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

Published online by Cambridge University Press:  08 October 2015

Claudianor O. Alves
Departamento de Matemática, Universidade Federal de Campina, Grande, CEP 58429-900, Campina Grande – Pb, Brazil (
Minbo Yang
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China (


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.

Research Article
Copyright © Royal Society of Edinburgh 2016 

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