Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Normalized solutions to the Chern-Simons-Schrödinger system: the supercritical case

Liejun Shen, Marco Squassina

TL;DR

The paper addresses the existence of normalized (mass-constrained) solutions to a generalized Chern-Simons-Schrödinger system in two dimensions with nonlinearities exhibiting subcritical, critical, and supercritical exponential growth. It develops a robust variational framework on the mass constraint $S(a)$, leveraging gauge-field estimates, the Trudinger-Moser inequality, and a natural constraint $\mathcal{M}(a)$ to obtain ground states via constrained minimization and Palais-Smale sequences; a homotopy stable family further guarantees the existence of a minimax level. In the supercritical regime, the authors introduce truncated nonlinearities $f^{R,\bar{\delta}}$ and $f^{R,2}$ to regain $C^1$-regularity and apply both constrained minimization (5.1) and mountain-pass (5.2) approaches to produce normalized solutions for small $a$, with uniform $L^{\infty}$ bounds. The work extends the theory of normalized CSS solutions in 2D without imposing radial symmetry or compact embeddings and provides several new mechanisms, including Moser-type compactness controls and cutoff strategies, to handle supercritical exponential growth.

Abstract

We are concerned with the existence of normalized solutions for a class of generalized Chern-Simons-Schrödinger type problems with supercritical exponential growth $$ -Δu +λu+A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u),\quad \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,\quad \partial_1A_1+\partial_2A_2=0,\quad \partial_1A_0=A_2|u|^2,\quad \partial_2A_0=-A_1|u|^2,\quad \int_{\mathbb{R}^2}|u|^2dx=a^2, $$ where $a\neq0$, $λ\in \mathbb{R}$ is known as the Lagrange multiplier and $f\in C^1(\mathbb{R})$ denotes the nonlinearity that fulfills the supercritical exponential growth in the Trudinger-Moser sense at infinity. Under suitable assumptions, combining the constrained minimization approach together with the homotopy stable family and elliptic regularity theory, we obtain that the problem has at least a ground state solution.

Normalized solutions to the Chern-Simons-Schrödinger system: the supercritical case

TL;DR

The paper addresses the existence of normalized (mass-constrained) solutions to a generalized Chern-Simons-Schrödinger system in two dimensions with nonlinearities exhibiting subcritical, critical, and supercritical exponential growth. It develops a robust variational framework on the mass constraint , leveraging gauge-field estimates, the Trudinger-Moser inequality, and a natural constraint to obtain ground states via constrained minimization and Palais-Smale sequences; a homotopy stable family further guarantees the existence of a minimax level. In the supercritical regime, the authors introduce truncated nonlinearities and to regain -regularity and apply both constrained minimization (5.1) and mountain-pass (5.2) approaches to produce normalized solutions for small , with uniform bounds. The work extends the theory of normalized CSS solutions in 2D without imposing radial symmetry or compact embeddings and provides several new mechanisms, including Moser-type compactness controls and cutoff strategies, to handle supercritical exponential growth.

Abstract

We are concerned with the existence of normalized solutions for a class of generalized Chern-Simons-Schrödinger type problems with supercritical exponential growth where , is known as the Lagrange multiplier and denotes the nonlinearity that fulfills the supercritical exponential growth in the Trudinger-Moser sense at infinity. Under suitable assumptions, combining the constrained minimization approach together with the homotopy stable family and elliptic regularity theory, we obtain that the problem has at least a ground state solution.
Paper Structure (7 sections, 30 theorems, 258 equations)

This paper contains 7 sections, 30 theorems, 258 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminary results
  3. The subcritical case
  4. The critical case
  5. The supercritical case
  6. The case $\bar{\delta}=\delta$
  7. The case $\bar{\delta}=2$

Key Result

Theorem 1.1

Suppose that $f$ satisfies definitionsubcriticalgrowth and $(f_1)-(f_3)$, then there is a small $a^*>0$ such that problems mainequation1-mainequation1a possess a couple weak solution $(u_0,\lambda_0)\in H^1(\mathbb{R}^2)\times\mathbb{R}^+$ for all $a\in(0,a^*]$, where $\lambda_0>0$, $u_0(x)>0$ for a Here the energy functional $J:S(a)\to\mathbb{R}$ is defined by

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 56 more