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arXiv · 1810.11186

Uniqueness and nondegeneracy of solutions for a critical nonlocal equation

Lele Du·Minbo Yang
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Abstract

The aim of this paper is to classify the positive solutions of the nonlocal critical equation: $$ -\Delta u=\left(I_{\mu}\ast u^{2^{\ast}_{\mu}}\right)u^{{2}^{\ast}_{\mu}-1},~~x\in\mathbb{R}^{N}, $$ where $0<\mu 0$ 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 and prove the nondegeneracy of the unique solutions for the equation when $\mu$ close to $N$.

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Lele Du, Minbo Yang. 2018-10-26. Uniqueness and nondegeneracy of solutions for a critical nonlocal equation. https://arxiv.org/abs/1810.11186

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