Bubbling along boundary geodesics near the second critical exponent

del 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 -