Existence and multiplicity of positive solutions to a perturbed singular elliptic system deriving from a strongly coupled critical potential

Boundary Value Problems20122012:116

DOI: 10.1186/1687-2770-2012-116

Received: 12 March 2012

Accepted: 3 October 2012

Published: 17 October 2012

Abstract

In this paper, we consider singular elliptic systems involving a strongly coupled critical potential and concave nonlinearities. By using variational methods and analytical techniques, the existence and multiplicity of positive solutions to the system are established.

MSC: 35J60, 35B33.

Keywords

Palais-Smale condition Nehari manifold strongly coupled elliptic system critical potential

1 Introduction and main results

In this paper, we consider the following elliptic system:
{ L u = η 1 α 1 2 | u | α 1 2 | v | β 1 u + η 2 α 2 2 | u | α 2 2 | v | β 2 u + σ 1 | u | q 2 u , L v = η 1 β 1 2 | u | α 1 | v | β 1 2 v + η 2 β 2 2 | u | α 2 | v | β 2 2 v + σ 2 | v | q 2 v , u , v H 0 1 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ1_HTML.gif
(1.1)

where Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq1_HTML.gif is a smooth bounded domain such that 0 Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq2_HTML.gif, L : = ( Δ μ | x | 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq3_HTML.gif, 2 : = 2 N N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq4_HTML.gif is the critical Sobolev exponent, μ ¯ : = ( N 2 2 ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq5_HTML.gif is the best Hardy constant and H : = H 0 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq6_HTML.gif denotes the completion of C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq7_HTML.gif with respect to the norm u 0 = ( Ω | u | 2 d x ) 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq8_HTML.gif and H μ = H μ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq9_HTML.gif is defined as the completion of the C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq7_HTML.gif with respect to the norm defined by u μ = ( Ω ( | u | 2 μ u 2 | x | 2 ) d x ) 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq10_HTML.gif for μ < μ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq11_HTML.gif.

Definitions of strongly and weakly coupled terms are as follows.

The terms | u | α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq12_HTML.gif and | v | β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq13_HTML.gif ( α , β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq14_HTML.gif) are weakly coupled, | L u | α | K v | β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq15_HTML.gif ( α , β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq14_HTML.gif) is strongly coupled when L or K is a derivative operator. Thus, η 1 | u | α 1 | v | β 1 + η 2 | u | α 2 | v | β 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq16_HTML.gif is strongly coupled when η 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq17_HTML.gif and η 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq18_HTML.gif are positive.

The parameters in (1.1) satisfy the following assumption.

(ℋ) N 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq19_HTML.gif, 0 μ < μ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq20_HTML.gif, 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq21_HTML.gif, η 1 + η 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq22_HTML.gif, 0 η i < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq23_HTML.gif, σ i > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq24_HTML.gif, α i , β i > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq25_HTML.gif, α i + β i = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq26_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq27_HTML.gif.

The corresponding energy functional of (1.1) is defined in H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq28_HTML.gif by
J ( u , v ) : = 1 2 Ω ( | u | 2 + | v | 2 μ | u | 2 + | v | 2 | x | 2 ) d x 1 2 Q ( u , v ) 1 q K ( u , v ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equa_HTML.gif
where Q ( u , v ) : = Ω ( η 1 | u | α 1 | v | β 1 + η 2 | u | α 2 | v | β 2 ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq29_HTML.gif and K ( u , v ) : = Ω ( σ 1 | u | q + σ 2 | v | q ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq30_HTML.gif. Then J C 1 ( H × H , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq31_HTML.gif and the duality product between H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq28_HTML.gif and its dual space ( H × H ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq32_HTML.gif is defined as
J ( u , v ) , ( ϕ 1 , ϕ 2 ) : = Ω ( u ϕ 1 + v ϕ 2 μ u ϕ 1 + v ϕ 2 | x | 2 ) d x Ω ( η 1 α 1 2 | u | α 1 2 | v | β 1 u ϕ 1 + η 2 α 2 2 | u | α 2 2 | v | β 2 u ϕ 1 ) d x Ω ( η 1 β 1 2 | u | α 1 | v | β 1 2 v ϕ 2 + η 2 β 2 2 | u | α 2 | v | β 2 2 v ϕ 2 ) d x Ω ( σ 1 | u | q 2 u ϕ 1 + σ 2 | v | q 2 v ϕ 2 ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equb_HTML.gif
where u , v , ϕ 1 , ϕ 2 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq33_HTML.gif and J ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq34_HTML.gif denotes the Fréchet derivative of J at ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq35_HTML.gif. A pair of functions ( u , v ) H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq36_HTML.gif is said to be a weak solution of (1.1) if
( u , v ) ( 0 , 0 ) , J ( u , v ) , ( ϕ 1 , ϕ 2 ) = 0 , ( ϕ 1 , ϕ 2 ) H × H . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equc_HTML.gif

Therefore, a weak solution of (1.1) is equivalent to a nonzero critical point of J ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq37_HTML.gif [1].

