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

Abstract

We proceed with the investigation of the problem $(P_{\lambda }):$ $-Δu = λb(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } Ω, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial Ω$, where $\Omega$ is a bounded smooth domain in ${\mathbb{R}}^N$ ($N\ge 2$), $1<q<2<p$, $\lambda \in \mathbb{R}$, and $a,b\in C^{\alpha }(\overline{\Omega })$ with $0<\alpha <1$. Dealing now with the case $b\ge 0$, $b\not\equiv 0$, 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.

15 pages, 3 figures