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### Open Access Journals

CPAA

A generic semilinear equation in a star-shaped ring is considered. Any
solution bounded between its boundary values is shown to be decreasing along
rays starting from the origin, provided that a structural condition is
satisfied. A corresponding property for the product between the solution and a
(positive) power of $|x|$ is also investigated. Applications to the
Emden-Fowler and the Liouville equation are developed.

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$\varrho$ |

$8 π$ |

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