The Nehari manifold and application to a semilinear elliptic system

In this paper, we study the Nehari manifold and its application to the following semilinear elliptic system: {−Δu+u=λf(x)|u|q−2u,x∈Ω,−Δv+v=μg(x)|v|q−2v,x∈Ω,∂u∂n=αα+βh(x)|u|α−2u|v|β,x∈∂Ω,∂v∂n=βα+βh(x)|u|α|v|β−2v,x∈∂Ω, where ΩΩ is a bounded domain in RNRN with smooth boundary ∂Ω∂Ω. We prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters (λ,μ)(λ,μ) belongs to a certain subset of R2R2.