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

@article{delPino2014,
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 $d=d_j\rightarrow 0$ and a positive solution $u_d$ such that $d^2 |\nabla u_\{d\} |^2\,\rightharpoonup \, S\,\delta _K \: \mathrm \{as\} \: d \rightarrow 0 $ in the sense of measures, where $\delta _K$ stands for the Dirac measure supported on $K$ and $S$ is a positive constant.},
author = {del Pino, Manuel, Mahmudi, Fethi, Musso, Monica},
journal = {Journal of the European Mathematical Society},
keywords = {critical Sobolev exponent; blowing-up solutions; nondegenerate minimal submanifolds; critical Sobolev exponent; blowing-up solutions; nondegenerate minimal submanifolds},
language = {eng},
number = {8},
pages = {1687-1748},
publisher = {European Mathematical Society Publishing House},
title = {Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents},
url = {http://eudml.org/doc/277661},
volume = {016},
year = {2014},
}

TY - JOUR
AU - del Pino, Manuel
AU - Mahmudi, Fethi
AU - Musso, Monica
TI - Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 8
SP - 1687
EP - 1748
AB - 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 $d=d_j\rightarrow 0$ and a positive solution $u_d$ such that $d^2 |\nabla u_{d} |^2\,\rightharpoonup \, S\,\delta _K \: \mathrm {as} \: d \rightarrow 0 $ in the sense of measures, where $\delta _K$ stands for the Dirac measure supported on $K$ and $S$ is a positive constant.
LA - eng
KW - critical Sobolev exponent; blowing-up solutions; nondegenerate minimal submanifolds; critical Sobolev exponent; blowing-up solutions; nondegenerate minimal submanifolds
UR - http://eudml.org/doc/277661
ER -