A non-existence theorem for a semilinear Dirichlet problem involving critical exponent on halfspaces of the Heisenberg group

Abstract.

Let $\Delta _ {{\Bbb H}^n}$ be the Kohn Laplacian on the Heisenberg group ${\Bbb H}^n$ and let Q = 2n + 2 be the homogeneous dimension of ${\Bbb H}^n$. In this note, completing a recent result obtained with E. Lanconelli [9], we prove that, if $\Pi$ is a halfspace of ${\Bbb H}^n$, then the critical Dirichlet problem ¶¶$(^*) \qquad - \Delta _{{\Bbb H}^n}u = u^{{Q+2} \over {Q-2}} \qquad {\rm in} \, \Pi, \qquad u = 0 \qquad {\rm in} \, \partial \Pi$,¶¶ has no nontrivial nonnegative weak solutions. This result enables to improve a representation theorem by Citti [2], for Palais-Smale sequences related to the equation in (*).