A priori bounds for positive solutions of subcritical elliptic equations

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Article - postprint


Harvey Mudd College

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We provide a-priori L∞ bounds for positive solutions to a class of subcritical elliptic problems in bounded C2 domains. Our arguments rely on the moving planes method applied on the Kelvin transform of solutions. We prove that locally the image through the inversion map of a neighborhood of the boundary contains a convex neighborhood; applying the moving planes method, we prove that the transformed functions have no extremal point in a neighborhood of the boundary of the inverted domain. Retrieving the original solution u, the maximum of any positive solution in the domain $\Om,$ is bounded above by a constant multiplied by the maximum on an open subset strongly contained in $\Om.$ The constant and the open subset depend only on geometric properties of $\Om,$ and are independent of the non-linearity and on the solution u. Our analysis answers a longstanding open problem.

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