Open Access

Existence of three solutions for a nonlocal elliptic system of ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq1_HTML.gif-Kirchhoff type

  • Guang-Sheng Chen1,
  • Hui-Yu Tang1Email author,
  • De-Quan Zhang2,
  • Yun-Xiu Jiao3 and
  • Hao-Xiang Wang4
Boundary Value Problems20132013:175

DOI: 10.1186/1687-2770-2013-175

Received: 1 May 2013

Accepted: 10 July 2013

Published: 25 July 2013

Abstract

In this paper, we study the solutions of a nonlocal elliptic system of ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq1_HTML.gif-Kirchhoff type on a bounded domain based on the three critical points theorem introduced by Ricceri. Firstly, we establish the existence of three weak solutions under appropriate hypotheses; then, we prove the existence of at least three weak solutions for the nonlocal elliptic system of ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq1_HTML.gif-Kirchhoff type.

Keywords

( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq1_HTML.gif-Kirchhoff type system multiple solutions three critical points theory

1 Introduction and main results

We consider the boundary problem involving ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq1_HTML.gif-Kirchhoff
{ [ M 1 ( Ω | u | p ) ] p 1 Δ p u = λ F u ( x , u , υ ) + μ G u ( x , u , υ ) , in  Ω , [ M 2 ( Ω | υ | q ) ] q 1 Δ q υ = λ F υ ( x , u , υ ) + μ G v ( x , u , υ ) , in  Ω , u = υ = 0 , on  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ1_HTML.gif
(1.1)
where Ω R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq2_HTML.gif ( N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq3_HTML.gif) is a bounded smooth domain, λ , μ [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq4_HTML.gif, p > N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq5_HTML.gif, q > N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq6_HTML.gif, Δ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq7_HTML.gif is the p-Laplacian operator Δ p u = div ( | u | p 2 u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq8_HTML.gif. F , G : Ω × R × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq9_HTML.gif are functions such that F ( , s , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq10_HTML.gif, G ( , s , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq11_HTML.gif are measurable in Ω for all ( s , t ) R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq12_HTML.gif and F ( x , , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq13_HTML.gif, G ( x , , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq14_HTML.gif are continuously differentiable in R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq15_HTML.gif for a.e. x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq16_HTML.gif. F i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq17_HTML.gif is the partial derivative of F with respect to i, i = u , v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq18_HTML.gif, so is G i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq19_HTML.gif. M i : R + R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq20_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq21_HTML.gif, are continuous functions which satisfy the following bounded conditions.
  1. (M)
    There exist two positive constants m 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq22_HTML.gif, m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq23_HTML.gif such that
    m 0 M i ( t ) m 1 , t 0 , i = 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ2_HTML.gif
    (1.2)
     
Here and in the sequel, X denotes the Cartesian product of two Sobolev spaces W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq24_HTML.gif and W 0 1 , q ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq25_HTML.gif, i.e., X = W 0 1 , p ( Ω ) × W 0 1 , q ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq26_HTML.gif. The reflexive real Banach space X is endowed with the norm
( u , υ ) = u p + υ q , u p = ( Ω | u | p ) 1 / p , υ q = ( Ω | υ | q ) 1 / q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equa_HTML.gif
Since p > N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq5_HTML.gif and q > N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq6_HTML.gif, W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq24_HTML.gif and W 0 1 , q ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq25_HTML.gif are compactly embedded in C 0 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq27_HTML.gif. Let
C = max { sup u W 0 1 , p ( Ω ) { 0 } max x Ω ¯ { | u ( x ) | p } u p p , sup v W 0 1 , q ( Ω ) { 0 } max x Ω ¯ { | υ ( x ) | q } υ q q } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ3_HTML.gif
(1.3)
then one has C < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq28_HTML.gif. Furthermore, it is known from [1] that
sup u W 0 1 , p ( Ω ) { 0 } max x Ω ¯ { | u ( x ) | p } u p N 1 / p π ( Γ ( 1 + N 2 ) ) 1 / N ( p 1 p N ) 1 1 / p | Ω | ( 1 / N ) ( 1 / p ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equb_HTML.gif
where Γ is the gamma function and | Ω | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq29_HTML.gif is the Lebesgue measure of Ω. As usual, by a weak solution of system (1.1), we mean any ( u , υ ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq30_HTML.gif such that
[ M 1 ( Ω | u | p ) ] p 1 Ω | u | p 2 u ϕ + [ M 2 ( Ω | υ | q ) ] q 1 Ω | υ | q 2 υ ψ λ Ω ( F u ϕ + F v ψ ) d x μ Ω ( G u ϕ + G v ψ ) d x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ4_HTML.gif
(1.4)

