Semilinear fractional elliptic equations with source term and
boundary measure data

J 2022

Semilinear fractional elliptic equations with source term and boundary measure data

BHAKTA, Mousomi a Phuoc Tai NGUYEN

Základní údaje

Originální název

Semilinear fractional elliptic equations with source term and boundary measure data

Autoři

BHAKTA, Mousomi a Phuoc Tai NGUYEN

Vydání

Pure and Applied Functional Analysis, Yokohama Publishers, 2022, 2189-3756

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Japonsko

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Označené pro přenos do RIV

Ano

Kód RIV

RIV/00216224:14310/22:00128818

Organizační jednotka

Přírodovědecká fakulta

EID Scopus

2-s2.0-85208447667

Klíčová slova anglicky

nonlocal; source terms; a priori estimate; singularity estimate; existence; boundary trace; boundary singularity; isolated singularity; fractional; minimal solution; mountain pass solution; gradient nonlinearity

Štítky

rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 30. 7. 2025 15:14, Mgr. Michal Petr

Anotace

V originále

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 $\mathbb{R}^N$ ($N>2s$), $s \in (\frac{1}{2},1)$, $f\in C^{\beta}_{loc}(\mathbb{R})$, 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 $p\geq p_s$, the problem does not admit any positive solution with $\nu$ being a Dirac mass. If $p\in(1,p_s)$ there exits a threshold value $\rho^*>0$ such that for $\rho\in (0, \rho^*]$, the problem admits a positive solution and for $\rho>\rho^*$, no positive solution exists. We also show that, for $\rho>0$ small enough, the problem admits at least two positive solutions.

Citovat

BHAKTA, Mousomi a Phuoc Tai NGUYEN. Semilinear fractional elliptic equations with source term and boundary measure data. Pure and Applied Functional Analysis. Yokohama Publishers, 2022, roč. 7, č. 3, s. 863-885. ISSN 2189-3756.

@article{2264917,
   author = {Bhakta, Mousomi and Nguyen, Phuoc Tai},
   article_number = {3},
   keywords = {nonlocal; source terms; a priori estimate; singularity estimate; existence; boundary trace; boundary singularity; isolated singularity; fractional; minimal solution; mountain pass solution; gradient nonlinearity},
   language = {eng},
   issn = {2189-3756},
   journal = {Pure and Applied Functional Analysis},
   title = {Semilinear fractional elliptic equations with source term and boundary measure data},
   url = {http://yokohamapublishers.jp/online2/oppafa/vol7/p863.html},
   volume = {7},
   year = {2022}
}

TY  - JOUR
ID  - 2264917
AU  - Bhakta, Mousomi - Nguyen, Phuoc Tai
PY  - 2022
TI  - Semilinear fractional elliptic equations with source term and boundary measure data
JF  - Pure and Applied Functional Analysis
VL  - 7
IS  - 3
SP  - 863-885
EP  - 863-885
PB  - Yokohama Publishers
SN  - 21893756
KW  - nonlocal
KW  - source terms
KW  - a priori estimate
KW  - singularity estimate
KW  - existence
KW  - boundary trace
KW  - boundary singularity
KW  - isolated singularity
KW  - fractional
KW  - minimal solution
KW  - mountain pass solution
KW  - gradient nonlinearity
UR  - http://yokohamapublishers.jp/online2/oppafa/vol7/p863.html
N2  - 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 $\mathbb{R}^N$ ($N>2s$), $s \in (\frac{1}{2},1)$, $f\in C^{\beta}_{loc}(\mathbb{R})$, 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 $p\geq p_s$, the problem does not admit any positive solution with $\nu$ being a Dirac mass. If $p\in(1,p_s)$ there exits a threshold value $\rho^*>0$ such that for $\rho\in (0, \rho^*]$, the problem admits a positive solution and for $\rho>\rho^*$, no positive solution exists. We also show that, for $\rho>0$ small enough, the problem admits at least two positive solutions.
ER  -

BHAKTA, Mousomi a Phuoc Tai NGUYEN. Semilinear fractional elliptic equations with source term and boundary measure data. \textit{Pure and Applied Functional Analysis}. Yokohama Publishers, 2022, roč.~7, č.~3, s.~863-885. ISSN~2189-3756.

Přehled o publikaci