In this paper we prove that,
given a compact four-dimensional smooth Riemannian manifold (M,g) with smooth
boundary, there exists a metric conformal to g with constant T-curvature, zero
Q-curvature and zero mean curvature under generic and conformally invariant
assumptions. The problem amounts to solving a fourth-order nonlinear elliptic
boundary value problem (BVP) with boundary conditions given by a third-order
pseudodifferential operator and homogeneous Neumann operator. It has a variational
structure, but since the corresponding Euler–Lagrange functional is in general
unbounded from below, we look for saddle points. We do this by using topological
arguments and min-max methods combined with a compactness result for the
corresponding BVP.
