Supercritical problems on manifolds

TL;DR

This paper proves the existence of solutions to supercritical elliptic equations on compact manifolds that concentrate along submanifolds as parameters tend to critical values, under symmetry conditions.

Contribution

It establishes the existence of solutions concentrating on submanifolds for supercritical problems on manifolds, extending previous results to more general geometric settings.

Findings

Solutions concentrate along (m-1)-dimensional submanifolds

Existence results for supercritical nonlinearities

Concentration phenomena depend on symmetry assumptions

Abstract

Let $(M,g)$ be a $m$-dimensional compact Riemannian manifold without boundary. Assume $\kappa\in C^2(M)$ is such that $-\Delta_g+\kappa$ is coercive. We prove the existence of a solution to the supercritical problems $$ -\Delta_gu+\kappa u= u^p,\ u>0\quad\hbox{in}\ (M,g)\quad\hbox{and}\quad-\Delta_gu+\kappa u=\lambda e^u\quad\hbox{in}\ (M,g) $$ which concentrate s along a $(m-1)-$dimensional submanifold of $M$ as $p\to\infty$ and $\lambda\to0$, respectively, under suitable symmetry assumptions on the manifold $M$.