LAUSR

In this work, we study the following Kirchhoff type problem $$\begin{aligned} \begin{gathered} -\Big (a+b\int _{\Omega }|\nabla u|^pdx\Big )\Delta _p u =g(x)u^{-\gamma }+\lambda f(x,u),\quad \text {in }\Omega , \\ u=0, \quad \text {on }\partial \Omega , \end{gathered} \end{aligned}$$-(a+b∫Ω|∇u|pdx)Δpu=g(x)u-γ+λf(x,u),inΩ,u=0,on∂Ω,where $$p\ge 2$$p≥2, $$\Omega $$Ω is a regular bounded domain in $$\mathbb {R}^N$$RN, $$(N\ge 3)$$(N≥3). Firstly, for $$p>2$$p>2, we prove under some appropriate conditions on the singularity and the nonlinearity the existence of nontrivial weak solution to this problem. For $$p=2$$p=2, we show, under supplementary condition, the positivity of this solution. Moreover, in the case $$\lambda =0$$λ=0 we prove an uniqueness result. We use the variational method to prove our main results.