L∞ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity

We present new L∞ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in combining elliptic regularity with Gagliardo-Nirenberg or Caf...

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Author: Pardo San Gil, Rosa María
Format: article
Publication Date:2022
Country:España
Institution:Universidad Complutense de Madrid (UCM)
Repository:Docta Complutense
Language:English
OAI Identifier:oai:docta.ucm.es:20.500.14352/72705
Online Access:https://hdl.handle.net/20.500.14352/72705
Access Level:Open access
Keyword:517.95
A priori estimates
Subcritical non-linearities
L∞ a priori bounds
Changing sign weights
Singular elliptic equations
Ecuaciones diferenciales
1202.07 Ecuaciones en Diferencias
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spelling L∞ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearityPardo San Gil, Rosa María517.95A priori estimatesSubcritical non-linearitiesL∞ a priori boundsChanging sign weightsSingular elliptic equationsEcuaciones diferenciales1202.07 Ecuaciones en DiferenciasWe present new L∞ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in combining elliptic regularity with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a semilinear boundary value problem −Δu=f(x,u), in Ω, with Dirichlet boundary conditions, where Ω⊂RN, with N>2, is a bounded smooth domain, and f is a subcritical Carathéodory non-linearity. We provide L∞ a priori estimates for weak solutions, in terms of their L2∗-norm, where 2∗=2N/N−2 is the critical Sobolev exponent. By a subcritical non-linearity we mean, for instance, |f(x,s)|≤|x|−μ˜f(s), where μ∈(0,2), and ˜f(s)/|s|2∗μ−1→0 as |s|→∞, here 2∗μ:=2(N−μ)/N−2 is the critical Sobolev-Hardy exponent. Our non-linearities includes non-power non-linearities. In particular we prove that when f(x,s)=|x−μ |s|2∗μ−2s/[log(e+|s|)]β, with μ∈[1,2), then, for any ε>0 there exists a constant Cε>0 such that for any solution u∈H10(Ω), the following holds [log(e+∥u∥∞)]β≤Cε(1+∥u∥2∗)(2∗μ−2)(1+ε).Universidad Complutense de Madrid20222022-01-0120222022-01-01journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/72705reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Atribución-NoComercial-SinDerivadas 3.0 Españahttps://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/727052026-06-02T12:44:21Z
dc.title.none.fl_str_mv L∞ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity
title L∞ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity
spellingShingle L∞ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity
Pardo San Gil, Rosa María
517.95
A priori estimates
Subcritical non-linearities
L∞ a priori bounds
Changing sign weights
Singular elliptic equations
Ecuaciones diferenciales
1202.07 Ecuaciones en Diferencias
title_short L∞ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity
title_full L∞ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity
title_fullStr L∞ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity
title_full_unstemmed L∞ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity
title_sort L∞ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity
dc.creator.none.fl_str_mv Pardo San Gil, Rosa María
author Pardo San Gil, Rosa María
author_facet Pardo San Gil, Rosa María
author_role author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 517.95
A priori estimates
Subcritical non-linearities
L∞ a priori bounds
Changing sign weights
Singular elliptic equations
Ecuaciones diferenciales
1202.07 Ecuaciones en Diferencias
topic 517.95
A priori estimates
Subcritical non-linearities
L∞ a priori bounds
Changing sign weights
Singular elliptic equations
Ecuaciones diferenciales
1202.07 Ecuaciones en Diferencias
description We present new L∞ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in combining elliptic regularity with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a semilinear boundary value problem −Δu=f(x,u), in Ω, with Dirichlet boundary conditions, where Ω⊂RN, with N>2, is a bounded smooth domain, and f is a subcritical Carathéodory non-linearity. We provide L∞ a priori estimates for weak solutions, in terms of their L2∗-norm, where 2∗=2N/N−2 is the critical Sobolev exponent. By a subcritical non-linearity we mean, for instance, |f(x,s)|≤|x|−μ˜f(s), where μ∈(0,2), and ˜f(s)/|s|2∗μ−1→0 as |s|→∞, here 2∗μ:=2(N−μ)/N−2 is the critical Sobolev-Hardy exponent. Our non-linearities includes non-power non-linearities. In particular we prove that when f(x,s)=|x−μ |s|2∗μ−2s/[log(e+|s|)]β, with μ∈[1,2), then, for any ε>0 there exists a constant Cε>0 such that for any solution u∈H10(Ω), the following holds [log(e+∥u∥∞)]β≤Cε(1+∥u∥2∗)(2∗μ−2)(1+ε).
publishDate 2022
dc.date.none.fl_str_mv 2022
2022-01-01
2022
2022-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14352/72705
url https://hdl.handle.net/20.500.14352/72705
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Atribución-NoComercial-SinDerivadas 3.0 España
https://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Atribución-NoComercial-SinDerivadas 3.0 España
https://creativecommons.org/licenses/by-nc-nd/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Docta Complutense
instname:Universidad Complutense de Madrid (UCM)
instname_str Universidad Complutense de Madrid (UCM)
reponame_str Docta Complutense
collection Docta Complutense
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300719

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