Open Access

The existence of positive solutions to an elliptic system with nonlinear boundary conditions

Boundary Value Problems20132013:159

DOI: 10.1186/1687-2770-2013-159

Received: 8 April 2013

Accepted: 16 June 2013

Published: 1 July 2013

Abstract

In this paper, we consider the following system:

{ Δ u = u , Δ v = v , x Ω , u ν = f ( x , v ) , v ν = g ( x , u ) , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equa_HTML.gif

where Ω is a bounded domain in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq1_HTML.gif ( N 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq2_HTML.gif) with smooth boundary, ν https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq3_HTML.gif is the outer normal derivative and f , g : Ω × R R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq4_HTML.gif are positive and continuous functions. Under certain assumptions on f ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq5_HTML.gif and g ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq6_HTML.gif, but without the usual (AR) condition, we prove that the problem has at least one positive strong pair solution ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq7_HTML.gif (see Definition 1.4 below) by applying a linking theorem for strong indefinite functional.

MSC:35A01, 35J20, 35J25.

Keywords

fractional Sobolev spaces linking theorem nonlinear boundary conditions strong solution

1 Introduction and main result

In this paper, we mainly study the following system:
{ Δ u = u , Δ v = v , x Ω , u ν = f ( x , v ) , v ν = g ( x , u ) , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ1_HTML.gif
(1.1)

where Ω is a bounded domain in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq1_HTML.gif ( N 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq2_HTML.gif) with smooth boundary, ν https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq3_HTML.gif is the outer normal derivative and f , g : Ω × R R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq4_HTML.gif are positive and continuous functions.

Existence results for nonlinear elliptic systems have received a lot of interest in recent years (see [112]), particularly when the nonlinear term appears as a source in the equation, complemented with Dirichlet boundary conditions. To our knowledge, about the system with nonlinear boundary conditions, there are not many results. Here we refer to [9, 13, 14].

We are mainly motivated by [12] and [14].

In [12], Li and one of the authors considered
{ Δ u + u = f ( x , v ) , x R N , Δ v + v = g ( x , u ) , x R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ2_HTML.gif
(1.2)

Under some given conditions, we proved that (1.2) had at least one positive solution pair ( u , v ) H 1 ( R N ) × H 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq8_HTML.gif.

In [14], Bonder, Pinasco, Rossi studied
{ Δ u = u , Δ v = v , x Ω , u ν = H v ( x , u , v ) , v ν = H u ( x , u , v ) , x Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ3_HTML.gif
(1.3)

They assumed that H satisfied the following conditions:

( H ˜ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq9_HTML.gif) | H ( x , u , v ) | C ( | u | p + 1 + | v | q + 1 + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq10_HTML.gif.

( H ˜ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq11_HTML.gif) The Ambrosetti-Rabinowitz type condition: For R large, if | ( u , v ) | R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq12_HTML.gif,
1 α H u ( x , u , v ) u + 1 β H v ( x , u , v ) u H ( x , u , v ) > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ4_HTML.gif
(1.4)
where p + 1 α > p > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq13_HTML.gif and q + 1 β > q > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq14_HTML.gif with
1 > 1 α + 1 β , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ5_HTML.gif
(1.5)
max { p α + q β ; q q + 1 p + 1 α + p p + 1 q + 1 β } < 1 + 1 N 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ6_HTML.gif
(1.6)
q q + 1 p + 1 α < 1 , and p p + 1 q + 1 β < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ7_HTML.gif
(1.7)
When N 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq15_HTML.gif, they also assumed
max { p α + q β ; q q + 1 p + 1 α + p p + 1 q + 1 β } < N + 1 2 ( N 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ8_HTML.gif
(1.8)

( H ˜ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq16_HTML.gif) | H u ( x , u , v ) | ( | u | p + | v | p ( q + 1 ) p + 1 + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq17_HTML.gif, | H v ( x , u , v ) | ( | u | q ( p + 1 ) q + 1 + | v | q + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq18_HTML.gif.

( H ˜ 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq19_HTML.gif) H ( x , u , v ) = H ( x , u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq20_HTML.gif.

They obtained infinitely many nontrivial solutions of (1.3) under the assumptions ( H ˜ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq9_HTML.gif) to ( H ˜ 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq19_HTML.gif) by using variational arguments and a fountain theorem. Note that (1.4) implies
| H ( x , u , v ) | c ( | u | α + | v | β ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ9_HTML.gif
(1.9)

(see Lemma 1.1 in [6]). Therefore it is not difficult to verify that any (PS) sequence (or (C) c sequence) of the corresponding functional is bounded in some suitable space.

The crucial part in the nonlinear boundary conditions case is to find the proper functional setting for (1.1) that allows us to treat our problem variationally. We accomplish this by defining a self-adjoint operator that takes into account the boundary conditions together with the equations and considering its fractional powers that satisfy a suitable ‘integration by parts’ formula. In order to obtain nontrivial solutions, we use a linking theorem (see [11]).

The assumptions we impose on f ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq5_HTML.gif and g ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq6_HTML.gif are as follows:

(H1) f , g C 0 ( Ω × R 1 , R 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq21_HTML.gif with f ( x , t ) = g ( x , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq22_HTML.gif for any ( x , t ) Ω × ( , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq23_HTML.gif,
f ( x , t ) > 0 and g ( x , t ) > 0 for any  ( x , t ) Ω × ( 0 , + ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equb_HTML.gif

(H2) lim t 0 f ( x , t ) t = lim t 0 g ( x , t ) t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq24_HTML.gif uniformly in x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq25_HTML.gif.

