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

Abstract

We proceed with the investigation of the problem \begin{equation} -\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, \tag{${\rm P}_\lambda$} \end{equation} where $\Omega$ is a bounded smooth domain in $\mathbb R^N$ ($N \geq 2$), $a,b \in C^\alpha(\overline{\Omega})$ 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 $({\rm P}_\lambda)$. Our approach is based on a priori bounds, a regularisation procedure, and Whyburn's topological method.