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The planar Schrodinger--Poisson system with exponential critical growth: The local well-posedness and standing waves with prescribed mass

Juntao Sun, Shuai Yao, Jian Zhang

TL;DR

This work studies the planar Schrödinger–Poisson system in $\mathbb{R}^2$ with exponential critical growth, establishing local well-posedness of the Cauchy problem in the energy space $X$ by decomposing the nonlocal logarithmic term and introducing the self-adjoint operator $\mathcal{L}=\Delta - m\ln(1+|x|)$. It then analyzes standing waves with prescribed mass by variational methods, proving the existence of two normalized solutions: a ground state with positive energy and a higher-energy mountain-pass state, and showing orbital stability of the ground-state set for small mass. The results are obtained under relaxed nonlinear growth conditions, notably allowing exponential critical growth with $\alpha_0=4\pi$, and rely on a careful treatment of the logarithmic interaction via $V(u)$, $V_1$, and $V_2$, together with constrained minimization and Jeanjean-type mountain-pass arguments. These contributions extend prior work by handling the 2D nonlocal log-term without the Ambrosetti–Rabinowitz condition and by offering a detailed dynamical perspective through orbital stability and normalized solutions. The findings have implications for the dynamics and stability of planar quantum systems with nonlocal interactions under critical exponential nonlinearities.

Abstract

In this paper, we investigate a class of planar Schrödinger-Poisson systems with critical exponential growth. We establish conditions for the local well-posedness of the Cauchy problem in the energy space, which seems innovative as it was not discussed at all in any previous results. By introducing some new ideas and relaxing some of the classical growth assumptions on the nonlinearity, we show that such system has at least two standing waves with prescribed mass, where one is a ground state standing waves with positive energy, and the other one is a high-energy standing waves with positive energy. In addition, with the help of the local well-posedness, we show that the set of ground state standing waves is orbitally stable.

The planar Schrodinger--Poisson system with exponential critical growth: The local well-posedness and standing waves with prescribed mass

TL;DR

This work studies the planar Schrödinger–Poisson system in with exponential critical growth, establishing local well-posedness of the Cauchy problem in the energy space by decomposing the nonlocal logarithmic term and introducing the self-adjoint operator . It then analyzes standing waves with prescribed mass by variational methods, proving the existence of two normalized solutions: a ground state with positive energy and a higher-energy mountain-pass state, and showing orbital stability of the ground-state set for small mass. The results are obtained under relaxed nonlinear growth conditions, notably allowing exponential critical growth with , and rely on a careful treatment of the logarithmic interaction via , , and , together with constrained minimization and Jeanjean-type mountain-pass arguments. These contributions extend prior work by handling the 2D nonlocal log-term without the Ambrosetti–Rabinowitz condition and by offering a detailed dynamical perspective through orbital stability and normalized solutions. The findings have implications for the dynamics and stability of planar quantum systems with nonlocal interactions under critical exponential nonlinearities.

Abstract

In this paper, we investigate a class of planar Schrödinger-Poisson systems with critical exponential growth. We establish conditions for the local well-posedness of the Cauchy problem in the energy space, which seems innovative as it was not discussed at all in any previous results. By introducing some new ideas and relaxing some of the classical growth assumptions on the nonlinearity, we show that such system has at least two standing waves with prescribed mass, where one is a ground state standing waves with positive energy, and the other one is a high-energy standing waves with positive energy. In addition, with the help of the local well-posedness, we show that the set of ground state standing waves is orbitally stable.
Paper Structure (7 sections, 28 theorems, 196 equations)

This paper contains 7 sections, 28 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminary results
  4. The local well-posedness for the Cauchy problem
  5. The ground state standing waves
  6. The high-energy standing waves
  7. Acknowledgments

Key Result

Theorem 1.1

Assume that $\Vert \nabla \psi _{0}\Vert _{L^{2}}^{2}<1$ and $f$ satisfies $(f_{1})$ and $(f_{7})$ for any $z_{1},z_{2}\in \mathbb{C}$ and $\varepsilon >0$, there exists $C_{\varepsilon }>0$ such that and Then there exist $T_{\max }>0$ and a unique solution $\psi \in C([0,T_{\max }],X)$ for the Cauchy problem (1.1).

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • ...and 23 more