Open Access

Existence result for semilinear elliptic systems involving critical exponents

Boundary Value Problems20122012:119

DOI: 10.1186/1687-2770-2012-119

Received: 14 July 2012

Accepted: 4 October 2012

Published: 24 October 2012

Abstract

In this paper we deal with the existence of a positive solution for a class of semilinear systems of multi-singular elliptic equations which involve Sobolev critical exponents. In fact, by the analytic techniques and variational methods, we prove that there exists at least one positive solution for the system.

MSC: 35J60, 35B33.

Keywords

semilinear elliptic system nontrivial solution critical exponent variational method

1 Introduction

We consider the following elliptic system:
{ L u = σ α 2 u | u | α 2 | ν | β + η u | u | 2 2 + a 1 u + a 2 ν , Ω , L ν = σ β 2 ν | ν | β 2 | u | α + λ ν | ν | 2 2 + a 2 u + a 3 ν , Ω , u = ν = 0 , Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ1_HTML.gif
(1.1)

where Ω R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq1_HTML.gif ( N 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq2_HTML.gif) is a smooth bounded domain such that ξ i Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq3_HTML.gif, i = 1 , 2 , , k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq4_HTML.gif, k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq5_HTML.gif, are different points, 0 μ i < μ ¯ : = ( N 2 2 ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq6_HTML.gif, L : = Δ i = 1 k μ i | x ξ i | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq7_HTML.gif, η , λ , σ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq8_HTML.gif, a 1 , a 2 , a 3 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq9_HTML.gif, 1 < α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq10_HTML.gif, β < 2 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq11_HTML.gif, α + β = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq12_HTML.gif.

We work in the product space H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq13_HTML.gif, where the space H : = H 0 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq14_HTML.gif is the completion of C 0 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq15_HTML.gif with respect to the norm ( Ω | | 2 d x ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq16_HTML.gif.

In resent years many publications [13] concerning semilinear elliptic equations involving singular points and the critical Sobolev exponent have appeared. Particularly in the last decade or so, many authors used the variational method and analytic techniques to study the existence of positive solutions of systems of the form of (1.1) or its variations; see, for example, [48].

Before stating the main result, we clarify some terminology. Since our method is variational in nature, we need to define the energy functional of (1.1) on H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq13_HTML.gif
J ( u , ν ) = 1 2 Ω ( | u | 2 + | ν | 2 i = 1 k μ i ( u 2 + ν 2 ) | x ξ i | 2 ) d x 1 2 Ω ( a 1 u 2 + 2 a 2 u ν + a 3 ν 2 ) d x σ σ 2 Ω | u | α | u | β d x η 2 Ω | u | 2 d x λ 2 Ω | ν | 2 d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equa_HTML.gif
Then J ( u , ν ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq17_HTML.gif belongs to C 1 ( H × H , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq18_HTML.gif. A pair of functions ( u 0 , ν 0 ) H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq19_HTML.gif is said to be a solution of (1.1) if ( u 0 , ν 0 ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq20_HTML.gif, and for all ( φ , ϕ ) H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq21_HTML.gif, we have
Ω ( u 0 φ + ν 0 ϕ i = 1 k μ i ( u 0 φ + ν 0 ϕ ) | x ξ i | 2 ( a 1 u 0 φ + a 2 φ ν 0 + a 2 ϕ u 0 + a 3 ν 0 ϕ ) σ ( α φ u 0 | u 0 | α 2 | ν 0 | β + β ϕ ν 0 | ν 0 | β 2 | u 0 | α ) η | u 0 | 2 2 u 0 φ λ | ν 0 | 2 2 ν 0 ϕ ) d x = J ( u 0 , ν 0 ) , ( φ , ϕ ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equb_HTML.gif
Standard elliptic arguments show that
u , ν C 2 ( Ω { ξ 1 , , ξ k } ) C 1 ( Ω ¯ { ξ 1 , , ξ k } ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equc_HTML.gif

The following assumptions are needed:

( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) η + λ + σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq23_HTML.gif, 0 μ 1 μ 2 μ k < μ ¯ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq24_HTML.gif and i = 1 k μ i < μ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq25_HTML.gif, α + β > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq26_HTML.gif, α + β = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq12_HTML.gif, a 1 , a 2 , a 3 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq27_HTML.gif, a 1 a 3 a 2 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq28_HTML.gif,

( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq29_HTML.gif) 0 < λ 1 λ 2 < Λ 1 ( μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq30_HTML.gif, where Λ 1 ( μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq31_HTML.gif is the first eigenvalue of L, λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq32_HTML.gif, λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq33_HTML.gif are the eigenvalues of the matrix A = ( a 1 a 2 a 2 a 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq34_HTML.gif.

