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Generalized Chern-Simons-Schrodinger system with critical exponential growth: the zero mass case

Liejun Shen, Marco Squassina

Abstract

We consider the existence of ground state solutions for a class of zero-mass Chern-Simons-Schrödinger systems \[ \left\{ \begin{array}{ll} \displaystyle -Δu +A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u)-a(x)|u|^{p-2}u, \newline \displaystyle \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,~\partial_1A_1+\partial_2A_2=0, \newline \displaystyle \partial_1A_0=A_2|u|^2,~ \partial_2A_0=-A_1|u|^2, \end{array} \right. \] where $a:\mathbb R^2\to\mathbb R^+$ is an external potential, $p\in(1,2)$ and $f\in \mathcal{C}(\mathbb R)$ denotes a nonlinearity that fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. By introducing an improvement of the version of Trudinger-Moser inequality, we are able to investigate the existence of positive ground state solutions for the given system using variational method.

Generalized Chern-Simons-Schrodinger system with critical exponential growth: the zero mass case

Abstract

We consider the existence of ground state solutions for a class of zero-mass Chern-Simons-Schrödinger systems where is an external potential, and denotes a nonlinearity that fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. By introducing an improvement of the version of Trudinger-Moser inequality, we are able to investigate the existence of positive ground state solutions for the given system using variational method.
Paper Structure (5 sections, 22 theorems, 116 equations)

This paper contains 5 sections, 22 theorems, 116 equations.

Table of Contents

  1. Introduction and main results
  2. Variational Framework and Preliminaries
  3. The limit problem \ref{['mainequation2']}
  4. Proof od Theorem \ref{['maintheorem2']}
  5. Proof od Theorem \ref{['maintheorem4']}

Key Result

Proposition 1.1

Suppose that $3<p<4$, then $(e^{\alpha u^2}-1-\alpha u^2)\in L^1(\mathbb{R}^2)$ for all $\alpha>0$ and $u\in E$. Moreover, if $u\in E$, $|\nabla u|_2^2\leq1$, $|u|_p^p\leq M>+\infty$ and $\alpha<4\pi$, then there exists a constant $C(M,\alpha)>0$, which depends only on $M$ and $\alpha$, such that

Theorems & Definitions (40)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 30 more