Axially symmetric solutions of Allen-Cahn equation with finite Morse index

TL;DR

This paper investigates axially symmetric solutions of the Allen-Cahn equation with finite Morse index, establishing nonexistence in higher dimensions, finiteness of ends in three dimensions, and characterizing solutions with Morse index one.

Contribution

It proves nonexistence of such solutions in dimensions 4 to 10, shows solutions in 3D have finitely many ends, and precisely two ends when Morse index is one.

Findings

No solutions in dimensions 4-10.

Solutions in 3D have finitely many ends.

Exactly two ends for Morse index 1 solutions.

Abstract

In this paper we study axially symmetric solutions of Allen-Cahn equation with finite Morse index. It is shown that there does not exist such a solution in dimensions between $4$ and $10$. In dimension $3$, we prove that these solutions have finitely many ends. Furthermore, the solution has exactly two ends if its Morse index equals $1$.