The quadratic from Q ( u , ν ) : = ( u , ν ) A ( u , ν ) T = a 1 u 2 + 2 a 2 u ν + a 3 ν 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq35_HTML.gif is positively defined and satisfies
λ 1 ( u 2 + ν 2 ) a 1 u 2 + 2 a 2 u ν + a 3 ν 2 λ 2 ( u 2 + ν 2 ) u , ν H . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ2_HTML.gif
(1.2)

Our main results are as follows.

Theorem 1.1 Suppose ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) holds. Then for any solution ( u , v ) H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq36_HTML.gif of problem (1.1), there exists a positive constant C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq37_HTML.gif such that
max { | u ( x ) | , | ν ( x ) | } C 1 | x ξ i | ( μ ¯ μ ¯ μ i ) , x B ρ 1 ( ξ i ) { ξ i } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equd_HTML.gif

where ρ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq38_HTML.gif and B ρ 1 ( ξ i ) Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq39_HTML.gif.

Theorem 1.2 Suppose ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) holds. Then for any positive solution ( u , v ) H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq36_HTML.gif of problem (1.1), there exists a positive constant C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq40_HTML.gif such that B ρ 2 ( ξ i ) Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq41_HTML.gif and
min { u ( x ) , ν ( x ) } C 2 | x ξ i | ( μ ¯ μ ¯ μ i ) , x B ρ 2 ( ξ i ) { ξ i } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Eque_HTML.gif

where ρ 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq42_HTML.gif.

Theorem 1.3 Suppose ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif), ( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq29_HTML.gif) hold. Then the problem (1.1) has a positive solution.

2 Preliminaries

On H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq43_HTML.gif, we use the norm
( u , ν ) H × H = ( Ω ( | u | 2 + | ν | 2 ) d x ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equf_HTML.gif
Using the Young inequality, the following best constant is well defined:
S μ i : = inf u D 1 , 2 ( R N ) { 0 } R N ( | u | 2 μ i u 2 | x ξ i | 2 ) d x ( R N | u | 2 d x ) 2 / 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ3_HTML.gif
(2.1)

where D 1 , 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq44_HTML.gif is the completion of C 0 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq45_HTML.gif with respect to the norm ( R N | | 2 d x ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq46_HTML.gif.

We infer that S μ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq47_HTML.gif is attained in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq48_HTML.gif by the functions
V μ i , ε ξ i ( x ) = ε 2 N 2 U μ i ( x ξ i ε ) , ε > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equg_HTML.gif
where
U μ i ( x ξ i ) = ( 2 N ( μ ¯ μ i ) μ ¯ ) μ ¯ / 2 | x ξ i | ( μ ¯ μ ¯ μ i ) ( 1 + | x ξ i | 2 μ ¯ μ i μ ¯ ) μ ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equh_HTML.gif
For all η , λ , σ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq8_HTML.gif, η + λ + σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq23_HTML.gif, α + β > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq26_HTML.gif, α + β = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq12_HTML.gif, by the Young and Hardy-Sobolev inequalities, the following constant is well defined on D : = ( D 1 , 2 ( R N ) { 0 } ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq49_HTML.gif:
S η , λ , σ ( μ i ) : = inf ( u , ν ) D R N ( | u | 2 + | ν | 2 μ i u 2 + ν 2 | x ξ i | 2 ) d x ( R N ( η | u | 2 + λ | ν | 2 + σ | u | α | ν | β ) d x ) 2 / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ4_HTML.gif
(2.2)
Set
u μ , ε ξ ( x ) = ψ ( x ) V μ , ε ξ ( x ) = ε 2 N 2 ψ ( x ) U μ ( x ξ ε ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equi_HTML.gif
where ξ Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq50_HTML.gif, 0 μ < μ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq51_HTML.gif, ψ C 0 ( B ρ ( ξ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq52_HTML.gif satisfies 0 ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq53_HTML.gif and ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq54_HTML.gif, x B ρ 2 ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq55_HTML.gif, for all ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq56_HTML.gif small. Then for any 0 < μ < μ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq57_HTML.gif, by [9] we have the following estimates:
Ω ( | u μ , ε ξ | 2 μ ( u μ , ε ξ ) 2 | x ξ | 2 ) d x = S μ N 2 + o ( ε 2 μ ¯ μ ) , Ω | u μ , ε ξ | 2 d x = S μ N 2 + o ( ε 2 N N 2 μ ¯ μ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equj_HTML.gif
and for any a R N { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq58_HTML.gif,
Ω | u μ , ε 0 | 2 | x + ξ | 2 d x = { ε 2 | ξ | 2 R N U μ 2 ( x ) d x + o ( ε 2 ) if  μ < μ ¯ 1 , C μ 2 ω N | ξ | 2 ε 2 | log ε | + o ( ε 2 ) if  μ = μ ¯ 1 , Ω | u μ , ε ξ | 2 d x = { o 1 ( ε 2 ) if  0 μ < μ ¯ 1 , o 1 ( ε 2 | log ε | ) if  μ = μ ¯ 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equk_HTML.gif