(H3) There is a positive constant C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq26_HTML.gif such that
| f ( x , t ) | C ( 1 + | t | p 1 ) , | g ( x , t ) | C ( 1 + | t | q 1 ) , ( x , t ) Ω × R 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ10_HTML.gif
(1.10)
where p , q > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq27_HTML.gif and satisfy
1 p + 1 q > 1 1 N 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ11_HTML.gif
(1.11)

(H4) lim | t | + F ( x , t ) t 2 = lim | t | + G ( x , t ) t 2 = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq28_HTML.gif uniformly in x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq25_HTML.gif, where F ( x , t ) : = 0 t f ( x , s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq29_HTML.gif, G ( x , t ) : = 0 t g ( x , s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq30_HTML.gif.

(H5) For all 0 < t < s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq31_HTML.gif, x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq25_HTML.gif or s < t < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq32_HTML.gif, x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq33_HTML.gif, there are two positive constants C 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq34_HTML.gif, C 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq35_HTML.gif such that
H 1 ( x , t ) H 1 ( x , s ) + C 1 , , H 2 ( x , t ) H 2 ( x , s ) + C 2 , , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ12_HTML.gif
(1.12)

where H 1 ( x , t ) = t f ( x , t ) 2 F ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq36_HTML.gif, H 2 ( x , t ) = t g ( x , t ) 2 G ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq37_HTML.gif with H 1 ( x , t ) , H 2 ( x , t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq38_HTML.gif for any t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq39_HTML.gif, x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq25_HTML.gif.

Remark 1.1 By (1.11), there exist l and m with l + m = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq40_HTML.gif, l , m > 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq41_HTML.gif such that
1 p > 1 2 2 l 1 2 N 1 , 1 q > 1 2 2 m 1 2 N 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ13_HTML.gif
(1.13)

Remark 1.2 (H5) was first introduced by Miyagaki and Souto in [15]. A typical pair of functions f ( x , t ) = t p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq42_HTML.gif, g ( x , t ) = t q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq43_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq39_HTML.gif, p , q > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq27_HTML.gif; f ( x , t ) = g ( x , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq22_HTML.gif, t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq44_HTML.gif satisfy (H1) to (H5). However, the pair of functions f ( x , t ) = t ( 2 ln t + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq45_HTML.gif, g ( x , t ) = t ( 2 ln t + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq46_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq39_HTML.gif; f ( x , t ) = g ( x , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq22_HTML.gif, t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq44_HTML.gif satisfy (H5) but do not satisfy the usual (AR) condition and ( H ˜ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq11_HTML.gif) in this paper.

Remark 1.3 The assumptions we impose on f and g are different from the assumptions in [14]. To our best knowledge, it is the first time the group assumptions have been used to deal with a system with nonlinear boundary conditions.

In order to state our main result, first we give a definition.

Definition 1.4 We say that ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq7_HTML.gif is a strong solution of (1.1) if
u W 2 , p p 1 ( Ω ) , v W 2 , q q 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equc_HTML.gif

and ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq7_HTML.gif satisfies (1.1) a.e. in Ω.

Our main results is as follows.

Theorem 1.5 Let (H1)-(H5) hold. Then system (1.1) possesses at least one positive strong solution pair z = ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq47_HTML.gif.

The main difficulties to deal with system (1.1) consist in at least three aspects. Firstly, due to the type of growth of the functions f and g, we cannot work with the usual H 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq48_HTML.gif, and then we need fractional Sobolev spaces. Secondly, although we have a variational problem, the functional associated to it always has a strong indefinite quadratic part. So, the functional possesses no mountain-pass structure but the linking geometric structure, which is more complicated to handle. Thirdly, as we do not assume that the functions f and g satisfy the (AR) conditions, it is much more difficult to show that any (C) c sequence is uniformly bounded in E (see Section 2).

To prove Theorem 1.5, we try to find a critical point of the functional Φ (see (2.5)) in E. We prove that Φ has a linking geometric structure and use a linking theorem under (C) c condition (see Theorem 2.1 in [11]) to get a (C) c -sequence { z n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq49_HTML.gif of Φ. The main difficulty now will be to prove that { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq50_HTML.gif is uniformly bounded in E without the (AR) condition. Then we prove that any (C) c -sequence { z n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq49_HTML.gif of Φ is bounded. To overcome this difficulty, we use some techniques used in [12, 16] for which the assumptions (H4), (H5) play important roles. As { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq50_HTML.gif is bounded, then we can prove that { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq50_HTML.gif has a subsequence which converges to a nontrivial critical point of Φ. Hence, by the strong maximum principle, we can prove that the pair solution ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq7_HTML.gif is positive.

The paper is organized as follows. In Section 2, we give some preliminaries. We prove our main result in Section 3.

2 Some preliminaries

In this section we mainly give some preliminaries which will be used in Section 3. We follow the structure in [13].

