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.