Symmetry and classification of solutions to an integral equation of the Choquard type

Phuong Le

doi:10.1016/j.crma.2019.11.005

Partial differential equations

Symmetry and classification of solutions to an integral equation of the Choquard type

Presented by: the Editorial Board

Phuong Le ^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

Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 878-888.

Abstract
Résumé

We study the integral equation

u (x) = \int_{R^{n}} \frac{u^{p} (y)}{| x - y |^{n - α}} \int_{R^{n}} \frac{u^{q} (z)}{| y - z |^{n - β}} d z d y, x \in R^{n},

where

0 < α, β < n

and

p + q = \frac{n + α + 2 β}{n - α}

. We prove that all positive

L^{\frac{2 n}{n - α}} (R^{n})

solutions to the equation are radially symmetric and monotone decreasing about some point, and we classify all such solutions when

p + 1 = q = \frac{n + β}{n - α}

. As a consequence, we derive similar results for positive

H^{\frac{α}{2}} (R^{n})

solutions to the higher-fractional-order Choquard-type equation

{(- Δ)}^{\frac{α}{2}} u = \frac{1}{R_{n, α}} (\frac{1}{| x |^{n - β}} ⁎ u^{q}) u^{p} in R^{n} .

Nous étudions l'équation intégrale

u (x) = \int_{R^{n}} \frac{u^{p} (y)}{| x - y |^{n - α}} \int_{R^{n}} \frac{u^{q} (z)}{| y - z |^{n - β}} d z d y, x \in R^{n},

où

0 < α, β < n

et

p + q = \frac{n + α + 2 β}{n - α}

. Nous démontrons que toute solution positive

L^{\frac{2 n}{n - α}} (R^{n})

de l'équation est à symétrie radiale et monotone décroissante autour d'un point. Nous classifions toutes les solutions telles que

p + 1 = q = \frac{n + β}{n - α}

. Nous en déduisons des résultats similaires pour les solutions positives

H^{\frac{α}{2}} (R^{n})

de l'équation de type Choquard d'ordre fractionnaire supérieur

{(- Δ)}^{\frac{α}{2}} u = \frac{1}{R_{n, α}} (\frac{1}{| x |^{n - β}} ⁎ u^{q}) u^{p} in R^{n} .

Received: 2018-10-30
Accepted: 2019-11-05
Published online: 2019-11-01

DOI: 10.1016/j.crma.2019.11.005

Author's affiliations:

Phuong Le ^{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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     author = {Phuong Le},
     title = {Symmetry and classification of solutions to an integral equation of the {Choquard} type},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {878--888},
     publisher = {Elsevier},
     volume = {357},
     number = {11-12},
     year = {2019},
     doi = {10.1016/j.crma.2019.11.005},
     language = {en},
}

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Phuong Le. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.11.005/

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