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Equivalence of Subcritical and Critical Adams Inequalities in \(W^{m,2}(\mathbb {R}^{2m})\) and Existence and Non-existence of Extremals for Adams Inequalities under Inhomogeneous Constraints

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Abstract

Though there have been extensive works on the existence of maximizers for sharp Trudinger-Moser inequalities under homogeneous and inhomogeneous constraints, and sharp Adams inequalities under homogeneous constraints, much less is known for that of the maximizers for Adams inequalities under inhomogeneous constraints. Furthermore, whether exists equivalence between subcritical and critical Adams inequalities in \(W^{m,2}(\mathbb {R}^{2m})\) also remains open. In this paper, we shall give partial answers to these problems. We first establish the equivalence of subcritical Adams inequalities and critical Adams inequalities under inhomogeneous constraints through exploiting the scaling invariance of Adams inequalities in \(W^{m,2}(\mathbb {R}^{2m})\) (see Theorem 1.1). Then we consider the existence and non-existence of extremals for sharp Adams inequalities under inhomogeneous constraints in Theorem 1.2, 1.3 and 1.4. Our methods are based on Fourier rearrangement inequality and careful analysis for vanishing phenomenon of radially maximizing sequence for Adams inequalities in \(W^{m,2}(\mathbb {R}^{2m})\).

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Acknowledgements

Caifeng Zhang was partly supported by grant from the NNSF of China (No.12001038) and Fundamental Research Funds for the Central Universities (No. FRF-IDRY-20-034), Lu Chen was partly supported by grant from the NNSF of China (No.11901031) and grant from Beijing Institute of Technology (No.3170012221903, No.2022CX01002).

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  1. Department of Applied Mathematics, School of Mathematics and Physics University of Science and Technology Beijing, Beijing, 100083, People’s Republic of China

    Caifeng Zhang

  2. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Lu Chen

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  1. Caifeng Zhang
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Zhang, C., Chen, L. Equivalence of Subcritical and Critical Adams Inequalities in \(W^{m,2}(\mathbb {R}^{2m})\) and Existence and Non-existence of Extremals for Adams Inequalities under Inhomogeneous Constraints. Potential Anal 59, 2071–2091 (2023). https://doi.org/10.1007/s11118-022-10040-x

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