A new proof of the boundedness results for stable solutions to semilinear elliptic equations
PublisherAmerican Institute of Mathematical Sciences
Rights accessOpen Access
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 pape
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CitationCabre, X. A new proof of the boundedness results for stable solutions to semilinear elliptic equations. "Discrete and continuous dynamical systems. Series A", 1 Desembre 2019, vol. 39, núm. 12, p. 7249-7264.