Normalized Solutions for Schrödinger Equations with Critical Exponential Growth in \(\boldsymbol{\mathbb{R}^2}\)

.For any \(a\gt 0\), we study the existence of normalized solutions and ground state solutions to the following Schrödinger equation with \(L^2\)-constraint: \(\left\{ \begin{array}{ll} -\Delta u+\lambda u=b(x)f(u) & x\in \mathbb{R}^2, \\ \int_{\mathbb{R}^2}u^2\mathrm{d}x=a, \\ \end{array} \right.\) where \(\lambda \in \mathbb{R}\) is a Lagrange multiplier, the potential \(b\in \mathcal{C}(\mathbb{R}^2, (0, \infty ))\) satisfies \(0\lt \lim_{|y|\to \infty }b(y)\leq \inf_{x\in \mathbb{R}^2}b(x)\) and appears as a converse direction of the Rabinowitz-type trapping potential, and the reaction \(f\in \mathcal{C}(\mathbb{R},\mathbb{R})\) enjoys critical exponential growth of Trudinger–Moser type. Under two different kinds of assumptions on \(f\), we prove several new existence results, which, in the context of normalized solutions, can be considered as both counterparts of planar unconstrained critical problems and extensions of unconstrained Schrödinger problems with Rabinowitz-type trapping potential. Especially, in this scenario, we develop some sharp estimates of energy levels and ingenious analysis techniques to restore the compactness which are novel even for \(b(x)\equiv\) constant. We believe that these techniques will allow not only treating other \(L^2\)-constrained problems in the Trudinger–Moser critical setting but also generalizing previous results to the case of variable potentials.KeywordsSchrödinger equationnormalized solutioncritical exponential growthTrudinger–Moser inequalityMSC codes35J2035J6235Q55