# Bubbling along boundary geodesics near the second critical exponent

Manuel del Pino; Monica Musso; Frank Pacard

Journal of the European Mathematical Society (2010)

- Volume: 012, Issue: 6, page 1553-1605
- ISSN: 1435-9855

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topdel Pino, Manuel, Musso, Monica, and Pacard, Frank. "Bubbling along boundary geodesics near the second critical exponent." Journal of the European Mathematical Society 012.6 (2010): 1553-1605. <http://eudml.org/doc/277608>.

@article{delPino2010,

abstract = {The role of the second critical exponent $p=(n+1)/(n-3)$, the Sobolev critical
exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem $\Delta u+u^p=0$, $u>0$ under zero Dirichlet boundary conditions, in a domain $\Omega $ in $\mathbb \{R\}^n$ with bounded, smooth boundary. Given $\Gamma $, a geodesic of the boundary with negative inner normal curvature we find that for $p=(n+1)/(n-3-\varepsilon )$, there exists a solution $u_\varepsilon $ such that $|\nabla u_\varepsilon |^2$ converges weakly to a Dirac measure on $\Gamma $ as $\varepsilon \rightarrow 0^+$, provided that $\Gamma $ is nondegenerate in the sense of second variations of length and $\varepsilon $ remains away from a certain explicit discrete set of values for which a resonance phenomenon takes place.},

author = {del Pino, Manuel, Musso, Monica, Pacard, Frank},

journal = {Journal of the European Mathematical Society},

keywords = {critical Sobolev exponent; blowing-up solution; non degenerate geodesic; critical Sobolev exponent; blowing-up solution; non degenerate geodesic},

language = {eng},

number = {6},

pages = {1553-1605},

publisher = {European Mathematical Society Publishing House},

title = {Bubbling along boundary geodesics near the second critical exponent},

url = {http://eudml.org/doc/277608},

volume = {012},

year = {2010},

}

TY - JOUR

AU - del Pino, Manuel

AU - Musso, Monica

AU - Pacard, Frank

TI - Bubbling along boundary geodesics near the second critical exponent

JO - Journal of the European Mathematical Society

PY - 2010

PB - European Mathematical Society Publishing House

VL - 012

IS - 6

SP - 1553

EP - 1605

AB - The role of the second critical exponent $p=(n+1)/(n-3)$, the Sobolev critical
exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem $\Delta u+u^p=0$, $u>0$ under zero Dirichlet boundary conditions, in a domain $\Omega $ in $\mathbb {R}^n$ with bounded, smooth boundary. Given $\Gamma $, a geodesic of the boundary with negative inner normal curvature we find that for $p=(n+1)/(n-3-\varepsilon )$, there exists a solution $u_\varepsilon $ such that $|\nabla u_\varepsilon |^2$ converges weakly to a Dirac measure on $\Gamma $ as $\varepsilon \rightarrow 0^+$, provided that $\Gamma $ is nondegenerate in the sense of second variations of length and $\varepsilon $ remains away from a certain explicit discrete set of values for which a resonance phenomenon takes place.

LA - eng

KW - critical Sobolev exponent; blowing-up solution; non degenerate geodesic; critical Sobolev exponent; blowing-up solution; non degenerate geodesic

UR - http://eudml.org/doc/277608

ER -

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