Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains: the subcritical case

Marcus, Moshe; Veron, Laurent

Marcus, Moshe ; Veron, Laurent

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 4, pp. 913-984.

Résumé

We study the generalized boundary value problem for nonnegative solutions of $- Δ u + g (u) = 0$ 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.

Publié le : 2018-06-21

MR 2932897 | Zbl 1243.35054 | 1 citation dans Numdam

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},
     pages = {913--984},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {4},
     year = {2011},
     zbl = {1243.35054},
     mrnumber = {2932897},
     language = {en},
     url = {archive.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, Série 5, Tome 10 (2011) no. 4, pp. 913-984. http://archive.numdam.org/item/ASNSP_2011_5_10_4_913_0/

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