Positive Blow-up Solutions for a Linearly Perturbed Boundary Yamabe Problem

2023/01/18 by Cruz-Blázquez, Sergio, Vaira, Giusi · 1 citation
#35B44 #53C21 #58J32 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2301.07396

Abstract

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 \mathfrak D_n=√(n(n-1))|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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