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

### Manuel del Pino

Universidad de Chile, Santiago, Chile### Fethi Mahmoudi

Universidad de Chile, Santiago, Chile### Monica Musso

Ponificia Universidad Catolica de Chile, Santiago, Chile

## Abstract

Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary $\partial \Omega$. We consider the equation $d^2\Delta u - u+u^{\frac{n-k+2}{n-k-2}} =0\,\hbox{ in }\,\Omega$ , 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.

## Cite this article

Manuel del Pino, Fethi Mahmoudi, Monica Musso, Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents. J. Eur. Math. Soc. 16 (2014), no. 8, pp. 1687–1748

DOI 10.4171/JEMS/473