On a Class of Critical p-Biharmonic Kirchhoff Type Problems with Indefinite Weights

Abstract

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, $$1<p<\frac{N}{2}$$ , $$p^{**} = \frac{Np}{N-2p}$$ is the critical exponent, $$\Delta $$ is the Laplace operator and $$\frac{\partial u}{\partial \nu }$$ is the outer normal derivative, $$\lambda $$ is a positive parameter, $$M: [0,+\infty ) \rightarrow \mathbb {R}$$ and $$f: \overline{\Omega }\times \mathbb {R}\rightarrow \mathbb {R}$$ are continuous functions, and $$a\in L^\infty (\Omega )$$ is a weight function.