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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

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.

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

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