Continuous dependence for p-Laplace equations with varying operators

TL;DR

This paper investigates how solutions to a nonlinear Neumann problem involving the p-Laplace operator depend continuously on the parameter p, establishing existence of nonconstant solutions under certain conditions.

Contribution

It proves continuous dependence of radially nondecreasing solutions on p for a class of p-Laplace equations and provides an existence result for nonconstant solutions when p is between 1 and 2.

Findings

Solutions depend continuously on p in the specified range.

Existence of nonconstant solutions for p in (1,2) and large q.

Characterization of solution behavior as p varies.

Abstract

For the following Neumann problem in a ball $$\begin{cases} -\Delta_p u+u^{p-1}=u^{q-1}\quad&\text{in }B,\\ u>0,\,u\text{ radial}\quad&\text{in }B,\\ \frac{\partial u}{\partial \nu}=0\quad&\text{on }\partial B, \end{cases}$$ with $1<p<q<\infty$, we prove continuous dependence on $p$, for radially nondecreasing solutions. As a byproduct, we obtain an existence result for nonconstant solutions in the case $p\in(1,2)$ and $q$ larger than an explicit threshold.