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

the Editorial Board

Phuong Le ^1,²

¹Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
²Faculty 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 (Original language)
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} .

Article information
Cite this article

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

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

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     title = {Symmetry and classification of solutions to an integral equation of the {Choquard} type},
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     pages = {878--888},
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     publisher = {Elsevier},
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     language = {en},
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