Throughout this paper, we consider the space L 2 ( Ω ) × L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq51_HTML.gif which is a Hilbert space with the inner product, which we denote by , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq52_HTML.gif, given by
( u , v ) , ( φ , ψ ) = Ω u φ d x + Ω v ψ d σ , for any  ( u , v ) , ( φ , ψ ) L 2 ( Ω ) × L 2 ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equd_HTML.gif
Now we let A : D ( A ) L 2 ( Ω ) × L 2 ( Ω ) L 2 ( Ω ) × L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq53_HTML.gif be the operator defined by
A ( u , u | Ω ) = ( Δ u + u , u ν ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Eque_HTML.gif
where D ( A ) = { ( u , u | Ω ) : u H 2 ( Ω ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq54_HTML.gif. It is not difficult to verify that D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq55_HTML.gif is dense in L 2 ( Ω ) × L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq56_HTML.gif. Note that A is invertible with its inverse given by
A 1 ( h 1 , h 2 ) = ( u , u | Ω ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equf_HTML.gif
where u is the solution of
{ Δ u + u = h 1 , in  Ω , u ν = h 2 , on  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ14_HTML.gif
(2.1)
By standard regularity (see [17]), it follows that A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq57_HTML.gif is bounded and compact. Hence, R ( A ) = L 2 ( Ω ) × L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq58_HTML.gif. Therefore, in order to see that A (hence A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq57_HTML.gif) is self-adjoint, it suffices to prove that A is symmetric ([18], p.512). In fact, for u , v D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq59_HTML.gif, applying Green’s formula, we obtain
A u , v = Ω ( Δ u + u ) v d x + Ω u ν v d σ = Ω ( Δ v + v ) u d x + Ω v ν u d σ = v , A u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equg_HTML.gif
Hence A is symmetric. Also we can check that A (and so A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq57_HTML.gif) is positive. For any u D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq60_HTML.gif and by Green’s formula again, we have
A u , u = Ω ( Δ u + u ) u d x + Ω u ν u d σ = Ω ( | u | 2 + u 2 ) d x 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equh_HTML.gif
Hence there is a sequence of eigenvalues ( λ n ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq61_HTML.gif with eigenfunctions ( φ n , ψ n ) L 2 ( Ω ) × L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq62_HTML.gif satisfying 0 < λ 1 λ 2 λ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq63_HTML.gif and φ n H 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq64_HTML.gif, φ n | Ω = ψ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq65_HTML.gif,
{ Δ φ n + φ n = λ n φ n , in  Ω , φ n ν = λ n ψ n , on  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ15_HTML.gif
(2.2)
Now we consider the following fractional powers of A, i.e., for 0 < l < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq66_HTML.gif,
A l : D ( A l ) L 2 ( Ω ) × L 2 ( Ω ) , with  A l u = n = 1 λ n l a n ( φ n , ψ n ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equi_HTML.gif
where u = n = 1 a n ( φ n , ψ n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq67_HTML.gif. Let E l = D ( A l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq68_HTML.gif, which is a Hilbert space under the inner product
( u , φ ) E l = A l u , A l φ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equj_HTML.gif
Note that E l H 2 l ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq69_HTML.gif. Indeed, if we define A 1 : H 2 ( Ω ) L 2 ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq70_HTML.gif by
A 1 u = Δ u + u , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equk_HTML.gif
and A 2 : H 2 ( Ω ) D ( A 2 ) L 2 ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq71_HTML.gif by
A 2 u = u ν , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equl_HTML.gif
then A ¯ = ( A 1 , A 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq72_HTML.gif satisfies
A = A ¯ | ( u , u ) , u D ( A 1 ) D ( A 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equm_HTML.gif
and hence
A l = A ¯ l | ( u , u ) , u D ( A 1 l ) D ( A 2 l ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equn_HTML.gif
Since D ( A 1 ) = H 2 ( Ω ) D ( A 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq73_HTML.gif, we have D ( A 1 l ) D ( A 2 l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq74_HTML.gif. Therefore,
E l = D ( A l ) = D ( A 1 l ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equo_HTML.gif
Noting that Ω is smooth, it follows from the results of p.187 in [19] (see also [18, 20]) that
E l = D ( A 1 l ) H 2 l ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equp_HTML.gif

The following compact result will be useful later.

Proposition 2.1 (Theorem 2.1, [14])

Given l > 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq75_HTML.gif and r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq76_HTML.gif so that 1 r 1 2 2 l 1 2 N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq77_HTML.gif, the inclusion map i : E l L r ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq78_HTML.gif is well defined and bounded. Moreover, if 1 r > 1 2 2 l 1 2 N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq79_HTML.gif, then the inclusion is compact.

Denote E = E l × E m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq80_HTML.gif, where l + m = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq40_HTML.gif, l, m are the same as in Remark 1.1 and define B : E × E R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq81_HTML.gif by
B ( ( u , v ) , ( φ , ψ ) ) = A l u , A m ψ + A l φ , A m v . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equq_HTML.gif
Associated to B, we have the quadratic form
Q ( z ) : = 1 2 B ( z , z ) = A l u , A m v . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equr_HTML.gif
It is easy to see (one can refer to [14]) that the bounded self-adjoint operator L : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq82_HTML.gif defined by ( L z , η ) : = B ( z , η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq83_HTML.gif has exactly two eigenvalues +1 and −1, and that the corresponding eigenvalues E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq84_HTML.gif and E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq85_HTML.gif are given by
E + = { ( u , A m A l u ) : u E l } and E = { ( u , A m A l u ) : u E l } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equs_HTML.gif
where we use the notation A m = ( A m ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq86_HTML.gif. Then E = E + E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq87_HTML.gif. The spaces E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq84_HTML.gif and E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq85_HTML.gif are orthogonal with respect to the bilinear B, that is,
B ( z + , z ) = 0 , z + E + , z E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equt_HTML.gif
Moreover, we have
1 2 z ± E 2 = Q ( z ± ) = Ω | A l u | 2 d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equu_HTML.gif
if z ± = ( u , ± ( A m A l u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq88_HTML.gif. We see also that for z = ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq47_HTML.gif, z = z + + z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq89_HTML.gif with z + = ( ( u + A l A m v ) 2 , ( A m A l u + v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq90_HTML.gif, z = ( ( u A l A m u v ) 2 , ( A m A l u + v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq91_HTML.gif and
A l u , A m v = 1 2 B ( z , z ) = 1 2 L z , z = 1 2 ( z + E 2 z E 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ16_HTML.gif
(2.3)
From (1.10), Remark 1.1 and Proposition 2.1, we can define the functional H : E R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq92_HTML.gif as
H ( u , v ) = Ω F ( x , v ) d σ + Ω G ( x , u ) d σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ17_HTML.gif
(2.4)
Lemma 2.2 The functional defined by (2.4) is of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq93_HTML.gif and its derivative is given by
H ( u , v ) ( φ , ψ ) = Ω f ( x , v ) ψ d σ + Ω g ( x , u ) φ d σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equv_HTML.gif

Moreover, H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq94_HTML.gif is compact.

