Bubbling solutions for supercritical problems on manifolds

TL;DR

This paper proves the existence of solutions to a supercritical nonlinear PDE on a compact manifold that concentrate along a nondegenerate closed geodesic as a parameter approaches zero, under certain curvature conditions.

Contribution

It establishes the existence of bubbling solutions concentrating along geodesics for supercritical problems on manifolds, linking PDE solutions to periodic ODEs with singularities.

Findings

Solutions concentrate along geodesics as epsilon approaches zero.

Conditions on curvature and function h ensure solution existence.

Connection to periodic ODEs with singularities is demonstrated.

Abstract

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold without boundary and $\Gamma$ be a non degenerate closed geodesic of $(M,g)$. We prove that the supercritical problem $$-\Delta_gu+h u=u^{\frac{n+1}{n-3}\pm\epsilon},\ u>0,\ \hbox{in}\ (M,g)$$ has a solution that concentrates along $\Gamma$ as $\epsilon$ goes to zero, provided the function $h$ and the sectional curvatures along $\Gamma$ satisfy a suitable condition. A connection with the solution of a class of periodic O.D.E.'s with singularity of attractive or repulsive type is established.