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A new proof of the boundedness results for stable solutions to semilinear elliptic equations

Dedicated to Luis Caffarelli, with friendship and great admiration

Xavier Cabré is supported by grants MTM2017-84214-C2-1-P and MdM-2014-0445 (Government of Spain), and is a member of the research group 2017SGR1392 (Government of Catalonia)

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  • We consider the class of stable solutions to semilinear equations $ -\Delta u = f(u) $ in a bounded smooth domain of $ \mathbb{R}^n $. Since 2010 an interior a priori $ L^\infty $ bound for stable solutions is known to hold in dimensions $ n\le 4 $ for all $ C^1 $ nonlinearities $ f $. In the radial case, the same is true for $ n\leq 9 $. Here we provide with a new, simpler, and unified proof of these results. It establishes, in addition, some new estimates in higher dimensions —for instance $ L^p $ bounds for every finite $ p $ in dimension 5.

    Since the mid nineties, the existence of an $ L^\infty $ bound holding for all $ C^1 $ nonlinearities when $ 5\leq n\leq 9 $ was a challenging open problem. This has been recently solved by A. Figalli, X. Ros-Oton, J. Serra, and the author, for nonnegative nonlinearities, in a forthcoming paper.

    Mathematics Subject Classification: 35K57, 35B65.


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