Abstract

Given an
$n$dimensional
compact Riemannian manifold
$(M,g)$
without boundary, we consider the nonlocal equation
$${\mathit{\epsilon}}^{2s}{P}_{g}^{s}u+u={u}^{p}\phantom{\rule{1em}{0ex}}in(M,g),$$ 
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
${2}_{s}^{\ast}:=\frac{2n}{n2s}$,
$n>2s$, represents the critical
Sobolev exponent and
$\mathit{\epsilon}>0$
is a small real parameter. We construct a family of positive solutions
${u}_{\mathit{\epsilon}}$ that
concentrate, as
$\mathit{\epsilon}\to 0$
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

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