Normalized solutions of mass supercritical Schr\"odinger equations with potential

Abstract

This paper is concerned with the existence of normalized solutions of the nonlinear Schr\"odinger equation \[ -\Delta u+V(x)u+\lambda u = |u|^{p-2}u \qquad\text{in $\mathbb{R}^N$} \] in the mass supercritical and Sobolev subcritical case $2+\frac{4}{N}<p<2^*$. We prove the existence of a solution $(u,\lambda)\in H^1(\mathbb{R}^N)\times\mathbb{R}^+$ with prescribed $L^2$-norm $\|u\|_2=\rho$ under various conditions on the potential $V:\mathbb{R}^N\to\mathbb{R}$, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.