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

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, 2023, 58 pp. ISSN 0308-2105. Available from: https://dx.doi.org/10.1017/prm.2023.122.

Basic information

• Original name: Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities
• Authors: GKIKAS, Konstantinos T and Phuoc-Tai NGUYEN.
• Edition: Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Cambridge University Press, 2023, 0308-2105.

Other information

• Original 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
• WWW: URL
• Impact factor: Impact factor: 1.300 in 2022
• Organization unit: Faculty of Science
• Doi: http://dx.doi.org/10.1017/prm.2023.122
• UT WoS: 001133169600001
• Keywords in English: Hardy potentials; boundary singularities; capacities; critical exponents; removable singularity; Keller-Osserman estimates; 35J60; 35J75; 35J10; 35J66
• Tags: International impact, Reviewed

Abstract

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.