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November/December 2016 Uniqueness and nondegeneracy of sign-changing radial solutions to an almost critical elliptic problem
Weiwei Ao, Juncheng Wei, Wei Yao
Adv. Differential Equations 21(11/12): 1049-1084 (November/December 2016).

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.

Citation

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Weiwei Ao. Juncheng Wei. Wei Yao. "Uniqueness and nondegeneracy of sign-changing radial solutions to an almost critical elliptic problem." Adv. Differential Equations 21 (11/12) 1049 - 1084, November/December 2016.

Information

Published: November/December 2016
First available in Project Euclid: 13 October 2016

zbMATH: 1375.35025
MathSciNet: MR3556760

Subjects:
Primary: 35B25 , 35J61 , 35Q55

Rights: Copyright © 2016 Khayyam Publishing, Inc.

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Vol.21 • No. 11/12 • November/December 2016
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