Proof From (1.10), Hölder’s inequality and Proposition 2.1, we have
| Ω f ( x , v ) ψ d σ | Ω | f ( x , v ) ψ | d σ Ω ( 1 + | v | p 1 ) | ψ | d σ v L p ( Ω ) p 1 ψ L p ( Ω ) + ψ L p ( Ω ) | Ω | p 1 p C ( v E m p 1 + 1 ) ψ E m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equw_HTML.gif
Similarly, we have
| Ω g ( x , u ) φ d σ | C ( u E l q 1 + 1 ) φ E l . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equx_HTML.gif

Hence H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq94_HTML.gif is well defined and bounded in E. A standard argument yields that is Fréchet differentiable with H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq94_HTML.gif continuous. By Proposition 2.1 we know that H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq94_HTML.gif is compact (see [21] for the details). □

Now we define the functional Φ : E R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq95_HTML.gif for (1.1) given by
Φ ( z ) = Q ( z ) H ( z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ18_HTML.gif
(2.5)

Moreover, Φ is class C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq96_HTML.gif.

Definition 2.3 We say that z = ( u , v ) E = E l × E m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq97_HTML.gif is an ( l , m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq98_HTML.gif-weak solution of (1.1) if z is a critical point of Φ. In other words, for every ( φ , ψ ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq99_HTML.gif, we have
A l u , A m ψ + A l φ , A m v Ω f ( x , v ) ψ d σ Ω g ( x , u ) φ d σ = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ19_HTML.gif
(2.6)

Now we give a regularity result of an ( l , m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq98_HTML.gif weak solution.

Proposition 2.4 If ( u , v ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq100_HTML.gif is an ( l , m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq98_HTML.gif-weak solution of (1.1), then u W 2 , p p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq101_HTML.gif, v W 2 , q q 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq102_HTML.gif and
Δ u = u in Ω , u ν = f ( x , v ) on Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ20_HTML.gif
(2.7)
Δ v = v in Ω , v ν = g ( x , u ) on Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ21_HTML.gif
(2.8)

In other words, ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq7_HTML.gif is a strong solution of (1.1).

Proof Although the proof is only needed to make some minor modifications as that of Theorem 2.2 in [13], for the readers’ convenience, we give its detailed proof.

Let us consider φ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq103_HTML.gif in (2.6), then
A l u , A m ψ Ω f ( x , v ) ψ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ22_HTML.gif
(2.9)

for all ψ E m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq104_HTML.gif.

If we take ψ H 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq105_HTML.gif, then we have
A l u , A m ψ = u , A ψ = Ω ( Δ ψ + ψ ) u d x + Ω ψ ν u d σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ23_HTML.gif
(2.10)
On the other hand, by (1.10) and Proposition 2.1, we have
Ω | f ( x , v ) | p p 1 d σ Ω ( 1 + | v | p 1 ) p p 1 d σ C Ω ( 1 + | v | p ) d σ C + C v E m p < + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ24_HTML.gif
(2.11)
i.e., f ( x , v ) L p p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq106_HTML.gif. Then from basic elliptic theory (Theorem 9.9, p.9, [17]) there exists one function w W 2 , p p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq107_HTML.gif such that
Δ w = w in  Ω , w ν = f ( x , v ) on  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equy_HTML.gif
Then we get
0 = Ω ( Δ w + w ) ψ d x = Ω ( Δ ψ + ψ ) w d x + Ω w ψ ν d σ Ω f ( x , v ) ψ d σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ25_HTML.gif
(2.12)
From (2.10), (2.11) and (2.12), we have
u w , A ψ = Ω ( u w ) ( Δ ψ + ψ ) d x + Ω ( u w ) ψ ν = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equz_HTML.gif

which implies that u = w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq108_HTML.gif. We have gotten that u W 2 , p p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq109_HTML.gif. Finally, since u = w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq108_HTML.gif, we conclude that u satisfies (2.7). We can make the same argument for v. □

3 The proof of our main result

In this section, we mainly want to prove Theorem 1.5. First we present a linking theorem from [11]. Then we prove that it can be applied to our functional setting stated in Section 2.

Suppose that f ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq5_HTML.gif, g ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq6_HTML.gif satisfy the assumptions (H1)-(H3), then it is easy to see that for any ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq110_HTML.gif there is a C ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq111_HTML.gif such that for ( x , t ) Ω × R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq112_HTML.gif we have
| f ( x , t ) | ϵ | t | + C ϵ | t | p 1 , | g ( x , t ) | ϵ | t | + C ϵ | t | q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ26_HTML.gif
(3.1)
and
| F ( x , t ) | ϵ | t | 2 + C ϵ | t | p , | G ( x , t ) | ϵ | t | 2 + C ϵ | t | q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ27_HTML.gif
(3.2)

Since f ( x , t ) = g ( x , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq22_HTML.gif when t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq44_HTML.gif, ( u , v ) = ( 0 , 0 ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq113_HTML.gif is a solution of (1.1). So we are interested in nontrivial and nonnegative solutions of (1.1).