Problem (1.1) is related to the well-known Hardy inequality [2]
R N | u | 2 | x | 2 d x 1 μ ¯ R N | u | 2 d x , u C 0 ( R N ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ2_HTML.gif
(1.2)
If μ < μ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq11_HTML.gif, by (1.2), Ω ( | u | 2 μ u 2 | x | 2 ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq38_HTML.gif is an equivalent norm of H, the operator L is positive and the first eigenvalue Λ 1 ( μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq39_HTML.gif of L and the following best constant are well defined:
S ( μ ) : = inf u D 1 , 2 ( R N ) { 0 } R N ( | u | 2 μ u 2 | x | 2 ) d x ( R N | u | 2 d x ) 2 2 , μ ( , μ ¯ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ3_HTML.gif
(1.3)
where D 1 , 2 ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq40_HTML.gif is the completion of C 0 ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq41_HTML.gif with respect to ( R N | u | 2 d x ) 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq42_HTML.gif. Note that S ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq43_HTML.gif is the well-known best Sobolev constant. For 0 μ < μ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq20_HTML.gif, the constant S ( μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq44_HTML.gif is achieved by the following extremal functions [3]:
V μ , ε ( x ) : = ε 2 N 2 U μ ( ε 1 x ) , ε > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ4_HTML.gif
(1.4)
where U μ ( x ) = U μ ( | x | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq45_HTML.gif is a radially symmetric function
U μ ( x ) = ( 2 N ( μ ¯ μ ) μ ¯ ) μ ¯ 2 ( | x | μ ¯ μ ¯ μ μ ¯ + | x | μ ¯ + μ ¯ μ μ ¯ ) μ ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equd_HTML.gif
On the other hand, for any μ < μ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq11_HTML.gif, η 1 + η 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq22_HTML.gif, 0 η i < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq46_HTML.gif, α i , β i > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq47_HTML.gif and α i + β i = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq26_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq27_HTML.gif, by the Young and Sobolev inequalities, the following best constants are well defined on the space D = ( D 1 , 2 ( R N ) { 0 } ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq48_HTML.gif:
S η , β ( μ ) : = inf ( u , v ) D R N ( | u | 2 + | v | 2 μ | u | 2 + | v | 2 | x | 2 ) d x ( R N ( η 1 | u | α 1 | v | β 1 + η 2 | u | α 2 | v | β 2 ) d x ) 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ5_HTML.gif
(1.5)
We define
f ( τ ) : = 1 + τ 2 ( η 1 τ β 1 + η 2 τ β 2 ) 2 2 , τ > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ6_HTML.gif
(1.6)
Since f is a continuous function on ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq49_HTML.gif such that lim τ 0 + f ( τ ) = lim τ + f ( τ ) = + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq50_HTML.gif. Then there exists τ 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq51_HTML.gif such that
f ( τ 0 ) : = min τ > 0 f ( τ ) > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ7_HTML.gif
(1.7)

Set α 1 = β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq52_HTML.gif, α 2 = β 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq53_HTML.gif, σ 1 = σ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq54_HTML.gif and u = v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq55_HTML.gif. Then (1.1) reduces to the semilinear scalar problems that have been extensively investigated by many authors. See [46] and the references therein.

Regular semilinear elliptic systems have been studied extensively and many conclusions have been established. For example, Alves et al. studied in [7] an elliptic system and some important conclusions had been obtained. However, the elliptic systems involving the Hardy inequality have seldom been studied and we only find some results in [816]. Thus it is necessary for us to investigate the related singular systems deeply. Among the references above, the elliptic systems involving the Hardy inequality and concave-convex nonlinearities had been studied only in [12]. In this paper, only the case 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq21_HTML.gif of (1.1) involving multiple strongly-coupled critical terms is considered.

Let | Ω | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq56_HTML.gif be the Lebesgue measure of Ω. We define the following constant:
ϒ 1 : = ( 2 q 2 2 Λ 1 q 2 | Ω | 1 q 2 ) 1 ( 2 q 2 q ( S η , β ( μ ) ) 2 2 ) 2 q 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ8_HTML.gif
(1.8)

Then the main results of this paper can be concluded in the following theorems and the conclusions are new to the best of our knowledge. It can be verified that the intervals in Theorems 1.1 and 1.2 for the parameters σ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq57_HTML.gif, σ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq58_HTML.gif, μ and q are allowable.

Theorem 1.1 Suppose that (ℋ) holds and σ 1 + σ 2 < ϒ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq59_HTML.gif. Then problem (1.1) has at least one positive solution.

Theorem 1.2 Suppose that (ℋ) holds, N > 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq60_HTML.gif, μ < μ ¯ ( N q μ ¯ ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq61_HTML.gif and N N 2 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq62_HTML.gif. Then there exists ϒ 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq63_HTML.gif such that problem (1.1) has at least two positive solutions for all σ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq57_HTML.gif and σ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq58_HTML.gif satisfying σ 1 + σ 2 < ϒ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq64_HTML.gif.

This paper is organized as follows. Some preliminary results and properties of the Nehari manifold are established in Sections 2 and 3, and Theorems 1.1 and 1.2 are proved in Section 4.

