Two Nontrivial Solutions for the Nonhomogenous Fourth Order Kirchhoff Equation

Abstract

In this paper, we consider the following nonhomogenous fourth order Kirchhoff equation

where ${\Delta }^2:=\Delta (\Delta )$, constants $a>0$, $b\ge 0$, $V\in C({\mathbb{R}}^N,\mathbb{R})$, $f\in C({\mathbb{R}}^N\times \mathbb{R},\mathbb{R})$ and $g\in L^2({\mathbb{R}}^N)$. Under more relaxed assumptions on the nonlinear term $f$ that are much weaker than those in L. Xu and H. Chen, using some new proof techniques especially the verification of the boundedness of Palais–Smale sequence, a new result is obtained.