Non-degeneracy and new type of cylindrial solutions for a critical Grushin-type problem

TL;DR

This paper investigates a critical Grushin-type problem related to CR sphere curvature, establishing non-degeneracy and constructing new cylindrically symmetric multi-bubbling solutions using Lyapunov-Schmidt reduction.

Contribution

It introduces a non-degeneracy result and constructs a novel type of multi-bubbling solutions with cylindrical symmetry for the critical Grushin problem.

Findings

Proved non-degeneracy via local Pohozaev identities.

Constructed new cylindrically symmetric multi-bubbling solutions.

Linked solutions to prescribed Webster scalar curvature problems.

Abstract

In this paper, we consider a critical Grushin-type problem, which is closely related to the prescribed Webster scalar curvature problems on the CR sphere with cylindrically symmetric curvature. We first prove a non-degeneracy result through local Pohozaev identities, then by using the Lyapunov-Schmidt reduction methods, we construct new type of multi-bubbling solutions with cylindrical symmetry.