Multiple positive bound state solutions for a critical Choquard equation

Claudianor O. Alves; Giovany M. Figueiredo; Riccardo Molle

doi:10.3934/dcds.2021061

Article Contents

2021, Volume 41, Issue 10: 4887-4919. Doi: 10.3934/dcds.2021061

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Multiple positive bound state solutions for a critical Choquard equation

1.
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, 58429-970, Campina Grande - PB, Brazil
2.
Departamento de Matemática, Universidade de Brasilia - UNB, 70910-900, Brasília-DF, Brazil
3.
Dipartimento di Matematica, Università di Roma "Tor Vergata", CEP: 00133, Roma, Italia

^* Corresponding author: R. Molle
^* Corresponding author: R. Molle

Received: August 2020

Early access: March 2021

Published: October 2021

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Abstract

In this paper we consider the problem

$ (P_{\lambda})\ \ \ \ \ \ \left\{ \begin{array}{rcl} -\Delta u+V_{\lambda}(x)u = (I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N}, \\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right. $

where $ V_{\lambda} = \lambda+V_{0} $ with $ \lambda \geq 0 $, $ V_0\in L^{N/2}({\mathbb{R}}^N) $, $ I_{\mu} = \frac{1}{|x|^\mu} $ is the Riesz potential with $ 0<\mu<\min\{N, 4\} $ and $ 2^{*}_{\mu} = \frac{2N-\mu}{N-2} $ with $ N\geq 3 $. Under some smallness assumption on $ V_0 $ and $ \lambda $ we prove the existence of two positive solutions of $ (P_\lambda) $. In order to prove the main results, we used variational methods combined with degree theory.

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Mathematics Subject Classification: Primary: 81Q05, 35A15, 35B33.

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