Large conformal metrics with prescribed scalar curvature

Abstract

Let $(M,g)$ be an $n-$dimensional compact Riemannian manifold. Let $h$ be a smooth function on $M$ and assume that it has a critical point $\xi \in M$ such that $h(\xi )=0$ and which satisfies a suitable flatness assumption. We are interested in finding conformal metrics $g_{\lambda }=u_{\lambda }^{\frac{4}{n-2}}g$, with $u>0$, whose scalar curvature is the prescribed function $h_{\lambda }={\lambda }^2+h$, where $\lambda$ is a small parameter. In the positive case, i.e. when the scalar curvature $R_g$ is strictly positive, we find a family of bubbling metrics $g_{\lambda }$, where $u_{\lambda }$ blows-up at the point $\xi$ and approaches zero far from $\xi$ as $\lambda$ goes to zero. In the general case, if in addition we assume that there exists a non-degenerate conformal metric $g_0=u_0^{\frac{4}{n-2}}g$, with $u_0>0$, whose scalar curvature is equal to $h$, then there exists a bounded family of conformal metrics $g_{0,\lambda }=u_{0,\lambda }^{\frac{4}{n-2}}g$, with $u_{0,\lambda }>0$, which satisfies $u_{0,\lambda }\to u_0$ uniformly as $\lambda \to 0$. Here, we build a second family of bubbling metrics $g_{\lambda }$, where $u_{\lambda }$ blows-up at the point $\xi$ and approaches $u_0$ far from $\xi$ as $\lambda$ goes to zero. In particular, this shows that this problem admits more than one solution.

30 pages, final version. Accepted for publication in J. Differential Equations

References in corpus (2)

• Large Conformal metrics with prescribed sign-changing Gauss curvature
• Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3