## Advances in Differential Equations

- Adv. Differential Equations
- Volume 21, Number 11/12 (2016), 1049-1084.

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

Weiwei Ao, Juncheng Wei, and Wei Yao

#### Abstract

We study uniqueness of 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. The existence of sign-changing radial solutions with exactly $k$ nodes is also known. However, the uniqueness of such solutions is open. In this paper, we show that such sign-changing radial solution is 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.

#### Article information

**Source**

Adv. Differential Equations, Volume 21, Number 11/12 (2016), 1049-1084.

**Dates**

First available in Project Euclid: 13 October 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1476369296

**Mathematical Reviews number (MathSciNet)**

MR3556760

**Zentralblatt MATH identifier**

1375.35025

**Subjects**

Primary: 35B25: Singular perturbations 35J61: Semilinear elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

#### Citation

Ao, Weiwei; Wei, Juncheng; Yao, Wei. Uniqueness and nondegeneracy of sign-changing radial solutions to an almost critical elliptic problem. Adv. Differential Equations 21 (2016), no. 11/12, 1049--1084. https://projecteuclid.org/euclid.ade/1476369296