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

## TL;DR

This paper investigates solutions to a nonlinear elliptic PDE with Neumann boundary conditions, establishing the existence of an unbounded continuum of non-negative solutions using topological methods and a priori bounds.

## Contribution

It extends previous work by analyzing the case where the coefficient b is non-negative and not identically zero, proving the existence of a large continuum of solutions.

## Key findings

- Existence of an unbounded subcontinuum of solutions.
- Properties of non-negative solutions under Neumann conditions.
- Application of topological methods to nonlinear PDEs.

## 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.