Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T17:43:54.142Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Partial differential equations Elliptic equations and systems

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 ([email protected])
Minbo Yang
Affiliation:
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)