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

Let $(M,g)$ be an analytic Riemannian manifold of dimension $n\ge 5$. In this paper, we consider the so-called constant $Q$-curvature equation $${\epsilon }^4{\mathrm{\Delta }}_g^{2}u-{\epsilon }^{2}b{\mathrm{\Delta }}_{g}u+au=u^p,\text{in }M,u>0,u\in H_g^2(M)$$ where $a,b$ are positive constants such that $b^2-4a>0$, $p$ is a sub-critical exponent $1<p<2^{\mathrm{\#}}-1=\frac{n+4}{n-4}$, ${\mathrm{\Delta }}_g$ denotes the Laplace-Beltrami operator and ${\mathrm{\Delta }}_g^2:={\mathrm{\Delta }}_g({\mathrm{\Delta }}_g)$ is the bilaplacian operator on $M$. We show that, if $\epsilon >0$ 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 $${\tau }_g(\xi ):=\sum _{i,j=1}^n\frac{{\mathrm{\partial }}^2{g}_{\xi }^{ii}}{\mathrm{\partial }{z}_j^2}(0),$$ where $g_{\xi }^{ij}$ 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 $(M\times X,g+{\epsilon }^{2}h)$ where $(M,g)$ is flat and closed, and $(X,h)$ any $m$-dimensional Einstein Riemannian manifold, $m\ge 3$.