where C μ = ( 4 N ( μ ¯ μ ) N 2 ) N 2 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq59_HTML.gif, ω N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq60_HTML.gif is the volume of the unit ball in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq48_HTML.gif.

3 Asymptotic behavior of solutions

Proof of Theorem 1.1 Suppose ( u 0 , ν 0 ) H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq61_HTML.gif is a nontrivial solution to problem (1.1). For all 0 μ i μ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq62_HTML.gif define
u ( x ) = | x ξ i | γ u 0 ( x ) and ν ( x ) = | x ξ i | γ ν 0 ( x ) , where  γ = ( μ ¯ μ ¯ μ i ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equl_HTML.gif
It is not difficult to verify that u , ν H 0 1 ( Ω , | x ξ i | 2 γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq63_HTML.gif and satisfy
{ div ( | x ξ i | 2 γ u ) = σ α 2 | x ξ i | 2 γ u | u | α 2 | ν | β + η | x ξ i | 2 γ u | u | 2 2 + a 1 | x ξ i | 2 γ u + a 2 | x ξ i | 2 γ ν + j = 1 j i k μ j | x ξ j | 2 | x ξ i | 2 γ u , div ( | x ξ i | 2 γ ν ) = σ β 2 | x ξ i | 2 γ ν | ν | β 2 | u | α + λ | x ξ i | 2 γ ν | ν | 2 2 + a 2 | x ξ i | 2 γ u + a 3 | x ξ i | 2 γ ν + j = 1 j i k μ j | x ξ j | 2 | x ξ i | 2 γ ν . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ5_HTML.gif
(3.1)
Let R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq64_HTML.gif small enough such that B R ( ξ i ) Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq65_HTML.gif and ξ i B R ( ξ i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq66_HTML.gif for j i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq67_HTML.gif. Also, let φ i C 0 ( B R ( ξ i ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq68_HTML.gif be a cut-off function. Set
u n : = min { | u | , n } ; ν n : = min { | ν | , n } ; ϕ 1 i : = φ i 2 u u n 2 ( s 1 ) , ϕ 2 i : = φ i 2 ν ν n 2 ( s 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equm_HTML.gif
where s , n > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq69_HTML.gif. Multiplying the first equation of (3.1) by ϕ 1 i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq70_HTML.gif and the second one by ϕ 2 i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq71_HTML.gif respectively and integrating, we have
Ω | x ξ i | 2 γ u ϕ 1 i = σ α 2 Ω | x ξ i | 2 γ u | u | α 2 | ν | β ϕ 1 i + η Ω | x ξ i | 2 γ × u | u | 2 2 ϕ 1 i + a 1 Ω | x ξ i | 2 γ u ϕ 1 i + a 2 Ω | x ξ i | 2 γ × ν ϕ 1 i + j = 1 j i k μ j | x ξ j | 2 | x ξ i | 2 γ u ϕ 1 i . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equn_HTML.gif

Note that ϕ 1 i = 2 φ i u u n 2 ( s 1 ) φ i + φ i 2 u n 2 ( s 1 ) u + 2 ( s 1 ) φ i 2 u n 2 ( s 1 ) u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq72_HTML.gif.

Then
Ω | x ξ i | 2 γ u ϕ 1 i = 2 Ω | x ξ i | 2 γ φ i u u n 2 ( s 1 ) φ i u + Ω | x ξ i | 2 γ φ i 2 × u n 2 ( s 1 ) | u | 2 + 2 ( s 1 ) Ω | x ξ i | 2 γ φ i 2 u n 2 ( s 1 ) | u n | 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equo_HTML.gif
By the Cauchy inequality and the Young inequality, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ6_HTML.gif
(3.2)

The same result holds for Ω | x ξ i | 2 γ ϕ 2 i ν https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq73_HTML.gif.

