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

J 2025

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

GKIKAS, Konstantinos T and Phuoc-Tai NGUYEN

Basic information

Original name

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

Authors

GKIKAS, Konstantinos T and Phuoc-Tai NGUYEN (704 Viet Nam, belonging to the institution)

Edition

Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Cambridge University Press, 2025, 0308-2105

Other information

Language

English

Type of outcome

Article in a journal

Field of Study

10101 Pure mathematics

Country of publisher

United Kingdom of Great Britain and Northern Ireland

Confidentiality degree

is not subject to a state or trade secret

References:

URL

Impact factor

Impact factor: 1.300 in 2023

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1017/prm.2023.122

UT WoS

001133169600001

EID Scopus

2-s2.0-85180930351

Keywords in English

Hardy potentials; boundary singularities; capacities; critical exponents; removable singularity; Keller-Osserman estimates; 35J60; 35J75; 35J10; 35J66

Abstract

In the original language

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 > 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.

Links

GA22-17403S, research and development project

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

Investor: Czech Science Foundation

Cite

GKIKAS, Konstantinos T and Phuoc-Tai NGUYEN. Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. Cambridge University Press, 2025, vol. 155, No 3, p. 993-1050. ISSN 0308-2105. Available from: https://dx.doi.org/10.1017/prm.2023.122.

@article{2386474,
   author = {Gkikas, Konstantinos T and Nguyen, PhuocandTai},
   article_number = {3},
   doi = {http://dx.doi.org/10.1017/prm.2023.122},
   keywords = {Hardy potentials; boundary singularities; capacities; critical exponents; removable singularity; Keller-Osserman estimates; 35J60; 35J75; 35J10; 35J66},
   language = {eng},
   issn = {0308-2105},
   journal = {Proceedings of the Royal Society of Edinburgh Section A: Mathematics},
   title = {Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities},
   url = {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},
   volume = {155},
   year = {2025}
}

TY  - JOUR
ID  - 2386474
AU  - Gkikas, Konstantinos T - Nguyen, Phuoc-Tai
PY  - 2025
TI  - Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities
JF  - Proceedings of the Royal Society of Edinburgh Section A: Mathematics
VL  - 155
IS  - 3
SP  - 993-1050
EP  - 993-1050
PB  - Cambridge University Press
SN  - 03082105
KW  - Hardy potentials
KW  - boundary singularities
KW  - capacities
KW  - critical exponents
KW  - removable singularity
KW  - Keller-Osserman estimates
KW  - 35J60
KW  - 35J75
KW  - 35J10
KW  - 35J66
UR  - 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
N2  - 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 > 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.
ER  -

GKIKAS, Konstantinos T and Phuoc-Tai NGUYEN. Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities. \textit{Proceedings of the Royal Society of Edinburgh Section A: Mathematics}. Cambridge University Press, 2025, vol.~155, No~3, p.~993-1050. ISSN~0308-2105. Available from: https://dx.doi.org/10.1017/prm.2023.122.

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