An indefinite concave-convex equation under a Neumann boundary condition II

Abstract

We proceed with the investigation of the problem $$ -\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \quad \mbox{in } \Omega, \qquad \frac{\partial u}{\partial \n} = 0\quad \mbox{on } \partial \Omega, \leqno{(\rom{P}_\lambda)} $$% where $\Omega$ is a bounded smooth domain in $\mathbb R^N$ ($N \geq2$), $1< q< 2< p$, $\lambda \in \mathbb R$, and $a,b \in C^\alpha(\overline{\Omega})$ with $0< \alpha< 1$. Dealing now with the case $b \geq 0$, $b \not \equiv 0$, we show the existence (and several properties) of an unbounded subcontinuum of nontrivial nonnegative solutions of $(\rom{P}_\lambda)$. Our approach is based on {\it a priori} bounds, a regularisation procedure, and Whyburn's topological method.