MULTIPLE SOLUTIONS FOR THE NONHOMOGENEOUS FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE

Abstract

This paper considers the following nonhomogeneous fourth order elliptic equations of Kirchhoff type:\begin{eqnarray*}\begin{cases}\displaystyle\triangle ^2u-(a+b\int_{\text R^N}|\nabla u|^2dx)\triangle u+V(x)u= f(x,u)+h(x),~\text{in}~ \text R^N,\\ \displaystyle u\in H^2(\text R^N), \end{cases}\end{eqnarray*}where constants $a\gt 0,~b\geq0$. Under certain assumptions on $V(x)$, $f(x,u)$ and $h(x)$, we show the existence and multiplicity of solutions by the Ekeland$^{,}$s variational principle and the Mountain Pass Theorem in the critical theory.