Abstract

We develop a new method for proving the existence of a boundary trace, in the class of Borel measures, of nonnegative solutions of $-\Delta u+g\ge 0$ in a smooth domain $\Omega $ under very general assumptions on $g$. This new definition which extends the previous notions of boundary trace is based upon a sweeping technique by solutions of Dirichlet problems with measure boundary data. We also prove a boundary pointwise blow-up estimate of any solution of such inequalities in terms of the Poisson kernel. If the nonlinearity is very degenerate near the boundary, for example if $g\approx \exp )u^q$, we exhibit a new full boundary blow-up phenomenon