Properties for Nonlinear Fractional SubLaplace Equations on the Heisenberg Group

sci, Xinjing Wang and Pengcheng Niu

doi:10.4208/jpde.v32.n1.5

JPDE

2019

DOI: 10.4208/jpde.v32.n1.5

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Properties for Nonlinear Fractional SubLaplace Equations on the Heisenberg Group

Xinjing Wang and Pengcheng Niu sci¹

Abstract: The aim of the paper is to study properties of solutions to the nonlinear fractional subLaplace equations on the Heisenberg group. Based on the method of moving planes to the Heisenberg group, we prove the Liouville property of solutions on a half space and the symmetry and monotonicity of the solutions on the whole group respectively. AMS Subject Classifications: 35A01, 35J57, 35D99Chinese Library Classifications: O175.2

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Compact embeddings, eigenvalue problems, and subelliptic Brezis–Nirenberg equations involving singularity on stratified Lie groups

2023

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The purpose of this paper is twofold: first we study an eigenvalue problem for the fractional p-sub-Laplacian over the fractional Folland–Stein–Sobolev spaces on stratified Lie groups. We apply variational methods to investigate the eigenvalue problems. We conclude the positivity of the first eigenfunction via the strong minimum principle for the fractional p-sub-Laplacian. Moreover, we deduce that the first eigenvalue is simple and isolated. Secondly, utilising established properties, we prove the existence of at least two weak solutions via the Nehari manifold technique to a class of subelliptic singular problems associated with the fractional p-sub-Laplacian on stratified Lie groups. We also investigate the boundedness of positive weak solutions to the considered problem via the Moser iteration technique. The results obtained here are also new even for the case $$p=2$$ p = 2 with $$\mathbb {G}$$ G being the Heisenberg group.

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Compact embeddings, eigenvalue problems, and subelliptic Brezis–Nirenberg equations involving singularity on stratified Lie groups

2023

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Properties for Nonlinear Fractional SubLaplace Equations on the Heisenberg Group

Cited by 1 publication

References 17 publications

Compact embeddings, eigenvalue problems, and subelliptic Brezis–Nirenberg equations involving singularity on stratified Lie groups

Compact embeddings, eigenvalue problems, and subelliptic Brezis–Nirenberg equations involving singularity on stratified Lie groups

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