LAUSR

Using the genus theory introduced by Krasnoselskii and a variant of the mountain pass theorem due to Rabinowitz [24], we study the existence of solutions for the following Kirchhoff type problem: $$\begin{aligned} {\left\{ \begin{array}{ll} M\left( \int _\Omega |\Delta u|^p\,\mathrm{d}x\right) \Delta \Big (|\Delta u|^{p{-}2}\Delta u\Big ) {=} \lambda |u|^{p^{**}{-}2}u{+}a(x)|u|^{p{-}2}u{+}f(x,u), \,\, x\in \Omega ,\\ u = \frac{\partial u}{\partial \nu } = 0,\,\,x\in \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega $$ is a bounded domain in $$\mathbb {R}^N$$ ( $$N\ge 3$$ ) with $$C^2$$ boundary, $$1