Published Paper

Inserted: 7 oct 2022
Last Updated: 19 aug 2024

Journal: Comm. Pure Appl. Math.
Year: 2024

Abstract:

We consider finite Morse index solutions to semilinear elliptic equations and investigate their smoothness. It is well-known that:

- For \( n = 2 \), there exist Morse index 1 solutions whose \( L^\infty \) norm goes to infinity. - For \( n \geq 3 \), 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 \( L^\infty \).

In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori \( L^\infty \) bound for finite Morse index solutions in the sharp dimensional range \( 3 \leq n \leq 9 \). 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