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The non-local mean-field equation on an interval. De la Torre, Azahara


We study the quantization properties for a non-local mean-field equation and give a necessary and sufficient condition for the existence of solution for a ``Mean Field''-type equation in an interval with Dirichlet-type boundary condition. We restrict the study to the $1$-dimensional case and consider the fractional mean-field equation on the interval $I = (-1, 1)$ $$(-\Delta)^\frac{1}{2} u=\rho \frac{e^{u}}{\int_I e^{u}dx},$$ subject to Dirichlet boundary conditions. As in the $2-$dimensional case, it is expected that for a sequence of solutions to our equation, either we get a $\mathcal{C}^{\infty}$ limiting solution or, after a suitable rescaling, we obtain convergence to the Liouville equation. Then, we can reduce the problem to the study of the non-local Liouville's equation. One of the key points here is to find an appropriate Pohozaev identity. We prove that existence holds if and only if $\rho < 2\pi$. This requires the study of blowing-up sequences of solutions. In particular, we provide a series of tools which can be used (and extended) to higher-order mean field equations of non-local type. We provide a completely non-local method for this study, since we do not use the localization through the extension method. Instead, we use the study of blowing-up sequences of solutions. This is a work done in collaboration with with A. Hyder, Y. Sire and L. Martinazzi.

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