Semilinear elliptic Schrödinger equations involving singular potentials and source terms

GKIKAS, Konstantinos T and Phuoc-Tai NGUYEN. Semilinear elliptic Schrödinger equations involving singular potentials and source terms. Nonlinear Analysis. Elsevier, 2024, vol. 238, January, p. 1-44. ISSN 0362-546X. Available from: https://dx.doi.org/10.1016/j.na.2023.113403.

Basic information

• Original name: Semilinear elliptic Schrödinger equations involving singular potentials and source terms
• Authors: GKIKAS, Konstantinos T and Phuoc-Tai NGUYEN (704 Viet Nam, belonging to the institution).
• Edition: Nonlinear Analysis, Elsevier, 2024, 0362-546X.

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: 1.400 in 2022
• Organization unit: Faculty of Science
• Doi: http://dx.doi.org/10.1016/j.na.2023.113403
• UT WoS: 001106766700001
• Keywords in English: Hardy potentials; Critical exponents; Source terms; Capacities; Measure data
• Tags: rivok
• Tags: International impact, Reviewed

Abstract

Let $Ω\subset \mathbb{R}^N$ ($N>2$) be a $C^2$ bounded domain and $Σ\subset Ω$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_μ= Δ+ μd_Σ^{-2}$ in $Ω\setminus Σ$, where $d_Σ(x) = \mathrm{dist}(x,Σ)$ and $μ$ is a parameter. We study the boundary value problem (P) $-L_μu = g(u) + τ$ in $Ω\setminus Σ$ with condition $u=ν$ on $\partial Ω\cup Σ$, where $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing, continuous function and $τ$ and $ν$ are positive measures. The interplay between the inverse-square potential $d_Σ^{-2}$, the nature of the source term $g(u)$ and the measure data $τ,ν$ yields substantial difficulties in the research of the problem. We perform a deep analysis based on delicate estimate on the Green kernel and Martin kernel and fine topologies induced by appropriate capacities to establish various necessary and sufficient conditions for the existence of a solution in different cases.

Links

• GA22-17403S, research and development project: Name: Nelineární Schrödingerovy rovnice a systémy se singulárním potenciálem (Acronym: NSESSP)

Investor: Czech Science Foundation