On the critical Schrödinger-Poisson system with $ p $-Laplacian

Yao Du; Jiabao Su; Cong Wang

doi:10.3934/cpaa.2022020

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2022, Volume 21, Issue 4: 1329-1342. Doi: 10.3934/cpaa.2022020

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On the critical Schrödinger-Poisson system with $ p $-Laplacian

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

^*Corresponding author
^*Corresponding author

Received: June 30, 2021

Revised: November 30, 2021

Early access: January 2022

Published: April 2022

Supported by KZ202010028048 and NSFC(11771302, 12171326)

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Abstract

In this paper we consider the critical quasilinear Schrödinger-Poisson system

$ \begin{eqnarray*} \left \{\begin{array}{ll} -\Delta_p u+|u|^{p-2}u+\mu\phi |u|^{p-2}u = \lambda|u|^{q-2}u+|u|^{p^*-2}u,&\mathrm{in} \ \mathbb{R}^3,\\ -\Delta \phi = |u|^p, &\mathrm{in}\ \mathbb{R}^3, \end{array} \right. \end{eqnarray*} $

where $ \frac{3}{2}<p<3 $, $ \Delta_p u = \hbox{div}(|\nabla u|^{p-2}\nabla u) $, $ p<q<p^*: = \frac{3p}{3-p} $ and $ \mu,\lambda>0 $. Based upon the variational approach, the ground state solutions and the nontrivial solutions are obtained depending on the parameters $ q $, $ \mu $ and $ \lambda $.

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Mathematics Subject Classification: 35J10, 35J50, 35J60, 35J92.

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