Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 82, pp. 1-14.

Title: Normalized solutions for biharmonic Schrodinger equations with potential and general nonlinearity

Authors: Fengwei Zou (Shandong Univ. of Technology, Shandong, Zibo, China)
         Shuai Yao   (Shandong Univ. of Technology, Shandong, Zibo, China)
         Juntao Sun  (Shandong Univ. of Technology, Shandong, Zibo, China)
  
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.