Accepted Paper

Inserted: 7 oct 2022
Last Updated: 7 oct 2022

Journal: Comm. Pure Appl. Math.
Year: 2022

Abstract:

We consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well-known that: - For n = 2, there exist Morse index 1 solutions whose L∞ norm goes to infinity. - For n ≥ 3, uniform boundedness holds in the subcritical case for power-type nonlinearities, while for critical nonlinearities the boundedness of the Morse index does not prevent blow-up in L∞. In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori L∞ bound for finite Morse index solution in the sharp dimensional range 3 ≤ n ≤ 9. As a corollary, we obtain uniform bounds for finite Morse index solutions to the Gelfand problem constructed via the continuity method.

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• Uniform-boundedness-for-finite-Morse-index-solutions-to-supercritical-semilinear-elliptic-equations.pdf