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

Abstract

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

where is a smooth bounded domain in and . We consider solutions satisfying

and we are interested in the shape and the asymptotic behavior as .

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 can be characterized as the difference of two Greenʼs functions and the nodal line intersects the boundary of , for large .

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