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Uniform boundedness for finite Morse index solutions to supercritical semilinear elliptic equations

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 $L^\infty$ 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 solution 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. [In the previous version, the proof of Proposition A.1 was not complete. Indeed, Lemma A.2 was proved only for estimates centered at the origin, and Footnote 9 claimed that the same estimates hold at every point; however, no complete proof was provided for this extension. In the current version, Lemma A.2 is restricted to estimates at the origin, and the boundedness at arbitrary points is obtained by combining the results in [9] with the new version of Lemma A.2. This avoids the need to extend Lemma A.2 to arbitrary centers and yields a complete proof of Proposition A.1.]