Existence of positive solutions for a class of semipositone problems with Kirchhoff operator

Abstract

In this work we show a result of existence of positive solution for the following nonlocal problem of Kirchhoff type \(-M\biggl(\int_{\Omega} |\nabla u|^{2} dx\biggl)\Delta u= f(u)-a\) in \(\Omega\), \(u=0\) on \(\partial\Omega,\) where \(\Omega \subset\mathbf{R}^{N}\) is a smooth bounded domain, \(M,f\) are continuous nonnegative functions and \(a>0\). By using mainly variational methods, we prove the existence of a solution for \(a\) small enough, under two different sets of hypotheses, which generalize the classical superlinear and sublinear problems.