Fengwei Zou, Shuai Yao, Juntao Sun
Abstract:
We study the existence and non-existence of normalized solutions to
the biharmonic equation
$$
\Delta ^2u-\Delta u+V(x)u+\lambda u=f(u) \quad \text{in }\mathbb{R}^N.
$$
where \(0\neq V(x)\leq V_{\infty }:=\lim_{|x|\to \infty }V(x)\in
(-\infty ,+\infty ]\)
and \(f\in C(\mathbb{R},\mathbb{R})\) is a nonlinearity.
For the trapping case of \(V_{\infty }=+\infty \), under some suitable
assumptions on \(f\), we prove that there exists a ground state as a global
minimizer of the corresponding energy functional.
For the case of \(V_{\infty }<+\infty \), under some other assumptions on
\(f\), we prove that there exists \(\bar{\alpha}\geq 0\) such that a global
minimizer exists if \(\alpha >\bar{\alpha}\) while no global minimizer
exists if \(\alpha <\bar{\alpha}\). Moreover, the size of \(\bar{\alpha}\) is
also explored, depending on the potential \(V\).
Submitted August 2, 2024. Published December 11, 2024.
Math Subject Classifications: 35J20, 35J60, 35J92.
Key Words: Biharmonic NLS; normalized solution; variational method.
DOI: 10.58997/ejde.2024.82
Show me the PDF file (383 KB), TEX file for this article.
 |
Fengwei Zou School of Mathematics and Statistics Shandong University of Technology Shandong, Zibo 255049, China email: zfw746265367@163.com |
|---|---|
 |
Shuai Yao School of Mathematics and Statistics Shandong University of Technology Shandong, Zibo 255049, China email: shyao@sdut.edu.cn |
 |
Juntao Sun School of Mathematics and Statistics Shandong University of Technology Shandong, Zibo 255049, China email: jtsun@sdut.edu.cn |
Return to the EJDE web page