Sharp weighted Trudinger–Moser–Adams inequalities on the whole space and the existence of their extremals

Abstract

Though there have been extensive works on the existence of maximizers for sharp first order Trudinger–Moser inequalities, much less is known for that of the maximizers for higher order Adams’ inequalities. In this paper, we mainly study the existence of extremals for sharp weighted Trudinger–Moser–Adams type inequalities with the Dirichlet and Sobolev norms (also known as the critical and subcritical Trudinger–Moser–Adams inequalities), see Theorems 1.1, 1.2, 1.3, 1.5, 1.7, 1.9 and 1.11. First, we employ the method based on level-sets of functions under consideration and Fourier transform to establish stronger weighted Trudinger–Moser–Adams type inequalities with the Dirichlet norm in $$W^{2,\frac{n}{2}}(\mathbb {R}^n)$$ and $$W^{m,2}(\mathbb {R}^{2m})$$ respectively. While the first order sharp weighted Trudinger–Moser inequality and its existence of extremal functions was established by Dong and the second author using a quasi-conformal type transform (Dong and Lu in Calc Var Partial Differ Equ 55:55–88, 2016), such a transform does not work for the Adams inequality involving higher order derivatives. Since the absence of the Polya–Szego inequality and the failure of change of variable method for higher order derivatives for weighted inequalities, we will need several compact embedding results (Lemmas 2.1, 3.1 and 5.2). Through the compact embedding and scaling invariance of the subcritical Adams inequality, we investigate the attainability of best constants. Second, we employ the method developed by Lam et al. (Adv Math 352:1253–1298, 2019) which uses the relationship between the supremums of the critical and subcritical inequalities (see also Lam in Proc Amer Math Soc 145:4885–4892, 2017) to establish the existence of extremals for weighted Adams’ inequalities with the Sobolev norm. Third, using the Fourier rearrangement inequality established by Lenzmann and Sok (A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order, arXiv:1805.06294v1 ), we can reduce our problem to the radial case and then establish the existence of the extremal functions for the non-weighted Adams inequalities. As an application, we derive new results on high-order critical Caffarelli–Kohn–Nirenberg interpolation inequalities whose parameters extend those proved by Lin (Commun Partial Differ Equ 11:1515–1538, 1986) (see Theorems 1.13 and 1.14). Furthermore, we also establish the relationship between the best constants of the weighted Trudinger–Moser–Adams type inequalities and the Caffarelli–Kohn–Nirenberg inequalities in the asymptotic sense (see Theorems 1.13 and 1.14).