A priori bounds and multiplicity of positive solutions for $p$-Laplacian Neumann problems with sub-critical growth

TL;DR

This paper establishes bounds and multiplicity results for positive radial solutions of a p-Laplacian Neumann problem with sub-critical growth, using shooting methods to analyze oscillatory behavior around the value 1.

Contribution

It provides new a priori bounds and multiplicity results for positive solutions of p-Laplacian Neumann problems with specific nonlinearities, extending previous analysis to sub-critical growth cases.

Findings

Established a priori bounds for radial solutions.

Proved existence and multiplicity of solutions using shooting techniques.

Analyzed oscillatory behavior of solutions around the value 1.

Abstract

Let $1<p<+\infty$ and let $\Omega\subset\mathbb R^N$ be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type \[ -\Delta_p u = f(u), \quad u>0 \mbox{ in } \Omega, \quad \partial_\nu u = 0 \mbox{ on } \partial\Omega. \] We suppose that $f(0)=f(1)=0$ and that $f$ is negative between the two zeros and positive after. In case $\Omega$ is a ball, we also require that $f$ grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focusing in particular on solutions which start above 1. As an application, we use the shooting technique to get existence, multiplicity and oscillatory behavior (around 1) of non-constant radial solutions.