Abstract
We consider the nonlinear elliptic equation Delta u + V(x)u + f(x, u(x)) = 0 on D \ {0}, where D is a bounded domain containing 0 in R-n, n >= 2, and V and f are Borel measurable functions. Under general conditions on the functions V and f, we prove the existence of positive singular solutions globally comparable to the Dirichlet Green's function of the Laplacian with pole at the origin. Our result applies to various types of semilinear equations, in particular to Delta u + W(x)u(p) = 0 for all real exponent p which was extensively studied for the range p > 1. Moreover for this equation with sign-unchanging function W our condition is the optimal one.