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A Semilinear Problem for the Heisenberg Laplacian on Unbounded Domains

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Published:2005-12-01
Printed: Dec 2005
• Sara Maad
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Abstract

We study the semilinear equation \begin{equation*} -\Delta_{\mathbb H} u(\eta) + u(\eta) = f(\eta, u(\eta)),\quad u \in \So(\Omega), \end{equation*} where $\Omega$ is an unbounded domain of the Heisenberg group $\mathbb H^N$, $N\ge 1$. The space $\So(\Omega)$ is the Heisenberg analogue of the Sobolev space $W_0^{1,2}(\Omega)$. The function $f\colon \overline{\Omega}\times \mathbb R\to \mathbb R$ is supposed to be odd in $u$, continuous and satisfy some (superlinear but subcritical) growth conditions. The operator $\Delta_{\mathbb H}$ is the subelliptic Laplacian on the Heisenberg group. We give a condition on $\Omega$ which implies the existence of infinitely many solutions of the above equation. In the proof we rewrite the equation as a variational problem, and show that the corresponding functional satisfies the Palais--Smale condition. This might be quite surprising since we deal with domains which are far from bounded. The technique we use rests on a compactness argument and the maximum principle.
 Keywords: Heisenberg group, concentration compactness, Heisenberg Laplacian
 MSC Classifications: 22E30 - Analysis on real and complex Lie groups [See also 33C80, 43-XX] 22E27 - Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)

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