Advances in Differential Equations

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

Weiwei Ao, Juncheng Wei, and Wei Yao

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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

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.


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