Existence of nontrivial solutions to Chern-Simons-Schrödinger system with indefinite potential

Jincai Kang; Chunlei Tang

doi:10.3934/dcdss.2021016

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2021, Volume 14, Issue 6: 1931-1944. Doi: 10.3934/dcdss.2021016

This issue Previous Article The Orlicz-Minkowski problem for polytopes Next Article Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms

Existence of nontrivial solutions to Chern-Simons-Schrödinger system with indefinite potential

Jincai Kang and
Chunlei Tang^,

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

^* Corresponding author: Chunlei Tang
^* Corresponding author: Chunlei Tang

Received: October 2020

Revised: January 2021

Early access: September 2018

Published: June 2021

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

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Abstract

We consider a class of Chern-Simons-Schrödinger system

$ \begin{align*} \begin{cases} -\Delta u+V(x) u+A_{0}u+\sum\limits_{j = 1}^{2} A_{j}^{2}u = g(u), \\ \partial_{1}A_{0} = A_{2}|u|^{2}, \ \ \partial_{2}A_{0} = -A_{1}|u|^{2}, \\ \partial_{1}A_{2}-\partial_{2}A_{1} = -\frac{1}{2}u^{2}, \ \ \partial_{1}A_{1}+\partial_{2}A_{2} = 0, \end{cases} \end{align*} $

where $ V $ is coercive sign-changing potential and $ f $ satisfies some suitable conditions. Due to lack of the mountain pass geometry and the link geometry for the corresponding variational functional, we obtain the existence of nontrivial solutions via the local link theorem.

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

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