Uniqueness and nondegeneracy of solutions for a critical nonlocal equation

Lele Du; Minbo Yang

doi:10.3934/dcds.2019219

Article Contents

2019, Volume 39, Issue 10: 5847-5866. Doi: 10.3934/dcds.2019219

This issue Previous Article Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems Next Article Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity

Uniqueness and nondegeneracy of solutions for a critical nonlocal equation

Lele Du and
Minbo Yang^,

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

^* Corresponding author: Minbo Yang, he was partially supported by NSFC (11571317) and ZJNSF(LD19A010001)
^* Corresponding author: Minbo Yang, he was partially supported by NSFC (11571317) and ZJNSF(LD19A010001)

Received: September 30, 2018

Revised: January 31, 2019

Published: July 2019

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Abstract

The aim of this paper is to classify the positive solutions of the nonlocal critical equation:

$ - \Delta u = \left( {{I_\mu }*{u^{2_\mu ^*}}} \right){u^{2_\mu ^* - 1}},x \in {{\mathbb{R}}^N}$

where $ 0<\mu<N $, if $ N = 3\ \hbox{or} \ 4 $ and $ 0<\mu\leq4 $ if $ N\geq5 $, $ I_{\mu} $ is the Riesz potential defined by

${I_\mu }(x) = \frac{{\Gamma \left( {\frac{\mu }{2}} \right)}}{{\Gamma \left( {\frac{{N - \mu }}{2}} \right){\pi ^{\frac{N}{2}}}{2^{N - \mu }}|x{|^\mu }}}$

with $ \Gamma(s) = \int^{+\infty}_{0}x^{s-1}e^{-x}dx $, $ s>0 $ and $ 2^{\ast}_{\mu} = \frac{2N-\mu}{N-2} $ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We apply the moving plane method in integral forms to prove the symmetry and uniqueness of the positive solutions. Moreover, we also prove the nondegeneracy of the unique solutions for the equation when $ \mu $ close to $ N $.

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Mathematics Subject Classification: 35J15, 35B06, 45G15.

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