Normalized ground state solutions of nonlinear Schrödinger equations involving exponential critical growth

Abstract

We are concerned with the following nonlinear Schrödinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -Δu+λu=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=ρ, \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 ${\mathbb{R}}^2$.

arXiv comment19 pages

References in corpus (4)

• Dark solitons in atomic Bose-Einstein condensates: from theory to experiments
• Mathematical theory and numerical methods for Bose-Einstein condensation
• Multiple normalized solutions for a Sobolev critical Schrödinger equation
• Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth

Cited by in corpus (4)

• Positive solutions with prescribed mass for a planar Choquard equation with critical growth
• Existence and dynamics of normalized solutions to Schrödinger equations with generic double-behaviour nonlinearities
• Normalized solutions to polyharmonic equations with Hardy-type potentials and exponential critical nonlinearities
• Normalized ground states for nonlinear Schrödinger equations with general Sobolev critical nonlinearities