Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains: the subcritical case
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 4, p. 913-984

We study the generalized boundary value problem for nonnegative solutions of -Δu+g(u)=0 in a bounded Lipschitz domain Ω, when g is continuous and nondecreasing. Using the harmonic measure of Ω, we define a trace in the class of outer regular Borel measures. We amphasize the case where g(u)=|u| q-1 u, q>1. When Ω is (locally) a cone with vertex y, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that Ω possesses a tangent cone at every boundary point and q is subcritical, we prove an existence and uniqueness result for positive solutions with arbitrary boundary trace.

Published online : 2018-06-21
Classification:  35K60,  31A20,  31C15,  44A25,  46E35
@article{ASNSP_2011_5_10_4_913_0,
     author = {Marcus, Moshe and Veron, Laurent},
     title = {Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains: the subcritical case},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {4},
     year = {2011},
     pages = {913-984},
     zbl = {1243.35054},
     mrnumber = {2932897},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2011_5_10_4_913_0}
}
Marcus, Moshe; Veron, Laurent. Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains: the subcritical case. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 4, pp. 913-984. http://www.numdam.org/item/ASNSP_2011_5_10_4_913_0/

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