Recall that { z n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq49_HTML.gif is called a Palais-Smale sequence of a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq93_HTML.gif functional I on E at level c ((PS) c -sequence for short) if I ( z n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq114_HTML.gif and I ( z n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq115_HTML.gif in E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq116_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq117_HTML.gif. If I ( z n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq114_HTML.gif and ( 1 + z n E ) I ( z n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq118_HTML.gif in E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq116_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq117_HTML.gif, then { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq50_HTML.gif will be called a Cerami sequence at level c ((C) c -sequence for short). A standard way to prove the existence of a positive solution to (1.1) is to get a (PS) c or (C) c sequence for Φ and then to prove that the sequence converges to a solution to (1.1). In this paper, we want to get a (C) c sequence by a linking theorem (Theorem 2.1, in [11]). So, we need to recall some terminology (see, e.g., [11, 22]).

Let H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq119_HTML.gif be a closed separable subspace of a Hilbert space H with the norm H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq120_HTML.gif and let H + : = ( H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq121_HTML.gif. For u H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq122_HTML.gif, we shall write u = u + + u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq123_HTML.gif, where u ± H ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq124_HTML.gif. On H we define a new norm
u τ : = max { u + H , k = 1 1 2 k | u , e k | } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equaa_HTML.gif
where { e k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq125_HTML.gif is a total orthonormal sequence in H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq119_HTML.gif. The topology generated by τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq126_HTML.gif will be called the τ-topology. Recall from [22] that a homotopy h = I g : A × [ 0 , 1 ] H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq127_HTML.gif, where A H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq128_HTML.gif, is called admissible if:
  1. (i)

    h is τ-continuous, i.e., h ( u n , s n ) h ( u , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq129_HTML.gif in τ-topology as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq117_HTML.gif whenever u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq130_HTML.gif in τ-topology and s n s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq131_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq117_HTML.gif.

     
  2. (ii)

    g is τ-locally finite-dimensional, i.e., for each ( u , s ) A × [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq132_HTML.gif, there is a neighborhood U of ( u , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq133_HTML.gif in the product topology of ( E , τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq134_HTML.gif and [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq135_HTML.gif such that g ( U ( A × [ 0 , 1 ] ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq136_HTML.gif is contained in a finite-dimensional subspace of H.

     

Admissible maps are defined similarly. Recall also that admissible maps and homotopies are necessarily continuous, and on bounded subsets of H the τ-topology coincides with the product topology of H weak https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq137_HTML.gif and H strong + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq138_HTML.gif.

Let Φ C 1 ( H , R 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq139_HTML.gif, R > r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq140_HTML.gif and z 0 H + { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq141_HTML.gif and define
M = { z = z + t z 0 : z R , t 0 } , N = { z H + : z = r } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equab_HTML.gif
and
Γ : = { h C ( M × [ 0 , 1 ] , H ) | h  is admissible , h ( u , 0 ) = u  and Φ ( h ( u , s ) ) max { Φ ( u ) , 1 }  for all  s [ 0 , 1 ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equac_HTML.gif

Proposition 3.1 (Theorem 2.1, [11])

Let H = H + H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq142_HTML.gif be a separable Hilbert space with H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq119_HTML.gif orthogonal to H + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq143_HTML.gif. Suppose that
  1. (i)

    Φ ( z ) = 1 2 ( z + z 2 ) Ψ ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq144_HTML.gif, where Ψ C 1 ( H , R 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq145_HTML.gif is bounded below, weakly sequentially lower semi-continuous and Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq146_HTML.gif is weakly sequentially continuous.

     
  2. (ii)

    There exist z 0 H + { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq147_HTML.gif, α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq148_HTML.gif, and R > r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq140_HTML.gif such that Φ | N α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq149_HTML.gif and Φ | M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq150_HTML.gif.

     
Then there exists a (C) c -sequence for Φ, where
c : = inf h Γ sup u M Φ ( h ( u , 1 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equad_HTML.gif

Moreover, c α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq151_HTML.gif.

For fixed z 0 E + { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq152_HTML.gif and R > r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq140_HTML.gif, let
M R = { z = z + ρ z 0 : z E , z E R , ρ 0 } , N r = { z E + : z E = r } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equae_HTML.gif

Lemma 3.2 There exist r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq153_HTML.gif and α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq148_HTML.gif such that Φ | N r α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq154_HTML.gif.

Proof For any z N r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq155_HTML.gif, z = ( u , v ) E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq156_HTML.gif, we know that v = A m A l u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq157_HTML.gif or, equivalently, u = A l A m v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq158_HTML.gif. By (3.2) and Proposition 2.1, we have
| Ω F ( x , v ) d σ + Ω G ( x , u ) d σ | Ω ( | F ( x , v ) | + | G ( x , u ) | ) d σ Ω [ ( ϵ | v | 2 + C ϵ | v | p ) + ( ϵ | u | 2 + C ϵ | u q ) ] d σ ϵ u E l 2 + C ϵ u E l p + ϵ v E m 2 + C ϵ v E m q ϵ z E 2 + C ϵ z E p + ϵ z E 2 + C ϵ z E q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equaf_HTML.gif
Since p , q > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq27_HTML.gif, we have if z E = r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq159_HTML.gif is small enough,
Φ ( z ) = Ω A l u A m v d x Ω F ( x , v ) d σ Ω G ( x , u ) d σ 1 2 z E 2 ϵ z E 2 C ϵ z E p C ϵ z E q α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equag_HTML.gif

for some α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq148_HTML.gif. □

Lemma 3.3 For the r given by Lemma 3.2 and any z 0 = ( u 0 , v 0 ) E + { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq160_HTML.gif with z 0 E = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq161_HTML.gif, there exists R > r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq162_HTML.gif such that Φ | M R 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq163_HTML.gif, where M R = { z = z + ρ z 0 : z E R , ρ 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq164_HTML.gif.