By letting ψ 1 ( x ) = φ i u u n ( s 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq74_HTML.gif, ψ 2 ( x ) = φ i ν ν n ( s 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq75_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ7_HTML.gif
(3.3)
Using Caffarelli-Kohn-Nirenberg inequality [10], we infer that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ8_HTML.gif
(3.4)
Define
ω ( x ) : = max { u ( x ) , ν ( x ) } , ω n ( x ) : = min { ω ( x ) , n } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equp_HTML.gif
Then ω n ( x ) : = max { u n ( x ) , ν n ( x ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq76_HTML.gif. Now, from the Hölder inequality, we deduce that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ9_HTML.gif
(3.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ10_HTML.gif
(3.6)
In the sequel, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ11_HTML.gif
(3.7)
By the choice of φ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq77_HTML.gif, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ12_HTML.gif
(3.8)
So, from (3.4) to (3.8) it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ13_HTML.gif
(3.9)
Take s = 2 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq78_HTML.gif and φ i ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq79_HTML.gif to be a constant near the zero. Letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq80_HTML.gif, we infer that ω L 2 2 2 ( B R ( ξ i ) , | x ξ i | 2 γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq81_HTML.gif and so
u , ν L 2 2 2 ( B R ( ξ i ) , | x ξ i | 2 γ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ14_HTML.gif
(3.10)

Suppose r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq82_HTML.gif is sufficiently small such that r + l < R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq83_HTML.gif and φ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq77_HTML.gif is a cut-off function with the properties | φ i | < 1 l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq84_HTML.gif and φ i ( x ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq85_HTML.gif in B r ( ξ i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq86_HTML.gif.

Set t : = 2 2 2 ( 2 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq87_HTML.gif, δ : = 2 γ t 2 γ ( t 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq88_HTML.gif.

Then we have the following results:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ15_HTML.gif
(3.11)
where we used the Hölder inequality. From (3.9) in combination with (3.11), it follows that
( B r + l ( ξ i ) | x ξ i | 2 γ | ω | 2 s ) 1 2 s C 1 2 s l 1 2 s ( B r + l ( ξ i ) | x ξ i | 2 γ | ω | p ¯ 0 s ) 1 p ¯ 0 s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ16_HTML.gif
(3.12)

where p ¯ 0 = 2 t t 1 < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq89_HTML.gif.

Denote s = χ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq90_HTML.gif, χ = p p ¯ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq91_HTML.gif and l = ρ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq92_HTML.gif, j 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq93_HTML.gif, where χ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq94_HTML.gif, 2 γ < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq95_HTML.gif and p ¯ 0 χ j = p χ j 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq96_HTML.gif. Using (3.12) recursively, we get
( B r ( ξ i ) | ω | 2 χ j ) 1 2 χ j r γ χ j ( B r ( ξ i ) | ω | 2 χ j | x ξ i | 2 γ ) 1 2 χ j r γ χ j C i = 1 j 1 2 χ j ρ i = 1 j i χ i ( B r + ρ ( ξ i ) | x ξ i | 2 γ | ω | 2 ) 1 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equq_HTML.gif
we have χ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq97_HTML.gif as j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq98_HTML.gif. Note that the infinite sums on the right-hand side converge, then we obtain that ω L ( B r ( ξ i ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq99_HTML.gif, particularly, we have u , ν L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq100_HTML.gif. Thus,
u 0 ( x ) = | x ξ i | γ u ( x ) M 1 | x ξ i | γ for  x B r ( ξ i ) { ξ i } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equr_HTML.gif
where M 1 = max { u L ( B r ( ξ i ) ) , 1 i k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq101_HTML.gif.
ν 0 ( x ) = | x ξ i | γ ν ( x ) M 2 | x ξ i | γ for  x B r ( ξ i ) { ξ i } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equs_HTML.gif

where M 1 = max { ν L ( B r ( ξ i ) ) , 1 i k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq102_HTML.gif. The proof is complete. □

