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A mean field equation on a torus: one-dimensional symmetry of solutions
Cabré Vilagut, Xavier; Lucia D'Agostino, Marcello; Sanchón Rodellar, Manuel
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
We study the equation $$-\Delta u=\lambda\left(\frac{e^u}{\int_\oep e^u}- \frac{1}{|\oep|}\right)\quad \text{in }\oep$$ for $u\in E$, where $E = \{ u \in H^1(\oep): u \hbox{ is doubly periodic}, \int_{\oep} u = 0 \}$ and $\oep$ is a rectangle of $\R^2$ with side lengths $1/\epsilon$ and $1$, $0< \epsilon \leq 1$. We establish that every solution depends only on the $x$--variable when $\lambda \leq \lambda^*(\epsilon)$, where $\lambda^*(\epsilon)$ is an explicit positive constant depending on the maximum conformal radius of the rectangle. As a consequence, we obtain an explicit range of parameters $\epsilon$ and $\lambda$ in which every solution is identically zero. This range is optimal for $\epsilon\leq1/2$.
Partial differential equations
mean field equation
Equacions en derivades parcials
Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type
Classificació AMS::35 Partial differential equations::35B Qualitative properties of solutions
Attribution-NonCommercial-NoDerivs 2.5 Spain

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