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 \[ ε^4Δg^2 u -ε^2 b Δg u +a u = uᵖ , in M, u>0, u∈ H^2_g(M) \] where $a,b$ are positive constants such that b²-4 a>0, p is a sub-critical exponent 1<p<2^#-1=n+4/n-4, Δg denotes the Laplace-Beltrami operator and Δg²:=Δg(Δg) is the bilaplacian operator on M. We show that, if ε>0 is small enough, then positive solutions to the above constant Q-curvature equation are generated by a maximum or minimum point of the function τg, given by \[ τ_g(ξ):= ∑i, j=1ⁿ {∂² gξi i}{∂ zⱼ²}(0), \] where gξi 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+ε² h) where $(M,g)$ is flat and closed, and $(X,h)$ any m-dimensional Einstein Riemannian manifold, m≥ 3.