2 The local Palais-Smale condition

Throughout this paper, we always assume that the assumption (ℋ) holds, u H : = u μ = ( Ω ( | u | 2 μ | u | 2 | x | 2 ) d x ) 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq65_HTML.gif denotes the norm of the space H, by the Hardy inequality μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq66_HTML.gif is equivalent to 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq67_HTML.gif, i.e.,
( 1 1 μ ¯ max ( μ , 0 ) ) 1 / 2 u 0 u μ ( 1 1 μ ¯ min ( μ , 0 ) ) 1 / 2 u 0 , u H . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Eque_HTML.gif

Λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq68_HTML.gif denotes the first eigenvalue of the operator L, z = z H × H = ( u , v ) H × H = ( u H 2 + v H 2 ) 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq69_HTML.gif means the norm of the space E : = H 0 1 ( Ω ) × H 0 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq70_HTML.gif, E 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq71_HTML.gif is the dual space of E. t z = t ( u , v ) = ( t u , t v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq72_HTML.gif for all z E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq73_HTML.gif and t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq74_HTML.gif. z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq75_HTML.gif is said to be nonnegative in Ω if u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq76_HTML.gif and v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq77_HTML.gif in Ω. z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq75_HTML.gif is said to be positive in Ω if u > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq78_HTML.gif and v > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq79_HTML.gif in Ω. B r ( 0 ) = { x R N | x | < r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq80_HTML.gif is a ball in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq81_HTML.gif. O ( ε t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq82_HTML.gif denotes a quantity satisfying | O ( ε t ) | / ε t C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq83_HTML.gif, o ( ε t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq84_HTML.gif means | o ( ε t ) | / ε t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq85_HTML.gif as ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq86_HTML.gif and o ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq87_HTML.gif is a generic infinitesimal value. In particular, the quantity O 1 ( ε t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq88_HTML.gif means that there exist the constants C 1 , C 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq89_HTML.gif such that C 1 ε t O 1 ( ε t ) C 2 ε t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq90_HTML.gif as ε is small. We always denote positive constants as C and omit dx in integrals for convenience.

Lemma 2.1 If { z n } E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq91_HTML.gif is a (PS) c -sequence of J with z n z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq92_HTML.gif in E, then J ( z ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq93_HTML.gif and J ( z ) F ( τ 0 ( 1 ) , τ 0 ( 2 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq94_HTML.gif, where
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equf_HTML.gif
Proof Let z n = ( u n , v n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq95_HTML.gif and z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq75_HTML.gif. Since { z n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq96_HTML.gif is a (PS) c -sequence of J with z n z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq92_HTML.gif in E, we can deduce that J ( z ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq93_HTML.gif, and therefore J ( z ) , z = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq97_HTML.gif, that is,
Q ( z ) = z 2 K ( z ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equg_HTML.gif
Consequently,
J ( z ) = ( 1 2 1 2 ) z 2 ( 1 q 1 2 ) K ( z ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equh_HTML.gif
From the Hölder inequality it follows that
J ( z ) ( 1 2 1 2 ) z 2 ( 1 q 1 2 ) | Ω | 1 q 2 ( σ 1 ( Ω | u | 2 ) q 2 + σ 2 ( Ω | u | 2 ) q 2 ) 1 N z 2 2 q 2 q Λ 1 q 2 | Ω | 1 q 2 ( σ 1 u H q + σ 2 v H q ) F ( τ 0 ( 1 ) , τ 0 ( 2 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equi_HTML.gif

Thus, the proof is complete. □

Lemma 2.2 If { z n } E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq91_HTML.gif is a (PS) c -sequence of the functional J, then { z n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq96_HTML.gif is bounded in E.

Proof See Hsu [[12], Lemma 2.2]. □

Lemma 2.3 Suppose that (ℋ) holds. Then J satisfies the (PS) c condition for all c < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq98_HTML.gif, where
c = 1 N ( S η , β ( μ ) ) N 2 + F ( τ 0 ( 1 ) , τ 0 ( 2 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ9_HTML.gif
(2.1)
Proof We follow the argument in [15]. Let { z n } E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq99_HTML.gif be a (PS) c -sequence of J with c < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq98_HTML.gif. Write z n = ( u n , v n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq95_HTML.gif. We know from Lemma 2.2 that { z n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq96_HTML.gif is bounded in E, and then z n z = ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq100_HTML.gif up to a subsequence, z is a critical point of J. Furthermore, we may assume that u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq101_HTML.gif, v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq102_HTML.gif weakly in H and u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq103_HTML.gif, v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq104_HTML.gif strongly in L s ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq105_HTML.gif for all 1 s < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq106_HTML.gif and u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq103_HTML.gif, v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq104_HTML.gif a.e. in Ω. Hence, we have that
J ( z ) = 0 and K ( z n ) = K ( z ) + o ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ10_HTML.gif
(2.2)
Set u ˜ n = u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq107_HTML.gif, v ˜ n = v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq108_HTML.gif and z ˜ n = ( u ˜ n , v ˜ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq109_HTML.gif. From the Brézis-Lieb lemma [17] it follows that
z ˜ n 2 = z n 2 z 2 + o ( 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ11_HTML.gif
(2.3)
and by Lemma 2.1 in [18] we have
Ω | u ˜ n | α i | v ˜ n | β i = Ω | u n | α i | v n | β i Ω | u | α i | v | β i + o ( 1 ) , i = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ12_HTML.gif
(2.4)
Since J ( z n ) = c + o ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq110_HTML.gif, J ( z n ) = o ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq111_HTML.gif and by (2.2) to (2.4), we can deduce that
1 2 z ˜ n 2 1 2 Q ( z ˜ n ) = c J ( z ) + o ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ13_HTML.gif
(2.5)
and
z ˜ n 2 Q ( z ˜ n ) = o ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equj_HTML.gif
Hence, we may assume that
lim n z ˜ n 2 = lim n Q ( z ˜ n ) = l . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ14_HTML.gif
(2.6)
If l = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq112_HTML.gif, the proof is complete. Assume l > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq113_HTML.gif; then from (2.6) and the definition of S η , β ( μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq114_HTML.gif it follows that
S η , β ( μ ) l 2 2 = lim n S η , β ( μ ) ( Q ( z ˜ n ) ) 2 2 lim n z ˜ n 2 = l , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equk_HTML.gif
which implies that
l ( S η , β ( μ ) ) N 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ15_HTML.gif
(2.7)
In addition, from (2.5) to (2.7) and Lemma 2.1, we get
c = ( 1 2 1 2 ) l + J ( z ) 1 N ( S η , β ( μ ) ) N 2 + F ( τ 0 ( 1 ) , τ 0 ( 2 ) ) = c , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equl_HTML.gif

