Geodesic
Boundary trace of positive solutions of nonlinear elliptic inequalities
Marcus, Moshe1  ; Véron, Laurent2 
1 Department of Mathematics, Israel Institute of Technology Technion, Haifa 32000, Israel
2 Université François Rabelais Tours 37200, France
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 3, pp. 481-533

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We develop a new method for proving the existence of a boundary trace, in the class of Borel measures, of nonnegative solutions of -Δu+g(x,u)≥0 in a smooth domain Ω 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)≈exp(-ρ ∂Ω -1 (x))u q , we exhibit a new full boundary blow-up phenomenon.

Classification : 35K60

Marcus, Moshe 1 ; Véron, Laurent 2

1 Department of Mathematics, Israel Institute of Technology Technion, Haifa 32000, Israel
2 Université François Rabelais Tours 37200, France 
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     title = {Boundary trace of positive solutions of nonlinear elliptic inequalities},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {3},
     year = {2004},
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Marcus, Moshe; Véron, Laurent. Boundary trace of positive solutions of nonlinear elliptic inequalities. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 3, pp. 481-533. http://geodesic.mathdoc.fr/item/ASNSP_2004_5_3_3_481_0/