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Existence Results for Critical Semi-linear Equations on Heisenberg Group Domains

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Abstract

Following the work of G. Citti and F. Uguzzoni who studied Yamabe type problems on Heisenberg group domains, we consider here the following critical semi-linear equation on domains of the Heisenberg group \({{\mathbb{H}^1}}\):

$$(P) \left\{\begin{array}{lll}-{\Delta_{H}}u\quad =\quad K{u^{3}}\quad\,{\rm in}\,\,\Omega,\\ \quad\quad\,{u}\quad > \quad0\qquad\,\,\,\,{\rm in}\,\,\Omega,\\ \quad\quad\,{u}\quad = \quad 0 \quad\quad\,\,\,{\rm on}\,\partial \Omega, \end{array}\right. $$

where Δ H is the sublaplacian on \({{\mathbb{H}^1}}\) and K is a C 3 positive function defined on Ω. Using a version of the Morse Lemma at infinity, we give necessary conditions on K to insure the existence of solutions for (P).

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Authors and Affiliations

  1. Campus universitaire, El Manar II - 2092, Tunis, Tunesia

    Najoua Gamara

  2. Institut d’informatique de Medenine, Medenine, 4100, Tunesia

    Habiba Guemri

  3. Institut d’informatique de Gabes, Gabes, 6000, Tunesia

    Amine Amri

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  1. Najoua Gamara
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  2. Habiba Guemri
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  3. Amine Amri
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Correspondence to Najoua Gamara.

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Gamara, N., Guemri, H. & Amri, A. Existence Results for Critical Semi-linear Equations on Heisenberg Group Domains. Mediterr. J. Math. 9, 803–831 (2012). https://doi.org/10.1007/s00009-011-0152-6

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  • DOI: https://doi.org/10.1007/s00009-011-0152-6

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