Normalized ground state solutions of nonlinear Schr\"odinger equations involving exponential critical growth

TL;DR

This paper proves the existence of normalized ground state solutions for a nonlinear Schrödinger equation with exponential critical growth in two dimensions, using constrained minimization and Trudinger-Moser inequality without the Ambrosetti-Rabinowitz condition.

Contribution

It establishes the existence of solutions for all positive mass values without relying on the Ambrosetti-Rabinowitz condition, expanding the understanding of such equations.

Findings

Existence of normalized ground state solutions for all >0

Solutions obtained via constrained minimization method

No need for Ambrosetti-Rabinowitz condition

Abstract

We are concerned with the following nonlinear Schr\"odinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -\Delta u+\lambda u=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=\rho, \end{cases} \end{aligned} \end{eqnarray*} where $\rho>0$ is given, $\lambda\in\mathbb{R}$ arises as a Lagrange multiplier and $f$ satisfies an exponential critical growth. Without assuming the Ambrosetti-Rabinowitz condition, we show the existence of normalized ground state solutions for any $\rho>0$. The proof is based on a constrained minimization method and the Trudinger-Moser inequality in $\mathbb{R}^2$.