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1. CJM 2005 (vol 57 pp. 1279)
A Semilinear Problem for the Heisenberg Laplacian on Unbounded Domains 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 Categories:22E30, 22E27 |