The Nehari manifold for a class of concave–convex elliptic systems involving the pp-Laplacian and nonlinear boundary condition

The existence and multiplicity of weak solutions is established for a class of concave–convex elliptic systems of the form: {−Δpu+m(x)|u|p−2u=λa(x)|u|γ−2u,x∈Ω,−Δpv+m(x)|v|p−2v=μb(x)|v|γ−2v,x∈Ω,|∇u|p−2∂u∂n=αα+β|u|α−2u|v|β,|∇v|p−2∂v∂n=βα+β|u|α|v|β−2v,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 positive bounded function and a(x),b(x)∈C(Ω)a(x),b(x)∈C(Ω) are functions which change sign in ΩΩ. Our technical approach is based on the Nehari manifold which is similar to the fibering method of Drabek and Pohozaev (1997) [29] together with the recent idea from Brown and Wu (2008) [10].