Ω , | x | 2 b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq62_HTML.gif.

     
Since { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif is bounded in D a 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq13_HTML.gif, we may suppose, without loss of generality, that there exists T ( L 2 ( Ω , | x | 2 a ) ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq63_HTML.gif such that
D u n T in  ( L 2 ( Ω , | x | 2 a ) ) n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equi_HTML.gif
On the other hand, | u n | 2 2 u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq64_HTML.gif is also bounded in L 2 ( Ω , | x | 2 b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq65_HTML.gif and
| u n | 2 2 u n | u | 2 2 u in  L 2 ( Ω , | x | 2 b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equj_HTML.gif
Note that
o ( 1 ) φ = J ( u n ) , φ = Ω | x | 2 a D u n D φ d x λ Ω | x | 2 b | u n | 2 2 u n φ d x β Ω | x | α | u n | q 2 u n φ d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ11_HTML.gif
(11)
taking n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq66_HTML.gif in (11), we have
Ω | x | 2 a T D φ d x = λ Ω | x | 2 b | u | 2 2 u φ d x + β Ω | x | α | u | q 2 u φ d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ12_HTML.gif
(12)
for any φ D a 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq67_HTML.gif. Let φ = ψ u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq68_HTML.gif in (12), where ψ C ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq69_HTML.gif, then it follows that
Ω | x | 2 a D u n u n D ψ d x + Ω | x | 2 a | D u n | 2 ψ d x = λ Ω | x | 2 b | u n | 2 ψ d x + β Ω | x | α | u n | q ψ d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ13_HTML.gif
(13)
Taking n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq66_HTML.gif in (13), we have
Ω | x | 2 a u T D ψ d x + Ω ψ d μ = λ Ω ψ d ν + β Ω | x | α | u | q ψ d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ14_HTML.gif
(14)
Let φ = ψ u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq70_HTML.gif in (12), then it follows that
Ω | x | 2 a T ψ u d x + Ω | x | 2 a T u d ψ = λ Ω | x | 2 b | u | 2 ψ d x + β Ω | x | α | u | q ψ d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ15_HTML.gif
(15)
Thus, it implies that
Ω ψ d μ = λ i I ν i ψ ( x i ) + Ω | x | 2 a T D u ψ d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ16_HTML.gif
(16)
which implies that
S ( ν i ) 2 / 2 μ i = λ ν i . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equk_HTML.gif

Hence, ν i ( λ 1 S ) n / 2 d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq71_HTML.gif if ν i 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq72_HTML.gif.

On the other hand,
0 > c = lim n ( J ( u n ) 1 2 J ( u n ) , u n ) = lim n ( d n u n 2 β ( 1 q 1 2 ) Ω | x | α | u n | q d x ) d n u 2 β C u q , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equl_HTML.gif
then u q C β q / ( 2 q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq73_HTML.gif, so that
0 > c = lim n ( J ( u n ) 1 2 J ( u n ) , u n ) = lim n ( d n u n 2 β ( 1 q 1 2 ) Ω | x | α | u n | q d x ) d n μ i β C β q / ( 2 q ) d n S n 2 d ( λ 1 ) n 2 d 2 d C β 2 2 q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equm_HTML.gif

However, if β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq9_HTML.gif is given, we can choose λ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq58_HTML.gif so small that for every 0 < λ < λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq59_HTML.gif, the last term on the right-hand side above is greater than 0, which is a contradiction. Similarly, if λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq10_HTML.gif is given, we can take β 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq55_HTML.gif so small that for every 0 < β < β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq56_HTML.gif, the last term on the right-hand side above is greater than 0. Then ν i = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq74_HTML.gif for each i.

