Blowing-up solutions concentrating along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds

Abstract

Let $(M,g)$ and $(K,\kappa )$ be two Riemannian manifolds of dimensions $m$ and $k,$ respectively. Let $\omega \in C^2(N),$ $\omega >0.$ The warped product $M{\times }_{\omega }K$ is the $(m+k)$-dimensional product manifold $M\times K$ furnished with metric $g+{\omega }^2\kappa .$ We prove that the supercritical problem $$-{\Delta }_{g+{\omega }^2\kappa }u+hu=u^{\frac{m+2}{m-2}\pm \epsilon },u>0,in(M{\times }_{\omega }K,g+{\omega }^2\kappa )$$ has a solution which concentrate along a $k$-dimensional minimal submanifold $\Gamma$ of $M{\times }_{\omega }N$ as the real parameter $\epsilon$ goes to zero, provided the function $h$ and the sectional curvatures along $\Gamma$ satisfy a suitable condition.