Blow-up rates of radially symmetric large solutions

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Abstract

This paper adapts a technical device going back to [J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385–439] to ascertain the blow-up rate of the (unique) radially symmetric large solution given through the main theorem of [J. López-Gómez, Uniqueness of radially symmetric large solutions, Discrete Contin. Dyn. Syst., Supplement dedicated to the 6th AIMS Conference, Poitiers, France, 2007, pp. 677–686]. The requested underlying estimates are based upon the main theorem of [S. Cano-Casanova, J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations 244 (2008) 3180–3203]. Precisely, we show that if Ω is a ball, or an annulus, fC[0,) is positive and non-decreasing, VC[0,)C2(0,) satisfies V(0)=0, V(u)>0, V(u)0, for every u>0, and V(u)Hup1 as u, for some H>0 and p>1, then, for each λ0,Δu=λuf(dist(x,Ω))V(u)u possesses a unique positive large solution in Ω, L, which must be radially symmetric, by uniqueness, and we can estimate the exact blow-up rate of L(x) at ∂Ω in terms of p, H and f (see Theorem 1.1).

Keywords

Large solutions
Keller–Osserman condition
Existence and uniqueness
Blow-up rates

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This work has been supported by the National Plan of Global Change and Bio-diversity of the Spanish Ministry of Education and Science under Research Grant CGL/2006-00524/BOS.