Up to now, we have shown that
lim n Ω | x | 2 b | u n | 2 d x = Ω | x | 2 b | u | 2 d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equn_HTML.gif
So, by the Breiz-Lieb lemma,
o ( 1 ) u n = u n 2 λ Ω | x | 2 b | u n | 2 d x β Ω | x | α | u n | q d x = u n u 2 u 2 λ Ω | x | 2 b | u | 2 d x β Ω | x | α | u | q d x = u n u 2 + o ( 1 ) u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equo_HTML.gif

since J ( u ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq75_HTML.gif. Thus, we prove that { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif strongly converges to u in D a 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq13_HTML.gif. □

3 Existence of infinitely many solutions

In this section, we use the minimax procedure to prove the existence of infinitely many solutions. Let Σ be the class of subsets of D a 1 , 2 ( Ω ) { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq76_HTML.gif, which are closed and symmetric with respect to the origin. For A Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq77_HTML.gif, we define the genus γ ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq78_HTML.gif by
γ ( A ) = min { k N : ϕ C ( A , R k { 0 } ) , ϕ ( x ) = ϕ ( x ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equp_HTML.gif
Assume that 1 < q < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq23_HTML.gif, then we obtain
J ( u ) = 1 2 Ω | x | 2 a | D u | 2 d x λ 2 Ω | x | 2 b | u | 2 d x λ q Ω | x | α | u | q d x 1 2 u 2 C b λ 2 u 2 β C α q u q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equq_HTML.gif
Define
h ( t ) = 1 2 t 2 λ C 1 t 2 β C 2 t q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equr_HTML.gif
Then, given β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq9_HTML.gif, there exists λ 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq79_HTML.gif so small that for every 0 < λ < λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq80_HTML.gif, there exists 0 < T 0 < T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq81_HTML.gif such that h ( t ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq82_HTML.gif for 0 < t < T 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq83_HTML.gif, h ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq84_HTML.gif for T 0 < t < T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq85_HTML.gif, h ( t ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq82_HTML.gif for t > T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq86_HTML.gif. Similarly, given λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq10_HTML.gif, we can choose β 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq87_HTML.gif with the property that T 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq88_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq89_HTML.gif as above exist for each 0 < β < β 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq90_HTML.gif. Clearly, h ( T 0 ) = h ( T 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq91_HTML.gif. Following the same idea as in [68], we consider the truncated functional
J ˜ ( u ) = 1 2 Ω | x | 2 a | D u | 2 d x λ 2 ψ ( u ) Ω | x | 2 b | u | 2 d x λ q Ω | x | α | u | q d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equs_HTML.gif

where ψ ( u ) = τ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq92_HTML.gif, and τ : R + [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq93_HTML.gif is a nonincreasing C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq94_HTML.gif function such that τ ( t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq95_HTML.gif if t T 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq96_HTML.gif and τ ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq97_HTML.gif if t T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq98_HTML.gif. The main properties of J ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif are the following.

Lemma 3.1
  1. (1)

    J ˜ C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq100_HTML.gif and J ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif is bounded below.

     
  2. (2)

    If J ˜ ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq101_HTML.gif, then u T 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq102_HTML.gif and J ˜ ( u ) = J ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq103_HTML.gif.

     
  3. (3)

    For any λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq10_HTML.gif, there exists β 0 = min { β 1 , β 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq104_HTML.gif such that if 0 < β < β 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq33_HTML.gif and c < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq54_HTML.gif, then J ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif satisfies (PS) c condition.

     
  4. (4)

    for any β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq9_HTML.gif, there exists λ 0 = min { λ 1 , λ 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq105_HTML.gif such that if 0 < λ < λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq26_HTML.gif and c < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq54_HTML.gif, then J ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif satisfies (PS) c condition.

     

Proof (1) and (2) are immediate. To prove (3) and (4), observe that all (PS) c sequences for J ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif with c < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq54_HTML.gif must be bounded. Similar to the proof of Lemma 2.3, there exists a convergent subsequence. □

