Monotonicity and symmetry of solutions of $ p $-Laplace equations, $ 1 < p < 2 $, via the moving plane method

Damascelli, Lucio; Pacella, Filomena

Monotonicity and symmetry of solutions of

p

-Laplace equations,

1 < p < 2

, via the moving plane method

Damascelli, Lucio ; Pacella, Filomena

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 9 (1998), p. 95-100 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

We present some monotonicity and symmetry results for positive solutions of the equation $- div ({|D u|}^{p - 2} D u) = f (u)$ satisfying an homogeneous Dirichlet boundary condition in a bounded domain $Ω$ . We assume 1 < p < 2 and $f$ locally Lipschitz continuous and we do not require any hypothesis on the critical set of the solution. In particular we get that if $Ω$ is a ball then the solutions are radially symmetric and strictly radially decreasing.

Dimostriamo alcuni risultati di monotonia e simmetria per soluzioni positive dell’equazione $- div ({|D u|}^{p - 2} D u) = f (u)$ con condizioni di Dirichlet omogenee sul bordo in un dominio limitato $Ω$ . Supponiamo che 1 < p < 2 e che $f$ sia localmente Lipschitziana e non facciamo alcuna ipotesi sui punti critici della soluzione. In particolare otteniamo che se $Ω$ e` una palla le soluzioni sono radiali e radialmente strettamente decrescenti.

Publié le : 1998-06-01

MR 1677254 | Zbl 0923.35013

@article{RLIN_1998_9_9_2_95_0,
     author = {Lucio Damascelli and Filomena Pacella},
     title = {Monotonicity and symmetry of solutions of \( p \)-Laplace equations, \( 1 < p < 2 \), via the moving plane method},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {9},
     year = {1998},
     pages = {95-100},
     zbl = {0923.35013},
     mrnumber = {1677254},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1998_9_9_2_95_0}
}

Damascelli, Lucio; Pacella, Filomena. Monotonicity and symmetry of solutions of \( p \)-Laplace equations, \( 1 < p < 2 \), via the moving plane method. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 9 (1998) pp. 95-100. http://gdmltest.u-ga.fr/item/RLIN_1998_9_9_2_95_0/

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