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 $\Delta u+u^{\frac{n-k+2}{n-k-2}-\epsilon }=0in\Omega$, under zero Dirichlet boundary condition, where $\epsilon$ 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 $\epsilon ={\epsilon }_j$ and a solution $u_{\epsilon }$ which concentrate along $K$, as $\epsilon \to 0^{+}$, in the sense that $$ |\nabla u_{\varepsilon} |^2\,\rightharpoonup \, S_{n-k}^{\frac{n-k}{2}} \,δ_K \quad \mbox{as} \ \ \varepsilon \to 0 $$ 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.

50 Pages. arXiv admin note: substantial text overlap with arXiv:1107.5566, arXiv:1409.7321, Critical Sobolev Exponent, Blowing-up Solutions, Nondegenerate minimal submanifolds