Axial Symmetry of Normalized Solutions for Magnetic Gross-Pitaevskii Equations with Anharmonic Potentials

This paper is concerned with normalized solutions of the magnetic focusing Gross-Pitaevskii equations with anharmonic potentials in ${\mathbb{R}}^N$, where $N=2$ or $3$. We construct axially symmetric normalized concentrating solutions as the parameter $a>0$ approaches $a_{\ast }(N)$, where $a_{\ast }(N)\ge 0$ is a critical constant depending only on $N$. We further prove that up to a constant phase (and a rotational transformation for $N=2$), normalized concentrating solutions are unique and axially symmetric as $a\to a_{\ast }(N)$. When $N=3$, we also prove that the corresponding unique normalized concentrating solution is free of vortices as $a\to a_{\ast }(3)$, even if the anharmonic potential is non-radially symmetric.