which is a contradiction. Therefore, the proof of Lemma 2.3 is complete. □

3 Nehari manifold

Since J is unbounded below on E, we need to consider J on the Nehari manifold
M σ = { z E { 0 } : J ( z ) , z = 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equm_HTML.gif
Thus, z M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq115_HTML.gif if and only if
J ( z ) , z = z 2 Q ( z ) K ( z ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ16_HTML.gif
(3.1)
By the Hölder inequality and the definition of Λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq68_HTML.gif it follows that
K ( z ) ( σ 1 ( Ω | u | 2 ) q 2 + σ 2 ( Ω | v | 2 ) q 2 ) | Ω | 1 q 2 ( σ 1 ( u H ) q + σ 2 ( v H ) q ) Λ 1 q 2 | Ω | 1 q 2 ( σ 1 + σ 2 ) Λ 1 q 2 | Ω | 1 q 2 z q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ17_HTML.gif
(3.2)

Lemma 3.1 The functional J is coercive and bounded below on M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq116_HTML.gif.

Proof Suppose that z = ( u , v ) M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq117_HTML.gif. From (3.1) and (3.2) we get
J ( z ) = ( 1 2 1 2 ) z 2 ( 1 q 1 2 ) K ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ18_HTML.gif
(3.3)
1 N z 2 ( 2 q 2 q ) ( σ 1 + σ 2 ) Λ 1 q 2 | Ω | 1 q 2 z q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ19_HTML.gif
(3.4)

Thus, J is coercive and bounded below on M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq116_HTML.gif. □

Define Φ ( z ) = J ( z ) , z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq118_HTML.gif. Then for all z = ( u , v ) M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq119_HTML.gif we have
Φ ( z ) , z = 2 z 2 2 Q ( z ) q K ( z ) = ( 2 q ) z 2 ( 2 q ) Q ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ20_HTML.gif
(3.5)
= ( 2 2 ) z 2 + ( 2 q ) K ( z ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ21_HTML.gif
(3.6)
We split M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq116_HTML.gif into three parts:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equn_HTML.gif

Lemma 3.2 Suppose that z E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq73_HTML.gif is a local minimizer of J on M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq116_HTML.gif and z M σ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq120_HTML.gif. Then J ( z ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq93_HTML.gif in E 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq71_HTML.gif.

Proof The proof is similar to that of [19] and the details are omitted. □

Lemma 3.3 M σ 0 = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq121_HTML.gif for all σ 1 + σ 2 ( 0 , ϒ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq122_HTML.gif.

Proof We argue by contradiction. Suppose that there exist σ 1 , σ 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq123_HTML.gif such that 0 < σ 1 + σ 2 < ϒ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq124_HTML.gif and M σ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq125_HTML.gif. Then the fact z = ( u , v ) M σ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq126_HTML.gif together with (3.5) and (3.6) imply that
z 2 = 2 q 2 q Q ( z ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ22_HTML.gif
(3.7)
and
z 2 = 2 q 2 2 K ( z ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ23_HTML.gif
(3.8)
By (1.5) and (3.7) we have
z 2 2 q 2 q ( S η , β ( μ ) ) 2 2 z 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equo_HTML.gif
which implies that
z ( 2 q 2 q ( S η , β ( μ ) ) 2 2 ) 1 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ24_HTML.gif
(3.9)
By (3.2) and (3.8) we have
z ( 2 q 2 2 Λ 1 q 2 | Ω | 1 q 2 ) 1 2 q ( σ 1 + σ 2 ) 1 2 q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ25_HTML.gif
(3.10)
From (3.9) and (3.10) it follows that
σ 1 + σ 2 ( 2 q 2 2 Λ 1 q 2 | Ω | 1 q 2 ) 1 ( 2 q 2 q ( S η , β ( μ ) ) 2 2 ) 2 q 2 2 = ϒ 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equp_HTML.gif

which is a contradiction. □

By Lemma 3.3, we write M σ = M σ + M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq127_HTML.gif and define
α σ = inf z M σ J ( z ) ; α σ ± = inf z M σ ± J ( z ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equq_HTML.gif
Lemma 3.4
  1. (i)

    α σ α σ + < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq128_HTML.gif for all σ 1 + σ 2 ( 0 , ϒ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq129_HTML.gif.

