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

Le, Phuong

doi:10.1016/j.crma.2019.11.005

Partial differential equations

Symmetry and classification of solutions to an integral equation of the Choquard type
[Symétrie et classification des solutions d'une équation intégrale de type Choquard]

Présenté par : the Editorial Board

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

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

Résumé
Abstract

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

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} .

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} .

Reçu le : 2018-10-30
Accepté le : 2019-11-05
Publié le : 2019-11-01

DOI : 10.1016/j.crma.2019.11.005

Affiliations des auteurs :

Le, Phuong ^{1, 2}

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     author = {Le, Phuong},
     title = {Symmetry and classification of solutions to an integral equation of the {Choquard} type},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {878--888},
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Le, Phuong. Symmetry and classification of solutions to an integral equation of the Choquard type. Comptes Rendus. Mathématique, Tome 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/

Bibliographie
Cité par

[1] Caffarelli, L.A.; Gidas, B.; Spruck, J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., Volume 42 (1989) no. 3, pp. 271-297

[2] Chen, W.; Li, C. Classification of solutions of some nonlinear elliptic equations, Duke Math. J., Volume 63 (1991) no. 3, pp. 615-622

[3] Chen, W.; Li, C. Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci. Ser. B (Engl. Ed.), Volume 29 (2009) no. 4, pp. 949-960

[4] Chen, W.; Li, C.; Li, Y. A direct method of moving planes for the fractional Laplacian, Adv. Math., Volume 308 (2017), pp. 404-437

[5] Chen, W.; Li, C.; Ou, B. Classification of solutions for an integral equation, Commun. Pure Appl. Math., Volume 59 (2006) no. 3, pp. 330-343

[6] Cotsiolis, A.; Tavoularis, N.K. Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., Volume 295 (2004) no. 1, pp. 225-236

[7] Dai, W.; Fang, Y.; Qin, G. Classification of positive solutions to fractional order Hartree equations via a direct method of moving planes, J. Differ. Equ., Volume 265 (2018) no. 5, pp. 2044-2063

[8] Dai, W.; Huang, J.; Qin, Y.; Wang, B.; Fang, Y. Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., Volume 39 (2019) no. 3, pp. 1389-1403

[9] Gidas, B.; Ni, W.M.; Nirenberg, L. Symmetry and related properties via the maximum principle, Commun. Math. Phys., Volume 68 (1979) no. 3, pp. 209-243

[10] Ginibre, J.; Velo, G. Long range scattering and modified wave operators for some Hartree type equations. II, Ann. Henri Poincaré, Volume 1 (2000) no. 4, pp. 753-800

[11] Le, P. Liouville theorem and classification of positive solutions for a fractional Choquard type equation, Nonlinear Anal., Volume 185 (2019), pp. 123-141

[12] Lei, Y. On the regularity of positive solutions of a class of Choquard type equations, Math. Z., Volume 273 (2013) no. 3–4, pp. 883-905

[13] Lei, Y. Liouville theorems and classification results for a nonlocal Schrödinger equation, Discrete Contin. Dyn. Syst., Volume 38 (2018) no. 11, pp. 5351-5377

[14] Li, C. Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., Volume 123 (1996) no. 2, pp. 221-231

[15] Li, D.; Miao, C.; Zhang, X. The focusing energy-critical Hartree equation, J. Differ. Equ., Volume 246 (2009) no. 3, pp. 1139-1163

[16] Li, Y.; Zhu, M. Uniqueness theorems through the method of moving spheres, Duke Math. J., Volume 80 (1995) no. 2, pp. 383-417

[17] Li, Y.Y. Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., Volume 6 (2004) no. 2, pp. 153-180

[18] Lieb, E.H. Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., Volume 57 (1976/1977) no. 2, pp. 93-105

[19] Lieb, E.H. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), Volume 118 (1983) no. 2, pp. 349-374

[20] Lieb, E.H.; Simon, B. The Hartree-Fock theory for Coulomb systems, Commun. Math. Phys., Volume 53 (1977) no. 3, pp. 185-194

[21] Lin, C.-S. A classification of solutions of a conformally invariant fourth order equation in $R^{n}$ , Comment. Math. Helv., Volume 73 (1998) no. 2, pp. 206-231

[22] Liu, S. Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., Volume 71 (2009) no. 5–6, pp. 1796-1806

[23] Ma, L.; Zhao, L. Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., Volume 195 (2010) no. 2, pp. 455-467

[24] Ma, P.; Zhang, J. Symmetry and nonexistence of positive solutions for fractional Choquard equations, Apr 2017 (arXiv e-prints) | arXiv

[25] Moroz, V.; Van Schaftingen, J. A guide to the Choquard equation, J. Fixed Point Theory Appl., Volume 19 (2017) no. 1, pp. 773-813

[26] Nakanishi, K. Energy scattering for Hartree equations, Math. Res. Lett., Volume 6 (1999) no. 1, pp. 107-118

[27] Serrin, J. A symmetry problem in potential theory, Arch. Ration. Mech. Anal., Volume 43 (1971), pp. 304-318

[28] Stein, E.M. Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, N.J., 1970

[29] Wang, Y.; Tian, L. Classification of weak solutions of stationary Hartree-type equations, Int. J. Nonlinear Sci., Volume 20 (2015) no. 2, pp. 123-128

[30] Xu, D.; Lei, Y. Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Appl. Math. Lett., Volume 43 (2015), pp. 85-89

[31] Zhang, Y.; Hao, J. A remark on an integral equation via the method of moving spheres, J. Math. Anal. Appl., Volume 344 (2008) no. 2, pp. 682-686

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