On a Yamabe-type problem on a three-dimensional thin annulus. (English) Zbl 1161.35380

Ben Ayed, M.; Hammami, M.; El Mehdi, K.; Ould Ahmedou, M.

Adv. Differ. Equ. 10, No. 7, 813-840 (2005).

The authors consider the problem \[ -\Delta u = u^5\;\text{in}\;\Omega,\quad u>0\;\text{in}\;\Omega,\quad u = 0\;\text{on}\;\partial\Omega, \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^3\). The above equation arises in many mathematical and physical contexts, but its greatest interest lies in its relation with the Yamabe problem.
In the paper, the authors obtain a nonexistence results for the case where \(\Omega = A_{\varepsilon}\) is an annulus-shaped domain such that \(A_{\varepsilon}\) becomes “thin” as \(\varepsilon\to0\). More precisely, they proved the following: for any given positive constant \(C>0\), there exists \(\varepsilon_0>0\) such that, for any \(\varepsilon<\varepsilon_0\), the problem has no solution such that \(\int_{A_{\varepsilon}} |\nabla u_{\varepsilon}|^2 \leq C\).
The proof relies on obtaining refined estimates of the asymptotic profile of solutions \(u_{\varepsilon}\) as \(\varepsilon \to 0\). Another ingredient in the proof is a careful expansion of the associated functional and its gradient near a small neighborhood of highly concentrated functions.

Reviewer: Marcelo Furtado (Brasília)

MSC:

35J60	Nonlinear elliptic equations
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J25	Boundary value problems for second-order elliptic equations
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Keywords:

Yamabe problem