Morse inequalities at infinity for a resonant mean field equation
| Autor: | Mohamed Ben Ayed, Mohameden Ould Ahmedou |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Surface (mathematics)
Applied Mathematics General Mathematics media_common.quotation_subject Astrophysics::Instrumentation and Methods for Astrophysics Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Unit volume Infinity Morse code law.invention Type equation Mathematics - Analysis of PDEs Mean field theory Mean field equation law FOS: Mathematics Computer Science::General Literature 35C60 58J60 35J91 Analysis of PDEs (math.AP) Mathematics Mathematical physics media_common Morse theory |
| Popis: | In this paper we study the following mean field type equation \begin{equation*} (MF) \qquad -\D_g u \, = \varrho ( \frac{K e^{u}}{\int_{\Sig} K e^{u} dV_g} \, - \, 1) \, \mbox{ in } \Sigma, \end{equation*} where $(\Sigma, g)$ is a closed oriented surface of unit volume $Vol_g(\Sigma)$ = 1, $K$ positive smooth function and $\varrho= 8 \pi m$, $ m \in \N$. Building on the critical points at infinity approach initiated in \cite{ABL17} we develop, under generic condition on the function $K$ and the metric $g$, a full Morse theory by proving Morse inequalities relating the Morse indices of the critical points, the indices of the critical points at infinity, and the Betti numbers of the space of formal barycenters $B_m(\Sigma)$.\\ We derive from these \emph{Morse inequalities at infinity} various new existence as well as multiplicity results of the mean field equation in the resonant case, i.e. $\varrho \in 8 \pi \N$. Comment: 31 pages. More details have been added |
| Databáze: | OpenAIRE |
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