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Spike solutions for a fractional elliptic equation in a compact Riemannian manifold

Imene Bendahou, Zied Khemiri and Fethi Mahmoudi

Vol. 324 (2023), No. 1, 1–47
DOI: 10.2140/pjm.2023.324.1
Abstract

Given an n-dimensional compact Riemannian manifold (M,g) without boundary, we consider the nonlocal equation

𝜀2sP gsu + u = upin (M,g),

where Pgs stands for the fractional Paneitz operator with principal symbol (Δg)s, s (0,1), p (1,2s 1) with 2s := 2n n2s, n > 2s, represents the critical Sobolev exponent and 𝜀 > 0 is a small real parameter. We construct a family of positive solutions u𝜀 that concentrate, as 𝜀 0 goes to zero, near critical points of the mean curvature H for 0 < s < 1 2 and near critical points of a reduced function involving the scalar curvature of the manifold  M for  1 2 s < 1.

Keywords
fractional Laplacian, fractional nonlinear Schrödinger equation, Lyapunov–Schmidt reduction, concentration phenomena
Mathematical Subject Classification
Primary: 35R11
Secondary: 35B33, 35B44, 58J05
Milestones
Received: 13 May 2021
Revised: 21 April 2022
Accepted: 11 November 2022
Published: 22 June 2023
Authors
Imene Bendahou
University Mustapha Stambouli of Mascara
Mascara
Algeria
Zied Khemiri
ESPRIT School of Engineering
Tunis
Tunisia
Fethi Mahmoudi
Department of Mathematics
Faculty of Sciences of Tunis
University Tunis El Manar
Tunis
Tunisia

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