Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient

We establish that for n >= 3 and p > 1, the elliptic equation Delta u + K(x)u(p) = 0 in R-n possesses a continuum of positive entire solutions with logarithmic decay at 00, provided that a locally Holder continuous function K >= 0 in R-n \ {0}, satisfies K(x) = O(vertical bar x vertical bar sigma) at x = 0 for some sigma > -2, and vertical bar x vertical bar(2) K (x) = c + O ([log vertical bar x vertical bar](-theta)) near infinity for some constants c > 0 and theta > 1. The continuum contains at least countably many solutions among which any two do not intersect. This is an affirmative answer to an open question raised in [S. Bae, T.K. Chang, On a class of semilinear elliptic equations in R-n, J. Differential Equations 185 (2002) 225-250]. The crucial observation is that in the radial case of K(r) = K(vertical bar x vertical bar I), two fundamental weights, (log r)(p-1)(p) and r(n-2)(log r)(-)(p)(p-1) appear in analyzing the asymptotic behavior of solutions. (C) 2007 Elsevier Inc. All rights reserved.