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Kirchhoff-type problems involving the fractional p-Laplacian on the Heisenberg group

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Abstract

In this paper, we are interested in the existence of solutions for a class of Kirchhoff-type problems driven by a non-local integro-differential operator with the homogeneous Dirichlet boundary conditions on the Heisenberg group as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} M(\iint _{{\mathbb {H}}^{2N}}|u(\xi )-u(\eta )|^{p}K(\eta ^{-1}\circ \xi )d\xi \,d\eta )\pounds ^{p}_{K}u=f(\xi ,u) &{} { \text{ in } } \Omega ,\\ u=0 &{} { \text{ in } } {\mathbb {H}}^N \setminus \Omega , \end{array} \right. \end{aligned}$$

where \(\pounds ^{p}_{K}\) is a non-local integro-differential operator with singular kernel \(K,\Omega\) is an open bounded subset of the Heisenberg group \({\mathbb {H}}^N\) with Lipshcitz boundary \(\partial \Omega\). Under some suitable assumptions on the functions M and f, together with the variational methods and the mountain pass theorem, we discuss the existence of weak solutions for the above problem on the Heisenberg group.

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Acknowledgements

B. Zhang was supported by National Natural Science Foundation of China (Nos. 11871199 and 12171152), Shandong Provincial Natural Science Foundation, PR China (No. ZR2020MA006), and Cultivation Project of Young and Innovative Talents in Universities of Shandong Province.

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  1. School of Mathematics and Statistics, Northeast Petroleum University, Daqing, 163318, People’s Republic of China

    Jieyu Zhou & Lifeng Guo

  2. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Binlin Zhang

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Zhou, J., Guo, L. & Zhang, B. Kirchhoff-type problems involving the fractional p-Laplacian on the Heisenberg group. Rend. Circ. Mat. Palermo, II. Ser 71, 1133–1157 (2022). https://doi.org/10.1007/s12215-022-00763-6

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