Positive solutions for a semipositone problem involving nonlocal operator

Abstract

In this article, we are interested in the existence of positive solutions for the following Kirchhoff type problems

where $\Omega$ is a bounded smooth domain of $R^N,1<p<N,M:R_0^{+}\to R^{+}$ is a continuous and increasing function, $\lambda ,\mu$ are two positive parameters, $a\in C(\overline{\Omega }),a(x)\ge a_0>0$ , and $f$ is a $C^1([0,\infty ))$ function such that $f(0)=0,f(t)>0$ for all $0<t<t_0$ and $f(t)\le 0$ for all $t\ge t_0$ , where $t_0>0$ .