Constant Q-curvature metrics on a product Riemannian manifold

Let (M, g) be an analytic Riemannian manifold of dimension n 5. In this paper, we consider the so-called constant Q-curvature equation \ ⁴₆² u -² b ₆ u +a u = u^p, in M, u>0, u H²g (M) \ where a, b are positive constants such that b²-4 a>0, p is a sub-critical exponent 10 is small enough, then positive solutions to the above constant Q-curvature equation are generated by a maximum or minimum point of the function g, given by \ g (): = ₈, ₉=₁^n ^2 g_{^i i} z₉^{2} (0), \ where g_^i j denotes the components of the inverse of the metric g in geodesic normal coordinates. This result shows that the geometry of M plays a crucial role in finding solutions to the equation above and provides a metric of constant Q-curvature on a product manifold of the form (M X, g+² h) where (M, g) is flat and closed, and (X, h) any m-dimensional Einstein Riemannian manifold, m 3.