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

Mohammed Ben Ayed; Khalil El Mehdi; Mohameden Ould Ahmedou; Filomena Pacella

Journal of the European Mathematical Society (2005)

- Volume: 007, Issue: 3, page 283-304
- ISSN: 1435-9855

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topBen 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 -

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