Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems

Daniele Cassani; Luca Vilasi; Jianjun Zhang

doi:10.3934/cpaa.2021039

Article Contents

2021, Volume 20, Issue 4: 1737-1754. Doi: 10.3934/cpaa.2021039

This issue Previous Article Extremal functions for a class of trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary Next Article

Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems

1.
Dip. di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, and RISM – Riemann International School of Mathematics, Villa Toeplitz, Via G.B. Vico, 46 – 21100 Varese, Italy
2.
College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

^* Corresponding author
^* Corresponding author

Received: June 30, 2020

Revised: December 31, 2020

Early access: March 2021

Published: April 2021

The third named author was supported by NSFC(No.11871123)

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Abstract

We consider in $ \mathbb R^3 $ the singularly perturbed Schrödinger-Poisson system

$ \begin{equation*} \begin{cases} - \varepsilon^2\Delta v+V(x)v+\phi(x)v = f(v)\\ - \Delta\phi = v^2 . \end{cases} \end{equation*} $

Using variational techniques, we construct solutions which concentrate around the saddle points of the external potential $ V $, as $ \varepsilon \rightarrow 0 $.

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Mathematics Subject Classification: Primary: 35J47, Secondary: 35J50, 35J61.

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