Large solutions of elliptic semilinear equations in the borderline case. An exhaustive and intrinsic point of view

We revisit the problem Δu=f(u)Δu=f(u) in Ω, u(x)→∞u(x)→∞ as x→∂Ωx→∂Ω, where Ω⊂RNΩ⊂RN, N>1N>1, is a bounded smooth domain and f   is an increasing and continuous function in R+R+ with f(0+)=0f(0+)=0 for which the Keller–Osserman condition holds. We study uniqueness of solutions, extending known results about the boundary blow-up behavior of solutions. Furthermore, we obtain explicit representations for the second order terms in the explosive boundary expansion of solutions under intrinsic and direct assumptions. Our study is exhaustive including both ordinary and borderline cases providing new and sharp results.