Regularity of stable solutions to semilinear elliptic equations on Riemannian models
We consider the reaction-diffusion problem -Δgu = f(u) in ℬR with zero Dirichlet boundary condition, posed in a geodesic ball ℬR with radius R of a Riemannian model (M,g). This class of Riemannian manifolds includes the classical space forms, i.e., the Euclidean, elliptic, and hyperbolic spaces. For...
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doaj-c6d2004b5803489299cccc0905425c212021-09-06T19:39:54ZengDe GruyterAdvances in Nonlinear Analysis2191-94962191-950X2015-11-014429530910.1515/anona-2015-0047Regularity of stable solutions to semilinear elliptic equations on Riemannian modelsCastorina Daniele0Sanchón Manel1Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, ItalyCentre de Recerca Matemàtica and Universitat Autònoma de Barcelona, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), SpainWe consider the reaction-diffusion problem -Δgu = f(u) in ℬR with zero Dirichlet boundary condition, posed in a geodesic ball ℬR with radius R of a Riemannian model (M,g). This class of Riemannian manifolds includes the classical space forms, i.e., the Euclidean, elliptic, and hyperbolic spaces. For the class of semistable solutions we prove radial symmetry and monotonicity. Furthermore, we establish L∞, Lp, and W1,p estimates which are optimal and do not depend on the nonlinearity f. As an application, under standard assumptions on the nonlinearity λf(u), we prove that the corresponding extremal solution u* is bounded whenever n ≤ 9. To establish the optimality of our regularity results we find the extremal solution for some exponential and power nonlinearities using an improved weighted Hardy inequality.https://doi.org/10.1515/anona-2015-0047semistable and extremal solutionselliptic and hyperbolic spacesa priori estimatesimproved hardy inequality 35k57 35b65 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Castorina Daniele Sanchón Manel |
spellingShingle |
Castorina Daniele Sanchón Manel Regularity of stable solutions to semilinear elliptic equations on Riemannian models Advances in Nonlinear Analysis semistable and extremal solutions elliptic and hyperbolic spaces a priori estimates improved hardy inequality 35k57 35b65 |
author_facet |
Castorina Daniele Sanchón Manel |
author_sort |
Castorina Daniele |
title |
Regularity of stable solutions to semilinear elliptic equations on Riemannian models |
title_short |
Regularity of stable solutions to semilinear elliptic equations on Riemannian models |
title_full |
Regularity of stable solutions to semilinear elliptic equations on Riemannian models |
title_fullStr |
Regularity of stable solutions to semilinear elliptic equations on Riemannian models |
title_full_unstemmed |
Regularity of stable solutions to semilinear elliptic equations on Riemannian models |
title_sort |
regularity of stable solutions to semilinear elliptic equations on riemannian models |
publisher |
De Gruyter |
series |
Advances in Nonlinear Analysis |
issn |
2191-9496 2191-950X |
publishDate |
2015-11-01 |
description |
We consider the reaction-diffusion problem -Δgu = f(u) in ℬR with zero
Dirichlet boundary condition, posed in a geodesic ball ℬR with radius
R of a Riemannian model (M,g). This class of Riemannian manifolds includes the
classical space forms, i.e., the Euclidean, elliptic, and hyperbolic spaces.
For the class of semistable solutions we prove radial symmetry and monotonicity.
Furthermore, we establish L∞, Lp, and W1,p estimates which are optimal and do not
depend on the nonlinearity f. As an application, under standard assumptions on the nonlinearity
λf(u), we prove that the corresponding extremal solution u* is bounded whenever
n ≤ 9. To establish the optimality of our regularity results we find the extremal solution
for some exponential and power nonlinearities using an improved weighted Hardy inequality. |
topic |
semistable and extremal solutions elliptic and hyperbolic spaces a priori estimates improved hardy inequality 35k57 35b65 |
url |
https://doi.org/10.1515/anona-2015-0047 |
work_keys_str_mv |
AT castorinadaniele regularityofstablesolutionstosemilinearellipticequationsonriemannianmodels AT sanchonmanel regularityofstablesolutionstosemilinearellipticequationsonriemannianmodels |
_version_ |
1717769831115128832 |