Proof of Theorem 1.2 Suppose ( u 0 , ν 0 ) H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq61_HTML.gif is a positive solution to problem (1.1). For all 0 μ i μ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq62_HTML.gif, set
u ( x ) = | x ξ i | γ u 0 ( x ) and ν ( x ) = | x ξ i | γ ν 0 ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equt_HTML.gif
Then
{ div ( | x ξ i | 2 γ u ) = σ α 2 | x ξ i | 2 γ u α 1 ν β + η | x ξ i | 2 γ u 2 1 + a 1 | x ξ i | 2 γ u + a 2 | x ξ i | 2 γ ν + j = 1 j i k μ j | x ξ j | 2 | x ξ i | 2 γ u , div ( | x ξ i | 2 γ ν ) = σ β 2 | x ξ i | 2 γ ν β 1 u α + λ | x ξ i | 2 γ ν 2 1 + a 2 | x ξ i | 2 γ u + a 3 | x ξ i | 2 γ ν + j = 1 j i k μ j | x ξ j | 2 | x ξ i | 2 γ ν . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ17_HTML.gif
(3.13)
Choose 0 < ρ 0 < ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq103_HTML.gif and define n ( t ) = min | x ξ i | = t u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq104_HTML.gif for ρ 0 t ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq105_HTML.gif. Let
n ( ρ 0 ) = A ρ 0 2 μ ¯ μ i + B , n ( ρ ) = A ρ 2 μ ¯ μ i + B , where A = n ( ρ ) n ( ρ 0 ) ρ 2 μ ¯ μ i ρ 0 2 μ ¯ μ i , B = n ( ρ 0 ) ρ 2 μ ¯ μ i n ( ρ ) ρ 0 2 μ ¯ μ i ρ 2 μ ¯ μ i ρ 0 2 μ ¯ μ i . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equu_HTML.gif
It is easy to verify that
div ( | x ξ i | 2 ( μ ¯ μ ¯ μ i ) ( A | x ξ i | 2 μ ¯ μ i + B ) ) = 0 x Ω { ξ i } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ18_HTML.gif
(3.14)
Combining (3.13) with (3.14), we get
div ( | x ξ i | 2 ( μ ¯ μ ¯ μ i ) ( u A | x ξ i | 2 μ ¯ μ i + B ) ) 0 , x B ρ ( ξ i ) B ρ 0 ( ξ i ) , u ( x ) ( A | x ξ i | 2 μ ¯ μ i + B ) 0 , x ( B ρ ( ξ i ) B ρ 0 ( ξ i ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equv_HTML.gif
Therefore, by the maximum principle in B ρ ( ξ i ) B ρ 0 ( ξ i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq106_HTML.gif, we obtain
u ( x ) ( A | x ξ i | 2 μ ¯ μ i + B ) 0 , x B ρ ( ξ i ) B ρ 0 ( ξ i ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equw_HTML.gif
Thus, for all x B ρ ( ξ i ) B ρ 0 ( ξ i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq107_HTML.gif,
u ( x ) A | x ξ i | 2 μ ¯ μ i + B = | x ξ i | 2 μ ¯ μ i ρ 2 μ ¯ μ i ρ 0 2 μ ¯ μ i ρ 2 μ ¯ μ i n ( ρ 0 ) + ρ 0 2 μ ¯ μ i | x ξ i | 2 μ ¯ μ i ρ 0 2 μ ¯ μ i ρ 2 μ ¯ μ i n ( ρ ) | x ξ i | 2 μ ¯ μ i ρ 0 2 μ ¯ μ i | x ξ i | 2 μ ¯ μ i ρ 0 2 μ ¯ μ i ρ 2 μ ¯ μ i | x ξ i | 2 μ ¯ μ i n ( ρ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equx_HTML.gif

Taking ρ 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq108_HTML.gif, we conclude u ( x ) n ( ρ ) = min | x ξ i | = ρ u ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq109_HTML.gif for all x B ρ ( ξ i ) { ξ i } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq110_HTML.gif.

Similar result also holds for ν ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq111_HTML.gif. Therefore, we have
u 0 ( x ) = | x ξ i | γ u ( x ) | x ξ i | γ min | x ξ i | = ρ u ( x ) = | x ξ i | γ C i | x ξ i | γ min i = 1 , 2 , , k C i = | x ξ i | γ N 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equy_HTML.gif
For any x B ρ ( ξ i ) { ξ i } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq110_HTML.gif,
ν 0 ( x ) = | x ξ i | γ ν ( x ) | x ξ i | γ min | x ξ i | = ρ ν ( x ) = | x ξ i | γ C i ´ | x ξ i | γ min i = 1 , 2 , , k C i ´ = | x ξ i | γ N 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equz_HTML.gif

For any x B ρ ( ξ i ) { ξ i } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq110_HTML.gif. This proves the theorem. □

4 Local ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq112_HTML.gif-condition and the existence of positive solutions

We first establish a compactness result.

Lemma 4.1 Suppose that ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) holds. Then J satisfies the ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq112_HTML.gif-condition for all
c < c : = 1 N min { S η , λ , σ N 2 ( μ 1 ) , , S η , λ , σ N 2 ( μ k ) , ( S 0 ) N 2 } = 1 N S η , λ , σ N 2 ( μ k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equaa_HTML.gif

Proof Suppose that { ( u n , ν n ) } H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq113_HTML.gif satisfies J ( u n , ν n ) c < c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq114_HTML.gif and J ( u n , ν n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq115_HTML.gif. The standard argument shows that { ( u n , ν n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq116_HTML.gif is bounded in H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq13_HTML.gif.

