Compactness of Green operators with applications to semilinear
nonlocal elliptic equations

J 2025

Compactness of Green operators with applications to semilinear nonlocal elliptic equations

HUYNH, Phuoc-Truong and Phuoc-Tai NGUYEN

Basic information

Original name

Compactness of Green operators with applications to semilinear nonlocal elliptic equations

Authors

HUYNH, Phuoc-Truong and Phuoc-Tai NGUYEN (704 Viet Nam, guarantor, belonging to the institution)

Edition

Journal of Differential Equations, Academic Press Inc. 2025, 0022-0396

Other information

Language

English

Type of outcome

Article in a journal

Field of Study

10101 Pure mathematics

Country of publisher

United States of America

Confidentiality degree

is not subject to a state or trade secret

References:

URL

Impact factor

Impact factor: 2.300 in 2024

Organization unit

Faculty of Science

DOI

https://doi.org/10.1016/j.jde.2024.11.019

UT WoS

001364125100001

EID Scopus

2-s2.0-85209567326

Keywords in English

Integro-differential operators; Compactness; Green function; Kato-type inequalities; Critical exponents; Weak-dual solutions

Abstract

In the original language

In this paper, we consider a class of integro-differential operators L posed on a C2 bounded domain Ω⊂RN with appropriate homogeneous Dirichlet conditions where each of which admits an inverse operator commonly known as the Green operator GΩ. Under mild conditions on L and its Green operator, we establish various sharp compactness of GΩ involving weighted Lebesgue spaces and weighted measure spaces. These results are then employed to prove the solvability for semilinear elliptic equation Lu+g(u)=μ in Ω with boundary condition u=0 on ∂Ω or exterior condition u=0 in RN∖Ω if applicable, where μ is a Radon measure on Ω and g:R→R is a nondecreasing continuous function satisfying a subcriticality integral condition. When g(t)=|t|p−1t with p>1, we provide a sharp sufficient condition expressed in terms of suitable Bessel capacities for the existence of a solution. The contribution of the paper consists of (i) developing novel unified techniques which allow to treat various types of fractional operators and (ii) obtaining sharp compactness and existence results in weighted spaces, which refine and extend several related results in the literature.

Links

GA22-17403S, research and development project

Name: Nelineární Schrödingerovy rovnice a systémy se singulárním potenciálem (Acronym: NSESSP)

Investor: Czech Science Foundation

Cite

HUYNH, Phuoc-Truong and Phuoc-Tai NGUYEN. Compactness of Green operators with applications to semilinear nonlocal elliptic equations. Journal of Differential Equations. Academic Press Inc., 2025, vol. 418, February, p. 97-141. ISSN 0022-0396. Available from: https://doi.org/10.1016/j.jde.2024.11.019.

@article{2465626,
   author = {Huynh, PhuocandTruong and Nguyen, PhuocandTai},
   article_number = {February},
   doi = {https://doi.org/10.1016/j.jde.2024.11.019},
   keywords = {Integro-differential operators; Compactness; Green function; Kato-type inequalities; Critical exponents; Weak-dual solutions},
   language = {eng},
   issn = {0022-0396},
   journal = {Journal of Differential Equations},
   title = {Compactness of Green operators with applications to semilinear nonlocal elliptic equations},
   url = {https://www.sciencedirect.com/science/article/pii/S0022039624007356},
   volume = {418},
   year = {2025}
}

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PY  - 2025
TI  - Compactness of Green operators with applications to semilinear nonlocal elliptic equations
JF  - Journal of Differential Equations
VL  - 418
IS  - February
SP  - 97-141
EP  - 97-141
PB  - Academic Press Inc.
SN  - 00220396
KW  - Integro-differential operators
KW  - Compactness
KW  - Green function
KW  - Kato-type inequalities
KW  - Critical exponents
KW  - Weak-dual solutions
UR  - https://www.sciencedirect.com/science/article/pii/S0022039624007356
N2  - In this paper, we consider a class of integro-differential operators L posed on a C2 bounded domain Ω⊂RN with appropriate homogeneous Dirichlet conditions where each of which admits an inverse operator commonly known as the Green operator GΩ. Under mild conditions on L and its Green operator, we establish various sharp compactness of GΩ involving weighted Lebesgue spaces and weighted measure spaces. These results are then employed to prove the solvability for semilinear elliptic equation Lu+g(u)=μ in Ω with boundary condition u=0 on ∂Ω or exterior condition u=0 in RN∖Ω if applicable, where μ is a Radon measure on Ω and g:R→R is a nondecreasing continuous function satisfying a subcriticality integral condition. When g(t)=|t|p−1t with p>1, we provide a sharp sufficient condition expressed in terms of suitable Bessel capacities for the existence of a solution. The contribution of the paper consists of (i) developing novel unified techniques which allow to treat various types of fractional operators and (ii) obtaining sharp compactness and existence results in weighted spaces, which refine and extend several related results in the literature.
ER  -

HUYNH, Phuoc-Truong and Phuoc-Tai NGUYEN. Compactness of Green operators with applications to semilinear nonlocal elliptic equations. \textit{Journal of Differential Equations}. Academic Press Inc., 2025, vol.~418, February, p.~97-141. ISSN~0022-0396. Available from: https://doi.org/10.1016/j.jde.2024.11.019.

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