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

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \Delta _ {{\Bbb H}^n}$\end{document} be the Kohn Laplacian on the Heisenberg group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {\Bbb H}^n $\end{document} and let Q = 2n + 2 be the homogeneous dimension of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {\Bbb H}^n $\end{document}. In this note, completing a recent result obtained with E. Lanconelli [9], we prove that, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \Pi $\end{document} is a halfspace of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {\Bbb H}^n $\end{document}, then the critical Dirichlet problem ¶¶\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ (^*) \qquad - \Delta _{{\Bbb H}^n}u = u^{{Q+2} \over {Q-2}} \qquad {\rm in} \, \Pi, \qquad u = 0 \qquad {\rm in} \, \partial \Pi$\end{document},¶¶ 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 (*).