Positive solutions for Kirchhoff problems with vanishing nonlocal term

TL;DR

This paper establishes the existence of multiple positive solutions for a Kirchhoff problem with a nonlocal term that can vanish, using variational methods and a priori estimates under an area condition.

Contribution

It introduces a novel approach to handle Kirchhoff problems with vanishing nonlocal terms, proving multiplicity of solutions with an ordered structure.

Findings

Multiple positive solutions are proven to exist.

Solutions are ordered in the $H_{0}^{1}( abla)$-norm.

The method applies to problems with vanishing nonlocal functions.

Abstract

In this paper we study the Kirchhoff problem \begin{equation*} \left \{ \begin{array}{ll} -m(\| u \|^{2})\Delta u = f(u) & \mbox{in $\Omega$,} u=0 & \mbox{on $\partial\Omega$,} \end{array}\right. \end{equation*} in a bounded domain, allowing the function $m$ to vanish in many different points. Under an appropriated {\sl area condition}, by using a priori estimates, truncation techniques and variational methods, we prove a multiplicity result of positive solutions which are ordered in the $H_{0}^{1}(\Omega)$-norm.