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 $-\Delta u+g\left(u\right)=0$ in a bounded Lipschitz domain $\Omega$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Omega$, we define a trace in the class of outer regular Borel measures. We amphasize the case where $g\left(u\right)={|u|}^{q-1}u$, $q>1$. When $\Omega$ is (locally) a cone with vertex $y$, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that $\Omega$ 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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