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.[
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