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 $-M({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^{2}dx)\mathrm{\Delta }u=f(u)-a$ in $\mathrm{\Omega }$, $u=0$ on $\partial \mathrm{\Omega },$ where $\mathrm{\Omega }\subset {\mathbf{R}}^N$ 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.