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We consider the Gelfand problem in a bounded smooth domain $\Omega\subset\mathbb{R}^N$ with the Dirichlet boundary condition. We are interested in theboundedness of the extremal solution $u^*$. When the dimension $N\ge10$, it isknown 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, andconvex.  In this paper, we succeed in generalizing their result to general $m$-convexnonlinearities. Moreover, we give a unified viewpoint on the results ofprevious studies by considering $m$-convexity. We provide a closedness resultfor stable solutions with $m$-convex nonlinearities. As a consequence, weprovide a Liouville-type result and by using a blow-up argument, we prove theboundedness of extremal solutions.

Regularity of extremal solutions of semilinear elliptic equations with m-convex nonlinearities