for all ( ϕ , ψ ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq31_HTML.gif.

System (1.1) is related to the stationary version of a model established by Kirchhoff [2]. More precisely, Kirchhoff proposed the following model:
ρ 2 u t 2 ( P 0 h + E 2 L 0 L | u x | 2 d x ) 2 u x 2 = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ5_HTML.gif
(1.5)

which extends D’Alembert’s wave equation with free vibrations of elastic strings, where ρ denotes the mass density, P 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq32_HTML.gif denotes the initial tension, h denotes the area of the cross-section, E denotes the Young modulus of the material, and L denotes the length of the string. Kirchhoff’s model considers the changes in length of the string produced during the vibrations.

Later, (1.1) was developed into the following form:
u t t M ( Ω | u | 2 ) Δ u = f ( x , u ) in  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ6_HTML.gif
(1.6)
where M : R + R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq33_HTML.gif is a given function. After that, many authors studied the following problem:
M ( Ω | u | 2 ) Δ u = f ( x , u ) in  Ω , u = 0 on  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ7_HTML.gif
(1.7)
which is the stationary counterpart of (1.6). By applying variational methods and other techniques, many results of (1.7) were obtained, the reader is referred to [313] and the references therein. In particular, Alves et al. [[3], Theorem 4] supposed that M satisfies bounded condition (M) and f ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq34_HTML.gif satisfies the condition
0 < υ F ( x , t ) f ( x , t ) t , | t | R , x Ω  for some  v > 2  and  R > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ8_HTML.gif
(AR)

where F ( x , t ) = 0 t f ( x , s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq35_HTML.gif; one positive solution for (1.7) was given.

In [14], using Ekeland’s variational principle, Corrêa and Nascimento proved the existence of a weak solution for the boundary problem associated with the nonlocal elliptic system of p-Kirchhoff type
{ [ M 1 ( Ω | u | p ) ] p 1 Δ p u = f ( u , υ ) + ρ 1 ( x ) , in  Ω , [ M 2 ( Ω | υ | p ) ] p 1 Δ p υ = g ( , u , υ ) + ρ 2 ( x ) , in  Ω , u η = υ η = 0 , on  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ9_HTML.gif
(1.8)

where η is the unit exterior vector on Ω, and M i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq36_HTML.gif, ρ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq37_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq21_HTML.gif), f, g satisfy suitable assumptions.

In [15], when μ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq38_HTML.gif in (1.1), Bitao Cheng et al. studied the existence of two solutions and three solutions of the following nonlocal elliptic system:
{ [ M 1 ( Ω | u | p ) ] p 1 Δ p u = λ F u ( x , u , υ ) , in  Ω , [ M 2 ( Ω | υ | q ) ] q 1 Δ q υ = λ F υ ( x , u , υ ) , in  Ω , u = υ = 0 , on  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ10_HTML.gif
(1.9)

In this paper, our objective is to prove the existence of three solutions of problem (1.1) by applying the three critical points theorem established by Ricceri [16]. Our result, under appropriate assumptions, ensures the existence of an open interval Λ [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq39_HTML.gif and a positive real number ρ such that, for each λ Λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq40_HTML.gif, problem (1.1) admits at least three weak solutions whose norms in X are less than ρ. The purpose of the present paper is to generalize the main result of [15].

