Abstract
We take up the existence and the asymptotic behavior of a classical solution to the following semilinear Dirichlet problem {,Delta u = a(x)g(u), x E'S2, u > 0 in Q, ups? = 0, where Q is a C-1.1-bounded domain in le, R-N > 2 and the function a belongs to c(loc)(gamma)(Omega), (0 < gamma < 1) such that there exist c(1), c(2) > 0 satisfying for each x E Omega, c1 delta(x)(-lambda 1)(z(1)(s) z(2)(s) ci3(x)-x1 exp (f. ds)) a(x) <= c23(x)-12 exp (f ds), 8 (x) S 8 (x) S where 77 > diam(S2), 8(x) = dist(x, as?), A <= A <= 2 and for i E {1, 2), z(i) is a continuous function on [0, eta] with z,(0) = 0. Our arguments are based on the s'ub-supersolution method with Karamata regular variation theory. (C) 2013 Elsevier Inc. All rights reserved.