Conformal metrics with constant $Q$-curvature for manifolds with boundary

Abstract

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 nonlinear elliptic boundary value problem (BVP) with boundary condition given by a third-order pseudodifferential operator, and homogeneous Neumann conditions. 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.