Qualitative properties of solutions to elliptic singular problems

We investigate the singular boundary value problem Delta u + u(-gamma) = 0 in D, u = 0 on partial derivative D, where gamma > 0. for gamma > 1, we fin d the estimate \u(x) -b(0)delta(2/(gamma+1)) (x)\ < beta delta((gamma-1)/(gamma+1)) (x), where b(0) depends on gamma only, delta(1) denotes the distance from x to p artial derivative D and beta is a suitable constant. For gamma > 0, we prov e that the function u((1+gamma)/2) is concave whenever D is convex. A simil ar result is well known for the equation Delta u + u(P) = 0, with 0 less th an or equal to p less than or equal to 1. For p = 0,p = 1 and gamma greater than or equal to 1 we prove convexity sharpness results.