Normalized solutions of mass supercritical Schrödinger equations with potential

Abstract

This paper is concerned with the existence of normalized solutions of the nonlinear Schrödinger equation \[ -Δu+V(x)u+λu = |u|^{p-2}u \qquad\text{in ${\mathbb{R}}^N$} \] in the mass supercritical and Sobolev subcritical case $2+\frac{4}{N}<p<2^{\ast }$. We prove the existence of a solution $(u,\lambda )\in H^1({\mathbb{R}}^N)\times {\mathbb{R}}^{+}$ with prescribed $L^2$-norm $\Vert u{\Vert }_2=\rho$ 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.

24 pages