In this article, we study the symmetry of positive solutions to a class of singular semilinear elliptic equations whose prototype is
where , , , 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
où , , , , 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).
Accepted:
Published online:
Rakesh Arora 1; Jacques Giacomoni 1; Divya Goel 2; Konijeti Sreenadh 2
@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/
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