For some ( u , ν ) H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq117_HTML.gif, we have
( u n , ν n ) ( u , ν ) weakly in  H × H , ( u n , ν n ) ( u , ν ) weakly in  L 2 ( Ω , | x ξ i | 2 ) × L 2 ( Ω , | x ξ i | 2 ) , ( u n , ν n ) ( u , ν ) weakly in  L 2 ( Ω ) × L 2 ( Ω ) , ( u n , ν n ) ( u , ν ) strongly in  L q 1 ( Ω ) × L q 2 ( Ω ) , q 1 , q 2 [ 1 , 2 ) , ( u n , ν n ) ( u , ν ) a.e. in  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equab_HTML.gif
Therefore, ( u , ν ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq118_HTML.gif is a solution to (1.1). Then by the concentration-compactness principle [1113] and up to a subsequence, there exist an at most countable set J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq119_HTML.gif, a set of different points { x j } j J Ω ξ i i = 1 k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq120_HTML.gif, nonnegative real numbers τ ˜ x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq121_HTML.gif, ν ˜ x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq122_HTML.gif, j J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq123_HTML.gif, and τ ˜ ξ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq124_HTML.gif, ν ˜ ξ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq125_HTML.gif, γ ˜ ξ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq126_HTML.gif ( 1 i k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq127_HTML.gif) such that the following convergence holds in the sense of measures:
| u n | 2 + | ν n | 2 d τ ˜ | u | 2 + | ν | 2 + j J τ ˜ x j δ x j + i = 1 k τ ˜ ξ i δ ξ i , u n 2 + ν n 2 | x ξ i | 2 d γ ˜ = u 2 + ν 2 | x ξ i | 2 + γ ˜ ξ i δ ξ i , η | u n | 2 + λ | ν n | 2 + σ | u n | α | ν n | β d ν ˜ = η | u | 2 + λ | ν | 2 + σ | u | α | ν | β + j J ν ˜ x j δ x j + i = 1 k ν ˜ ξ i δ ξ i . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equac_HTML.gif
By the Sobolev inequalities [10], we have
S μ i ν ˜ ξ i 2 2 τ ˜ ξ i μ i γ ˜ ξ i , 1 i k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ19_HTML.gif
(4.1)

We claim that J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq119_HTML.gif is finite, and for any j J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq123_HTML.gif, ν ˜ x j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq128_HTML.gif or ν ˜ x j S 0 N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq129_HTML.gif.

In fact, let ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq130_HTML.gif be small enough for any 1 i k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq127_HTML.gif, ξ i B ε ( x j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq131_HTML.gif and B ε ( x i ) B ε ( x j ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq132_HTML.gif for i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq133_HTML.gif, i , j J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq134_HTML.gif. Let ϕ ε j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq135_HTML.gif be a smooth cut-off function centered at x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq136_HTML.gif such that 0 ϕ ε j 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq137_HTML.gif, ϕ ε j = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq138_HTML.gif for | x x j | ε 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq139_HTML.gif, ϕ ε j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq140_HTML.gif for | x x j | ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq141_HTML.gif and | ϕ ε j | 4 ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq142_HTML.gif. Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equad_HTML.gif
Then we have
0 = lim ε 0 lim n J λ ( u n , ν n ) , ( u n ϕ ε j , ν n ϕ ε j ) τ ˜ x j ν ˜ x j . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equae_HTML.gif

By the Sobolev inequality, S 0 ν ˜ x j 2 2 τ ˜ x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq143_HTML.gif; and then we deduce that ν ˜ x j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq128_HTML.gif or ν ˜ x j S 0 N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq129_HTML.gif, which implies that J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq119_HTML.gif is finite.

