2007 Conference on Variational and Topological Methods: Theory, Applications,
Numerical Simulations, and Open Problems. 
Electron. J. Diff. Eqns., Conference 18 (2010), pp. 23-31.

Title: Oddness of least energy nodal solutions on radial domains

Authors: Christopher Grumiau  (Univ. de Mons, Belgium)
         Christophe Troestler (Univ. de Mons, Belgium)
 
Abstract: 
  In this article, we consider the Lane-Emden problem
 $$\displaylines{
 \Delta u(x) + |{u(x)}\mathclose|^{p-2}u(x)=0, \quad
 \hbox{for } x\in\Omega,\cr
 u(x)=0, \quad \hbox{for } x\in\partial\Omega,
 }$$
  where $2 < p < 2^{*}$ and $\Omega$ is a ball or an annulus in
  $\mathbb{R}^{N}$, $N\geq 2$.  We show that, for p
  close to 2, least  energy nodal solutions are odd with
  respect to an hyperplane --
  which is their nodal surface.  The proof ingredients are a
  constrained implicit function theorem and the fact that the second
  eigenvalue is simple up to rotations.

Published July 10, 2010.
Math Subject Classifications: 35J20, 35A30.
Key Words:  Variational method; least energy nodal solution; symmetry;
  oddness; (nodal) Nehari manifold; Bessel functions; Laplace-Beltrami
  operator on the sphere; implicit function theorem.