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

Abstract

We consider the Gelfand problem in a bounded smooth domain with the Dirichlet boundary condition. We are interested in the boundedness of the extremal solution. When the dimension $N\ge10$, it is known that a singular extremal solution is obtained for the exponential nonlinearity if the domain is the unit ball. When $3\le N\le 9$, Cabré, Figalli, Ros-Oton and Serra (2020) proved the following surprising result: the extremal solution is bounded if the nonlinearity 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 obtain a closedness result for stable solutions with $m$-convex nonlinearities. As a consequence, we get a Liouville-type result and by using a blow-up argument, we prove the boundedness of extremal solutions.