Symmetry of solutions to singular fractional elliptic equations and applications

Rakesh Arora; Jacques Giacomoni; Divya Goel; Konijeti Sreenadh

doi:10.5802/crmath.58

Analyse mathématique, Équations aux dérivées partielles

Symmetry of solutions to singular fractional elliptic equations and applications
[Symétrie radiale des solutions d’équations elliptiques fractionnaires singulières et quelques applications]

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

Comptes Rendus. Mathématique, Volume 358 (2020) no. 2, pp. 237-243.

Résumé
Abstract

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

\begin{matrix} (P) \{\begin{matrix} {(- Δ)}^{s} u = \frac{1}{u^{δ}} + f (u), u > 0 & in Ω; \\ u = 0 & in ℝ^{n} ∖ Ω, \end{matrix} \end{matrix}

où $0 < s < 1$ , $Ω = B_{r} (0) \subset ℝ^{n}$ , $n \geq 2 s$ , $δ > 0$ , $f : ℝ^{+} \to ℝ^{+}$ 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).

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

\begin{matrix} (P) \{\begin{matrix} {(- Δ)}^{s} u = \frac{1}{u^{δ}} + f (u), u > 0 & in Ω; \\ u = 0 & in ℝ^{n} ∖ Ω, \end{matrix} \end{matrix}

where $0 < s < 1$ , $n \geq 2 s$ , $Ω = B_{r} (0) \subset ℝ^{n}, δ > 0$ , $f (u)$ 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.

Reçu le : 2019-09-06
Accepté le : 2020-04-24
Publié le : 2020-06-15

DOI : 10.5802/crmath.58

Classification : 35B40, 35B45, 35J75, 35B06

Affiliations des auteurs :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2020__358_2_237_0,
     author = {Rakesh Arora and Jacques Giacomoni and Divya Goel and Konijeti Sreenadh},
     title = {Symmetry of solutions to singular fractional elliptic equations and applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {237--243},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {2},
     year = {2020},
     doi = {10.5802/crmath.58},
     language = {en},
}

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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/

Bibliographie
Cité par

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[2] Rakesh Arora; Jacques Giacomoni; Divya Goel; Konijeti Sreenadh Positive solutions of $1$ -D half Laplacian equation with singular and critical exponential nonlinearity (to appear in Asymptotic Anal.) | DOI

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[8] Jacques Giacomoni; Pawan K. Mishra; Konijeti Sreenadh Critical growth problems for $\frac{1}{2}$ -Laplacian in $ℝ$ , Differ. Equ. Appl., Volume 8 (2016) no. 3, pp. 295-317 | MR | Zbl

[9] Jacques Giacomoni; Tuhina Mukherjee; Konijeti Sreenadh Positive solutions of fractional elliptic equation with critical and singular nonlinearty, Adv. Nonlinear Anal., Volume 6 (2017) no. 3, pp. 327-354 | DOI | Zbl

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