Multiple solutions for a Neumann system involving subquadratic nonlinearities

TL;DR

This paper proves the existence of multiple solutions for a Neumann boundary value problem involving subquadratic nonlinearities, identifying parameter ranges where solutions are trivial or multiple nontrivial solutions exist.

Contribution

It establishes explicit bounds on the parameter for the existence of multiple solutions in a Neumann system with subquadratic nonlinearities.

Findings

No nontrivial solutions for small parameters below 

Multiple solutions appear when the parameter exceeds a certain threshold

Explicit bounds < and < are derived

Abstract

In this paper we consider the model semilinear Neumann system $$\left\{ \begin{array}{lll} -\Delta u+a(x)u=\lambda c(x) F_u(u,v)& {\rm in} & \Omega,\\ -\Delta v+b(x)v=\lambda c(x) F_v(u,v)& {\rm in} & \Omega,\\ \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0 & {\rm on} & \partial\Omega, \end{array}\right.$$ where $\Omega\subset \mathbb R^N$ is a smooth open bounded domain, $\nu$ denotes the outward unit normal to $\partial \Omega$, $\lambda\geq 0$ 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\leq \overline \lambda$, such that for every $0\leq \lambda<\underline \lambda$, 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.