Let (M, g) be a smooth, compact Riemannian manifold of dimension N≥3. We consider the almost critical problem where Δg denotes the Laplace-Beltrami operator, Scalg is the scalar curvature of g and ε a R is a small parameter. It is known that problem (Pε) does not have any blowing-up solutions when ε ↗ 0, at least for N≤24 or in the locally conformally flat case, and this is not true anymore when ε ↘ 0. Indeed, we prove that, if N≥7 and the manifold is not locally conformally flat, then problem (Pε) does have a family of solutions which blow-up at a maximum point of the function ℰ → |Weylg(ℰ)|g as ε ↘ 0. Here Weylg denotes the Weylg curvature tensor of g:
Blowing-up solutions for the Yamabe equation / P., Esposito; Pistoia, Angela. - In: PORTUGALIAE MATHEMATICA. - ISSN 0032-5155. - STAMPA. - 71:3-4(2014), pp. 249-276. [10.4171/pm/1952]
Blowing-up solutions for the Yamabe equation
PISTOIA, Angela
2014
Abstract
Let (M, g) be a smooth, compact Riemannian manifold of dimension N≥3. We consider the almost critical problem where Δg denotes the Laplace-Beltrami operator, Scalg is the scalar curvature of g and ε a R is a small parameter. It is known that problem (Pε) does not have any blowing-up solutions when ε ↗ 0, at least for N≤24 or in the locally conformally flat case, and this is not true anymore when ε ↘ 0. Indeed, we prove that, if N≥7 and the manifold is not locally conformally flat, then problem (Pε) does have a family of solutions which blow-up at a maximum point of the function ℰ → |Weylg(ℰ)|g as ε ↘ 0. Here Weylg denotes the Weylg curvature tensor of g:I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
