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 $3leq nleq 9$. As a corollary, we obtain uniform bounds for finite Morse index solutions to the Gelfand problem constructed via the continuity method.