Existence and Multiplicity Results for Nonlocal Lane-Emden
Systems

J 2023

Existence and Multiplicity Results for Nonlocal Lane-Emden Systems

ARORA, Rakesh a Phuoc-Tai NGUYEN

Základní údaje

Originální název

Existence and Multiplicity Results for Nonlocal Lane-Emden Systems

Autoři

ARORA, Rakesh a Phuoc-Tai NGUYEN

Vydání

Acta Mathematica Vietnamica, Springer, 2023, 0251-4184

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Singapur

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Impakt faktor

Impact factor: 0.300

Označené pro přenos do RIV

Ano

Kód RIV

RIV/00216224:14310/23:00134083

Organizační jednotka

Přírodovědecká fakulta

Klíčová slova anglicky

Nonlocal elliptic systems; Weak-dual solutions; Measure data; Green function; Multiplicity; Palais-smale sequences

Štítky

14311010, rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 11. 5. 2023 15:27, Mgr. Marie Novosadová Šípková, DiS.

Anotace

V originále

In this work, we show the existence and multiplicity for the nonlocal Lane-Emden system of the form \begin{array}{@{}rcl@{}} \left\{ \begin{aligned} \mathbb L u &= v^{p} + \rho \nu \quad &&\text{in } {\varOmega}, \\ \mathbb L v &= u^{q} + \sigma \tau \quad &&\text{in } {\varOmega},\\ u&=v = 0 \quad &&\text{on } \partial {\varOmega} \text{ or in } {\varOmega}^{c} \text{ if applicable}, \end{aligned} \right. \end{array} where {\varOmega } \subset \mathbb {R}^{N} is a C2 bounded domain, \mathbb L is a nonlocal operator, ν,τ are Radon measures on Ω, p,q are positive exponents, and ρ,σ > 0 are positive parameters. Based on a fine analysis of the interaction between the Green kernel associated with \mathbb L, the source terms uq,vp and the measure data, we prove the existence of a positive minimal solution. Furthermore, by analyzing the geometry of Palais-Smale sequences in finite dimensional spaces given by the Galerkin type approximations and their appropriate uniform estimates, we establish the existence of a second positive solution, under a smallness condition on the positive parameters ρ,σ and superlinear growth conditions on source terms. The contribution of the paper lies on our unifying technique that is applicable to various types of local and nonlocal operators.

Návaznosti

GA22-17403S, projekt VaV

Název: Nelineární Schrödingerovy rovnice a systémy se singulárním potenciálem (Akronym: NSESSP)

Investor: Grantová agentura ČR, Nonlinear Schrödinger equations and systems with singular potentials

Citovat

ARORA, Rakesh a Phuoc-Tai NGUYEN. Existence and Multiplicity Results for Nonlocal Lane-Emden Systems. Acta Mathematica Vietnamica. Springer, 2023, roč. 48, č. 1, s. 3-28. ISSN 0251-4184. Dostupné z: https://doi.org/10.1007/s40306-022-00485-y.

@article{2264919,
   author = {Arora, Rakesh and Nguyen, PhuocandTai},
   article_number = {1},
   doi = {https://doi.org/10.1007/s40306-022-00485-y},
   keywords = {Nonlocal elliptic systems; Weak-dual solutions; Measure data; Green function; Multiplicity; Palais-smale sequences},
   language = {eng},
   issn = {0251-4184},
   journal = {Acta Mathematica Vietnamica},
   title = {Existence and Multiplicity Results for Nonlocal Lane-Emden Systems},
   url = {https://doi.org/10.1007/s40306-022-00485-y},
   volume = {48},
   year = {2023}
}

TY  - JOUR
ID  - 2264919
AU  - Arora, Rakesh - Nguyen, Phuoc-Tai
PY  - 2023
TI  - Existence and Multiplicity Results for Nonlocal Lane-Emden Systems
JF  - Acta Mathematica Vietnamica
VL  - 48
IS  - 1
SP  - 3-28
EP  - 3-28
PB  - Springer
SN  - 02514184
KW  - Nonlocal elliptic systems
KW  - Weak-dual solutions
KW  - Measure data
KW  - Green function
KW  - Multiplicity
KW  - Palais-smale sequences
UR  - https://doi.org/10.1007/s40306-022-00485-y
N2  - In this work, we show the existence and multiplicity for the nonlocal Lane-Emden system of the form \begin{array}{@{}rcl@{}} \left\{ \begin{aligned} \mathbb L u &= v^{p} + \rho \nu \quad &&\text{in } {\varOmega}, \\ \mathbb L v &= u^{q} + \sigma \tau \quad &&\text{in } {\varOmega},\\ u&=v = 0 \quad &&\text{on } \partial {\varOmega} \text{ or in } {\varOmega}^{c} \text{ if applicable}, \end{aligned} \right. \end{array} where {\varOmega } \subset \mathbb {R}^{N} is a C2 bounded domain, \mathbb L is a nonlocal operator, ν,τ are Radon measures on Ω, p,q are positive exponents, and ρ,σ > 0 are positive parameters. Based on a fine analysis of the interaction between the Green kernel associated with \mathbb L, the source terms uq,vp and the measure data, we prove the existence of a positive minimal solution. Furthermore, by analyzing the geometry of Palais-Smale sequences in finite dimensional spaces given by the Galerkin type approximations and their appropriate uniform estimates, we establish the existence of a second positive solution, under a smallness condition on the positive parameters ρ,σ and superlinear growth conditions on source terms. The contribution of the paper lies on our unifying technique that is applicable to various types of local and nonlocal operators.
ER  -

ARORA, Rakesh a Phuoc-Tai NGUYEN. Existence and Multiplicity Results for Nonlocal Lane-Emden Systems. \textit{Acta Mathematica Vietnamica}. Springer, 2023, roč.~48, č.~1, s.~3-28. ISSN~0251-4184. Dostupné z: https://doi.org/10.1007/s40306-022-00485-y.

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