Multi-bump solutions for fractional Nirenberg problem

TL;DR

This paper constructs multi-bump solutions for a fractional Nirenberg problem with periodic coefficients, demonstrating the existence of solutions with bumps clustered on lattice points in certain subspaces of ^n.

Contribution

The paper establishes the existence of multi-bump solutions for the fractional Nirenberg problem with periodic coefficients using Lyapunov-Schmidt reduction, including infinitely many solutions clustered on lattice points.

Findings

Existence of multi-bump solutions with bumps on lattice points in ^k.

Construction of infinitely many solutions with bumps on lattice points.

Application of Lyapunov-Schmidt reduction to fractional PDEs.

Abstract

We consider the multi-bump solutions of the following fractional Nirenberg problem \begin{equation}\label{01} (-\Delta)^s u=K(x)u^{\frac{n+2s}{n-2s}}, \;\;\;\;u>0\;\;\text{ in }\mathbb{R}^n, \end{equation} where $s\in (0,1)$ and $n>2+2s$. If $K$ is a periodic function in some $k$ variables with $1\leq k<\frac{n-2s}2$, we proved that \eqref{01} has multi-bump solutions with bumps clustered on some lattice points in $\mathbb{R}^k$ via Lyapunov-Schmidt reduction. It is also established that the equation \eqref{01} has an infinite-many-bump solutions with bumps clustered on some lattice points in $\mathbb{R}^n$ which is isomorphic to $\mathbb{Z}_+^k$.