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

Abstract

A priori bounds for positive, very weak solutions of semilinear elliptic boundary value problems − Δ u = f ( x , u ) on a bounded domain Ω ⊂ R n with u = 0 on ∂ Ω are studied, where the nonlinearity 0 ⩽ f ( x , s ) grows at most like s p . 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 ) s p 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].