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

2022/03/29 by Kumagai, K.
#35B35 #35B65 #35K57 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2203.15286

Abstract

We consider the Gelfand problem in a bounded smooth domain Ω⊂ ℝ^N with the Dirichlet boundary condition. We are interested in the boundedness of the extremal solution u^*. When the dimension N≥10, it is known that a singular extremal solution can be constructed for the nonlinearity f(u)=e^u and Ω=B₁. When 3≤ N≤ 9, Cabré, 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.

Related

• Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains
• Regularity of the extremal solutions associated to elliptic systems
• Regularity of minimizers of semilinear elliptic problems up to dimension four
• Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities
• Boundedness of the extremal solutions in dimension 4