     
  2. (ii)

    There exists a positive constant d 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq130_HTML.gif depending on σ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq57_HTML.gif, σ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq58_HTML.gif, q, N, S η , β ( μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq114_HTML.gif, Λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq68_HTML.gif and | Ω | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq56_HTML.gif such that α σ > d 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq131_HTML.gif for all σ 1 + σ 2 ( 0 , q 2 ϒ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq132_HTML.gif.

     
Proof (i) Let z = ( u , v ) M σ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq133_HTML.gif. By (3.1) and (3.6) it follows that
2 q 2 q z 2 > Q ( z ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ26_HTML.gif
(3.11)
According to (3.1) and (3.11), we have that
J ( z ) = ( 1 2 1 q ) z 2 + ( 1 q 1 2 ) Q ( z ) < [ ( 1 2 1 q ) + ( 1 q 1 2 ) 2 q 2 q ] z 2 = 2 q q N z 2 < 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equr_HTML.gif
which implies that α σ α σ + < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq128_HTML.gif.
  1. (ii)
    Suppose that σ 1 + σ 2 ( 0 , q 2 ϒ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq134_HTML.gif and z = ( u , v ) M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq135_HTML.gif. By (1.7), (3.1) and (3.5) we have that
    2 q 2 q z < Q ( z ) ( S η , β ( μ ) ) 2 2 z 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equs_HTML.gif
     
which implies that
z > ( 2 q 2 q ) 1 2 2 ( S η , β ( μ ) ) 2 2 ( 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ27_HTML.gif
(3.12)
From (3.4) and (3.12) it follows that
J ( z ) z q ( 1 N z 2 q ( 2 q 2 q ) Λ 1 q 2 | Ω | 1 q 2 ( σ 1 + σ 2 ) ) d 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equt_HTML.gif

where d 0 = d 0 ( σ 1 , σ 2 , q , N , Λ 1 , S η , β ( μ ) , | Ω | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq136_HTML.gif is a positive constant. □

Lemma 3.5 Suppose that σ 1 + σ 2 ( 0 , ϒ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq129_HTML.gif and z E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq73_HTML.gif with Q ( z ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq137_HTML.gif. Then there exist unique t + , t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq138_HTML.gif such that t + z M σ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq139_HTML.gif and t z M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq140_HTML.gif. In particular, we have
t > t max : = ( ( 2 q ) z 2 ( 2 q ) Q ( z ) ) 1 2 2 > t + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equu_HTML.gif

J ( t + z ) = min 0 t t max J ( t z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq141_HTML.gif and J ( t z ) = max t t max J ( t z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq142_HTML.gif.

Proof The proof is similar to that of [20] and is omitted. □

For each z E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq73_HTML.gif with K ( z ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq143_HTML.gif, we write
t ¯ max = ( ( 2 q ) K ( z ) ( 2 2 ) z 2 ) 1 2 q > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equv_HTML.gif

Then we have the following lemma.

Lemma 3.6 Suppose that σ 1 + σ 2 ( 0 , ϒ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq129_HTML.gif and z E { ( 0 , 0 ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq144_HTML.gif with K ( z ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq143_HTML.gif. Then there exist unique 0 < t + < t ¯ max < t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq145_HTML.gif such that t + z M σ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq146_HTML.gif, t z M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq140_HTML.gif and
J ( t + z ) = inf 0 t t ¯ max J ( t z ) ; J ( t z ) = sup t 0 J ( t z ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equw_HTML.gif

Proof The proof is almost the same as that in [[20], Lemma 2.7] and is omitted here. □

4 Proof of Theorems 1.1 and 1.2

Lemma 4.1
  1. (i)

    If σ 1 + σ 2 ( 0 , ϒ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq129_HTML.gif, then the functional J has a (PS) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq147_HTML.gif -sequence { z n } M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq148_HTML.gif.

     
  2. (ii)

    If σ 1 + σ 2 ( 0 , q 2 ϒ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq149_HTML.gif, then the functional J has a ( PS ) α σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq150_HTML.gif-sequence { z n } M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq151_HTML.gif.

     

Proof The proof is similar to that of [21] and is omitted. □

Lemma 4.2 Suppose that σ 1 + σ 2 ( 0 , ϒ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq122_HTML.gif. Then J has a minimizer z ( 1 ) M σ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq152_HTML.gif such that z ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq153_HTML.gif is a positive solution of (1.1) and J ( z ( 1 ) ) = α σ = α σ + < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq154_HTML.gif.

