Electron. J. Differential Equations, Vol. 2016 (2016), No. 289, pp. 1-16.

Multiple solutions for biharmonic elliptic problems with the second Hessian

Fei Fang, Chao Ji, Binlin Zhang

In this article, we study the biharmonic elliptic problem with the secondnd Hessian
 \Delta^2u =S_2(D^2u)+\lambda f(x) |u|^{p-1}u,\quad \text{in }
 \Omega \subset \mathbb{R}^3,    \cr
  u =\frac{\partial u}{\partial n}=0,   \quad \text{on } \partial\Omega,
where $f(x)\in C(\bar{\Omega})$ is a sign-changing weight function. By using variational methods and some properties of the Nehari manifold, we prove that the biharmonic elliptic problem has at least two nontrivial solutions.

Submitted August 18, 2016. Published October 26, 2016.
Math Subject Classifications: 35J50, 35J60, 35J62, 35J96.
Key Words: Biharmonic elliptic problem; second Hessian; variational methods; Nehari manifold.

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Fei Fang
Department of Mathematics
Beijing Technology and Business University
Beijing 100048, China
email: fangfei68@163.com
Chao Ji
Department of Mathematics
East China University of Science and Technology
Shanghai 200237, China
email: jichao@ecust.edu.cn
Binlin Zhang
Department of Mathematics
Heilongjiang Institute of Technology
Harbin 150050, China
email: zhangbinlin2012@163.com

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