Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth

Jincai Kang; Chunlei Tang

doi:10.3934/cpaa.2020235

Article Contents

2020, Volume 19, Issue 11: 5239-5252. Doi: 10.3934/cpaa.2020235

This issue Previous Article On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum Next Article Liouville type theorem for Fractional Laplacian system

Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth

Jincai Kang and
Chunlei Tang^,

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

^* Corresponding author
^* Corresponding author

Received: February 29, 2020

Revised: May 31, 2020

Early access: September 2020

Published: November 2020

This work is supported by National Natural Science Foundation of China (No. 11971393)

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Abstract

We investigate the following gauged nonlinear Schrödinger equation

$ \begin{equation*} \begin{cases} -\Delta u+\omega u+\lambda\bigg(\dfrac{h_{u}^{2}(|x|)}{|x|^{2}}+ \int_{|x|}^{+\infty}\dfrac{h_{u}(s)}{s}u^{2}(s)ds\bigg)u = f(u) \ \ \ \ \ \mbox{in}\ \mathbb{R}^{2},\\ u\in H_r^1(\mathbb{R}^{2}), \end{cases} \end{equation*} $

where $ \omega,\lambda>0 $ and $ h_{u}(s) = \frac{1}{2}\int_{0}^{s}ru^{2}(r)dr $. When $ f $ has exponential critical growth, by using the constrained minimization method and Trudinger-Moser inequality, it is proved that the equation has a ground state radial sign-changing solution $ u_{\lambda} $ which changes sign exactly once. Moreover, the asymptotic behavior of $ u_{\lambda} $ as $ \lambda\rightarrow0 $ is analyzed.

Keywords:

Gauged Schrödinger equation,
critical growth,
variational methods,
ground state radial sign-changing solution,
asymptotic behavior.

Mathematics Subject Classification: Primary: 35J20; Secondary: 35A15, 35D30, 35J91.

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