Bound state with prescribed angular momentum and mass

Abstract

As a continuation of Wang (2024), in the present paper, we consider the following problem in $R^N$ $\left(\begin{array}{c}-\Delta u+V\left(x\right)u=\lambda (-i{x}^{\perp }\cdot \nabla u)+\mu u+{\left|u\right|}^{p}u,u\in H^1(R^N,ℂ),\\ {\int }_{R^N}{\left|u\right|}^{2}dx=m>0,\\ Re{\int }_{R^N}(-i{x}^{\perp }\cdot \nabla u\overline{u})dx=l\in R,\end{array}\right)$where $N=2$ or $N=3$, $x^{\perp }$ is the magnetic potential (see Introduction). When $\frac{l}{m}\notin Z$, $2<p+2<2+\frac{4}{N}$, under suitable assumptions for $V$, the existence of bound state is given via a double constrained energy minimization. And the Pohozaev identity is given. $V$ grows super-quadratically at infinity is needed in our proof.