Existence of extremal functions and Wulff symmetry for anisotropic Trudinger-Moser inequalities

In this paper, we investigate the extremal functions for anisotropic Trudinger-Moser inequalities. Our method uses convex symmetrization, the continuity of the supremum function, together with the relation between the supremums of the subcritical and the critical anisotropic Trudinger-Moser inequality, we give some results of existence and symmetry about the extremal functions for several different types of anisotropic Trudinger-Moser inequalities.