Multiplicity and concentration of solutions for Choquard equation via Nehari method and pseudo-index theory

Min Liu; Zhongwei Tang

doi:10.3934/dcds.2019139

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2019, Volume 39, Issue 6: 3365-3398. Doi: 10.3934/dcds.2019139

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Multiplicity and concentration of solutions for Choquard equation via Nehari method and pseudo-index theory

Min Liu and
Zhongwei Tang^,

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

^* Corresponding author: Zhongwei Tang
^* Corresponding author: Zhongwei Tang

Received: July 31, 2018

Published: February 2019

This work is supported by NSFC (11571040, 11671331)

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Abstract

This paper concerns the following nonlinear Choquard equation:

$ \begin{equation} -\varepsilon^{2}\Delta w+V(x)w = \varepsilon^{-\theta}W(x)(I_\theta*(W|w|^p))|w|^{p-2}w,\quad x\in\mathbb{R}^N, ~~~~~~~~~~~~(*)\end{equation} $

where $ \varepsilon>0,\ N>2,\ I_\theta $ is the Riesz potential with order $ \theta\in(0,N),\ p\in\big[2,\frac{N+\theta}{N-2}\big),\ \min V>0 $ and $ \inf W>0 $. Under proper assumptions, we explore the existence, concentration, convergence and decay estimate of semiclassical solutions for $ (\ast) $. The multiplicity of solutions is established via pseudo-index theory. The existence of sign-changing solutions is achieved by minimizing the energy on Nehari nodal set.

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Mathematics Subject Classification: Primary: 35J20, 35R09; Secondary: 35J61.

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