Supercritical problems on manifolds

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

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