Normalized Ground State Solutions of Nonlinear Schrödinger Equations Involving Exponential Critical Growth

We are concerned with the following nonlinear Schrödinger equation: -Δu+λu=f(u)inR2,u∈H1(R2),∫R2u2dx=ρ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda u=f(u) \ \ \textrm{in}\ \mathbb {R}^{2},\\ u\in H^{1}(\mathbb {R}^{2}),~~~ \int _{\mathbb {R}^2}u^2dx=\rho , \end{array}\right. } \end{aligned} \end{aligned}$$\end{document}where ρ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho >0$$\end{document} is given, λ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in \mathbb {R}$$\end{document} 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 ρ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho >0$$\end{document}. The proof is based on a constrained minimization method and the Trudinger–Moser inequality in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document}.