Normalized solutions to the biharmonic nonlinear Schrödinger equation with combined nonlinearities

Abstract

In this article, we study the existence of normalized ground state solutions for the following biharmonic nonlinear Schrödinger equation with combined nonlinearities \begin{equation*} Δ^2u=λu+μ|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 $N\ge 2$, $\mu \in \mathbb{R}$, $a>0$, $2<q<p<\infty$ if $2\le N\le 4$, $2<q<p\le 4^{\ast }$ if $N\ge 5$, and $4^{\ast }=\frac{2N}{N-4}$ is the Sobolev critical exponent and $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier. By using the Sobolev subcritical approximation method, we prove the second critical point of mountain pass type for the case $N\ge 5$, $\mu >0$, $p=4^{\ast }$, and $2<q<2+\frac{8}{N}$. Moreover, we also consider the case $\mu =0$ and $\mu <0$.