Abstract
In this paper, we give an exact behavior on the boundary and at infinity of the unique solution to the following singular boundary value problem -Lambda u = a(x)g(u), x is an element of Omega, u > 0, in Omega, u(vertical bar partial derivative Omega) = 0 and lim(vertical bar x vertical bar ->infinity) u(x) = 0. Here Omega is an exterior domain in R-n (n >= 3) with compact C-2-boundary, g is an element of C-1((0,infinity), (0,infinity)) is nonincreasing on (0,infinity) with lim(t -> 0) g'(t) integral(t)(0)ds/g(s) = -C-g <= 0 and the function a is in C-loc(alpha)(Omega), 0 < alpha < 1 satisfying
0 < a(1) = lim(d(x)-> 0)inf a(x)/h(d(x)) <= lim(d(x)-> 0)sup a(x)/h(d(x)) = a(2) < infinity,
and
0 < b(1) = lim(vertical bar x vertical bar ->infinity)inf a(x/k(vertical bar x vertical bar)) <= lim(vertical bar x vertical bar ->infinity)sup a(x)/k(vertical bar x vertical bar) = b(2) < infinity,
where d(x) is the Euclidean distance from x is an element of Omega to the boundary partial derivative Omega, h(t) = c(1)t(-lambda) exp (integral(n)(t) z(s)/s ds), lambda <= 2, c(1) > 0 and z is continuous on [0, eta] for some eta > 0 such that z(0) = 0 and k(t) = c(2)t(-mu) exp (integral(t)(1) y(s)/s ds), mu >= 2, c(2) > 0 and y is continuous on [1,infinity) such that lim(t -> 0)y(t) = 0. (C) 2014 Elsevier Ltd. All rights reserved.