Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we consider equations like P_g u = u^{2#-1} where P_g u = (Δ_g)^2 u + α Δ_g u + a_α u is a Paneitz-Branson type operator with constant coefficients α and a_α, u is required to be positive, and 2# = 2n/(n-4) is critical from the Sobolev viewpoint. We define the energy function E_m as the infimum of E (u) = ||u||_{2#}^{2#} over the u's which are solutions of the above equation. We prove that E_m (α ) →+∞ as α →+∞ . In particular, for any Λ > 0, there exists α_0 >0 such that for α ≥ α_0, the above equation does not have a solution of energy less than or equal to Λ.

Felli, V., Hebey, E., Robert, F. (2005). Fourth order equations of critical Sobolev growth. Energy function and solutions of bounded energy in the conformally flat case. NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, 12(2), 171-213 [10.1007/s00030-005-0011-x].

Fourth order equations of critical Sobolev growth. Energy function and solutions of bounded energy in the conformally flat case

FELLI, VERONICA;
2005

Abstract

Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we consider equations like P_g u = u^{2#-1} where P_g u = (Δ_g)^2 u + α Δ_g u + a_α u is a Paneitz-Branson type operator with constant coefficients α and a_α, u is required to be positive, and 2# = 2n/(n-4) is critical from the Sobolev viewpoint. We define the energy function E_m as the infimum of E (u) = ||u||_{2#}^{2#} over the u's which are solutions of the above equation. We prove that E_m (α ) →+∞ as α →+∞ . In particular, for any Λ > 0, there exists α_0 >0 such that for α ≥ α_0, the above equation does not have a solution of energy less than or equal to Λ.
Articolo in rivista - Articolo scientifico
Paneitz-Branson type operators; energy function
English
2005
12
2
171
213
open
Felli, V., Hebey, E., Robert, F. (2005). Fourth order equations of critical Sobolev growth. Energy function and solutions of bounded energy in the conformally flat case. NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, 12(2), 171-213 [10.1007/s00030-005-0011-x].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/1578
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