A CONVEX-CONCAVE ELLIPTIC PROBLEM WITH A PARAMETER ON THE BOUNDARY CONDITION

In this paper we study existence and multiplicity of nonnegative solutions to {Delta u = u(p) + u(q) in Omega, partial derivative u/partial derivative v = lambda u on partial derivative Omega. Here Omega is a smooth bounded domain of R-N, nu stands for the outward unit normal and p, q are in the convex-concave case, that is 0 < q < 1 < p. We prove that there exists Lambda* > 0 such that there are no nonnegative solutions for lambda < Lambda*, and there is a maximal nonnegative solution for lambda >= Lambda*. If lambda is large enough, then there exist at least two nonnegative solutions. We also study the asymptotic behavior of solutions when lambda -> infinity and the occurrence of dead cores. In the particular case where Omega is the unit ball of R-N we show exact multiplicity of radial nonnegative solutions when lambda is large enough, and also the existence of nonradial nonnegative solutions.