Normalized ground states for nonlinear Schr\"{o}dinger equations with general Sobolev critical nonlinearities

TL;DR

This paper proves the existence of normalized ground state solutions for a class of nonlinear Schrödinger equations with Sobolev critical nonlinearities, using a constrained minimization approach without the Ambrosetti-Rabinowitz condition.

Contribution

It introduces a new method to find normalized solutions for Sobolev critical nonlinear Schrödinger equations without relying on the Ambrosetti-Rabinowitz condition.

Findings

Existence of normalized ground states for all positive c

Development of a constrained minimization approach

Analysis of ground state energy monotonicity

Abstract

In this paper, we study the existence of normalized solutions to the following nonlinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{aligned} &-\Delta u=f(u)+ \lambda u\quad \mbox{in}\ \mathbb{R}^{N},\\ &u\in H^1(\mathbb{R}^N), ~~~\int_{\mathbb{R}^N}|u|^2dx=c, \end{aligned} \right. \end{equation*} where $N\ge3$, $c>0$, $\lambda\in \mathbb{R}$ and $f$ has a Sobolev critical growth at infinity but does not satisfies the Ambrosetti-Rabinowitz condition. By analysing the monotonicity of the ground state energy with respect to $c$, we develop a constrained minimization approach to establish the existence of normalized ground state solutions for all $c>0$.