JournalszaaVol. 36, No. 2pp. 191–207

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
Two Nontrivial Solutions for the Nonhomogenous Fourth Order Kirchhoff Equation cover
Download PDF

A subscription is required to access this article.

Abstract

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

Δ2u(a+bRNu2dx)Δu+V(x)u=f(x,u)+g(x),xRN,\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 Δ2:=Δ(Δ)\Delta^2 := \Delta(\Delta), constants a>0a > 0, b0b \geq 0, VC(RN,R)V \in C(\mathbb{R}^N, \mathbb{R}), fC(RN×R,R)f \in C(\mathbb{R}^N \times \mathbb{R}, \mathbb{R}) and gL2(RN)g \in L^2(\mathbb{R}^N). Under more relaxed assumptions on the nonlinear term ff 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