Abstract
In this article, we are concerned with the asymptotic behavior of the classical solution to the semilinear boundary-value problem
Delta u + a(x)u(sigma) = 0
in R-n, u > 0, lim(vertical bar x vertical bar -> infinity) u(x) = 0, where sigma < 1. The special feature is to consider the function a in C-loc(alpha)(R-n), 0 < alpha < 1, such that there exists c > 0 satisfying
1/c L(vertical bar x vertical bar + 1)/(1 + vertical bar x vertical bar)(lambda) <= a(x) <= c L(vertical bar x vertical bar + 1)/(1 + vertical bar x vertical bar)(lambda),
where L(t) := exp (integral(t)(1) z(s)/s ds), with z is an element of C([1, infinity)) such that lim(t -> infinity) z(t) = 0. The comparable asymptotic rate of a(x) determines the asymptotic behavior of the solution.