A Necessary and Sufficient Condition for Existence of Large Solutions to Semilinear Elliptic Equations

We consider the semilinear equation Dusp x.f u. on a domain V:Rn, nG3, where f is a nonnegative, nondecreasing continuous function which vanishes at the origin, and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in V such that p is positive on its boundary. For V bounded, we show that a nonnegative solution u satisfying u x.ª` as xª­V exists if and only if the function c s.ʹH0s f t. dt satisfies ` ..y1r2 n. H cs ds- 1 `. For V unbounded including VsR , we show that a similar result holds where u x.ª` as < x<ª` within V and u x.ª` as xª­V if p x. decays to zero rapidly as < x<ª`.