Gamma-star-shapedness for semilinear elliptic equations
Antonio Greco Marcello Lucia
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.
keywords: Comparison principle Star-shapedness Monotone iteration scheme.
On a resonant mean field type equation: A "critical point at Infinity" approach
Mohameden Ahmedou Mohamed Ben Ayed Marcello Lucia
We consider the following mean field type equations on domains of
$\mathbb R^2$
under Dirichlet boundary conditions:
$\left\{ \begin{array}{l} - \Delta u = \varrho \frac{{K {e^u}}}{{\int_\Omega {K {e^u}} }}\;\;\;\;\;{\rm{in}}\;\Omega ,\\\;\;\;\;u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{on}}\;\partial \Omega ,\end{array} \right.$
is a smooth positive function and
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
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].
keywords: Mean field equation critical points at infinity infinite dimensional Morse theory variational and topological methods

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