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 infinity 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 infinity In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori L infinity 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.