Energy and Morse index of solutions of Yamabe type problems on thin annuli

Ben Ayed, Mohammed, et al. "Energy and Morse index of solutions of Yamabe type problems on thin annuli." Journal of the European Mathematical Society 007.3 (2005): 283-304. <http://eudml.org/doc/277341>.

@article{BenAyed2005,
abstract = {We consider the Yamabe type family of problems $(P_\varepsilon ):−\Delta u_\varepsilon =u_\varepsilon ^\{(n+2)/(n−2)\}$, $u_\varepsilon >0$ in $A_\varepsilon $, $u_\varepsilon =0$ on $\partial A_\varepsilon $, where $A_\varepsilon $ is an annulus-shaped domain of $\mathbb \{R\}^n$, $n\ge 3$, which becomes thinner as $\varepsilon \rightarrow 0$. We show that for every solution $u_\varepsilon $, the energy $\int _\{A_\varepsilon \}|\nabla u_|^2$ as well as the Morse index tend to infinity as $\varepsilon \rightarrow 0$. This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on $\mathbb \{R\}^n$, a half-space or an infinite strip. Our argument also involves a Liouville type theorem for regular solutions on an infinite strip.},
author = {Ben Ayed, Mohammed, El Mehdi, Khalil, Ould Ahmedou, Mohameden, Pacella, Filomena},
journal = {Journal of the European Mathematical Society},
keywords = {elliptic PDE; critical Sobolev exponent; blow up analysis; Liouville type theorem; Elliptic PDE, critical Sobolev exponent, blow up analysis, Liouville type theorem},
language = {eng},
number = {3},
pages = {283-304},
publisher = {European Mathematical Society Publishing House},
title = {Energy and Morse index of solutions of Yamabe type problems on thin annuli},
url = {http://eudml.org/doc/277341},
volume = {007},
year = {2005},
}

TY - JOUR
AU - Ben Ayed, Mohammed
AU - El Mehdi, Khalil
AU - Ould Ahmedou, Mohameden
AU - Pacella, Filomena
TI - Energy and Morse index of solutions of Yamabe type problems on thin annuli
JO - Journal of the European Mathematical Society
PY - 2005
PB - European Mathematical Society Publishing House
VL - 007
IS - 3
SP - 283
EP - 304
AB - We consider the Yamabe type family of problems $(P_\varepsilon ):−\Delta u_\varepsilon =u_\varepsilon ^{(n+2)/(n−2)}$, $u_\varepsilon >0$ in $A_\varepsilon $, $u_\varepsilon =0$ on $\partial A_\varepsilon $, where $A_\varepsilon $ is an annulus-shaped domain of $\mathbb {R}^n$, $n\ge 3$, which becomes thinner as $\varepsilon \rightarrow 0$. We show that for every solution $u_\varepsilon $, the energy $\int _{A_\varepsilon }|\nabla u_|^2$ as well as the Morse index tend to infinity as $\varepsilon \rightarrow 0$. This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on $\mathbb {R}^n$, a half-space or an infinite strip. Our argument also involves a Liouville type theorem for regular solutions on an infinite strip.
LA - eng
KW - elliptic PDE; critical Sobolev exponent; blow up analysis; Liouville type theorem; Elliptic PDE, critical Sobolev exponent, blow up analysis, Liouville type theorem
UR - http://eudml.org/doc/277341
ER -