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

del Pino, Manuel, Mahmudi, Fethi, and Musso, Monica. "Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents." Journal of the European Mathematical Society 016.8 (2014): 1687-1748. <http://eudml.org/doc/277661>.

@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 -