Blow-up rates of radially symmetric large solutions

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, f∈C[0,∞)f∈C[0,∞) is positive and non-decreasing, V∈C[0,∞)∩C2(0,∞)V∈C[0,∞)∩C2(0,∞) satisfies V(0)=0V(0)=0, V′(u)>0V′(u)>0, V″(u)⩾0V″(u)⩾0, for every u>0u>0, and V(u)∼Hup−1V(u)∼Hup−1 as u↑∞u↑∞, for some H>0H>0 and p>1p>1, then, for each λ⩾0λ⩾0,−Δu=λu−f(dist(x,∂Ω))V(u)u−Δu=λu−f(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)L(x) at ∂Ω in terms of p, H and f (see Theorem 1.1).