A POSITIVE SOLUTION FOR QUASILINEAR SCHRÖODINGER-POISSON SYSTEM WITH CRITICAL EXPONENT

doi:10.1007/s10473-025-0402-5

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (4): 1247-1264.doi: 10.1007/s10473-025-0402-5

A POSITIVE SOLUTION FOR QUASILINEAR SCHRÖODINGER-POISSON SYSTEM WITH CRITICAL EXPONENT

Lanxin HUANG^1,2,3, Jiabao SU^3,*

1. School of Mathematical Sciences, Shenzhen University, Shenzhen 518060, China;
2. College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China;
3. School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

收稿日期:2024-03-13 修回日期:2024-07-03 出版日期:2025-10-10 发布日期:2025-10-10

A POSITIVE SOLUTION FOR QUASILINEAR SCHRÖODINGER-POISSON SYSTEM WITH CRITICAL EXPONENT

Lanxin HUANG^1,2,3, Jiabao SU^3,*

1. School of Mathematical Sciences, Shenzhen University, Shenzhen 518060, China;
2. College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China;
3. School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received:2024-03-13 Revised:2024-07-03 Online:2025-10-10 Published:2025-10-10
Contact: ^*Jiabao SU, E-mail: sujb@cnu.edu.cn
About author:Lanxin HUANG, E-mail: 812419761@qq.com
Supported by:
The research was supported by the National Natural Science Foundation of China (12271373, 12171326).

摘要/Abstract

摘要： In this paper, we study the quasilinear Schrödinger-Poisson system with critical Sobolev exponent $$\begin{aligned} \begin{cases} -\Delta_{p} u+|u|^{p-2}u+l(x)\phi |u|^{p-2}u=|u|^{p^{*}-2}u+\mu h(x)|u|^{q-2}u & \ \ \ \mathrm{in}\ \mathbb{R}^{3},\\ -\Delta \phi=l(x)|u|^{p} &\ \ \ \mathrm{in}\ \mathbb{R}^{3}, \end{cases} \end{aligned}$$ where $\mu >0$, $\frac{3}{2}<p<3$, $p\leqslant q<p^{*}=\frac{3p}{3-p}$ and $\Delta_{p} u= \hbox{div}(|\nabla u|^{p-2}\nabla u)$. Under certain assumptions on the functions $l$ and $h$, we employ the mountain pass theorem to establish the existence of positive solutions for this system.

关键词: quasilinear Schröodinger-Poisson system, variational methods, critical growth

Abstract: In this paper, we study the quasilinear Schrödinger-Poisson system with critical Sobolev exponent $$\begin{aligned} \begin{cases} -\Delta_{p} u+|u|^{p-2}u+l(x)\phi |u|^{p-2}u=|u|^{p^{*}-2}u+\mu h(x)|u|^{q-2}u & \ \ \ \mathrm{in}\ \mathbb{R}^{3},\\ -\Delta \phi=l(x)|u|^{p} &\ \ \ \mathrm{in}\ \mathbb{R}^{3}, \end{cases} \end{aligned}$$ where $\mu >0$, $\frac{3}{2}<p<3$, $p\leqslant q<p^{*}=\frac{3p}{3-p}$ and $\Delta_{p} u= \hbox{div}(|\nabla u|^{p-2}\nabla u)$. Under certain assumptions on the functions $l$ and $h$, we employ the mountain pass theorem to establish the existence of positive solutions for this system.

Key words: quasilinear Schröodinger-Poisson system, variational methods, critical growth

Lanxin HUANG, Jiabao SU. A POSITIVE SOLUTION FOR QUASILINEAR SCHRÖODINGER-POISSON SYSTEM WITH CRITICAL EXPONENT[J]. 数学物理学报(英文版), 2025, 45(4): 1247-1264.

Lanxin HUANG, Jiabao SU. A POSITIVE SOLUTION FOR QUASILINEAR SCHRÖODINGER-POISSON SYSTEM WITH CRITICAL EXPONENT[J]. Acta mathematica scientia,Series B, 2025, 45(4): 1247-1264.

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A POSITIVE SOLUTION FOR QUASILINEAR SCHRÖODINGER-POISSON SYSTEM WITH CRITICAL EXPONENT

A POSITIVE SOLUTION FOR QUASILINEAR SCHRÖODINGER-POISSON SYSTEM WITH CRITICAL EXPONENT

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