Now, for every x 0 Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq41_HTML.gif and choosing R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq42_HTML.gif, R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq43_HTML.gif with R 2 > R 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq44_HTML.gif, such that B ( x 0 , R 2 ) Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq45_HTML.gif, where B ( x , R ) = { y R N : | y x | < R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq46_HTML.gif, put
α 1 = α 1 ( N , p , R 1 , R 2 ) = C 1 / p ( R 2 N R 1 N ) 1 / p R 2 R 1 ( π N / 2 Γ ( 1 + N / 2 ) ) 1 / p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ11_HTML.gif
(1.10)
α 2 = α 2 ( N , q , R 1 , R 2 ) = C 1 / q ( R 2 N R 1 N ) 1 / q R 2 R 1 ( π N / 2 Γ ( 1 + N / 2 ) ) 1 / q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ12_HTML.gif
(1.11)
Moreover, let a, c be positive constants and define
y ( x ) = a R 2 R 1 ( R 2 { i = 1 N ( x i x 0 i ) 2 } 1 / 2 ) , x B ( x 0 , R 2 ) B ( x 0 , R 1 ) , A ( c ) = { ( s , t ) R × R : | s | p + | t | q c } , M + = max { m 1 p 1 p , m 1 q 1 q } , M = min { m 0 p 1 p , m 0 q 1 q } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equc_HTML.gif

Our main result is stated as follows.

Theorem 1.1 Assume that R 2 > R 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq47_HTML.gif such that B ( x 0 , R 2 ) Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq48_HTML.gif, and suppose that there exist four positive constants a, b, γ and β with γ < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq49_HTML.gif, β < q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq50_HTML.gif, ( a α 1 ) p + ( a α 2 ) q > b M + / M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq51_HTML.gif, and a function α ( x ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq52_HTML.gif such that

  1. (j1)

    F ( x , s , t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq53_HTML.gif for a.e. x Ω B ( x 0 , R 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq54_HTML.gif and all ( s , t ) [ 0 , a ] × [ 0 , a ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq55_HTML.gif;

     
  2. (j2)

    [ ( a α 1 ) p + ( a α 2 ) q ] | Ω | sup ( x , s , t ) Ω × A ( b M + / M ) F ( x , s , t ) < b B ( x 0 , R 1 ) F ( x , a , a ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq56_HTML.gif;

     
  3. (j3)

    F ( x , s , t ) α ( x ) ( 1 + | s | γ + | t | β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq57_HTML.gif for a.e. x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq58_HTML.gif and all ( s , t ) R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq59_HTML.gif;

     
  4. (j4)

    F ( x , 0 , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq60_HTML.gif for a.e. x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq58_HTML.gif.

     

Then there exist an open interval Λ [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq61_HTML.gif and a positive real number ρ with the following property: for each λ Λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq62_HTML.gif and for two Carathéodory functions G u , G v : Ω × R × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq63_HTML.gif satisfying

  1. (j5)

    sup { | s | ξ , | t | ξ } ( | G u ( , s , t ) | + | G v ( , s , t ) | ) L 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq64_HTML.gif for all ξ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq65_HTML.gif,

     

there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq66_HTML.gif such that, for each μ [ 0 , δ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq67_HTML.gif, problem (1.1) has at least three weak solutions w i = ( u i , υ i ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq68_HTML.gif ( i = 1 , 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq69_HTML.gif) whose norms w i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq70_HTML.gif are less than ρ.

2 Proof of the main result

First we recall the modified form of Ricceri’s three critical points theorem (Theorem 1 in [16]) and Proposition 3.1 of [17], which is our primary tool in proving our main result.

Theorem 2.1 ([16], Theorem 1)

Suppose that X is a reflexive real Banach space and that Φ : X R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq71_HTML.gif is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq72_HTML.gif, and that Φ is bounded on each bounded subset of X; Ψ : X R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq73_HTML.gif is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact; I R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq74_HTML.gif is an interval. Suppose that
lim x + ( Φ ( x ) + λ Ψ ( x ) ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equd_HTML.gif
for all λ I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq75_HTML.gif, and that there exists h R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq76_HTML.gif such that
sup λ I inf x X ( Φ ( x ) + λ ( Ψ ( x ) + h ) ) < inf x X sup λ I ( Φ ( x ) + λ ( Ψ ( x ) + h ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ13_HTML.gif
(2.1)
Then there exist an open interval Λ I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq77_HTML.gif and a positive real number ρ with the following property: for every λ Λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq78_HTML.gif and every C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq79_HTML.gif functional J : X R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq80_HTML.gif with compact derivative, there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq66_HTML.gif such that, for each μ [ 0 , δ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq67_HTML.gif, the equation
Φ ( x ) + λ Ψ ( x ) + μ J ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Eque_HTML.gif

has at least three solutions in X whose norms are less than ρ.

