Boundary estimates for solutions of weighted semilinear elliptic equations

Let b(x) be a positive function in a bounded smooth domain Ω ⊂ RN, and let f(t) be a positive non decreasing function on (0,∞) such that limt→∞ f(t) = ∞. We investigate boundary blow-up solutions of the equation ∆u = b(x)f(u). Under appropriate conditions on b(x) as x approaches ∂Ω and on f(t) as t goes to infinity, we find a second order approximation of the solution u(x) as x goes to ∂Ω. We also investigate positive solutions of the equation ∆u+(δ(x))2lu−q = 0 in Ω with u = 0 on ∂Ω, where l ≥ 0, q > 3 + 2l and δ(x) denotes the distance from x to ∂Ω. We find a second order approximation of the solution u(x) as x goes to ∂Ω.