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

### Mohammed Ben Ayed

Faculté des Sciences de Sfax, Tunisia### Khalil El Mehdi

Université de Nouakchott, Mauritania### Mohameden Ould Ahmedou

Universität Tübingen, Germany### Filomena Pacella

Università di Roma La Sapienza, Italy

## Abstract

In this paper we consider the following Yamabe type family of problem $(P_\e) : \quad -\D u_\e = u_\e ^{\frac{n+2}{n-2}}, \, \, u_\e > 0$ in $A_\e$, $u_\e =0$ on $\partial A_\e$, where $A_\e$ is an annulus-shaped domain of $\R^n$, $n\geq 3$, which becomes thinner when $\e\to 0$. We show that for every solution $u_{\e}$, the energy $\int_{A_{\e}} \, |\n u_{\e}|^2$, as well as the Morse index tends to infinity as $\e\to 0$. Such a result is proved through a fine blow-up analysis of some appropriate scalings of solutions whose limiting profiles are regular as well as singular solutions of some elliptic problem on $\R^n$, a half space or an infinite strip. Our argument involves also a Liouville-type theorem for regular solutions on the infinite strip.

## Cite this article

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella, Energy and Morse index of solutions of Yamabe type problems on thin annuli. J. Eur. Math. Soc. 7 (2005), no. 3, pp. 283–304

DOI 10.4171/JEMS/29