Proposition 2.1 ([17], Proposition 3.1)

Assume that X is a nonempty set and Φ, Ψ are two real functions on X. Suppose that there are r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq81_HTML.gif and x 0 , x 1 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq82_HTML.gif such that
Φ ( x 0 ) = Ψ ( x 0 ) = 0 , Φ ( x 1 ) > 1 , sup x Φ 1 ( [ , r ] ) Ψ ( x ) < r Ψ ( x 1 ) Φ ( x 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equf_HTML.gif
Then, for each h satisfying
sup x Φ 1 ( [ , r ] ) Ψ ( x ) < h < r Ψ ( x 1 ) Φ ( x 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equg_HTML.gif
one has
sup λ 0 inf x X ( Φ ( x ) + λ ( Ψ ( x ) + h ) ) < inf x X sup λ 0 ( Φ ( x ) + λ ( Ψ ( x ) + h ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equh_HTML.gif

Before proving Theorem 1.1, we define a functional and give a lemma.

The functional H : X R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq83_HTML.gif is defined by
H ( u , v ) = Φ ( u , v ) + λ J ( u , v ) + μ ψ ( u , v ) = 1 p M ˆ 1 ( Ω | u | p ) + 1 q M ˆ 2 ( Ω | υ | q ) λ Ω F ( x , u , v ) d x μ Ω G ( x , u , v ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ14_HTML.gif
(2.2)
for all ( u , υ ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq30_HTML.gif, where
M ˆ 1 = 0 t [ M 1 ( s ) ] p 1 d s , M ˆ 2 = 0 t [ M 2 ( s ) ] q 1 d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ15_HTML.gif
(2.3)

By conditions (M) and (j3), it is clear that H C 1 ( X , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq84_HTML.gif and a critical point of H corresponds to a weak solution of system (1.1).

Lemma 2.2 Assume that there exist two positive constants a, b with ( a α 1 ) p + ( a α 2 ) q > b M + / M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq85_HTML.gif such that

  1. (j1)

    F ( x , s , t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq53_HTML.gif, for a.e. x Ω B ( x 0 , R 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq54_HTML.gif and all ( s , t ) [ 0 , a ] × [ 0 , a ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq55_HTML.gif;

     
  2. (j2)

    [ ( a α 1 ) p + ( a α 2 ) q ] | Ω | sup ( x , s , t ) Ω × A ( b M + / M ) F ( x , s , t ) < b B ( x 0 , R 1 ) F ( x , a , a ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq56_HTML.gif.

