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Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  08 October 2015

Claudianor O. Alves  and
Minbo Yang
Claudianor O. Alves
Affiliation:
Departamento de Matemática, Universidade Federal de Campina, Grande, CEP 58429-900, Campina Grande – Pb, Brazil (coalves@dme.ufcg.edu.br)
Minbo Yang
Affiliation:
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China (mbyang@zjnu.edu.cn)
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Abstract

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We study the multiplicity and concentration behaviour of positive solutions for a quasi-linear Choquard equation

where Δp is the p-Laplacian operator, 1 < p < N, V is a continuous real function on ℝN, 0 < μ < N, F(s) is the primitive function of f(s), ε is a positive parameter and * represents the convolution between two functions. The question of the existence of semiclassical solutions for the semilinear case p = 2 has recently been posed by Ambrosetti and Malchiodi. We suppose that the potential satisfies the condition introduced by del Pino and Felmer, i.e.V has a local minimum. We prove the existence, multiplicity and concentration of solutions for the equation by the penalization method and Lyusternik–Schnirelmann theory and even show novel results for the semilinear case p = 2.

Keywords

Choquard equationnon-local nonlinearitiesconcentrationLyusternik–Schnirelmann theoryvariational methods

MSC classification

Primary: 35J50: Variational methods for elliptic systems 35J60: Nonlinear elliptic equations 35A15: Variational methods
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 1 , February 2016 , pp. 23 - 58
Copyright
Copyright © Royal Society of Edinburgh 2016