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

Copyright 2019 American Meteorological Society (AMS). Permission to use figures, tables, and brief excerpts from this work in scientific and educational works is hereby granted provided that the source is acknowledged. Any use of material in this work that is determined to be “fair use” under Section 107 of the U.S. Copyright Act September 2010 Page 2 or that satisfies the conditions specified in Section 108 of the U.S. Copyright Act (17 USC §108, as revised by P.L. 94-553) does not require the AMS’s permission. Republication, systematic reproduction, posting in electronic form, such as on a website or in a searchable database, or other uses of this material, except as exempted by the above statement, requires written permission or a license from the AMS. All AMS journals and monograph publications are registered with the Copyright Clearance Center (http://www.copyright.com). Questions about permission to use materials for which AMS holds the copyright can also be directed to the AMS Permissions Officer at permissions@ametsoc.org. Additional details are provided in the AMS Copyright Policy statement, available on the AMS website (http://www.ametsoc.org/CopyrightInformation).We consider the class of stable solutions to semilinear equations -¿u=f(u) in a bounded smooth domain of Rn. Since 2010 an interior a priori L8 bound for stable solutions is known to hold in dimensions n=4 for all C1 nonlinearities f. In the radial case, the same is true for n=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 Lp bounds for every finite p in dimension 5. Since the mid nineties, the existence of an L8 bound holding for all C1 nonlinearities when 5=n=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 papePeer ReviewedPostprint (author's final draft