Concentration Phenomena for Conformal Metrics with Constant $Q$-Curvature

Abstract

Let $(M,g)$ be an analytic Riemannian manifold of dimension $n \geq 5$. In this paper, we consider the so-called constant $Q$-curvature equation \[ \varepsilon^4\Delta_{g}^2 u -\varepsilon^2 b \Delta_{g} u +a u = u^{p} , \qquad \text{in } M, \quad u>0, \quad u\in H^2_g(M) \] where $a,b$ are positive constants such that $b^2-4 a>0$, $p$ is a sub-critical exponent $1<p<2^\#-1=\frac{n+4}{n-4}$, $\Delta_g$ denotes the Laplace-Beltrami operator and $\Delta_g^2:=\Delta_{g}(\Delta_{g})$ is the bilaplacian operator on $M$. We show that, if $\varepsilon>0$ is small enough, then positive solutions to the above constant $Q$-curvature equation are generated by a maximum or minimum point of the function $\tau_g$, given by \[ \tau_g(\xi):= \sum_{i, j=1}^{n} \frac{\partial^{2} g_{\xi}^{i i}}{\partial z_{j}^{2}}(0), \] where $g_{\xi}^{i j}$ 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 $(M\times X, g+\varepsilon^2 h)$ where $(M,g)$ is flat and closed, and $(X,h)$ any $m$-dimensional Einstein Riemannian manifold, $m\geq 3$.