Partial differential equations
Symmetry and classification of solutions to an integral equation of the Choquard type
Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 878-888.

We study the integral equation

u(x)=Rnup(y)|xy|nαRnuq(z)|yz|nβdzdy,xRn,
where 0<α,β<n and p+q=n+α+2βnα. We prove that all positive L2nnα(Rn) solutions to the equation are radially symmetric and monotone decreasing about some point, and we classify all such solutions when p+1=q=n+βnα. As a consequence, we derive similar results for positive Hα2(Rn) solutions to the higher-fractional-order Choquard-type equation
(Δ)α2u=1Rn,α(1|x|nβuq)upin Rn.

Nous étudions l'équation intégrale

u(x)=Rnup(y)|xy|nαRnuq(z)|yz|nβdzdy,xRn,
0<α,β<n et p+q=n+α+2βnα. Nous démontrons que toute solution positive L2nnα(Rn) de l'équation est à symétrie radiale et monotone décroissante autour d'un point. Nous classifions toutes les solutions telles que p+1=q=n+βnα. Nous en déduisons des résultats similaires pour les solutions positives Hα2(Rn) de l'équation de type Choquard d'ordre fractionnaire supérieur
(Δ)α2u=1Rn,α(1|x|nβuq)upin Rn.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2019.11.005
Le, Phuong 1, 2

1 Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
2 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
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Le, Phuong. Symmetry and classification of solutions to an integral equation of the Choquard type. Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 878-888. doi : 10.1016/j.crma.2019.11.005. http://www.numdam.org/articles/10.1016/j.crma.2019.11.005/

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