Abstract

This paper is concerned with the existence of normalized solutions of the nonlinear Schrödinger equation $$-\Delta u+V(x)u+\lambda u=|u{|}^{p-2}u\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\text{in}\mathrm{}{\mathbb{R}}^N$$ 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 L²-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.

中文翻译：

---

具有势能的质量超临界薛定谔方程的归一化解

摘要

本文关注非线性薛定谔方程归一化解的存在性 $$-\Delta 你+伏(X)你+\lambda 你=|你{|}^{磷-2}你\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\text{在}\mathrm{}{\mathbb{电阻}}^N$$ 在质量超临界和 Sobolev 亚临界情况下 $2+\frac{4}{N}<磷<2^{*}.$ 我们证明解的存在 $(你,\lambda )\in H^1({\mathbb{电阻}}^N)\times {\mathbb{电阻}}^{+}$具有规定的L ^{2 -}范数$\Vert 你{\Vert }_2=\rho$ 在各种条件下对电势 $伏:{\mathbb{电阻}}^N\to \mathbb{电阻},$为正且在无穷远处消失，包括具有奇点的势。证明基于新的最小-最大参数。