Proof By Lemma 4.1(i), there exists a (PS) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq155_HTML.gif -sequence { z n } M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq148_HTML.gif of J such that
J ( z n ) = α σ + o ( 1 ) and J ( z n ) = o ( 1 ) in  E 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ28_HTML.gif
(4.1)
Since J is coercive on M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq116_HTML.gif (see Lemma 3.1), we get that { z n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq96_HTML.gif is bounded in E. Passing to a subsequence (still denoted by { z n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq96_HTML.gif), we can assume that there exists z ( 1 ) = ( u ( 1 ) , v ( 1 ) ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq156_HTML.gif such that
{ u n u ( 1 ) , v n v ( 1 ) weakly in  H , u n u ( 1 ) , v n v ( 1 ) a.e. in  Ω , u n u ( 1 ) , v n v ( 1 ) strongly in  L s ( Ω )  for all  1 s < 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ29_HTML.gif
(4.2)
which implies that
K ( z n ) = K ( z ( 1 ) ) + o ( 1 ) as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ30_HTML.gif
(4.3)
First, we claim that z ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq153_HTML.gif is a solution of (1.1). By (4.1) and (4.2), it is easy to see that z ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq153_HTML.gif is a solution of (1.1). Furthermore, from { z n } M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq148_HTML.gif and (3.3), we deduce that
K ( z n ) = q ( 2 2 ) 2 ( 2 q ) z n 2 2 q 2 q J ( z n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ31_HTML.gif
(4.4)
Taking n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq157_HTML.gif in (4.4), by (4.1), (4.2) and the fact α σ < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq158_HTML.gif, we get
K ( z ( 1 ) ) 2 q 2 q α σ > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equx_HTML.gif

Therefore, z ( 1 ) M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq159_HTML.gif is a nontrivial solution of (1.1).

Next, we prove that z n z ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq160_HTML.gif strongly in E and J ( z ( 1 ) ) = α σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq161_HTML.gif. Noting z ( 1 ) M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq159_HTML.gif and applying the Fatou lemma, we have
α σ J ( z ( 1 ) ) = 1 N z ( 1 ) 2 2 q 2 q K ( z ( 1 ) ) lim inf n ( 1 N z n 2 2 q 2 q K ( z n ) ) = lim inf n J ( z n ) = α σ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equy_HTML.gif
Therefore, J ( z ( 1 ) ) = α σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq162_HTML.gif and lim n z n 2 = z ( 1 ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq163_HTML.gif. Set z ˜ n = z n z ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq164_HTML.gif. By the Brézis-Lieb lemma [17], we get
z ˜ n 2 = z n 2 z ( 1 ) 2 + o ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equz_HTML.gif
Then standard argument shows that z n z ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq160_HTML.gif strongly in E. Moreover, we have z ( 1 ) M σ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq152_HTML.gif. Otherwise, if z ( 1 ) M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq165_HTML.gif, then by Lemma 3.5 there exist unique t 0 ± http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq166_HTML.gif such that t 0 ± z ( 1 ) M σ ± http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq167_HTML.gif and t 0 + < t 0 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq168_HTML.gif. Since
d d t J ( t 0 + z ( 1 ) ) = 0 and d 2 d t 2 J ( t 0 + z ( 1 ) ) > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equaa_HTML.gif
there exists t ¯ ( t 0 + , t 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq169_HTML.gif such that J ( t 0 + z ( 1 ) ) < J ( t ¯ z ( 1 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq170_HTML.gif. By Lemma 3.5 we get that
J ( t 0 + z ( 1 ) ) < J ( t ¯ z ( 1 ) ) J ( t 0 z ( 1 ) ) = J ( z ( 1 ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equab_HTML.gif

which is a contradiction. Since J ( z ( 1 ) ) = J ( | z ( 1 ) | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq171_HTML.gif and | z ( 1 ) | M σ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq172_HTML.gif, by Lemma 3.2 we may assume that z ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq153_HTML.gif is a nontrivial nonnegative solution of (1.1).

In particular u ( 1 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq173_HTML.gif, v ( 1 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq174_HTML.gif. Indeed, without loss of generality, we may assume that v ( 1 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq175_HTML.gif. Then as u ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq176_HTML.gif is a nontrivial nonnegative solution of
{ Δ u μ u | x | 2 = σ 1 | u | q 2 u in  Ω , u = 0 on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equac_HTML.gif
by the standard regularity theory, we have u ( 1 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq177_HTML.gif in Ω and
( u ( 1 ) , 0 ) 2 = K ( u ( 1 ) , 0 ) > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equad_HTML.gif
Moreover, we may choose w H 0 1 ( Ω ) { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq178_HTML.gif such that
( 0 , w ) 2 = K ( 0 , w ) > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equae_HTML.gif
Now,
K ( u ( 1 ) , w ) = K ( u ( 1 ) , 0 ) + K ( 0 , w ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equaf_HTML.gif
and so by Lemma 3.6 there is unique 0 < t + < t ¯ max http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq179_HTML.gif such that ( t + u ( 1 ) , t + w ) M σ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq180_HTML.gif. Moreover,
t ¯ max = ( ( 2 q ) K ( u ( 1 ) , w ) ( 2 2 ) ( u ( 1 ) , w ) 2 ) = ( 2 q 2 2 ) 1 / ( 2 q ) > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equag_HTML.gif
and
J ( t + u ( 1 ) , t + w ) = inf 0 t t ¯ max J ( t u ( 1 ) , t w ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equah_HTML.gif
This implies
α σ + J ( t + u ( 1 ) , t + w ) J ( u ( 1 ) , w ) < J ( u ( 1 ) , 0 ) = α σ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equai_HTML.gif

which is a contradiction.

