Normalized solutions to the biharmonic nonlinear Schr\"{o}dinger equation with combined nonlinearities

Abstract

In this article, we study the existence of normalized ground state solutions for the following biharmonic nonlinear Schr\"{o}dinger equation with combined nonlinearities \begin{equation*} \Delta^2u=\lambda u+\mu|u|^{q-2}u+|u|^{p-2}u,\quad \text {in $\mathbb{R}^N$} \end{equation*} having prescribed mass \begin{equation*} \int_{\mathbb{R}^N}|u|^2dx=a^2, \end{equation*} where $N\geq2$, $\mu\in \mathbb{R}$, $a>0$, $2 0$, $p=4^*$, and $2<q<2+\frac{8}{N}$. Moreover, we also consider the case $\mu=0$ and $\mu<0$.