Lane–Emden problems: Asymptotic behavior of low energy nodal solutions

Grossi, Massimo; Grumiau, Christopher; Pacella, Filomena

Grossi, Massimo ; Grumiau, Christopher ; Pacella, Filomena

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 121-140 / Harvested from Numdam

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Résumé

We study the nodal solutions of the Lane–Emden–Dirichlet problem ${\begin{matrix} - Δ u = {| u |}^{p - 1} u, & in Ω, \\ u = 0, & on \partial Ω, \end{matrix}$ where Ω is a smooth bounded domain in $ℝ^{2}$ and $p > 1$ . We consider solutions $u_{p}$ satisfying $p \int_{Ω} {| \nabla u_{p} |}^{2} \to 16 π e as p \to + \infty$ and we are interested in the shape and the asymptotic behavior as $p \to + \infty$ .First we prove that (⁎) holds for least energy nodal solutions. Then we obtain some estimates and the asymptotic profile of this kind of solutions. Finally, in some cases, we prove that $p u_{p}$ can be characterized as the difference of two Greenʼs functions and the nodal line intersects the boundary of Ω, for large p.

Publié le : 2013-01-01

MR 3011294 | Zbl 1266.35106

DOI : https://doi.org/10.1016/j.anihpc.2012.06.005
Classification: 35J91, 35B32

@article{AIHPC_2013__30_1_121_0,
     author = {Grossi, Massimo and Grumiau, Christopher and Pacella, Filomena},
     title = {Lane--Emden problems: Asymptotic behavior of low energy nodal solutions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {30},
     year = {2013},
     pages = {121-140},
     doi = {10.1016/j.anihpc.2012.06.005},
     mrnumber = {3011294},
     zbl = {1266.35106},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2013__30_1_121_0}
}

Grossi, Massimo; Grumiau, Christopher; Pacella, Filomena. Lane–Emden problems: Asymptotic behavior of low energy nodal solutions. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) pp. 121-140. doi : 10.1016/j.anihpc.2012.06.005. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_1_121_0/

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