     
Then there exist r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq86_HTML.gif and u 0 W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq87_HTML.gif, υ 0 W 0 1 , q ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq88_HTML.gif such that
Φ ( u 0 , v 0 ) > r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equi_HTML.gif
and
| Ω | sup ( x , s , t ) Ω × A ( b M + / M ) F ( x , s , t ) b M + C Ω F ( x , u 0 , υ 0 ) d x Φ ( u 0 , υ 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equj_HTML.gif
Proof We put
w 0 ( x ) = { 0 , x Ω ¯ B ( x 0 , R 2 ) , a R 2 R 1 ( R 2 { i = 1 N ( x i x 0 i ) } 1 / 2 ) , x B ( x 0 , R 2 ) B ( x 0 , R 1 ) , a , x B ( x 0 , R 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equk_HTML.gif
and u 0 ( x ) = υ 0 ( x ) = w 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq89_HTML.gif. Then we can verify easily ( u 0 , υ 0 ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq90_HTML.gif and, in particular, we have
u 0 p p = ( R 2 N R 1 N ) π N / 2 Γ ( 1 + N / 2 ) ( a R 2 R 1 ) p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ16_HTML.gif
(2.4)
and
υ 0 q q = ( R 2 N R 1 N ) π N / 2 Γ ( 1 + N / 2 ) ( a R 2 R 1 ) q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ17_HTML.gif
(2.5)
Hence, we obtain from (1.10), (1.11), (2.4) and (2.5) that
u 0 p p = w 0 p p = ( a α 1 ) p C , υ 0 q q = w 0 q q = ( a α 2 ) q C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ18_HTML.gif
(2.6)
Under condition (M), by a simple computation, we have
M ( u p p + υ q q ) Φ ( u , υ ) M + ( u p p + υ q q ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ19_HTML.gif
(2.7)
Setting r = b M + C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq91_HTML.gif and applying the assumption of Lemma 2.2
( a α 1 ) p + ( a α 2 ) q > b M + / M , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equl_HTML.gif
from (2.6) and (2.7), we obtain
Φ ( u 0 , v 0 ) M ( u 0 p p + v 0 q q ) = M C [ ( a α 1 ) p + ( a α 2 ) q ] > M C b M + M = r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equm_HTML.gif
Since, 0 u 0 a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq92_HTML.gif, 0 v 0 a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq93_HTML.gif for each x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq58_HTML.gif, from condition (j1) of Lemma 2.2, we have
Ω B ( x 0 , R 2 ) F ( x , u 0 , υ 0 ) d x + B ( x 0 , R 2 ) B ( x 0 , R 1 ) F ( x , u 0 , υ 0 ) d x 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equn_HTML.gif
Hence, based on condition (j2), we get
| Ω | sup ( x , s , t ) Ω × A ( b M + / M ) F ( x , s , t ) < b ( a α 1 ) p + ( a α 2 ) q B ( x 0 , R 1 ) F ( x , a , a ) d x = b M + C B ( x 0 , R 1 ) F ( x , a , a ) d x M + ( ( a α 1 ) p + ( a α 2 ) q ) / C b M + C Ω B ( x 0 , R 1 ) F ( x , u 0 , υ 0 ) d x + B ( x 0 , R 1 ) F ( x , u 0 , υ 0 ) d x M + ( u 0 p p + υ 0 q q ) b M + C Ω F ( x , u 0 , υ 0 ) d x Ψ ( u 0 , υ 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equo_HTML.gif

 □

Now, we can prove our main result.