Finally, from the maximum principle [22] we deduce that z ( 1 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq181_HTML.gif in Ω and z ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq153_HTML.gif is thus a positive solution of (1.1). □

Let V μ , ε ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq182_HTML.gif be defined as in (1.4) and set u ε ( x ) = ψ ( x ) V μ , ε ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq183_HTML.gif, where ψ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq184_HTML.gif is a cut-off function:
ψ ( x ) D ( Ω ) : = { ψ C 0 ( Ω ) : ψ ( x ) 1  in a neighborhood of  x = 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equaj_HTML.gif

The following results are already known.

Lemma 4.3 [4]

As ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq86_HTML.gif we have the following estimates:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ32_HTML.gif
(4.5)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ33_HTML.gif
(4.6)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ34_HTML.gif
(4.7)

Lemma 4.4 [11]

Suppose that (ℋ) holds, f ( τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq185_HTML.gif is defined as in (1.6) and V μ , ε ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq182_HTML.gif are the minimizers of S ( μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq44_HTML.gif defined as in (1.4). Then S η , β ( μ ) = f ( τ 0 ) S ( μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq186_HTML.gif and has the minimizers ( V μ , ε ( x ) , τ 0 V μ , ε ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq187_HTML.gif, where f ( τ 0 ) : = min τ 0 f ( τ ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq188_HTML.gif.

Lemma 4.5 Under the assumptions of Theorem 1.2, there exist z ˜ E { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq189_HTML.gif and Λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq190_HTML.gif such that for all σ 1 + σ 2 < Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq191_HTML.gif there holds
sup t 0 J ( t z ˜ ) < c = 1 N ( S η , β ( μ ) ) N 2 + F ( τ 0 ( 1 ) , τ 0 ( 2 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ35_HTML.gif
(4.8)

In particular, α σ < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq192_HTML.gif for all σ 1 + σ 2 < Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq191_HTML.gif.

Proof For all t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq193_HTML.gif, define the functions g 1 ( t ) : = J ( t u ε , t τ 0 u ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq194_HTML.gif and
g 2 ( t ) : = t 2 2 ( 1 + ( τ 0 ) 2 ) u ε H 2 t 2 2 ( η 1 ( τ 0 ) β 1 + η 2 ( τ 0 ) β 2 ) Ω | u ε | 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equak_HTML.gif

Note that lim t + g 2 ( t ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq195_HTML.gif and g 2 ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq196_HTML.gif as t is closed to 0. Thus, sup t 0 g 2 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq197_HTML.gif is attained at some finite t ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq198_HTML.gif with g 2 ( t ε ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq199_HTML.gif. Furthermore, C < t ε < C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq200_HTML.gif, where C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq201_HTML.gif and C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq202_HTML.gif are the positive constants independent of ε.

Choose δ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq203_HTML.gif small enough such that c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq204_HTML.gif for all σ 1 + σ 2 < δ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq205_HTML.gif. Set z ε = ( u ε , τ 0 u ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq206_HTML.gif. Then J ( t z ε ) t 2 2 z ε 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq207_HTML.gif for all t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq193_HTML.gif and σ 1 , σ 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq123_HTML.gif, which implies that there exists t 0 ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq208_HTML.gif satisfying sup 0 t t 0 J ( t z ε ) < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq209_HTML.gif, for all σ 1 + σ 2 < δ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq205_HTML.gif. Note that
max t 0 ( t 2 2 B 1 t 2 2 B 2 ) = 1 N ( B 1 B 2 2 2 ) N 2 , B 1 , B 2 > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ36_HTML.gif
(4.9)
From (4.9) and Lemmas 4.3, 4.4 it follows that
g 2 ( t ε ) 1 N ( ( 1 + ( τ 0 ) 2 ) u ε H 2 ( ( η 1 ( τ 0 ) β 1 + η 2 ( τ 0 ) β 2 ) Ω | u ε | 2 ) 2 2 ) N 2 1 N ( f ( τ 0 ) S ( μ ) N 2 + O ( ε 2 μ ¯ μ ) ( S ( μ ) N 2 + O ( ε 2 μ ¯ μ ) ) 2 2 ) N 2 1 N ( f ( τ 0 ) S ( μ ) ) N 2 + O ( ε 2 μ ¯ μ ) = 1 N ( S η , β ( μ ) ) N 2 + O ( ε 2 μ ¯ μ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equal_HTML.gif
Consequently,
sup t t 0 g 1 ( t ) < sup t t 0 ( g 2 ( t ) t q q K ( z ε ) ) 1 N ( S η , β ( μ ) ) N 2 + O ( ε 2 μ ¯ μ ) t 0 q q ( σ 1 + ( τ 0 ) q σ 2 ) Ω | u ε | q 1 N ( S η , β ( μ ) ) N 2 + O ( ε 2 μ ¯ μ ) C ( σ 1 + σ 2 ) Ω | u ε | q 1 N ( S η , β ( μ ) ) N 2 + O ( ε 2 μ ¯ μ ) ( σ 1 + σ 2 ) O 1 ( ε N q μ ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equam_HTML.gif
and
( N q μ ¯ ) 2 q q < 2 μ ¯ μ ( N q μ ¯ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equan_HTML.gif

where we have used the assumption μ < μ ¯ ( N q μ ¯ ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq210_HTML.gif.