Now, we consider the possibility of concentration at points ξ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq144_HTML.gif ( 1 i k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq127_HTML.gif), for ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq130_HTML.gif small enough that x j B ε ( ξ i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq145_HTML.gif for all j J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq123_HTML.gif and B ε ( ξ i ) B ε ( ξ j ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq146_HTML.gif for i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq133_HTML.gif and 1 i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq147_HTML.gif, j k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq148_HTML.gif. Let φ ε i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq149_HTML.gif be a smooth cut-off function centered at ξ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq144_HTML.gif such that 0 φ ε i 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq150_HTML.gif, φ ε i = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq151_HTML.gif for | x ξ i | ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq152_HTML.gif and | φ ε i | 4 ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq153_HTML.gif. Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equaf_HTML.gif
Thus, we have
0 = lim ε 0 lim n J λ ( u n , ν n ) , ( u n φ ε i , ν n φ ε i ) τ ˜ ξ i μ i γ ˜ ξ i ν ˜ ξ i . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ20_HTML.gif
(4.2)
From (4.1) and (4.2) we derive that S μ i ν ˜ ξ i 2 2 ν ˜ ξ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq154_HTML.gif, 1 i k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq127_HTML.gif, and then either ν ˜ ξ i = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq155_HTML.gif or ν ˜ ξ i S μ i N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq156_HTML.gif. On the other hand, from the above arguments, we conclude that
c = lim n ( J ( u n , ν n ) 1 2 J ( u n , ν n ) , ( u n , ν n ) ) = 1 N lim n Ω ( η | u n | 2 + λ | ν n | 2 + σ | u n | α | ν n | β ) d x = 1 N ( Ω ( η | u | 2 + λ | ν | 2 + σ | u | α | ν | β ) d x + j J ν ˜ x j + i = 1 k ν ˜ ξ i ) = 1 N ( j J ν ˜ x j + i = 1 k ν ˜ ξ i ) + J ( u , ν ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equag_HTML.gif
If ν ˜ ξ i = ν ˜ x j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq157_HTML.gif for all i { 1 , , k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq158_HTML.gif and j J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq123_HTML.gif, then c = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq159_HTML.gif, which contradicts the assumption that c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq160_HTML.gif. On the other hand, if there exists an i { 1 , , k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq158_HTML.gif such that ν ˜ ξ i 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq161_HTML.gif or there exists a j J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq123_HTML.gif with ν ˜ x j 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq162_HTML.gif, then we infer that
c 1 N min { ( S η , λ , σ ( 0 ) ) N / 2 , ( S η , λ , σ ( μ 1 ) ) N / 2 , , ( S η , λ , σ ( μ k ) ) N / 2 } = 1 N ( S η , λ , σ ( μ k ) ) N / 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equah_HTML.gif

which contradicts our assumptions. Hence, ( u n , ν n ) ( u , ν ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq163_HTML.gif, as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq164_HTML.gif in H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq43_HTML.gif. □

First, under the assumptions ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif), ( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq29_HTML.gif), we have the following notations:
f η , λ , σ ( τ ) = ( 1 + τ 2 ) S μ k ( η + σ τ β + λ τ 2 ) 2 2 , τ > 0 ; f η , λ , σ ( τ min ) : = min τ > 0 f η , λ , σ ( τ ) > 0 , σ > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equai_HTML.gif
where τ min > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq165_HTML.gif is a minimal point of f η , λ , σ ( τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq166_HTML.gif, and therefore a root of the equation
α σ τ β σ β τ β 2 2 λ τ 2 2 + 2 η = 0 , τ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equaj_HTML.gif
Lemma 4.2 Suppose that ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) holds. Then we have
  1. (i)

    S η , λ , σ ( μ ) = f η , λ , σ ( τ min ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq167_HTML.gif

     
  2. (ii)

    S η , λ , σ ( μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq168_HTML.gif has the minimizers ( V μ , ε ξ ( x ) , τ min V μ , ε ξ ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq169_HTML.gif, ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq170_HTML.gif, where V μ , ε ξ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq171_HTML.gif are the extremal functions of S η , λ , σ ( μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq172_HTML.gif defined as in (2.2).

     

Proof The argument is similar to that of [6]. □

Lemma 4.3 Under the assumptions of ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif), we have
sup J ( t u ε , μ k , t ( τ min u ε , μ k ) ) < c = 1 N ( S η , λ , σ ( μ k ) ) N / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equak_HTML.gif
Proof Suppose ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) holds. Define the function
g ( t ) : = J ( t u ε , μ k , t ( τ min u ε , μ k ) ) , t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equal_HTML.gif
Note that lim t + g ( t ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq173_HTML.gif and g ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq174_HTML.gif as t is close to 0. Thus, sup t 0 g ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq175_HTML.gif is attained at some finite t ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq176_HTML.gif with g ( t ε ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq177_HTML.gif. Furthermore, c < t ε < c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq178_HTML.gif, where c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq179_HTML.gif and c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq180_HTML.gif are the positive constants independent of ε. By using (1.2), we have
g ( t ) t 2 2 ( 1 + τ min 2 ) ( Ω ( | u ε , μ k | 2 μ k u ε , μ k 2 | x ξ i | 2 λ 1 u ε , μ k 2 ) d x ) t 2 2 ( σ τ min β + η + λ τ min 2 ) Ω | u ε , μ k | 2 d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equam_HTML.gif
Note that
max ( t 2 2 B 1 t 2 2 B 2 ) = 1 N ( B 1 B 2 2 / 2 ) N / 2 , B 1 > 0 , B 2 > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ21_HTML.gif
(4.3)

and 0 μ μ ¯ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq181_HTML.gif and so 2 < 2 μ ¯ μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq182_HTML.gif.

