A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains

摘要

A priori bounds for positive, very weak solutions of semilinear elliptic boundary value problems -Delta u = f (x, u) on a bounded domain Omega subset of R-n with u = 0 on delta Omega are studied, where the nonlinearity 0 <= f (x, u) grows at most like s(p). If Omega is a Lipschitz domain we exhibit two exponents p(*) and p*, which depend on the boundary behavior of the Green function and on the smallest interior opening angle of delta Omega. We prove that for 1 < p < p(*) all positive very weak solutions are a priori bounded in L-infinity. For p > p* we construct a nonlinearity f (x, s) = a(x)s(p) together with a positive very weak solution which does not belong to L-infinity. Finally we exhibit a class of domains for which p* = p*. For such domains we have found a true critical exponent for very weak solutions. In the case of smooth domains p(*) = p* = n+1-1 is an exponent which is well known from classical work of Brezis, Turner [H. Brezis, R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601-614] and from recent work of Quittner, Souplet [P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal. 174 (2004) 49-81]. (c) 2006 Elsevier Inc. All rights reserved.