  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.
