Solutions for fourth-order Kirchhoff type elliptic equations involving concave-convex nonlinearities in ${\mathbb{R}}^N$

Dong-Lun Wu, Fengying Li

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations \begin{eqnarray*} Δ^{2}u-M(\|\nabla u\|_{2}^{2})…