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 $$-{\Delta }_{g}u+\kappa u=u^p,u>0in(M,g)and-{\Delta }_{g}u+\kappa u=\lambda e^{u}in(M,g)$$ which concentrate s along a $(m-1)-$dimensional submanifold of $M$ as $p\to \infty$ and $\lambda \to 0$, respectively, under suitable symmetry assumptions on the manifold $M$.