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Normalized solutions to mass supercritical Schrödinger equations with radial potentials

P. Carrillo, L. Jeanjean

Abstract

We study the stationary nonlinear Schrödinger equation \begin{equation}-Δu+V(x)u+λu=|u|^{q-2}u,\quad u \in H^1(\mathbb{R}^N), \quad N \geq 2\end{equation} where $V \in L^{\infty}(\mathbb{R}^N)$ is a radial potential. In the $L^2$-supercritical regime, we show the existence of an explicit $μ_0 >0$ such that, for any $μ\in (0, μ_0)$, the equation admits two solutions having $L^2$ norm $μ$. The potential $V$ is not assumed to have a sign, nor a specific behavior at infinity and only a low regularity is required. Our proof relies on the use of Morse type information, on some spectral arguments, and on a blow-up analysis developed in a radial setting.

Normalized solutions to mass supercritical Schrödinger equations with radial potentials

Abstract

We study the stationary nonlinear Schrödinger equation \begin{equation}-Δu+V(x)u+λu=|u|^{q-2}u,\quad u \in H^1(\mathbb{R}^N), \quad N \geq 2\end{equation} where is a radial potential. In the -supercritical regime, we show the existence of an explicit such that, for any , the equation admits two solutions having norm . The potential is not assumed to have a sign, nor a specific behavior at infinity and only a low regularity is required. Our proof relies on the use of Morse type information, on some spectral arguments, and on a blow-up analysis developed in a radial setting.
Paper Structure (4 sections, 18 theorems, 229 equations)

This paper contains 4 sections, 18 theorems, 229 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['theo main aproximation']}, Theorem \ref{['theo: local min']} and conditional proof of Theorem \ref{['theo:main']}.
  4. A blow-up analysis and proof of Theorem \ref{['theo maxima to 0']}

Key Result

Theorem 1.1

Assume that $(V0)-(V2)$ holds. Then, there exists an explicit value $\mu_0 >0$ depending only on $N,q$ and $(\underline{V} - \lambda_{ess}(V))$ such that, for all $\mu \in (0, \mu_0),$ there exists a positive radial solution to Equation EQ:Rn under the mass constraint eq:mass constraint lying at a m

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 44 more