Positive solutions of the p-Kirchhoff problem with degenerate and sign-changing nonlocal term

Abstract

We establish the existence and multiplicity of positive solutions of the p-Kirchhoff problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -m\left( \int \limits _\Omega |\nabla u|^p \mathrm{d}x\right) \Delta _p u = f(u) &{}\text { in } \Omega ,\\ u=0 &{}\text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

where $p>1$, $\Omega$ is a smooth bounded domain of ${\mathbb {R}}^N$, and $f\in C({\mathbb {R}}_0^+)\cap L^1({\mathbb {R}}_0^+)$ is subcritical and positive in a right neighborhood of zero. The main feature of our problem is that $m:{\mathbb {R}}_0^+\rightarrow {\mathbb {R}}$ may be any continuous function such that the integral of m in each connected component of $m^{-1}((0,+\infty ))$ is controlled by p, f and $\Omega$. Therefore, in our paper m may be degenerate, i.e., it could vanish, and sign-changing at any number of different points.