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

$${𝜀}^{2s}{P}_g^{s}u+u=u^{p}in(M,g),$$

where $P_g^s$ stands for the fractional Paneitz operator with principal symbol ${(-{\mathrm{\Delta }}_g)}^s$, $s\in (0,1)$, $p\in (1,2_s^{\ast }-1)$ with $2_s^{\ast }:=\frac{2n}{n-2s}$, $n>2s$, represents the critical Sobolev exponent and $𝜀>0$ is a small real parameter. We construct a family of positive solutions $u_{𝜀}$ that concentrate, as $𝜀\to 0$ goes to zero, near critical points of the mean curvature $H$ for $0<s<\frac{1}{2}$ and near critical points of a reduced function involving the scalar curvature of the manifold  $M$ for  $\frac{1}{2}\le s<1$.

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