We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.This section follows notations used in [20] and [21], where Bott-Morse foliations on dimension 3 were described.Let M m be a closed, orientable, smooth manifold of dimension m, for m ≥ 3. Let F be a codimension-one smooth foliation with singularities on M . Denote by Sing(F) the set of singular points of F.

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We show an iterative method to solve a Dirichlet problem for a Yamabe-type equation in small convex domains in R 3 and small balls in R n .

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We consider the non-trivial Ricci soliton on CP 2 #CP 2 constructed by Koiso and Cao. It is a Kähler metric invariant by the U (2) action on CP 2 #CP 2 . We study its Yamabe equation and prove it has exactly one U (2)−invariant solution up to homothecies.