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

A priori bounds for positive, very weak solutions of semilinear elliptic boundary value problems − u = f (x,u) on a bounded domain Ω ⊂ Rn with u = 0 on ∂Ω are studied, where the nonlinearity 0 f (x,s) grows at most like sp. If Ω 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 ∂Ω. We prove that for 1 < p < p∗ all positive very weak solutions are a priori bounded in L∞. For p > p∗ we construct a nonlinearity f (x,s) = a(x)sp together with a positive very weak solution which does not belong to L∞. 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 n−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]. © 2006 Elsevier Inc. All rights reserved.