A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system

In this work, motivated by [T.F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight function, Nonlinear. Anal. 68 (2008) 1733–1745], and using recent ideas from Brown and Wu [K.J. Brown, T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign changing weight function, J. Math. Anal. Appl. 337 (2008) 1326–1336], we prove the existence of nontrivial nonnegative solutions to the nonlinear elliptic system {−Δpu+m(x)|u|p−2u=λ|u|γ−2u+αα+βc(x)|u|α−2u|v|β,x∈Ω,−Δpv+m(x)|v|p−2v=μ|v|γ−2v+βα+βc(x)|u|α|v|β−2v,x∈Ω,u=0=v,x∈∂Ω.Here ΔpΔp denotes the pp-Laplacian operator defined by Δpz=div(|∇z|p−2∇z), p>2,Ω⊂RNp>2,Ω⊂RN is a bounded domain with smooth boundary, α>1,β>1,2<α+β1,β>1,2<α+βpN>p, p∗=∞p∗=∞ if N≤pN≤p),∂∂n is the outer normal derivative, (λ,μ)∈R2∖{(0,0)}(λ,μ)∈R2∖{(0,0)}, the weight m(x)m(x) is a bounded function with ‖m‖∞>0‖m‖∞>0, and c(x)c(x) is a continuous function which changes sign in Ω¯.