Local uniqueness problem for a nonlinear elliptic equation

Miao Chen; Youyan Wan; Chang-Lin Xiang

doi:10.3934/cpaa.2020048

Article Contents

2020, Volume 19, Issue 2: 1037-1055. Doi: 10.3934/cpaa.2020048

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Local uniqueness problem for a nonlinear elliptic equation

1.
School of Information and Mathematics, Yangtze University, Jingzhou 434023, China
2.
Department of Mathematics, Jianghan University, Wuhan, Hubei, 430056, China

^* Corresponding author
^* Corresponding author

Received: March 2019

Revised: March 2019

Published: October 2019

Wan is supported by Scientific Research Fund of Hubei Provincial Education Department (B2013155). The corresponding author Xiang is financially supported by NSFC (No. 11701045) and the Yangtze Youth Fund (No. 2016cqn56).

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Abstract

In this paper, we consider the following nonlinear Schrödinger equation

$ \begin{eqnarray*} - \varepsilon^{2}\Delta u_{ \varepsilon}+u_{ \varepsilon} = K(x)u_{ \varepsilon}^{p-1} & & {\rm{in\;}}\mathbb{R}^{N}, \end{eqnarray*} $

where $ N\ge3 $ and $ 2<p<2N/(N-2) $. Under mild assumptions on the function $ K $ and using the local Pohozaev identity method developed by Deng, Lin and Yan [10], we show that multi-peak solutions to the above equation are unique for $ \varepsilon>0 $ sufficiently small.

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Mathematics Subject Classification: Primary: 35A02; Secondary: 35J20, 35J61.

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