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DCDS

We consider the following mean field type equations on domains of

under Dirichlet boundary conditions:

$\mathbb R^2$ |

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

where

is a smooth positive function and

is a positive real parameter.

$K$ |

$\varrho$ |

A "critical point theory at Infinity" approach of A. Bahri to the above problem is developed for the resonant case,

is a multiple of

. 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 ].

*i*.*e*. when the parameter$\varrho$ |

$8 π$ |

DCDS

We consider the existence and multiplicity of riemannian metrics of
prescribed mean curvature and zero boundary mean curvature on the
three dimensional half sphere $(S^3_+,g_c)$ endowed with its
standard metric $g_c$. Due to Kazdan-Warner type obstructions,
conditions on the function to be realized as a scalar curvature have
to be given. Moreover the existence of

*critical point at infinity*for the associated Euler Lagrange functional makes the existence results harder to be proved. However it turns out that such noncompact orbits of the gradient can be treated as a usual critical point once a*Morse Lemma at infinity*is performed. In particular their topological contribution to the level sets of the functional can be computed. In this paper we prove that, under generic conditions on $K$, this*topology at infinity*is a lower bound for the number of metrics in the conformal class of $g_c$ having prescribed scalar curvature and zero boundary mean curvature.## Year of publication

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