Recently, U. Dierkes and G. Huisken (Math. Ann. 2024) studied a generalized Euler's variational problem, a family of natural non-compact stationary surfaces are minimal cones, stability and minimizing properties for minimal hypercones are studied in their paper. In this talk, we recall some classical research on area-minimizing cones. Then, they will be used to generalize Dierkes and Huisken's results. This talk is based on two joint works with Professor Xiaowei Xu, and Professor Qing Chen.