Proof of Theorem 1.1 For each ( u , v ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq94_HTML.gif, let
Φ ( u , v ) = M ˆ 1 ( u p p ) p + M ˆ 2 ( v q q ) q , Ψ ( u , υ ) = Ω F ( x , u , υ ) d x , J ( u , v ) = Ω G ( x , u , v ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equp_HTML.gif
From the assumption of Theorem 1.1, we know that Φ is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional. Additionally, the Gâteaux derivative of Φ has a continuous inverse on X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq72_HTML.gif. Since p > N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq5_HTML.gif, q > N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq6_HTML.gif, Ψ and J are continuously Gâteaux differential functionals whose Gâteaux derivatives are compact. Obviously, Φ is bounded on each bounded subset of X. In particular, for each ( u , v ) , ( ξ , η ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq95_HTML.gif,
Φ ( u , v ) , ( ξ , η ) = [ M 1 ( Ω | u | p ) ] p 1 Ω | u | p 2 u ξ + [ M 2 ( Ω | υ | q ) ] q 1 Ω | υ | q 2 υ η , Ψ ( u , v ) , ( ξ , η ) = Ω F u ( x , u , v ) ξ d x Ω F v ( x , u , v ) η d x , J ( u , v ) , ( ξ , η ) = Ω G u ( x , u , v ) ξ d x Ω G v ( x , u , v ) η d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equq_HTML.gif
Hence, the weak solutions of problem (1.1) are exactly the solutions of the following equation:
Φ ( u , v ) + λ Ψ ( u , v ) + μ J ( u , v ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equr_HTML.gif
From (j3), for each λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq96_HTML.gif, one has
lim ( u , v ) + ( λ Φ ( u , v ) + μ Ψ ( u , v ) ) = + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ20_HTML.gif
(2.8)
and so the first condition of Theorem 2.1 is satisfied. By Lemma 2.2, there exists ( u 0 , υ 0 ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq97_HTML.gif such that
Φ ( u 0 , v 0 ) = M ˆ 1 ( u 0 p p ) p + M ˆ 2 ( v 0 q q ) q M ( u 0 p p + v 0 q q ) = M C [ ( a α 1 ) p + ( a α 2 ) q ] > M C b M + M = b M + C > 0 = Φ ( 0 , 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ21_HTML.gif
(2.9)
and
| Ω | sup ( x , s , t ) Ω × A ( b M + / M ) F ( x , s , t ) b M + C Ω F ( x , u 0 , υ 0 ) d x Φ ( u 0 , υ 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ22_HTML.gif
(2.10)
From (1.3), we have
max x Ω ¯ { | u ( x ) | p } C u p p , max x Ω ¯ { | υ ( x ) | q } C υ q q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equs_HTML.gif
for each ( u , υ ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq98_HTML.gif. We obtain
max x Ω ¯ { | u ( x ) | p p + | v ( x ) | q q } C { u p p p + v q q q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ23_HTML.gif
(2.11)
for each ( u , υ ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq98_HTML.gif. Let r = b M + C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq91_HTML.gif for each ( u , υ ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq99_HTML.gif such that
Φ ( u , υ ) = M ˆ 1 ( u p p ) p + M ˆ 2 ( v q q ) q r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equt_HTML.gif
From (2.11), we get
| u ( x ) | p + | υ ( x ) | q C ( u p p + υ q q ) C r M = C M b M + C = b M + M . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ24_HTML.gif
(2.12)
Then, from (2.10) and (2.12), we find
sup ( u , υ ) Φ 1 ( , r ) ( Ψ ( u , υ ) ) = sup { ( u , υ ) | Φ ( u , υ ) r } Ω F ( x , u , υ ) d x sup { ( u , υ ) | | u ( x ) | p + | υ ( x ) | q b M + / M } Ω F ( x , u , υ ) d x Ω sup ( s , t ) A ( b M + / M ) F ( x , s , t ) d x | Ω | sup ( x , s , t ) Ω × A ( b M + / M ) F ( x , s , t ) b M + C Ω F ( x , u 0 , υ 0 ) d x Φ ( u 0 , υ 0 ) = r Ψ ( u 0 , υ 0 ) Φ ( u 0 , υ 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equu_HTML.gif
Hence, we have
sup { ( u , v ) | Φ ( u , v r } ( Ψ ( u , v ) ) < r Ψ ( u 0 , υ 0 ) Φ ( u 0 , υ 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ25_HTML.gif
(2.13)
Fix h such that
sup { ( u , v ) | Φ ( u , v r } ( Ψ ( u , v ) ) < h < r Ψ ( u 0 , υ 0 ) Φ ( u 0 , υ 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equv_HTML.gif
by (2.9), (2.13) and Proposition 2.1, with ( u 1 , v 1 ) = ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq100_HTML.gif and ( u , v ) = ( u 0 , v 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq101_HTML.gif, we obtain
sup λ 0 inf x X ( Φ ( x ) + λ ( h + Ψ ( x ) ) ) < inf x X sup λ 0 ( Φ ( x ) + λ ( h + Ψ ( x ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_Equ26_HTML.gif
(2.14)

and so assumption (2.1) of Theorem 2.1 is satisfied.

Now, with I = [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-175/MediaObjects/13661_2013_Article_426_IEq102_HTML.gif, from (2.8) and (2.14), all the assumptions of Theorem 2.1 hold. Hence, our conclusion follows from Theorem 2.1. □

Declarations

Acknowledgements

The authors would like to thank the editors and the referees for their valuable suggestions to improve the quality of this paper. This work was supported by the Scientific Research Project of Guangxi Education Department (no. 201204LX672).

Authors’ Affiliations

(1)
Department of Construction and Information Engineering, Guangxi Modern Vocational Technology College
(2)
Faculty of Science, Guilin University of Aerospace Industry
(3)
Department of Common Courses, Xinxiang Polytechnic College
(4)
School of Electronic Engineering, Xidian University

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