Boundary trace of positive solutions of nonlinear elliptic inequalities

Marcus, Moshe; Véron, Laurent

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 3, pp. 481-533.

We develop a new method for proving the existence of a boundary trace, in the class of Borel measures, of nonnegative solutions of $-\Delta u+g(x,u)\ge 0$ 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 $g(x,u)\approx exp(-{\rho }_{\partial \Omega }^{-1}\left(x\right))u^q$, we exhibit a new full boundary blow-up phenomenon.

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