On the solvability of asymptotically linear systems at resonance

This paper is concerned with the solvability of the system−Δu−ν1θ1v=f(x,u,v)+h1(x) in Ω;−Δv−ν1θ2u=g(x,u,v)+h2(x) in Ω;u=v=0 on ∂Ω, at resonance at the simple eigenvalue ν1ν1 of the corresponding linear eigenvalue problem. Here Ω⊂RNΩ⊂RN (N≥1N≥1) is a bounded domain with C2,ηC2,η-boundary ∂Ω, η∈(0,1)η∈(0,1) (a bounded interval if N=1N=1) and θ1θ1, θ2θ2 are positive constants. The nonlinear perturbations f(x,u,v),g(x,u,v):Ω×R2→Rf(x,u,v),g(x,u,v):Ω×R2→R are Carathéodory functions that are sublinear at infinity. We employ the Lyapunov–Schmidt method to provide sufficient conditions on h1,h2∈Lr(Ω)h1,h2∈Lr(Ω); r>Nr>N, to guarantee the solvability of the system.