Abstract
In this paper, we give an exact asymptotic of the unique solution to the following singular boundary value problem -Delta u = a(x)g(u), x is an element of Omega, u > 0, in Omega, u(vertical bar partial derivative Omega) = 0. Here Omega is a C-2-bounded domain in R-n (n >= 2), 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 inf(d(x)-> 0) a(x)/h(d(x)) <= lim sup(d(x)-> 0) a(x)/h(d(x)) = a(2) < infinity,
where h(t) = c(t-lambda) exp(integral(eta)(t) z(s)/s ds), lambda <= 2, c > 0 and z continuous on [0, eta] for some eta > 0 such that z(0) = 0. Two applications of this result are also given. The first concerns the boundary behavior of the unique solution of -Delta u + beta/u vertical bar del u vertical bar(2) = a(x)g(u) that vanishes on the boundary and the second concerns the behavior of u in the case where the open set Omega is an annular and the behaviors of the function a on the interior boundary and the exterior boundary may be different. (C) 2013 Elsevier Ltd. All rights reserved.