Multiple solutions for a Neumann system involving subquadratic nonlinearities

In this paper we consider the model semilinear Neumann system $\left\{ \begin{array}{lll} -Δu+a(x)u=λc(x) F_u(u,v)& {\rm in} & Ω,\\ -Δv+b(x)v=λc(x) F_v(u,v)& {\rm in} & Ω,\\ \frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0 & {\rm on} & \partialΩ, \end{array}\right.$ where $Ω\subset \mathbb R^N$ is a smooth open bounded domain, $ν$ denotes the outward unit normal to $\partial Ω$, $λ\geq 0$ is a parameter, $a,b,c\in L_+^\infty(Ω)\setminus\{0\},$ 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, $\underline λ$ and $\overline λ$ with $0<\underlineλ\leq \overline λ$, such that for every $0\leq λ<\underline λ$, system $(N_λ)$ has only the trivial pair of solution, while for every $λ>\overline λ$, system $(N_λ)$ has at least two distinct nonzero pairs of solutions.

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