Existence of three solutions for fourth-order Kirchhoff type elliptic problems with Hardy potential

In this work, we establish existence results for the following fourth-order Kirchhoff-type elliptic problem with Hardy potential\begin{equation*}\begin{gathered}M \Big(\int_{\Omega} |\Delta u|^p dx\Big) \Delta_p^۲ u -\frac{a}{|x|^{p}} |u|^{p-۲} u = \lambda f(x, u), \quad \text{in } \Omega, \\u = \Delta u = ۰, \quad \text{on } \partial \Omega.\end{gathered}\end{equation*}Precisely, by using the classical Hardy inequality and critical point theory, we prove the existence of multiple weak solutions for the fourth-order Kirchhoff-type elliptic problem with Hardy potential.

Keywords:

Kirchhoff-type , Multiple solutions , Critical points theory , Hardy potential , p-biharmonic type operator