A-priori bounds for semilinear elliptic equations in non-smooth domains

I will begin by giving a survey of some known methods for a-priori bounds of positive solutions of semilinear elliptic equations on smooth bounded domains with zero Dirichlet boundary conditions. I will then focus on a recent result of Souplet and Quittner, where they show that the old critical exponent pBT = (n+1)/(n−1) of Brezis and Turner is sharp in the sense that below pBT a-priori bounds hold whereas above pBT there exist unbounded very weak solutions. Finally I will explain how one can obtain sharp critical exponents on certain Lipschitz domains with corners, which generalize the BrezisTurner exponent. This work is in collaboration with P.J. McKenna(Univ. of Connecticut).