On the solvability of asymptotically linear systems at resonance

This paper is concerned with the solvability of the system -Delta u - nu(1)theta(1)v= f (x, u, v) + h(1)(x) in Omega; -Delta u - nu(1)theta(2)u= f (x, u, v) + h(2)(x) in Omega; u= v = 0 partial derivative Omega; at resonance at the simple eigenvalue vi of the corresponding linear eigenvalue problem. Here Omega subset of R-N (N >= 1) is a bounded domain with C-2,C-eta-boundary partial derivative Omega, eta is an element of (0,1) (a bounded interval if N = 1) and theta(1), theta(2) are positive constants. The nonlinear perturbations f (x, u, v), g(x, u, v) : Omega x R-2 -> R are Caratheodory functions that are sublinear at infinity. We employ the Lyapunov-Schmidt method to provide sufficient conditions on h(1), h(2) is an element of L-r(Omega); r > N, to guarantee the solvability of the system. (C) 2016 Elsevier Inc. All rights reserved.