Normalized solutions for nonlinear Schr\"{o}dinger equation involving potential and Sobolev critical exponent

Abstract

In this paper, we consider the existence of positive solutions with prescribed $L^2$-norm for the following nonlinear Schr\"{o}dinger equation involving potential and Sobolev critical exponent \begin{equation*} \begin{cases} -\Delta u+V(x)u=\lambda u+\mu |u|^{p-2}u+|u|^{\frac{4}{N-2}}u \;\;\text { in } \mathbb{R}^N, \\ \|u\|_2=a>0,\\ \end{cases} \end{equation*} where $N\ge 3$, $\mu>0$, $p\in [2+\frac{4}{N}, \frac{2N}{N-2})$ and $V\in C^1(\mathbb{R}^N)$. Under different assumptions on $V$, we derive two different Pohozaev identities. Based on these two cases, we respectively obtain the existence of positive solution. As far as we are aware, we did not find any works on normalized solutions with Sobolev critical growth and potential $V \not\equiv 0$. Our results extend some results of Wei and Wu [J. Funct. Anal. 283(2022)] to the potential case.