Asymptotically linear systems near and at resonance

This paper deals with an elliptic system of the form -Delta u = lambda theta(1)a(x)v + f(lambda,x,v) in Omega, -Delta v = lambda theta(2)a(x)u + g(lambda,x,u) in Omega, u = 0 = v on partial derivative Omega, where lambda is an element of R is a parameter and Omega subset of R-N (N >= 1) is a bounded domain with C-2,C-xi-boundary partial derivative Omega, xi is an element of (0, 1) (a bounded open interval if N = 1). Here a(x) is an element of L-infinity(Omega) with a(x) > 0 a.e. in Omega and theta(1),theta(2) > 0 are constants. The nonlinear perturbations f,g : R x Omega x R -> R are Caratheodory functions that are sublinear at infinity. We provide sufficient conditions for determining the lambda-direction to which a continuum of positive and negative solutions emanates from infinity at the first eigenvalue of the associated linear problem. Furthermore, as a consequence of main results, we also provide sufficient condition for the solvability of a class of asymptotically linear system near and at resonance satisfying Landesman-Lazer type conditions.