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

Abstract

We proceed with the investigation of the problem $(P_λ): $ $-Δu = λb(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } Ω, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial Ω$, where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq2$), $1<q<2<p$, $λ\in \mathbb{R}$, and $a,b \in C^α(\overlineΩ)$ with $0<α<1$. 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 $(P_λ)$. Our approach is based on a priori bounds, a regularization procedure, and Whyburn's topological method.

Figures (2)

• Figure 1: A unbounded subcontinuum of nontrivial non-negative solutions in the case $\int_\Omega a \geq 0$.
• Figure 2: A unbounded subcontinuum of nontrivial non-negative solutions in the case $\int_\Omega a < 0$.

Theorems & Definitions (23)

• Remark 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 3.1