Proof If z M R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq165_HTML.gif, then z = z + ρ z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq166_HTML.gif with either : z E = R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq167_HTML.gif, ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq168_HTML.gif or : z E < R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq169_HTML.gif, ρ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq170_HTML.gif.
  1. (i)
    If ρ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq170_HTML.gif, then we have z E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq171_HTML.gif, z = ( u , A m A l u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq172_HTML.gif and
    Φ ( z ) = 1 2 z E 2 Ω F ( x , v ) Ω G ( x , u ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equah_HTML.gif
     
since F ( x , t ) , G ( x , t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq173_HTML.gif for any ( x , t ) Ω × R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq112_HTML.gif.
  1. (ii)
    Assume that ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq174_HTML.gif. We argue by contradiction. Suppose that there exists a sequence { z n } M n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq175_HTML.gif, z n = ρ n z 0 + z n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq176_HTML.gif, ρ n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq177_HTML.gif, z 0 E = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq161_HTML.gif, z n E = n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq178_HTML.gif such that Φ ( z n ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq179_HTML.gif. If z n = ( u n , v n ) : = ( ρ n u 0 + φ n , ρ n v 0 + ψ n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq180_HTML.gif, then by the definitions of E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq84_HTML.gif and E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq85_HTML.gif, we have
    A l u n , A m v n = Ω A l u n A m v n d x = Ω A l ( ρ n u 0 + φ n ) A m ( ρ n v 0 + ψ n ) d x = Ω ( ρ n A l u 0 + A l φ n ) ( ρ n A m v 0 + A m ψ n ) d x = Ω ( ρ n 2 | A l u 0 | 2 ρ n A l u 0 A l φ n + ρ n A l u 0 A l φ n | A m ψ n | 2 ) d x = ρ n 2 2 z 0 E 2 1 2 z n E 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equai_HTML.gif
     
Hence,
Φ ( z n ) = 1 2 ( ρ n 2 z 0 E 2 z n E 2 ) Ω F ( x , v n ) d σ Ω G ( x , u n ) d σ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equaj_HTML.gif
Therefore,
Φ ( z n ) z n E 2 = 1 2 ( ρ n 2 z n E 2 z 0 E 2 z n E 2 z n E 2 ) Ω F ( x , v n ) + G ( x , u n ) z n E 2 d σ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equak_HTML.gif
Denote δ n : = ρ n z n E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq181_HTML.gif, w n : = z n z n E = ( φ ˜ n , ψ ˜ n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq182_HTML.gif. Then
Φ ( z n ) z n E 2 = 1 2 ( δ n 2 w n E 2 ) Ω F ( x , v n ) + G ( x , u n ) z n E 2 d σ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ28_HTML.gif
(3.3)

Since F ( x , t ) , G ( x , t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq173_HTML.gif for any ( x , t ) Ω × R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq112_HTML.gif, by (3.3) we know that δ n w n E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq183_HTML.gif.

On the other hand, δ n 2 + w n E 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq184_HTML.gif, which implies that δ n 2 δ 2 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq185_HTML.gif for some δ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq186_HTML.gif and w n w = ( φ ˜ , ψ ˜ ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq187_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq117_HTML.gif, where denotes the weak convergence in E.

If δ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq188_HTML.gif, then from (3.3) we get
w n 2 0 , Ω F ( x , v n ) z n E 2 d σ 0 , Ω G ( x , u n ) z n E 2 d σ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equal_HTML.gif
Therefore,
1 = δ n 2 + w n E 2 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equam_HTML.gif

which is impossible.

If δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq189_HTML.gif, since δ n 2 δ 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq190_HTML.gif and z n E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq191_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq117_HTML.gif, it follows that ρ n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq192_HTML.gif. If x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq25_HTML.gif is such that δ u 0 + φ ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq193_HTML.gif, we have
lim n + ρ n u 0 + φ n z n E = δ u 0 + φ ˜ ( x ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equan_HTML.gif
thus,
u n = ρ n u 0 + φ n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ29_HTML.gif
(3.4)

as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq117_HTML.gif.

Similarly, if δ u 0 ( x ) + ψ ˜ ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq194_HTML.gif, we have
v n = ρ n v 0 + ψ n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ30_HTML.gif
(3.5)

as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq117_HTML.gif.

Since Φ ( z n ) z n E 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq195_HTML.gif and F ( x , t ) , G ( x , t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq173_HTML.gif, we get
0 < 1 2 ( δ n 2 w n E 2 ) Ω [ F ( x , v n ) v n 2 ( v n z n E ) 2 + G ( x , u n ) u n 2 ( u n z n E ) 2 ] d σ 1 2 ( δ n 2 w n E 2 ) { δ u 0 + ψ ˜ 0 } F ( x , v n ) v n 2 ( v n z n E ) 2 d σ { δ u 0 + φ ˜ 0 } G ( x , u n ) u n 2 ( u n z n E ) 2 d σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equao_HTML.gif
Note that
u n z n E = ρ n u 0 + φ n z n E δ u 0 + φ ˜ in  E l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equap_HTML.gif
and
v n z n E = ρ n v 0 + ψ n z n E δ v 0 + ψ ˜ in  E m , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equaq_HTML.gif
as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq196_HTML.gif. Hence, by Proposition 2.1 we may assume, passing to a subsequence, that
u n z n E = ρ n u 0 + φ n z n E δ u 0 + φ ˜ , v n z n E = ρ n v 0 + ψ n z n E δ v 0 + ψ ˜ a.e. in  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equar_HTML.gif
as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq196_HTML.gif. By (3.8), (3.9) and (H4), taking limit in (3.10), using Fatou’s lemma and the fact that lim inf n w n E 2 w E 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq197_HTML.gif, we obtain
0 1 2 ( δ 2 w E 2 ) { δ v 0 + ψ ˜ 0 } ( + ) ( δ v 0 + ψ ˜ ) 2 d σ { δ u 0 + φ ˜ 0 } ( + ) ( δ u 0 + φ ˜ ) 2 d σ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equas_HTML.gif

