Critical gauged Schrödinger equations in $ \mathbb{R}^2 $ with vanishing potentials

Liejun Shen; Marco Squassina; Minbo Yang

doi:10.3934/dcds.2022059

Article Contents

2022, Volume 42, Issue 9: 4415-4438. Doi: 10.3934/dcds.2022059

This issue Previous Article Global existence and blow up for systems of nonlinear wave equations related to the weak null condition Next Article Long-time asymptotics for the modified complex short pulse equation

Critical gauged Schrödinger equations in $ \mathbb{R}^2 $ with vanishing potentials

1.
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
2.
Dipartimento di Matematica e Fisica Università Cattolica del Sacro Cuore, Via della Garzetta 48, 25133, Brescia, Italy

^*Corresponding author: Marco Squassina
^*Corresponding author: Marco Squassina

Received: September 30, 2021

Revised: February 28, 2022

Early access: May 2022

Published: August 2022

Marco Squassina is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Minbo Yang was partially supported by NSFC (11971436, 12011530199) and ZJNSF(LZ22A010001, LD19A010001)

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Abstract

We study a class of gauged nonlinear Schrödinger equations in the plane

$ \left\{ \begin{array}{l} -\Delta u+V(|x|) u+\lambda\bigg(\int_{|x|}^\infty \frac{h_u(s)}{s}u^2(s)ds+\frac{h_u^2(|x|)}{|x|^2} \bigg)u\\\qquad \, = K(|x|)f(u)+\mu g(|x|)|u|^{q-2}u, \\ u(x) = u(|x|) \; {\rm{in}}\; \mathbb{R}^2, \\\\ \end{array} \right. $

where $ h_u(s) = \int_0^s\frac{r}{2}u^2(r)dr $, $ \lambda,\mu>0 $ are constants, $ V(|x|) $ and $ K(|x|) $ are continuous functions vanishing at infinity. Assume that $ f $ is of critical exponential growth and $ g(x) = g(|x|) $ satisfies some technical assumptions with $ 1\leq q<2 $, we obtain the existence of two nontrivial solutions via the Mountain-Pass theorem and Ekeland's variational principle. Moreover, with the help of the genus theory, we prove the existence of infinitely many solutions if $ f $ in addition is odd.

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35Q55.

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