Uniqueness and nondegeneracy of sign-changing radial solutions to an almost critical elliptic problem

TL;DR

This paper investigates the uniqueness and nondegeneracy of sign-changing radial solutions to a semi-linear elliptic equation near the critical exponent, establishing their precise properties and kernel dimension.

Contribution

It proves the uniqueness and nondegeneracy of sign-changing radial solutions for the elliptic problem when the exponent is close to the critical value.

Findings

Sign-changing solutions are unique near the critical exponent.

Solutions are non-degenerate with an N-dimensional kernel.

Results extend understanding of solution structure in almost critical regimes.

Abstract

We study sign-changing radial solutions for the following semi-linear elliptic equation \begin{align*} \Delta u-u+|u|^{p-1}u=0\quad{\rm{in}}\ \mathbb{R}^N,\quad u\in H^1(\mathbb{R}^N), \end{align*} where $1<p<\frac{N+2}{N-2}$, $N\geq3$. It is well-known that this equation has a unique positive radial solution and sign-changing radial solutions with exactly $k$ nodes. In this paper, we show that such sign-changing radial solution is also unique when $p$ is close to $\frac{N+2}{N-2}$. Moreover, those solutions are non-degenerate, i.e., the kernel of the linearized operator is exactly $N$-dimensional.