Semilinear nonlocal elliptic equations with source term and measure data

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F23%3A00134082" target="_blank" >RIV/00216224:14310/23:00134082 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1007/s11854-022-0245-0" target="_blank" >https://doi.org/10.1007/s11854-022-0245-0</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s11854-022-0245-0" target="_blank" >10.1007/s11854-022-0245-0</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Semilinear nonlocal elliptic equations with source term and measure data

Popis výsledku v původním jazyce

Recently, several works have been undertaken in an attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In this paper, we study the Dirichlet problem for superlinear equation (E) Lu=uP+δμ in a bounded domain Ω with homogeneous boundary or exterior Dirichlet condition, where p &gt; 1 and λ &gt; 0. The operator L belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum μ is taken in the optimal weighted measure space. The interplay between the operator L , the source term up and the datum μ yields substantial difficulties and reveals the distinctive feature of the problem. We develop a unifying technique based on a fine analysis on the Green kernel, which enables us to construct a theory for semilinear equation (E) in measure frameworks. A main thrust of the paper is to provide a fairly complete description of positive solutions to the Dirichlet problem for (E). In particular, we show that there exist a critical exponent p* and a threshold value λ* such that the multiplicity holds for 1 &lt; p &lt; p* and 0 &lt;λ &lt; λ*, the uniqueness holds for 1 &lt; p &lt; p* and λ = λ*, and the nonexistence holds in other cases. Various types of nonlocal operators are discussed to exemplify the wide applicability of our theory.

Název v anglickém jazyce

Semilinear nonlocal elliptic equations with source term and measure data

Popis výsledku anglicky

Recently, several works have been undertaken in an attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In this paper, we study the Dirichlet problem for superlinear equation (E) Lu=uP+δμ in a bounded domain Ω with homogeneous boundary or exterior Dirichlet condition, where p &gt; 1 and λ &gt; 0. The operator L belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum μ is taken in the optimal weighted measure space. The interplay between the operator L , the source term up and the datum μ yields substantial difficulties and reveals the distinctive feature of the problem. We develop a unifying technique based on a fine analysis on the Green kernel, which enables us to construct a theory for semilinear equation (E) in measure frameworks. A main thrust of the paper is to provide a fairly complete description of positive solutions to the Dirichlet problem for (E). In particular, we show that there exist a critical exponent p* and a threshold value λ* such that the multiplicity holds for 1 &lt; p &lt; p* and 0 &lt;λ &lt; λ*, the uniqueness holds for 1 &lt; p &lt; p* and λ = λ*, and the nonexistence holds in other cases. Various types of nonlocal operators are discussed to exemplify the wide applicability of our theory.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ19-14413Y" target="_blank" >GJ19-14413Y: Lineární a nelineární eliptické rovnice se singulárními daty a související problémy</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2023

Kód důvěrnosti údajů

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

Název periodika

Journal d'Analyse Mathématique

ISSN

0021-7670

e-ISSN

1565-8538

Svazek periodika

149

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

IL - Stát Izrael

Počet stran výsledku

63

Strana od-do

49-111

Kód UT WoS článku

000904992900007

EID výsledku v databázi Scopus

2-s2.0-85144934310