Second-order boundary estimates for solutions to singular elliptic equations in borderline cases

Let Ω ⊂ RN be a bounded smooth domain. We investigate the effect of the mean curvature of the boundary ∂Ω on the behaviour of the solution to the homogeneous Dirichlet boundary value problem for the equation ∆u + f(u) = 0. Under appropriate growth conditions on f(t) as t approaches zero, we find asymptotic expansions up to the second order of the solution in terms of the distance from x to the boundary ∂Ω.