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Positive solutions of semilinear elliptic problems with a Hardy potential

Part of: Elliptic equations and systems Partial differential equations Boundary value problems

Published online by Cambridge University Press:  15 February 2019

Catherine Bandle  and
Maria Assunta Pozio
Catherine Bandle
Affiliation:
Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051Basel, Switzerland (catherine.bandle@unibas.ch)
Maria Assunta Pozio
Affiliation:
Dipartimento di Matematica, Sapienza Università di Roma, P.le A. Moro 5, I-00185Roma, Italy
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Abstract

Let Ω ⊂ ℝN be a bounded domain and δ(x) be the distance of a point x ∈ Ω to the boundary. We study the positive solutions of the problem Δu + (μ/(δ(x)2))u = up in Ω, where p > 0, p ≠ 1 and μ ∈ ℝ, μ ≠ 0 is smaller than the Hardy constant. The interplay between the singular potential and the nonlinearity leads to interesting structures of the solution sets. In this paper, we first give the complete picture of the radial solutions in balls. In particular, we establish for p > 1 the existence of a unique large solution behaving like δ−(2/(p−1)) at the boundary. In general domains, we extend the results of Bandle and Pozio and show that there exists a unique singular solutions u such that $u/\delta ^{\beta _-}\to c$ on the boundary for an arbitrary positive function $c \in C^{2+\gamma }(\partial \Omega ) \, (\gamma \in (0,1)), c \ges 0$. Here β− is the smaller root of β(β − 1) + μ = 0.

Keywords

Elliptic problemsHardy potentialpower nonlinearitiesdead core and blowup solutionssingular boundary data

MSC classification

Primary: 35J75: Singular elliptic equations
Secondary: 35B09: Positive solutions 35B51: Comparison principles 34B16: Singular nonlinear boundary value problems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 1791 - 1804
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Copyright
Copyright © Royal Society of Edinburgh 2019

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