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Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions

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Abstract

Let \(\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}\) be the n-dimensional Heisenberg group, \(Q=2n+2\) be the homogeneous dimension of \(\mathbb {H}^{n}\). We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of Lions (Rev Mat Iberoam 1:145–201, 1985) to the setting of the Heisenberg group \(\mathbb {H}^{n}\). Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space \({ HW}^{1,Q}(\mathbb {H}^{n}) \) on the entire Heisenberg group \(\mathbb {H}^{n}\). Our results improve the sharp Trudinger–Moser inequality on domains of finite measure in \(\mathbb {H}^{n}\) by Cohn and Lu (Indiana Univ Math J 50(4):1567–1591, 2001) and the corresponding one on the whole space \(\mathbb {H}^n\) by Lam and Lu (Adv Math 231:3259–3287, 2012). All the proofs of the concentration-compactness principles for the Trudinger–Moser inequalities in the literature even in the Euclidean spaces use the rearrangement argument and the Polyá–Szegö inequality. Due to the absence of the Polyá–Szegö inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of Q- Laplacian subelliptic equations on \(\mathbb {H}^{n}\):

$$\begin{aligned} -\mathrm {div}\left( \left| \nabla _{\mathbb {H}}u\right| ^{Q-2} \nabla _{\mathbb {H}}u\right) +V(\xi ) \left| u\right| ^{Q-2}u=\frac{f(u) }{\rho (\xi )^{\beta }} \end{aligned}$$

with nonlinear terms f of maximal exponential growth \(\exp (\alpha t^{\frac{Q}{Q-1}})\) as \(t\rightarrow +\infty \). All the results proved in this paper hold on stratified groups with the same proofs. Our method in this paper also provide a new proof of the classical concentration-compactness principle for Trudinger-Moser inequalities in the Euclidean spaces without using the symmetrization argument.

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Notes

  1. The sequence \(\left\{ u_{k}\right\} \) constructed in [17] cannot show that the supremum (1.4) is infinite for \(p=\tilde{M}_{n,u}\) (see Remark 1).

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Acknowledgements

The authors would also like to thank the referee for his or her comments which improve its exposition of the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Jungang Li & Guozhen Lu

  2. Faculty of Science, Jiangsu University, Zhenjiang, 212013, China

    Maochun Zhu

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  1. Jungang Li
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  2. Guozhen Lu
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Correspondence to Maochun Zhu.

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Communicated by A. Malchiodi.

M. Zhu was partly supported by Natural Science Foundation of China (11601190), Natural Science Foundation of Jiangsu Province (BK20160483) and Jiangsu University Foundation Grant (16JDG043). J. Li and G. Lu was partly supported by a US National Science Foundation Grant DMS No. 1700918, a Collaboration Grant No. 519099 and a Simons Fellowship from the Simons Foundation.

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Li, J., Lu, G. & Zhu, M. Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions. Calc. Var. 57, 84 (2018). https://doi.org/10.1007/s00526-018-1352-8

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