Therefore we can choose σ 1 = O 1 ( ε r 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq211_HTML.gif, σ 2 = O 1 ( ε r 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq212_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equao_HTML.gif
The definition of F ( τ 0 ( 1 ) , τ 0 ( 2 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq213_HTML.gif in Lemma 2.1 implies that
F ( τ 0 ( 1 ) , τ 0 ( 2 ) ) = O 1 ( ε 2 2 q min ( r 1 , r 2 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equap_HTML.gif
Note that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equaq_HTML.gif
Taking ε small enough, there exists δ 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq214_HTML.gif such that for all 0 < σ 1 + σ 2 < δ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq215_HTML.gif,
O ( ε 2 μ ¯ μ ) ( σ 1 + σ 2 ) O 1 ( ε N q μ ¯ ) < F ( τ 0 ( 1 ) , τ 0 ( 2 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ37_HTML.gif
(4.10)
Choose Λ = min { δ 1 , δ 2 } > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq216_HTML.gif. Then for all σ 1 + σ 2 ( 0 , Λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq217_HTML.gif there holds
sup t 0 J ( t z ε ) < c . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equ38_HTML.gif
(4.11)
Finally, we prove that α σ < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq218_HTML.gif for all σ 1 + σ 2 < Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq191_HTML.gif. Recall that z ε = ( u ε , τ 0 u ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq206_HTML.gif. By Lemma 3.5, the definition of α σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq219_HTML.gif and (4.11), we can deduce that there exists t ˜ 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq220_HTML.gif such that t ˜ 0 z ε M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq221_HTML.gif and
α σ J ( t ˜ 0 z ε ) sup t 0 J ( t ˜ 0 z ε ) < c . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equar_HTML.gif

The proof is thus complete by taking z ˜ = z ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq222_HTML.gif. □

Lemma 4.6 Set ϒ 2 : = min { Λ , q 2 ϒ 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq223_HTML.gif. Then for all σ 1 + σ 2 ( 0 , ϒ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq224_HTML.gif, problem (1.1) has a positive solution z ( 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq225_HTML.gif such that z ( 2 ) M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq226_HTML.gif and J ( z ( 2 ) ) = α σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq227_HTML.gif.

Proof By Lemma 4.1, there exists a (PS) α σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq228_HTML.gif-sequence { z n } M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq229_HTML.gif of J for all σ 1 + σ 2 < q 2 ϒ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq230_HTML.gif. From Lemmas 2.3, 3.4 and 4.5, it follows that α σ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq231_HTML.gif and J satisfies the (PS) α σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq232_HTML.gif condition for all σ 1 + σ 2 < ϒ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq64_HTML.gif. Since J is coercive on M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq116_HTML.gif, we get that { z n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq96_HTML.gif is bounded in E. Therefore, there exist a subsequence (still denoted by { z n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq96_HTML.gif) and z ( 2 ) = ( u ( 2 ) , v ( 2 ) ) M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq233_HTML.gif such that z n z ( 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq234_HTML.gif strongly in E and J ( z ( 2 ) ) = α σ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq235_HTML.gif for all σ 1 + σ 2 < ϒ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq64_HTML.gif. Since J ( z ( 2 ) ) = J ( | z ( 2 ) | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq236_HTML.gif and | z ( 2 ) | M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq237_HTML.gif, by Lemma 3.2 we may assume that z ( 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq225_HTML.gif is a nontrivial nonnegative solution of (1.1). Moreover, by (3.7) and z ( 2 ) = ( u ( 2 ) , v ( 2 ) ) M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq238_HTML.gif, we get
Q ( z ( 2 ) ) = 2 q 2 q z ( 2 ) 2 > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_Equas_HTML.gif

This implies that u ( 2 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq239_HTML.gif and v ( 2 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq240_HTML.gif. From the strong maximum principle [22] it follows that z ( 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq225_HTML.gif is a positive solution of (1.1). □

Proof of Theorems 1.1 and 1.2. By Lemma 4.2, we obtain that (1.1) has a positive solution z ( 1 ) M σ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq241_HTML.gif for all σ 1 + σ 2 < ϒ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq59_HTML.gif. On the other hand, from Lemma 4.6, we can get the second positive solution z ( 2 ) M σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq226_HTML.gif for all σ 1 + σ 2 < ϒ 2 < ϒ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq242_HTML.gif. Since M σ + M σ = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq243_HTML.gif, this implies that z ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq153_HTML.gif and z ( 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-116/MediaObjects/13661_2012_Article_216_IEq225_HTML.gif are distinct. □

Declarations

Acknowledgements

The author was grateful for the referee’s helpful suggestions and comments.

Authors’ Affiliations

(1)
Department of Natural Sciences in the Center for General Education, Chang Gung University

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