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Planar Schrödinger-Poisson system with steep potential well: supercritical exponential case

Liejun Shen, Marco Squassina

Abstract

We study a class of planar Schrödinger-Poisson systems $$ -Δu+λV(x)u+φu=f(u) , \quad x\in{\mathbb R}^2,\qquad Δφ=u^2, \quad x\in{\mathbb R}^2, $$ where $λ>0$ is a parameter, $V\in C({\mathbb R}^2,{\mathbb R}^+)$ has a potential well $Ω\triangleq\text{int}\, V^{-1}(0)$ and the nonlinearity $f$ fulfills the supercritical exponential growth at infinity in the Trudinger-Moser sense. By exploiting the mountain-pass theorem and elliptic regular theory, we establish the existence and concentrating behavior of ground state solutions for sufficiently large $λ$.

Planar Schrödinger-Poisson system with steep potential well: supercritical exponential case

Abstract

We study a class of planar Schrödinger-Poisson systems where is a parameter, has a potential well and the nonlinearity fulfills the supercritical exponential growth at infinity in the Trudinger-Moser sense. By exploiting the mountain-pass theorem and elliptic regular theory, we establish the existence and concentrating behavior of ground state solutions for sufficiently large .
Paper Structure (11 sections, 38 theorems, 193 equations)

This paper contains 11 sections, 38 theorems, 193 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. On problem \ref{['mainequation2']}: the (sub)critical case
  4. Proofs of Theorems \ref{['maintheorem1']} and \ref{['maintheorem3']}
  5. The Case (I) in \ref{['definition2']}$: \bar{\delta}=\delta\in(0,2)$
  6. The Case (II) in \ref{['definition2']}$: \bar{\delta}=2$
  7. Some further remarks
  8. Proofs of Theorems \ref{['maintheorem6']} and \ref{['maintheorem7']}
  9. Proof of Theorem \ref{['maintheorem6']}
  10. Proof of Theorem \ref{['maintheorem7']}
  11. Proofs of Theorems \ref{['maintheorem4']} and \ref{['maintheorem5']}

Key Result

Theorem 1.1

Let $V$ satisfy $(V_1)-(V_3)$. Suppose that the nonlinearity $f$ defined in form requires $(h_1)-(h_3)$, then for each $\tau>2$, there are $\alpha^*=\alpha^*(\tau)>0$ and $\lambda_0>0$ such that Eq. mainequation1 has a nonnegative ground state solution in $X_\lambda$ for all $\alpha \in (0, \alpha^* then for every $\alpha>0$, there exist $\tau_*=\tau_*(\alpha)>2$, $\lambda_0^\prime>0$ and ${\xi}_0

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 65 more