which is impossible, thus the lemma is proved. □

Lemma 3.4 If { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq50_HTML.gif is a (C) c -sequence of Φ, then { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq50_HTML.gif is bounded in E.

Proof Suppose that { z n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq198_HTML.gif is a (C) c sequence for Φ, that is,
Φ ( z n ) c , Φ ( z n ) E ( 1 + z n E ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equat_HTML.gif
which shows that
c + o ( 1 ) = Φ ( z n ) , Φ ( z n ) , z n = o ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ31_HTML.gif
(3.6)

where o ( 1 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq199_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq200_HTML.gif.

We suppose, by contradiction, that
z n E + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ32_HTML.gif
(3.7)
and let w n = z n z n E : = ( w n 1 , w n 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq201_HTML.gif. Then w n E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq202_HTML.gif with
w n E = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equau_HTML.gif
By Proposition 2.1, { w n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq203_HTML.gif contains a subsequence, denoted again by { w n } = { ( w n 1 , w n 2 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq204_HTML.gif such that we may assume that
{ w n 1 ( x ) w 1 ( x )  in  E l , w n 2 ( x ) w 2 ( x )  in  E m ; w n 1 ( x ) w 1 ( x ) , w n 2 ( x ) w 2 ( x )  a.e.  in  Ω ; w n 1 w  in  L α ( Ω ) ( 2 α < 2 ( N 1 ) N 4 l ) ; w n 2 w  in  L β ( Ω ) ( 2 β < 2 ( N m ) N 4 m ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ33_HTML.gif
(3.8)
Let Ω = { x Ω , w ( x ) ( 0 , 0 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq205_HTML.gif. Then we have
lim n + w n ( x ) = lim n + z n ( x ) z n E = w ( x ) ( 0 , 0 ) in  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equav_HTML.gif
and (3.7) implies that
| z n | + a.e. in  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equaw_HTML.gif
We may assume, without loss of generality, that
| u n | + a.e. in  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ34_HTML.gif
(3.9)
By (H4), we see that
lim n + G ( x , u n ( x ) ) | u n ( x ) | 2 = + a.e. in  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equax_HTML.gif
This means that
lim n + G ( x , u n ( x ) ) | u n ( x ) | 2 | w n 1 ( x ) | 2 = + a.e. in  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ35_HTML.gif
(3.10)
By (H4), there is an N 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq206_HTML.gif such that
G ( x , s ) | s | 2 > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ36_HTML.gif
(3.11)
for any x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq207_HTML.gif and s R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq208_HTML.gif with | s | N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq209_HTML.gif. Since G ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq210_HTML.gif is continuous on Ω × [ N 0 , N 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq211_HTML.gif, there is an M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq212_HTML.gif such that
| G ( x , s ) | M , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ37_HTML.gif
(3.12)
for ( x , t ) Ω ¯ × [ N 0 , N 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq213_HTML.gif. From (3.11) and (3.12), we see that there is a constant C, such that for any ( x , s ) Ω × R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq214_HTML.gif, we have
G ( x , s ) C , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equay_HTML.gif
which shows that
G ( x , u n ( x ) ) C z n E 2 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equaz_HTML.gif
This means that
G ( x , u n ( x ) ) | u n ( x ) | 2 | w n 1 ( x ) | 2 C z n E 2 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ38_HTML.gif
(3.13)
Since by (3.6) we have that
c + o ( 1 ) = Φ ( z n ) = A l u n , A m v n Ω [ F ( x , v n ) + G ( x , u n ) ] d σ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equba_HTML.gif
which shows that
0 c + o ( 1 ) z n E 2 = A l w n 1 , A m w n 2 Ω [ F ( x , v n ) + G ( x , u n ) ] z n E 2 d σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ39_HTML.gif
(3.14)
Since F ( x , t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq215_HTML.gif, we have
1 2 + o ( 1 ) = o ( 1 ) + 1 2 Ω ( | A l w n 1 | 2 + | A m w n 2 | 2 ) d x o ( 1 ) + A l w n 1 , A m w n 2 = Ω [ F ( x , v n ) + G ( x , u n ) ] z n E 2 d σ Ω G ( x , u n ) z n E 2 d σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ40_HTML.gif
(3.15)

We claim that | Ω | = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq216_HTML.gif.

