# Two Nontrivial Solutions for the Nonhomogenous Fourth Order Kirchhoff Equation

### Ling Ding

Hubei University of Arts and Science, China### Lin Li

Chongqing Technology and Business University, China

## Abstract

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

$\Delta^2 u - \left( a + b \int_{\mathbb{R}^N} |\nabla u|^2 dx \right) \Delta u + V(x) u = f(x,u) + g(x), \quad x \in \mathbb{R}^N,$

where $\Delta^2 := \Delta(\Delta)$, constants $a > 0$, $b \geq 0$, $V \in C(\mathbb{R}^N, \mathbb{R})$, $f \in C(\mathbb{R}^N \times \mathbb{R}, \mathbb{R})$ 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.

## Cite this article

Ling Ding, Lin Li, Two Nontrivial Solutions for the Nonhomogenous Fourth Order Kirchhoff Equation. Z. Anal. Anwend. 36 (2017), no. 2, pp. 191–207

DOI 10.4171/ZAA/1585