The authors study properties of positive solutions of
in a (possibly) nonsmooth -dimensional domain , , subject to the condition
Here and , , are continuous in with and . Positive solutions of (1) satisfying (2) are called large solutions. A central point of this paper is the following localization principle: let be a (not necessarily bounded) domain having the graph property and suppose is a positive solution of (1) satisfying locally uniformly as , where is relatively open. If is a large solution, then locally uniformly as . Closely related to this is a uniqueness result for large solutions in bounded domains having the graph property. For bounded Lipschitz domains the authors prove the existence of positive constants such that the (unique) large solution of (1) satisfies for all . This is also a consequence of the localization principle and an existence theorem, obtained for large solutions in bounded domains satisfying the exterior cone condition.
If the domain is not Lipschitz, the rate of blow-up at the boundary may be lower. This is proved for domains having a re-entrant cusp in the case . Finally, the authors discuss the dependence of large solutions on the function and the domain .