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.
