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

Filomena Pacella; Massimo Grossi; Christopher Grumiau

doi:10.1016/j.anihpc.2012.06.005

JournalsaihpcVol. 30, No. 1pp. 121–140

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

Filomena Pacella
Dipartimento di Matematica, Universitaʼ di Roma “Sapienza”, P. le A. Moro 2, 00185 Roma, Italy
Massimo Grossi
Dipartimento di Matematica, Universitaʼ di Roma “Sapienza”, P. le A. Moro 2, 00185 Roma, Italy
Christopher Grumiau
Institut de Mathématique, Université de Mons, 20, Place du Parc, B-7000 Mons, Belgium

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Abstract

We study the nodal solutions of the Lane–Emden–Dirichlet problem

{- Δ u = ∣ u ∣^{p - 1} u, u = 0, in Ω, on \partial Ω,

where $Ω$ is a smooth bounded domain in $R^{2}$ and $p > 1$ . We consider solutions $u_{p}$ satisfying

p Ω \int ∣ \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 $p$ .

Cite this article

Filomena Pacella, Massimo Grossi, Christopher Grumiau, Lane–Emden problems: Asymptotic behavior of low energy nodal solutions. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 1, pp. 121–140

DOI 10.1016/J.ANIHPC.2012.06.005