From (4.3), Lemma 4.2 and Lemma 4.3, it follows that
g ( t ε ) 1 N ( ( 1 + τ min 2 ) Ω ( | u ε , μ k | 2 μ k u ε , μ k 2 | x ξ k | 2 λ 1 u ε , μ k 2 ) d x ( ( σ τ min β + η + λ τ min 2 ) Ω | u ε , μ k | 2 d x ) 2 / 2 ) N / 2 1 N ( f ( τ min ) S ( μ k ) × ( S ( μ k ) ) N / 2 + O ( ε 2 μ ¯ μ ) C ε 2 ( S ( μ k ) ) ( N 2 ) / 2 + O ( ε 2 μ ¯ μ ) ) 1 N ( f ( τ min ) ) N / 2 + O ( ε 2 μ ¯ μ ) C ε 2 < 1 N ( S η , λ , σ ( μ k ) ) N / 2 = c , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equan_HTML.gif
so g ( t ε ) < c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq183_HTML.gif. Hence, g ( t ε ) < c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq183_HTML.gif, t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq184_HTML.gif and
sup t 0 g ( t ) = sup t 0 J ( t u ε , μ k , t ( τ min u ε , μ k ) ) < c , if  μ < μ ¯ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ22_HTML.gif
(4.4)

 □

Proof of Theorem 1.3 Set c : = inf h Γ max t [ 0 , 1 ] J ( h ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq185_HTML.gif, where
Γ = { h C ( [ 0 , 1 ] , H × H ) | h ( 0 ) = ( 0 , 0 ) , J ( h ( 1 ) ) < 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equao_HTML.gif
Suppose that ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) holds. For all ( u , ν ) H × H { ( 0 , 0 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq186_HTML.gif, from the Young and Hardy-Sobolev inequalities, it follows that
J ( u , ν ) C ( u 2 + ν 2 ) C ( u 2 + ν 2 ) C ( u , ν ) 2 C ( u , ν ) 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equap_HTML.gif
and there exists a constant ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq56_HTML.gif small such that
b : = inf ( u , ν ) = ρ J ( u , ν ) > 0 = J ( 0 , 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equaq_HTML.gif

Since J ( t u , t ν ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq187_HTML.gif as t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq188_HTML.gif, there exists t 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq189_HTML.gif such that ( t 0 u , t 0 ν ) > ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq190_HTML.gif and J ( t 0 u , t 0 ν ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq191_HTML.gif. By the mountain-pass theorem [14], there exists a sequence { ( u n , ν n ) } H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq192_HTML.gif such that J ( u n , ν n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq193_HTML.gif and J ( u n , ν n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq194_HTML.gif, as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq164_HTML.gif.

From Lemma 4.2 it follows that
0 < c sup t [ 0 , 1 ] J ( t t 0 u ε , μ k , t t 0 τ min u ε , μ k ) sup t 0 J ( t u ε , μ k , t τ min u ε , μ k ) < c . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equar_HTML.gif

By Lemma 4.1 there exists a subsequence of { ( u n , ν n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq116_HTML.gif, still denoted by { ( u n , ν n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq116_HTML.gif, such that ( u n , ν n ) ( u , ν ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq195_HTML.gif strongly in H × H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq43_HTML.gif. Thus, we get a critical point ( u , ν ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq118_HTML.gif of J satisfying (1.1), and c is a critical value. Set u + = max { u , 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq196_HTML.gif.

Replacing respectively u, ν with u + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq197_HTML.gif and ν + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq198_HTML.gif in terms of the right-hand side of (1.1) and repeating the above process, we can get a nonnegative nontrivial solution ( u , ν ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq118_HTML.gif of (1.1). If u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq199_HTML.gif, we get ν 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq200_HTML.gif by (1.1) and the assumption a 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq201_HTML.gif. Similarly, if ν 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq200_HTML.gif, we also have u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq199_HTML.gif. There, u , ν 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq202_HTML.gif. From the maximum principle, it follows that u , ν > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq203_HTML.gif in Ω. □

Declarations

Authors’ Affiliations

(1)
Department of Basic Sciences, Babol University of Technology

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