Supercritical problems on manifolds

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 $M$.