Fading absorption in non-linear elliptic equations

Marcus, Moshe; Shishkov, Andrey

doi:10.1016/j.anihpc.2012.08.002

Marcus, Moshe ; Shishkov, Andrey

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 2, pp. 315-336.

corrigé par Erratum to “Fading absorption in non-linear elliptic equations” [Ann. I. H. Poincaré – AN 30 (2) (2013) 315–336]

Résumé

We study the equation $- Δ u + h (x) {| u |}^{q - 1} u = 0$ , $q > 1$ , in $ℝ_{+}^{N} = ℝ^{N - 1} \times ℝ_{+}$ where $h \in C (\bar{ℝ_{+}^{N}})$ , $h ⩾ 0$ . Let $(x_{1}, \dots, x_{N})$ be a coordinate system such that $ℝ_{+}^{N} = [x_{N} > 0]$ and denote a point $x \in ℝ^{N}$ by $(x^{'}, x_{N})$ . Assume that $h (x^{'}, x_{N}) > 0$ when $x^{'} \neq 0$ but $h (x^{'}, x_{N}) \to 0$ as $| x^{'} | \to 0$ . For this class of equations we obtain sharp necessary and sufficient conditions in order that singularities on the boundary do not propagate in the interior.

MR Zbl

DOI : 10.1016/j.anihpc.2012.08.002

@article{AIHPC_2013__30_2_315_0,
     author = {Marcus, Moshe and Shishkov, Andrey},
     title = {Fading absorption in non-linear elliptic equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {315--336},
     publisher = {Elsevier},
     volume = {30},
     number = {2},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.08.002},
     mrnumber = {3035979},
     zbl = {1295.35208},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2012.08.002/}
}

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Marcus, Moshe; Shishkov, Andrey. Fading absorption in non-linear elliptic equations. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 2, pp. 315-336. doi : 10.1016/j.anihpc.2012.08.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2012.08.002/

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