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

Abstract

We proceed with the investigation of the problem $(P_\lambda): $ $-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } \Omega, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial \Omega$, 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 a unbounded subcontinuum of nontrivial non-negative solutions of $(P_\lambda)$. Our approach is based on a priori bounds, a regularization procedure, and Whyburn's topological method.