Multiple Solutions for a Nonlinear Elliptic System Subject to Nonautonomous Perturbations

Abstract

In this paper we consider the following Neumann problem $$ \left\{ \begin{array}{lll} -\Delta u = \alpha(x)(F_u(u,v)-u) + \lambda G_u(x,u,v) \ in \; \Omega \\ -\Delta v = \alpha(x) (F_v(u,v)-v) + \lambda G_v(x,u,v) \ in \; \Omega \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0 \ on \ \partial \Omega \end{array} \right. $$ In particular, by means of a multiplicity theorem obtained by Ricceri, we establish that if the set of all global minima of the function $\mathbb{R}^2 \ni y \longmapsto \frac{|y|^2}{2} - F(y)$ (where $F \in C^1(\mathbb{R}^2)$ and it satisfies the condition $F(0,0) = 0$) has at least $k \geq 2$ connected components, then the above Neumann problem admits at least $k+1$ weak solutions, $k$ of which are lying in a given set.