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 $$-\Delta _{g+\omega^2 \kappa}u+h u=u^{ {m+2\over m-2} \pm\varepsilon},\ u>0,\ \hbox{in}\ (M\times _\omega K,g+\omega^2 \kappa)$$ has a solution which concentrate along a $k$-dimensional minimal submanifold $\Gamma$ of $M\times _\omega N$ as the real parameter $\varepsilon$ goes to zero, provided the function $h$ and the sectional curvatures along $\Gamma$ satisfy a suitable condition.