Blow-up rates of radially symmetric large solutions

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, f ∈ C [ 0 , ∞ ) is positive and non-decreasing, V ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) satisfies V ( 0 ) = 0 , V ′ ( u ) > 0 , V ″ ( u ) ⩾ 0 , for every u > 0 , and V ( u ) ∼ H u p − 1 as u ↑ ∞ , for some H > 0 and p > 1 , then, for each λ ⩾ 0 , − Δ u = λ u − 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 ) at ∂ Ω in terms of p, H and f (see Theorem 1.1).