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

TL;DR

This paper constructs solutions to a supercritical elliptic PDE on warped product Riemannian manifolds that concentrate along minimal submanifolds as a parameter approaches zero, revealing geometric and analytical interactions.

Contribution

It demonstrates the existence of solutions concentrating along minimal submanifolds for supercritical elliptic problems on warped product manifolds, extending previous results to more complex geometric settings.

Findings

Solutions concentrate along minimal submanifolds as epsilon approaches zero.

Existence of solutions depends on curvature and potential conditions.

Addresses supercritical nonlinearities on warped product 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.