# Regularity of extremal solutions of semilinear elliptic equations with m-convex nonlinearities Source: [Regularity of extremal solutions of semilinear elliptic equations with m-convex nonlinearities]() Collection index: [llms.txt]() Data updated: 2025-10-25T15:43:24.976Z Authors: K. Kumagai Published: 2022-03-29T07:00:49.000Z ## Abstract 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()