This paper is concerned with the problems of existence, uniqueness and behavior near the boundary for the boundary blow-up problem
where the boundary condition means that
is not necessarily smooth and there is no a priori assumption on the behavior of near . Assume that is a bounded open subset of satisfying a uniform external sphere condition. The existence of a solution is proved if for some and if there exists a constant such that for all . If is only continuous on and satisfies for all , then is uniformly bounded on and the boundary blow-up problem has at most one solution.
The uniqueness is also proved in a less regular case, when is a bounded star-shaped domain in (and when is continuous and strictly positive on ). The proofs are based on dilatation arguments near the boundary, the comparison with solutions in balls and annuli and the maximum principle. Existence is given by sub- and super-solutions techniques and appropriate regularizations of the domain.