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 (*).
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Received January 19, 1998
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Uguzzoni, F. A non-existence theorem for a semilinear Dirichlet problem involving critical exponent on halfspaces of the Heisenberg group. NoDEA, Nonlinear differ. equ. appl. 6, 191–206 (1999). https://doi.org/10.1007/s000300050072
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DOI: https://doi.org/10.1007/s000300050072