On the Boundary Behaviour, Including Second Order Effects, of Solutions to Singular Elliptic Problems

For . . 1 we consider the solution u = u(x) of the Dirichlet boundary value problem .u + u .. = 0 in ., u = 0 on ... For . = 1 we find the estimate u(x)=p(.(x))[1+A(x)(log1.(x))..], where p(r).r2log(1/r)......... near r = 0, .(x) denotes the distance from x to .., 0 < . < 1/2, and A(x) is a bounded function. For 1 < . < 3 we find u(x)=(.+12(..1).........(x))2.+1[1+A(x)(.(x))2..1.+1]. For . = 3 we prove that u(x)=(2.(x))12[1+A(x).(x)log1.(x)].