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On a resonant mean field type equation: A "critical point at Infinity" approach

This work was partially supported by a grant from the Simons Foundation (Nr. 210368 to Marcello Lucia) and the grant MTM2014-52402-C3-1-P (Spain).
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  • We consider the following mean field type equations on domains of $\mathbb R^2$ under Dirichlet boundary conditions:

    $\left\{ \begin{array}{l} - \Delta u = \varrho \frac{{K {e^u}}}{{\int_\Omega {K {e^u}} }}\;\;\;\;\;{\rm{in}}\;\Omega ,\\\;\;\;\;u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{on}}\;\partial \Omega ,\end{array} \right.$

    where $K$ is a smooth positive function and $\varrho$ is a positive real parameter.

    A "critical point theory at Infinity" approach of A. Bahri to the above problem is developed for the resonant case, i.e. when the parameter $\varrho$ is a multiple of $8 π$ . Namely, we identify the so-called "critical points at infinity" of the associated variational problem and compute their Morse indices. We then prove some Bahri-Coron type results which can be seen as a generalization of a degree formula in the non-resonant case due to C.C.Chen and C.S.[18].

    Mathematics Subject Classification: Primary:35C60, 35J20;Secondary:35Q70.


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