Existence of Positive Solution for Kirchhoff Problems

Abstract

In this work, we study the following Kirchhoff type problem

$$\begin{aligned} \begin{gathered} -\Big (a+b\int _{\Omega }|\nabla u|^pdx\Big )\Delta _p u =g(x)u^{-\gamma }+\lambda f(x,u),\quad \text {in }\Omega , \\ u=0, \quad \text {on }\partial \Omega , \end{gathered} \end{aligned}$$

where $p\ge 2$, $\Omega$ is a regular bounded domain in $\mathbb {R}^N$, $(N\ge 3)$. Firstly, for $p>2$, we prove under some appropriate conditions on the singularity and the nonlinearity the existence of nontrivial weak solution to this problem. For $p=2$, we show, under supplementary condition, the positivity of this solution. Moreover, in the case $\lambda =0$ we prove an uniqueness result. We use the variational method to prove our main results.