Abstract
We study nonlinear second-order equations of the form \(-\Delta u + X^2 u + i\lambda X u + \sigma u = K|u|^{p - 1}u\) on \(S^n\) with the usual round metric, where X is a Killing field. The case of principal interest is when X has length \({\le }1\), which leads to hypoelliptic operators with loss of one derivative. After establishing existence of solutions via variational methods, we carry out the subelliptic analysis on \(S^n\) utilizing their homogeneous coset space properties. Writing \(L_\alpha = \Delta - X^2 - i\alpha X\), we establish the optimal range of p such that the embedding \(\mathcal {D}((-L_\alpha )^{1/2}) \hookrightarrow L^p\) is compact, which gives sharp versions of results in Taylor (Houst J Math 42(1):143–165, 2016) for all dimensions. Such subelliptic phenomena have no parallel in the setting of flat spaces.
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Mukherjee, M. A special class of nonlinear hypoelliptic equations on spheres. Nonlinear Differ. Equ. Appl. 24, 15 (2017). https://doi.org/10.1007/s00030-017-0439-9
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DOI: https://doi.org/10.1007/s00030-017-0439-9