Semilinear fractional elliptic equations with source term and boundary measure data

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F22%3A00128818" target="_blank" >RIV/00216224:14310/22:00128818 - isvavai.cz</a>

Výsledek na webu

<a href="http://yokohamapublishers.jp/online2/oppafa/vol7/p863.html" target="_blank" >http://yokohamapublishers.jp/online2/oppafa/vol7/p863.html</a>

DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Semilinear fractional elliptic equations with source term and boundary measure data

Popis výsledku v původním jazyce

A notion of s-boundary trace recently introduced by Nguyen and Véron (Adv. Nonlinear Stud. 18, 237-267, 2018) is an efficient tool to study boundary value problems with measure data for fractional elliptic equations with an absorption nonlinearity. In this paper, we investigate a fractional equation with a source term $(-Delta)^s u=f(u)$ in $Omega$ with a prescribed s-boundary trace $rho nu$, where $Omega$ is a $C^2$ bounded domain of $mathbb{R}^N$ ($N&gt;2s$), $s in (frac{1}{2},1)$, $fin C^{beta}_{loc}(mathbb{R})$, for some $beta in(0,1)$, $nu$ is a positive Radon measure on $partial Omega$ with total mass 1 and $rho$ is a positive parameter. We provide an existence result for the above equation and discuss regularity property of solutions. When $f(u)=u^p$, we prove that there exists a critical exponent $p_s:=frac{N+s}{N-s}$ in the following sense. If $pgeq p_s$, the problem does not admit any positive solution with $nu$ being a Dirac mass. If $pin(1,p_s)$ there exits a threshold value $rho^*&gt;0$ such that for $rhoin (0, rho^*]$, the problem admits a positive solution and for $rho&gt;rho^*$, no positive solution exists. We also show that, for $rho&gt;0$ small enough, the problem admits at least two positive solutions.

Název v anglickém jazyce

Semilinear fractional elliptic equations with source term and boundary measure data

Popis výsledku anglicky

A notion of s-boundary trace recently introduced by Nguyen and Véron (Adv. Nonlinear Stud. 18, 237-267, 2018) is an efficient tool to study boundary value problems with measure data for fractional elliptic equations with an absorption nonlinearity. In this paper, we investigate a fractional equation with a source term $(-Delta)^s u=f(u)$ in $Omega$ with a prescribed s-boundary trace $rho nu$, where $Omega$ is a $C^2$ bounded domain of $mathbb{R}^N$ ($N&gt;2s$), $s in (frac{1}{2},1)$, $fin C^{beta}_{loc}(mathbb{R})$, for some $beta in(0,1)$, $nu$ is a positive Radon measure on $partial Omega$ with total mass 1 and $rho$ is a positive parameter. We provide an existence result for the above equation and discuss regularity property of solutions. When $f(u)=u^p$, we prove that there exists a critical exponent $p_s:=frac{N+s}{N-s}$ in the following sense. If $pgeq p_s$, the problem does not admit any positive solution with $nu$ being a Dirac mass. If $pin(1,p_s)$ there exits a threshold value $rho^*&gt;0$ such that for $rhoin (0, rho^*]$, the problem admits a positive solution and for $rho&gt;rho^*$, no positive solution exists. We also show that, for $rho&gt;0$ small enough, the problem admits at least two positive solutions.

Druh

J_ost - Ostatní články v recenzovaných periodicích

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

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Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Pure and Applied Functional Analysis

ISSN

2189-3756

e-ISSN

2189-3764

Svazek periodika

7

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

JP - Japonsko

Počet stran výsledku

23

Strana od-do

863-885

Kód UT WoS článku

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EID výsledku v databázi Scopus

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