Existence Results for Critical Semi-linear Equations on Heisenberg Group Domains |
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Authors: | Najoua Gamara Habiba Guemri Amine Amri |
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Institution: | 1. Campus universitaire, El Manar II - 2092, Tunis, Tunesia 2. Institut d’informatique de Medenine, Medenine, 4100, Tunesia 3. Institut d’informatique de Gabes, Gabes, 6000, Tunesia
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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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