Normalized solutions for Schrödinger equations with mixed dispersion and critical exponential growth in ${\mathbb {R}}^2$

Abstract

By developing new mathematical strategies and analytical techniques, we prove the existence of normalized ground states for the following Schrödinger equation with mixed dispersion:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda u=\mu |u|^{p-2}u+\left( e^{u^2}-1-u^2\right) u, &{} x\in {\mathbb {R}}^2, \\ \int _{{\mathbb {R}}^2}u^2\textrm{d}x=c, \\ \end{array} \right. \end{aligned}$$

where $c>0$, $\lambda \in {\mathbb {R}}$, p is allowed to be $L^2$-subcritical $2<p<4$, $L^2$-critical $p=4$ or $L^2$-supercritical $4< p<+\infty$, and the mixed nonlinearity has critical exponential growth of Trudinger–Moser type which is a novelty for $L^2$-constrained problems. To restore the compactness, some ingenious analyses and sharp energy estimates are introduced. Our study achieves a significant extension from the Sobolev critical growth for the higher dimensions to the critical exponential growth for the planar dimension in the context of normalized solutions, and seems to be the first contribution in this direction. We believe that our approaches may be adapted and modified to attack more planar $L^2$-constrained problems with critical exponential growth, and hope to stimulate further research on this topic like that by Soave (J Funct Anal 279:108610, 2020) for the higher dimensional Sobolev critical case.