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

Manuel del Pino; Fethi Mahmudi; Monica Musso

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 8, page 1687-1748
- ISSN: 1435-9855

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topdel 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 -

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