Boundary trace of positive solutions of nonlinear elliptic inequalities
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 3, p. 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 $-\Delta u+g\left(x,u\right)\ge 0$ 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 $g\left(x,u\right)\approx exp\left(-{\rho }_{\partial \Omega }^{-1}\left(x\right)\right){u}^{q}$, we exhibit a new full boundary blow-up phenomenon.

Classification:  35K60
@article{ASNSP_2004_5_3_3_481_0,
author = {Marcus, Moshe and V\'eron, Laurent},
title = {Boundary trace of positive solutions of nonlinear elliptic inequalities},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 3},
number = {3},
year = {2004},
pages = {481-533},
zbl = {1121.35057},
mrnumber = {2099247},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2004_5_3_3_481_0}
}

Marcus, Moshe; Véron, Laurent. Boundary trace of positive solutions of nonlinear elliptic inequalities. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 3, pp. 481-533. http://www.numdam.org/item/ASNSP_2004_5_3_3_481_0/

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