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S. Cruz and G. Vaira have been supported by INdAM - GNAMPA Project 2022 "Fenomeni di blow-up per equazioni non lineari", E55F22000270001S and by PRIN 2017JPCAPN.

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POSITIVE BLOW-UP SOLUTIONS FOR A LINEARLY PERTURBED BOUNDARY YAMABE PROBLEM

Publicado en:DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - 2024-12-24 (), DOI: 10.3934/dcds.2024174

Autores: Cruz-blazquez, Sergio; Vaira, Giusi

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We consider the problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a n - dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature K and boundary mean curvature H of arbitrary sign which are non - constant and D-n = root n( n - 1)H |K|(- 1/2) > 1 at some point of the boundary. It is known that this problem admits a positive mountain pass solution if n = 3, while no existence results are known for n >= 4. We will consider a perturbation of the geometric problem and show the existence of a positive solution which blows-up at a boundary point which is critical for both prescribed curvatures.

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