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

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.