Sign-changing solutions for fourth-order elliptic equations of Kirchhoff type with critical exponent

Abstract

In this paper, we study the existence of ground state sign-changing solutions for the following fourth-order elliptic equations of Kirchhoff type with critical exponent. More precisely, we consider \begin{equation*} \begin{cases} \Delta^2u - \left(1 + b\int_{\Omega} |\nabla u|^2 dx\right)\Delta u = \lambda f(x,u) + |u|^{2^{\ast\ast}-2}u &\text{in } \Omega,\\ u = \Delta u = 0 & \mbox{on}\ \partial\Omega, \end{cases} \end{equation*} where $\Delta^2$ is the biharmonic operator, $N=\{5, 6, 7\}$, $2^{\ast\ast}=2N/(N-4)$ is the Sobolev critical exponent and $\Omega \subset \mathbb{R}^N$ is an open bounded domain with smooth boundary and $b, \lambda$ are some positive parameters. By using constraint variational method, topological degree theory and the quantitative deformation lemma, we prove the existence of ground state sign-changing solutions with precisely two nodal domains.

Keywords

• kirchhoff type problem
• fourth-order elliptic equation
• critical growth
• sign-changing solution