Existence results for mean field type equations on line bundles
Volume 276 / 2024
Abstract
Suppose that there exists a global unit frame on the set of all smooth sections of the line bundles we consider. In a recent joint work of J. Yang and the second named author (2022), a mean field type equation was introduced on line bundles, and several existence results were obtained in both the subcritical case and the critical case. In the current paper, we obtain an existence result, which is an analog of that of Djadli, in the supercritical case under the assumption that all parallel sections are zero. Our approach comes from the min-max scheme of Djadli. However, we need to establish all estimates in the line bundle setting. Firstly, we prove the existence of Green sections with respect to the bundle Laplace–Beltrami operator, and use the Green representation formula to get a compactness result for sequences of solutions of the mean field type equation. Secondly, a spreading Trudinger–Moser inequality for line bundles is derived. Thirdly, continuous maps between barycenters and sublevels of the associated functionals are constructed.
