On the existence of a maximal weak solution for a semilinear elliptic equation

Abstract

Consider the semilinear problem { -~u = f(x, u) u=O inn, on an, (0.1)where n is a bounded domain in IRn and f : n X IR -+ IR.It is well known, see e.g.[16], that for f E C 1 (!'2 x IR) and hence solutions in C2+ 19 (!1), there is a solution in between a suband a supersolution.(The supersolution has to lie above the subsolution) There the superand subsolutions are assumed to be in C 2 (!1).Similar results for sub-and supersolutions in W 2 •P(fl) are shown in [5, 6].A first place where a weaker supersolution is used is [13].Deuel and Hess established existence of a solution between weaker suband supersolutions in [10].Amann showed in [3,4] for the classical case (u E C 2 (!1) n C 19 (!'2)) in fact the existence of a minimal and a maximal solution between a sub-and a supersolution in C2+ 19 (!'2).The classical proofs can be extended to functions f which are Lipschitz.In this note we will show that the result is still true even if f is not Lipschitz.In section 1 we will use super (sub) solutions in C(i1).In section 2 we will use super (sub) solutions in W 1 • 2 (!1) and allow general bounded domains.Neither definition of super (sub) solution is included in the other even for regular domains, though a C0 (!'2)-solution is necessarily a W~' 2 (!1)-solution.Thus neither of our two main results is included in the other.