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 and Phuoc Tai NGUYEN

Basic information

Original name

Semilinear fractional elliptic equations with source term and boundary measure data

Authors

BHAKTA, Mousomi and Phuoc Tai NGUYEN (704 Viet Nam, guarantor, belonging to the institution)

Edition

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

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

Japan

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

RIV identification code

RIV/00216224:14310/22:00128818

Organization unit

Faculty of Science

Keywords in English

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

Abstract

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 and Phuoc Tai NGUYEN. Semilinear fractional elliptic equations with source term and boundary measure data. Pure and Applied Functional Analysis. Yokohama Publishers, 2022, vol. 7, No 3, p. 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 and Phuoc Tai NGUYEN. Semilinear fractional elliptic equations with source term and boundary measure data. \textit{Pure and Applied Functional Analysis}. Yokohama Publishers, 2022, vol.~7, No~3, p.~863-885. ISSN~2189-3756.

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