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

Oddness of least energy nodal solutions on radial domains

Christopher Grumiau, Christophe Troestler

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.

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Christopher Grumiau
Institut de Mathématique, Université de Mons
Place du Parc, 20, B-7000 Mons, Belgium
email: Christopher.Grumiau@umons.ac.be
Christophe Troestler
Institut de Mathématique, Université de Mons
Place du Parc, 20, B-7000 Mons, Belgium
email: Christophe.Troestler@umons.ac.be

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