If | Ω | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq217_HTML.gif, then by Fatou’s lemma, (H4) and Hölder’s inequality, we get
+ = + | Ω | = [ Ω lim inf n + G ( x , u n ( x ) ) | u n ( x ) | 2 | w n 1 ( x ) | 2 d σ Ω lim sup n + C z n E 2 d σ ] = Ω lim inf n + ( G ( x , u n ( x ) ) | u n ( x ) | 2 | w n 1 ( x ) | 2 C z n E 2 ) d σ lim inf n + Ω ( G ( x , u n ( x ) ) | u n ( x ) | 2 | w n 1 ( x ) | 2 C z n E 2 ) d σ lim inf n + Ω ( G ( x , u n ( x ) ) | u n ( x ) | 2 | w n 2 ( x ) | 2 C z n E 2 ) d σ = lim inf n + Ω G ( x , u n ( x ) ) u n 2 d σ lim n + Ω C z n E 2 d σ = lim inf n + Ω G ( x , u n ( x ) ) z n E 2 d σ 1 2 + o ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equbb_HTML.gif

which is impossible.

This shows that
| Ω | = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equbc_HTML.gif

Hence w ( x ) = ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq218_HTML.gif a.e. in Ω.

Since Φ ( t z n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq219_HTML.gif is continuous in t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq220_HTML.gif, there exists t n [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq221_HTML.gif ( n = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq222_HTML.gif), such that
Φ ( t n z n ) = max 0 t 1 Φ ( t z n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equbd_HTML.gif
As Φ ( z n ) , z n = o ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq223_HTML.gif, we see that
Φ ( t n z n ) , t n z n = o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Eqube_HTML.gif
By (H5), then we get for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq220_HTML.gif that
2 Φ ( t z n ) 2 Φ ( t n u n ) = 2 Φ ( t n z n ) Φ ( t n z n ) , t n z n + o ( 1 ) = Ω [ ( t n v n f ( x , t n v n ) 2 F ( x , t n v n ) ) + ( t n u n g ( x , t n u n ) 2 G ( x , t n u n ) ) ] d σ + o ( 1 ) Ω [ ( v n f ( x , v n ) 2 F ( x , v n ) + C 1 , ) + ( u n g ( x , u n ) 2 G ( x , u n ) + C 2 , ) ] d σ + o ( 1 ) = 2 c + ( C 1 , + C 2 , ) | Ω | + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equ41_HTML.gif
(3.16)
On the other hand, taking t = R z n E [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq224_HTML.gif and z n = ( u n , u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq225_HTML.gif, by (3.8) then w n = ( u n , u n ) z n E : = ( θ n , θ n ) E 1 × E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq226_HTML.gif and θ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq227_HTML.gif in L r ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq228_HTML.gif ( 1 r < 2 ( N 1 ) N 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq229_HTML.gif). From (3.2) and θ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq227_HTML.gif in L r ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq230_HTML.gif ( 1 r < 2 ( N 1 ) N 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq231_HTML.gif) as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq117_HTML.gif, we obtain
2 Φ ( R w n ) = 2 R 2 2 Ω [ F ( x , R θ n ) + G ( x , R θ n ) ] d σ = 2 R 2 + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equbf_HTML.gif
So we have
2 R 2 + o ( 1 ) = 2 Φ ( R w n ) 2 c + ( C 1 , + C 2 , ) | Ω | + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equbg_HTML.gif
Letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq232_HTML.gif, we get
2 R 2 ( C 1 , + C 2 , ) | Ω | + 2 c . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equbh_HTML.gif

Letting R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq233_HTML.gif, we get a contradiction. This proves that z n E C < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq234_HTML.gif for some constant C. □

Proof of Theorem 1.5 Under the assumptions (H1)-(H5), we know that the functional Φ given by (2.5) is in C 1 ( E , R 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq235_HTML.gif. By Lemma 3.2, there exist r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq153_HTML.gif and α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq148_HTML.gif such that Φ | N r α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq154_HTML.gif, where N r = { z E + : z E = r } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq236_HTML.gif. By Lemma 3.3, for such an r, there exist R > r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq162_HTML.gif and suitable z 0 E + { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq152_HTML.gif such that Φ | M R 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq163_HTML.gif, where M R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq237_HTML.gif was given before Lemma 3.3. Note that E = E + E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq87_HTML.gif and for z = ( u , v ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq238_HTML.gif, we have
Φ ( z ) = 1 2 z + E 2 1 2 z E 2 Ω [ F ( x , v ) + G ( x , u ) ] d σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equbi_HTML.gif
Since from Proposition 2.1 and Remark 1.1 we know that E L q ( Ω ) × L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq239_HTML.gif, from (3.2) and Fatou’s lemma, we know that
H ( z ) = Ω [ F ( x , v ) + G ( x , u ) ] d σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_Equbj_HTML.gif

is C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq93_HTML.gif and H 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq240_HTML.gif is weakly sequentially lower semicontinuous and H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq94_HTML.gif is weakly sequentially continuous in E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq116_HTML.gif. Hence by Proposition 3.1 there exists a (C) c -sequence { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq50_HTML.gif for Φ, where c α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq241_HTML.gif. By Lemma 3.4, { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq50_HTML.gif is bounded in E. So, up to a subsequence, we may assume that z n z ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq242_HTML.gif in E, as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq117_HTML.gif. From Lemma 2.2, we know that H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq243_HTML.gif is compact. So it is easy to check that Φ ( z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq244_HTML.gif in E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq116_HTML.gif. Hence z is a nontrivial solution pair of (1.1). Obviously, z = ( u , v ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq238_HTML.gif is a nonnegative solution pair of (1.1). Applying the strong maximum principle, we obtain that u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq245_HTML.gif and v > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-159/MediaObjects/13661_2013_Article_407_IEq246_HTML.gif. This completes the proof. □

Declarations

Acknowledgements

The authors were partially supported by NSFC (No. 11071092; No. 11071095; No. 11101171), the PhD specialized grant of the Ministry of Education of China (20110144110001).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Huazhong Normal University

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