Multiplicity of nontrivial solutions for system of nonhomogenous Kirchhoff‐type equations in R N

In this paper, we are interested in looking for multiple solutions for the following system of nonhomogenous Kirchhoff‐type equations: 1.1− a + b ∫R N| ∇ u | 2 dx △ u + V ( x ) u = F u ( x , u , v ) + λf ( x ) , x ∈ R N ,− c + d ∫R N| ∇ v | 2 dx △ v + V ( x ) v = F v ( x , u , v ) + λg ( x ) , x ∈ R N ,u ( x ) → 0 , v ( x ) → 0 , as | x | → ∞ ,where constants a , c > 0; b , d , λ ≥0, N = 1,2 or 3, f , g ∈ L 2 ( R N ) and f , g ≢0, F ∈ C 1 ( R N × R 2 , R ),F u = ∂F ∂u ,F v = ∂F ∂v , V ∈ C ( R N , R ) satisfy some appropriate conditions. Under more relaxed assumptions on the nonlinear term F , the existence of one negative energy solution and one positive energy solution for the nonhomogenous system 2.1 is obtained by Ekeland's variational principle and Mountain Pass Theorem, respectively. Copyright © 2014 John Wiley & Sons, Ltd.