Normalized solutions for biharmonic Schrodinger equations with potential and general nonlinearity

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 \(V\).  For more information see https://ejde.math.txstate.edu/Volumes/2024/83/abstr.html