Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents

Abstract

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^n$ with smooth boundary $\partial \Omega$. We consider the equation $d^2\Delta u-u+u^{\frac{n-k+2}{n-k-2}}=0in\Omega$ , under zero Neumann boundary conditions, where $\Omega$ is open, smooth and bounded and $d$ 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 certain weighted average of sectional curvatures of $\partial \Omega$ is positive along $K$. Then we prove the existence of a sequence $d=d_j\to 0$ and a positive solution $u_d$ such that

in the sense of measures, where ${\delta }_K$ stands for the Dirac measure supported on $K$ and $S$ is a positive constant.