Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (284 KB) | Metadata | Table of Contents | ZAA summary
Volume 36, Issue 2, 2017, pp. 191–207
DOI: 10.4171/ZAA/1585

Published online: 2017-03-29

Two Nontrivial Solutions for the Nonhomogenous Fourth Order Kirchhoff Equation

Ling Ding[1] and Lin Li[2]

(1) Hubei University of Arts and Science, China
(2) Chongqing Technology and Business University, China

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.

Keywords: Fourth order Kirchhoff equation, variational methods, critical point theorem

Ding Ling, Li Lin: Two Nontrivial Solutions for the Nonhomogenous Fourth Order Kirchhoff Equation. Z. Anal. Anwend. 36 (2017), 191-207. doi: 10.4171/ZAA/1585