Positive solutions for Kirchhoff problems with vanishing nonlocal term

Abstract

In this paper we study the Kirchhoff problem \begin{equation*} \left \{ \begin{array}{ll} -m(\| u \|^{2})Δu = 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.

arXiv comment8 pages