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Trudinger–Moser Type Inequalities with Vanishing Weights in the Unit Ball

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Abstract

Let \(\mathbf{B} \) denote the unit ball in \({\mathbb {R}}^n\) with \(n\ge 2\). In this paper, we present the balance conditions on the nonlinearity function F and the weight function h such that the weighted Trudinger–Moser type inequalities

$$\begin{aligned} \sup _{u \in W^{1,n}_{0}(\mathbf{B }),\, u \text { is radial}, \Vert \nabla u\Vert _{L^n(\mathbf{B })} \le 1} \int _{\mathbf{B }} F(u) h(|x|) dx < \infty \end{aligned}$$

holds. We also study the attainability of these inequalities. These results generalizes the ones obtained by De Figueiredo et al. [15] to the higher dimension \(n\ge 3\) as well as weaken the conditions on F and h given in [15]. Our results also extend the ones of Yang and Zhu [29] to more general cases of the nonlinearity function F and the weight function h.

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  1. Department of Mathematics, FPT University, Hanoi, Vietnam

    Van Hoang Nguyen

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  1. Van Hoang Nguyen
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Communicated by Luis Vega.

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Nguyen, V.H. Trudinger–Moser Type Inequalities with Vanishing Weights in the Unit Ball. J Fourier Anal Appl 26, 77 (2020). https://doi.org/10.1007/s00041-020-09789-9

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  • DOI: https://doi.org/10.1007/s00041-020-09789-9

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