Lemma 3.2 Given m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq106_HTML.gif, there is ε m < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq107_HTML.gifsuch that
γ ( { u D a 1 , 2 ( Ω ) : J ˜ ( u ) ε m } ) m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equt_HTML.gif
Proof Fix m and let H m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq108_HTML.gif be an m-dimensional subspace of D a 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq13_HTML.gif. Take u H m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq109_HTML.gif, u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq110_HTML.gif, write u = r m v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq111_HTML.gif with v H m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq112_HTML.gif, v = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq113_HTML.gif and r m = u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq114_HTML.gif. Thus, for 0 < r m < T 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq115_HTML.gif, since all the norms are equivalent, we have
J ˜ ( u ) = J ( u ) = 1 2 Ω | x | 2 a | D u | 2 d x λ 2 Ω | x | 2 b | u | 2 d x λ q Ω | x | α | u | q d x 1 2 u 2 λ C 1 2 u 2 λ C 2 q u q = 1 2 r m 2 λ C 1 2 r m 2 λ C 2 q r m q : = ε m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equu_HTML.gif
Therefore, we can choose r m ( 0 , T 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq116_HTML.gif so small that J ˜ ( u ) ε m < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq117_HTML.gif. Let S r m = { u D a 1 , 2 ( Ω ) : u = r m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq118_HTML.gif, then S r m H m J ˜ ε m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq119_HTML.gif. Hence, γ ( J ˜ ε m ) γ ( S r m H m ) = m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq120_HTML.gif. Denote Γ m = { A Σ : γ ( A ) m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq121_HTML.gif and let
c m = inf A Γ m sup u A J ˜ ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equv_HTML.gif

Then < c m ε m < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq122_HTML.gif because J ˜ ε m Γ m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq123_HTML.gif and J ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif is bounded from below. □

Lemma 3.3 Let λ, β be as in (3) or (4) of Lemma  3.1. Then all c m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq124_HTML.gifare critical values of J ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gifas c m 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq125_HTML.gif.

Proof It is clear that c m c m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq126_HTML.gif, c m < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq127_HTML.gif. Hence, c m c ¯ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq128_HTML.gif. Moreover, since all c m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq124_HTML.gif are critical values of J ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif, we claim that c ¯ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq129_HTML.gif. If c ¯ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq130_HTML.gif, because K c ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq131_HTML.gif is compact and K c ¯ Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq132_HTML.gif, it follows that γ ( K c ¯ ) = N 0 < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq133_HTML.gif and there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq134_HTML.gif such that γ ( K c ¯ ) = γ ( N δ ( K c ¯ ) ) = N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq135_HTML.gif. By the deformation lemma there exist ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq136_HTML.gif ( c ¯ + ε < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq137_HTML.gif) and an odd homeomorphism η such that
η ( J ˜ c ¯ + ε N δ ( K c ¯ ) ) J ˜ c ¯ ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equw_HTML.gif
Since c m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq124_HTML.gif is increasing and converges to c ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq138_HTML.gif, there exists m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq106_HTML.gif such that c m > c ¯ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq139_HTML.gif and c m + N 0 c ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq140_HTML.gif and there exists A Γ m + N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq141_HTML.gif such that sup u A J ˜ ( u ) < c ¯ + ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq142_HTML.gif. By the properties of γ, we have
γ ( A N δ ( K c ¯ ) ¯ ) γ ( A ) γ ( N δ ( K c ¯ ) ) m , γ ( A N δ ( K c ¯ ) ¯ ) m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equx_HTML.gif
Therefore,
η ( A N δ ( K c ¯ ) ¯ ) Γ m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equy_HTML.gif
Consequently,
sup u η ( A N δ ( K c ¯ ) ¯ ) J ˜ ( u ) c m > c ¯ ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equz_HTML.gif

a contradiction, hence c m 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq125_HTML.gif. □

With Lemma 3.1 to Lemma 3.3, we have proved Theorem 1.1.

Declarations

Acknowledgements

Project is supported by National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11226116), the China Scholarship Council, Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK2012109), the Fundamental Research Funds for the Central Universities (No. JUSRP11118, JUSRP211A22) and Foundation for young teachers of Jiangnan University (No. 2008LQN008).

Authors’ Affiliations

(1)
School of Science, Jiangnan University
(2)
School of Mathematical Science, Nanjing Normal University
(3)
Institute of Science, PLA University of Science and Technology

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