Concentration at submanifolds for an elliptic Dirichlet problem near high critical exponents

TL;DR

This paper proves the existence of solutions to a near-critical elliptic PDE that concentrate along a minimal submanifold of the boundary of a domain, generalizing previous work for the case where the submanifold is one-dimensional.

Contribution

It extends prior results by establishing concentration phenomena along higher-dimensional minimal submanifolds for a class of elliptic equations near critical exponents.

Findings

Solutions concentrate along the submanifold as the parameter approaches zero.

The concentration occurs in the sense of measures, with the gradient squared converging to a Dirac measure.

The result generalizes previous work from one-dimensional to higher-dimensional submanifolds.

Abstract

Let $\Omega$ be a open bounded domain in $\mathbb{R}^n $ with smooth boundary $\partial\Omega$. We consider the equation $ \Delta u + u^{\frac{n-k+2}{n-k-2}-\varepsilon} =0\,\hbox{ in }\,\Omega $, under zero Dirichlet boundary condition, where $\varepsilon$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partial\Omega$, which is non-degenerate, and along which a certain weighted average of sectional curvatures of $\partial\Omega$ is negative. Under these assumptions, we prove existence of a sequence $\varepsilon=\varepsilon_j$ and a solution $u_{\varepsilon}$ which concentrate along $K$, as $\varepsilon \to 0^+$, in the sense that $$ |\nabla u_{\varepsilon} |^2\,\rightharpoonup \, S_{n-k}^{\frac{n-k}{2}} \,\delta_K \quad \mbox{as} \ \ \varepsilon \to 0 $$ where $\delta_K $ stands for the Dirac measure supported on $K$ and $S_{n-k}$ is an explicit positive constant. This result generalizes the one obtained by del Pino-Musso-Pacard, where the case $k=1$ is considered.