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.

Abstract (VO)
Résumé

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.

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).

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},
}

TY  - JOUR
AU  - Rakesh Arora
AU  - Jacques Giacomoni
AU  - Divya Goel
AU  - Konijeti Sreenadh
TI  - Symmetry of solutions to singular fractional elliptic equations and applications
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 237
EP  - 243
VL  - 358
IS  - 2
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.58
LA  - en
ID  - CRMATH_2020__358_2_237_0
ER  -

%0 Journal Article
%A Rakesh Arora
%A Jacques Giacomoni
%A Divya Goel
%A Konijeti Sreenadh
%T Symmetry of solutions to singular fractional elliptic equations and applications
%J Comptes Rendus. Mathématique
%D 2020
%P 237-243
%V 358
%N 2
%I Académie des sciences, Paris
%R 10.5802/crmath.58
%G en
%F CRMATH_2020__358_2_237_0

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

[1] Adimurthi; Jacques Giacomoni; Sanjiban Santra Positive solutions to a fractional equation with singular nonlinearity, J. Differ. Equations, Volume 265 (2018) no. 4, pp. 1191-1226 | DOI | MR

[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

[3] Begoña Barrios; Ida De Bonis; María Medina; Ireneo Peral Semilinear problems for the fractional laplacian with a singular nonlinearity, Open Math., Volume 13 (2015), pp. 390-407 | MR | Zbl

[4] Haïm Brézis; Pierre-Louis Lions A note on isolated singularities for linear elliptic equations, Mathematical analysis and applications. Part A. (Advances in Mathematics. Supplementary Studies), Volume 7a, Academic Press Inc., 1981, pp. 263-266 | Zbl

[5] Huyuan Chen; Alexander Quaas Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results, J. Lond. Math. Soc., Volume 97 (2018) no. 2, pp. 196-221 | DOI | MR | Zbl

[6] Wenxiong Chen; Congming Li; Yan Li A direct method of moving planes for the fractional Laplacian, Adv. Math., Volume 308 (2017), pp. 404-437 | DOI | MR | Zbl

[7] Djairo G. de Figueiredo; Pierre-Louis Lions; Roger D. Nussbaum Apriori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., Volume 61 (1982), pp. 263-275 | Zbl

[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

[10] Sven Jarohs; Tobias Weth Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl., Volume 195 (2016) no. 1, pp. 273-291 | DOI | MR | Zbl

[11] Tuhina Mukherjee; Konijeti Sreenadh Fractional elliptic equations with critical growth and singular nonlinearities, Electron. J. Differ. Equ., Volume 2016 (2016), 54, 23 pages | MR | Zbl

[12] Xavier Ros-Oton; Joaquim Serra The extremal solution for the fractional Laplacian, Calc. Var. Partial Differ. Equ., Volume 50 (2014) no. 3-4, pp. 723-750 | DOI | MR | Zbl

[13] Futoshi Takahashi Critical and subcritical fractional Trudinger-–Moser-type inequalities on $ℝ$ , Adv. Nonlinear Anal., Volume 8 (2019), pp. 868-884 | DOI | MR | Zbl

Masoud Bayrami; Mahmoud Hesaaraki Entropy Solution to a Singular Fractional Problem, Fractional Calculus - From Theory to Applications (2025) | DOI:10.5772/intechopen.1007358

Cité par 1 document. Sources : Crossref

Commentaires - Politique