Conformal metrics with constant Q-curvature for manifolds with boundary

摘要

In this paper we prove that, given a compact four-dimensional smooth Riemannian manifold (M, g) with smooth boundary, there exists a metric in the conformal class [g] of the background metric g with constant Q-curvature, zero T-curvature and zero mean curvature under generic conformally invariant assumptions. The problem is equivalent to solving a fourth-order non-linear elliptic boundary value problem (BVP) with boundary condition given by a third-order pseudodifferential operator and homogeneous Neumann condition. It has a variational structure, but since the corresponding Euler-Lagrange functional is in general unbounded from above and below, we need to use min-max methods combined with a new topological argument and a compactness result for the above BVP.