Marcelo
F. Furtado*, A Note on the Number of Nodal Solutions of an
Elliptic Equation with Symmetry*

Abstract:

We consider the semilinear problem $-\Delta u + \lambda u =|u|^{p-2}u$ in $\Omega$, $u=0$ on $\partial \Omega$ where $\Omega \subset \mathbb{R}^N$ is a bounded smooth domain and $2<p<2^*=2N/(N-2)$. We show that if $\Omega$ is invariant by a nontrivial orthogonal involution then, for $\lambda>0$ sufficiently large, the equivariant topology of $\Omega$ is related with the number of solutions which change sign exactly once.

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We consider the semilinear problem $-\Delta u + \lambda u =|u|^{p-2}u$ in $\Omega$, $u=0$ on $\partial \Omega$ where $\Omega \subset \mathbb{R}^N$ is a bounded smooth domain and $2<p<2^*=2N/(N-2)$. We show that if $\Omega$ is invariant by a nontrivial orthogonal involution then, for $\lambda>0$ sufficiently large, the equivariant topology of $\Omega$ is related with the number of solutions which change sign exactly once.

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November 19, 2004