  We consider the Gelfand problem in a bounded smooth domain $\Omega\subset
\mathbb{R}^N$ with the Dirichlet boundary condition. We are interested in the
boundedness of the extremal solution $u^*$. When the dimension $N\ge10$, it is
known that a singular extremal solution can be constructed for the nonlinearity
$f(u)=e^u$ and $\Omega=B_1$. When $3\le N\le 9$, Cabr\'e, Figalli, Ros-Oton,
and Serra (2020) proved the following surprising result: the extremal solution
$u^*$ is bounded if the nonlinearity $f$ is positive, nondecreasing, and
convex.
  In this paper, we succeed in generalizing their result to general $m$-convex
nonlinearities. Moreover, we give a unified viewpoint on the results of
previous studies by considering $m$-convexity. We provide a closedness result
for stable solutions with $m$-convex nonlinearities. As a consequence, we
provide a Liouville-type result and by using a blow-up argument, we prove the
boundedness of extremal solutions.
