*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.

**Download:**