Multiplicity results for a subcritical Hamiltonian system with concave-convex nonlinearities

Abstract

We study the Hamiltonian elliptic system { -Delta u=lambda|v|(r-1)v+|v|(p-1)v in Omega, -Delta v=mu|u|(s-1)u+|u|(q-1)u in Omega, (0.1) u>0,v>0 in Omega u=v=0 on partial derivative Omega where Omega subset of R-N is a smooth bounded domain, lambda and mu are nonnegative parameters and r, s, p, q > 0. Our study includes the case in which the nonlinearities in (0.1) are concave near the origin and convex near infinity, and we focus on the region of non-negative pairs of parameters(lambda, mu) that guarantee existence and multiplicity of solutions of (0.1). In particular, we show the existence of a strictly decreasing curve lambda(& lowast;)(mu) on an interval [0, mu] with lambda(& lowast;)(0) > 0, lambda(& lowast;)(mu) = 0 and such that the system has two solutions for (lambda, mu) below the curve, one solution for (lambda, mu) on the curve and no solution for (lambda, mu) above the curve. A similar statement holds reversing lambda and mu.