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Positive solutions of semilinear elliptic problems with a Hardy potential

Published online by Cambridge University Press:  15 February 2019

Catherine Bandle
Affiliation:
Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051Basel, Switzerland ([email protected])
Maria Assunta Pozio
Affiliation:
Dipartimento di Matematica, Sapienza Università di Roma, P.le A. Moro 5, I-00185Roma, Italy

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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

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