On Schrödinger-Poisson equations with a critical nonlocal term

Xinyi Zhang; Jian Zhang; Xinyi Zhang; Jian Zhang

doi:10.3934/math.2024545

AIMS Mathematics

2024, Volume 9, Issue 5: 11122-11138. doi: 10.3934/math.2024545

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Research article

On Schrödinger-Poisson equations with a critical nonlocal term

Xinyi Zhang ,
Jian Zhang ^,

College of Science, China University of Petroleum, Qingdao 266580, Shandong, China

Received: 23 January 2024 Revised: 05 March 2024 Accepted: 11 March 2024 Published: 21 March 2024
MSC : 35A15, 35J60

In this paper, we study the following non-autonomous Schrödinger-Poisson equation with a critical nonlocal term and a critical nonlinearity:

$ \begin{equation*} \left\{\begin{aligned} & -\Delta u +V(x) u + \lambda \phi |u|^3 u = f(u) + (u^+)^5,\ \ {\rm in } \ \ \ \ \mathbb{R}^3,\\ & -\Delta \phi = |u|^5, \ \ {\rm in } \ \ \ \ \mathbb{R}^3. \end{aligned}\right. \end{equation*} $

First, we consider the case that the nonlinearity satisfies the Berestycki-Lions type condition with critical growth. Second, we consider the case that $ \mathrm{int}V^{-1}(0) $ is contained in a spherical shell. By using variational methods, we obtain the existence and asymptotic behavior of positive solutions.
- Schrödinger-Poisson equation,
- critical nonlocal term,
- critical nonlinearity,
- variational method
Citation: Xinyi Zhang, Jian Zhang. On Schrödinger-Poisson equations with a critical nonlocal term[J]. AIMS Mathematics, 2024, 9(5): 11122-11138. doi: 10.3934/math.2024545

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Abstract

In this paper, we study the following non-autonomous Schrödinger-Poisson equation with a critical nonlocal term and a critical nonlinearity:

$ \begin{equation*} \left\{\begin{aligned} & -\Delta u +V(x) u + \lambda \phi |u|^3 u = f(u) + (u^+)^5,\ \ {\rm in } \ \ \ \ \mathbb{R}^3,\\ & -\Delta \phi = |u|^5, \ \ {\rm in } \ \ \ \ \mathbb{R}^3. \end{aligned}\right. \end{equation*} $

First, we consider the case that the nonlinearity satisfies the Berestycki-Lions type condition with critical growth. Second, we consider the case that $ \mathrm{int}V^{-1}(0) $ is contained in a spherical shell. By using variational methods, we obtain the existence and asymptotic behavior of positive solutions.

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