Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F25%3A00144214" target="_blank" >RIV/00216224:14310/25:00144214 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/semilinear-elliptic-equations-involving-power-nonlinearities-and-hardy-potentials-with-boundary-singularities/1FA7BF7DD7DB916CFDAC66C169C4" target="_blank" >https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/semilinear-elliptic-equations-involving-power-nonlinearities-and-hardy-potentials-with-boundary-singularities/1FA7BF7DD7DB916CFDAC66C169C4</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1017/prm.2023.122" target="_blank" >10.1017/prm.2023.122</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities

Popis výsledku v původním jazyce

Let Ω ⊂ RN (N ≽ 3) be a C2 bounded domain and Σ ⊂ ∂Ω be a C2 compact submanifold without boundary, of dimension k, 0 ≼ k ≼ N - 1. We assume that Σ = {0} if k = 0 and Σ = ∂Ω if k = N - 1. Let dΣ(x) = dist (x, Σ) and Lμ = Δ + μ d-Σ2, where μ ∈ R. We study boundary value problems (P±) -Lμu ± |u|p-1u = 0 in Ω and trμ,Σ(u) = ν on ∂Ω, where p &gt; 1, ν is a given measure on ∂Ω and trμ,Σ(u) denotes the boundary trace of u associated to Lμ. Different critical exponents for the existence of a solution to (P±) appear according to concentration of ν. The solvability for problem (P+) was proved in [3, 29] in subcritical ranges for p, namely for p smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of -Lμ, we provide conditions on ν expressed in terms of capacities for the existence of a (unique) solution to (P+) in supercritical ranges for p, i.e. for p equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to (P-) under a smallness assumption on ν.

Název v anglickém jazyce

Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities

Popis výsledku anglicky

Let Ω ⊂ RN (N ≽ 3) be a C2 bounded domain and Σ ⊂ ∂Ω be a C2 compact submanifold without boundary, of dimension k, 0 ≼ k ≼ N - 1. We assume that Σ = {0} if k = 0 and Σ = ∂Ω if k = N - 1. Let dΣ(x) = dist (x, Σ) and Lμ = Δ + μ d-Σ2, where μ ∈ R. We study boundary value problems (P±) -Lμu ± |u|p-1u = 0 in Ω and trμ,Σ(u) = ν on ∂Ω, where p &gt; 1, ν is a given measure on ∂Ω and trμ,Σ(u) denotes the boundary trace of u associated to Lμ. Different critical exponents for the existence of a solution to (P±) appear according to concentration of ν. The solvability for problem (P+) was proved in [3, 29] in subcritical ranges for p, namely for p smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of -Lμ, we provide conditions on ν expressed in terms of capacities for the existence of a (unique) solution to (P+) in supercritical ranges for p, i.e. for p equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to (P-) under a smallness assumption on ν.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA22-17403S" target="_blank" >GA22-17403S: Nelineární Schrödingerovy rovnice a systémy se singulárním potenciálem</a><br>

Návaznosti

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

Rok uplatnění

2025

Kód důvěrnosti údajů

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

Název periodika

Proceedings of the Royal Society of Edinburgh Section A: Mathematics

ISSN

0308-2105

e-ISSN

1473-7124

Svazek periodika

155

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

58

Strana od-do

993-1050

Kód UT WoS článku

001133169600001

EID výsledku v databázi Scopus

2-s2.0-85180930351