Comptes Rendus
Mathematical Analysis, Partial Differential Equations
Symmetry of solutions to singular fractional elliptic equations and applications
Comptes Rendus. Mathématique, Volume 358 (2020) no. 2, pp. 237-243.

In this article, we study the symmetry of positive solutions to a class of singular semilinear elliptic equations whose prototype is

(P)(-Δ) s u=1 u δ +f(u),u>0inΩ;u=0in n Ω,

where 0<s<1, n2s, Ω=B r (0) n ,δ>0, f(u) is a locally Lipschitz function. We prove that classical solutions are radial and radially decreasing (see Theorem 1). The proof uses the moving plane method adapted to the non local setting. We then give two applications of this main result: Theorem 2 establishes the uniform apriori bound for classical solutions in case of polynomial growth nonlinearities whereas Theorem 3 ensures in case of exponential growth nonlinearities the convergence of large solutions with unbounded energy to a singular solution.

Dans cet article, nous étudions la symétrie et la monotonie des solutions positives d’une équation elliptique semi-linéaire singulière dont le modèle type est

(P)(-Δ) s u=1 u δ +f(u),u>0inΩ;u=0in n Ω,

0<s<1, Ω=B r (0) n , n2s, δ>0, f: + + est localement Lipschitz. Nous démontrons que les solutions classiques de ce problème type sont à symétrie radiale et radialement décroissantes (Théorème 1). Pour cela, nous mettons en oeuvre la méthode du “moving plane”. Nous utilisons ensuite ce résultat général de symétrie pour étudier le comportement global de solutions d’équations elliptiques singulières non locales  : existence d’estimations a priori uniformes (Théorème 2), convergence de solutions à énergie non bornée vers une solution singulière (Théorème 3).

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.58
Classification: 35B40, 35B45, 35J75, 35B06

Rakesh Arora 1; Jacques Giacomoni 1; Divya Goel 2; Konijeti Sreenadh 2

1 LMAP (UMR E2S-UPPA CNRS 5142) Bat. IPRA, Avenue de l’Université 64013 Pau, France
2 Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khaz, New Delhi-110016, India
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Symmetry of solutions to singular fractional elliptic equations and applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {237--243},
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Rakesh Arora; Jacques Giacomoni; Divya Goel; Konijeti Sreenadh. Symmetry of solutions to singular fractional elliptic equations and applications. Comptes Rendus. Mathématique, Volume 358 (2020) no. 2, pp. 237-243. doi : 10.5802/crmath.58. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.58/

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