Multiplicity of positive solutions for an equation with degenerate nonlocal diffusion

TL;DR

This paper establishes the existence of multiple positive solutions for a boundary value problem involving degenerate nonlocal diffusion, without relying on variational methods, by analyzing solutions with ordered $L^p$-norms.

Contribution

It provides a novel multiplicity result for positive solutions to a nonlocal degenerate diffusion equation without using variational techniques.

Findings

Multiple positive solutions with ordered $L^p$-norms are proven to exist.

The results apply even when the diffusion coefficient vanishes at multiple points.

The approach does not depend on variational methods, broadening applicability.

Abstract

Even without a variational background, a multiplicity result of positive solutions with ordered $L^{p}(\Omega)$-norms is provided to the following boundary value problem \begin{equation*} \left \{ \begin{array}{ll} -a(\int_{\Omega}u^{p}dx)\Delta u = f(u) & \mbox{in $\Omega$,}\\ u=0 & \mbox{on $\partial\Omega$,} \end{array}\right. \end{equation*} where $\Omega$ is a bounded domain and $a$, $f$ are continuous real functions with $a$ vanishing in many positive points.