On a resonant mean field type equation: A "critical point at Infinity" approach
Mohameden Ahmedou Mohamed Ben Ayed Marcello Lucia
Discrete & Continuous Dynamical Systems - A 2017, 37(4): 1789-1818 doi: 10.3934/dcds.2017075
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
On the prescribed scalar curvature on $3$-half spheres: Multiplicity results and Morse inequalities at infinity
M. Ben Ayed Mohameden Ould Ahmedou
Discrete & Continuous Dynamical Systems - A 2009, 23(3): 655-683 doi: 10.3934/dcds.2009.23.655
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.
keywords: Prescribed scalar curvature Topology at Infinity Gradient flow Morse inequalities Critical point at infinity

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