Soluções blow-up para uma classe de problemas do tipo Yamabe em variedades com bordo

Abstract

In this work, we study the existence of blow-up solutions to the following class of problems ( −∆gu + a(x)u = 0 in M, ∂u ∂ν + n−2 2 b(x)u = (n − 2)u n n−2 ±ε on ∂M, (P±ε) where (M, g) is a compact Riemannian manifold of dimension n ≥ 5 with a boundary ∂M, ∆g is the Laplace-Beltrami operator with respect to the metric g, a ∈ C 1 (M), b ∈ C 1 (∂M), ν is the outward-pointing unit vector normal to ∂M and ε is a positive parameter. Assuming also that there is a real number θ > 0 such that Z M (|∇gu| 2 + a(x)u 2 )dµg + n − 2 2 Z ∂M b(x)u 2 dσ ≥ θ Z M (|∇gu| 2 + u 2 )dµg, we show that it is possible to build blow-up positive solutions to the case (P−ε) or to the case (P+ε), depending on the behavior of the function ϕ : ∂M → R, given by ϕ(q) = b(q) − H(q), where H is the boundary mean curvature.