Multiple solutions for a Neumann system involving subquadratic nonlinearities

Abstract

In this paper we consider the model semilinear Neumann system $$\{\begin{array}{lll}-\Delta u+a(x)u=\lambda c(x)F_u(u,v)& \mathrm{in}& \Omega ,\\ -\Delta v+b(x)v=\lambda c(x)F_v(u,v)& \mathrm{in}& \Omega ,\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0& \mathrm{on}& \partial \Omega ,\end{array}$$ where $\Omega \subset {\mathbb{R}}^N$ is a smooth open bounded domain, $\nu$ denotes the outward unit normal to $\partial \Omega$, $\lambda \ge 0$ is a parameter, $a,b,c\in L_{+}^{\infty }(\Omega )\setminus \{0\},$ and $F\in C^1({\mathbb{R}}^2,\mathbb{R})\setminus \{0\}$ is a nonnegative function which is subquadratic at infinity. Two nearby numbers are determined in explicit forms, $\underline{\lambda }$ and $\overline{\lambda }$ with $0<\underline{\lambda }\le \overline{\lambda }$, such that for every $0\le \lambda <\underline{\lambda }$, system $(N_{\lambda })$ has only the trivial pair of solution, while for every $\lambda >\overline{\lambda }$, system $(N_{\lambda })$ has at least two distinct nonzero pairs of solutions.