Annali di Matematica Pura ed Applicata

, Volume 169, Issue 1, pp 375–392

Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian


  • G. Citti
    • Univ. di Bologna

DOI: 10.1007/BF01759361

Cite this article as:
Citti, G. Annali di Matematica pura ed applicata (1995) 169: 375. doi:10.1007/BF01759361


The paper is concerned with the Dirichlet problem
$$ - \Delta _H u + au = u^{{{\left( {q + 2} \right)} \mathord{\left/ {\vphantom {{\left( {q + 2} \right)} {\left( {q - 2} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {q - 2} \right)}}} in \Omega , u = 0 on \partial \Omega $$
where Ω is a smooth, bounded domain in R2n+1H is the Kohn-Laplacian on the Heisenberg group Hn, and q=2n+2 is the homogeneous dimension of Hn. We first prove a representation formula for the Palais Smale sequences of the functional naturally associated to (P). Then we use this expression to prove that, if 0⩾a> - λ11 is the smallest eigenvalue of ΔH), then (P) has at least a nonnegative solution. This theorem extends to this setting a previous result of Brezis and Niremberg for the classical Laplacian.

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© Fondazione Annali di Matematica Pura ed Applicata 1995