Compactness results for linearly perturbed Yamabe problem on manifolds with boundary

Marco Ghimenti; Anna Maria Micheletti

doi:10.3934/dcdss.2020453

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2021, Volume 14, Issue 5: 1757-1778. Doi: 10.3934/dcdss.2020453

This issue Previous Article On a class of semipositone problems with singular Trudinger-Moser nonlinearities Next Article Sign-changing solutions for a parameter-dependent quasilinear equation

Compactness results for linearly perturbed Yamabe problem on manifolds with boundary

Marco Ghimenti^, and
Anna Maria Micheletti

Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5, 56126 Pisa, Italy

^* Corresponding author: Marco Ghimenti
^* Corresponding author: Marco Ghimenti

Received: February 2020

Revised: June 2020

Early access: November 2020

Published: May 2021

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Abstract

Let $ (M,g) $ a compact Riemannian $ n $-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of $ g $ there are scalar-flat metrics that have $ \partial M $ as a constant mean curvature hypersurface. Also, under certain hypothesis, it is known that these metrics are a compact set. In this paper we prove that, both in the case of umbilic and non-umbilic boundary, if we linearly perturb the mean curvature term $ h_{g} $ with a negative smooth function $ \alpha, $ the set of solutions of Yamabe problem is still a compact set.

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Mathematics Subject Classification: 35J65, 53C21.

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