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In this paper, we consider the prescribed Gauduchon scalar curvature problem on almost Hermitian manifolds. By deducing the expression of the Gauduchon scalar curvature under the conformal variation, we reduce the problem to solving a semi-linear partial differential equation with exponential nonlinearity. Using the super- and sub-solutions method, we show that the existence of the solution to this semi-linear equation depends on the sign of a constant associated with the Gauduchon degree. When the sign is negative, we give both necessary and sufficient conditions such that a prescribed function is the Gauduchon scalar curvature of a conformal Hermitian metric. Besides, this paper recovers the Chern-Yamabe problem, the Lichnerowicz-Yamabe problem, and the Bismut-Yamabe problem. [ABSTRACT FROM AUTHOR]

We consider the prescribed scalar curvature problem on S N Δ S N v - N (N - 2) 2 v + K ~ (y) v N + 2 N - 2 = 0 on S N , v > 0 in S N , under the assumptions that the scalar curvature K ~ is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of O(3) obtained doubling the equatorial. We use the finite dimensional Lyapunov–Schmidt reduction method. [ABSTRACT FROM AUTHOR]

We consider the following prescribed curvature problem involving polyharmonic operator: D m u = Q (| y ′ | , y ′ ′) u m ∗ - 1 , u > 0 , u ∈ H m (S N) , where m ∗ = 2 N N - 2 m , N ≥ 4 m + 1 , m ∈ N + , (y ′ , y ′ ′) ∈ R 2 × R N - 2 , and Q (| y ′ | , y ′ ′) is a bounded nonnegative function in R + × R N - 2 . S N is the unit sphere with induced Riemannian metric g, D m is the polyharmonic operator given by D m = ∏ k = 1 m (- Δ g + 1 4 (N - 2 k) (N + 2 k - 2)) , where Δ g is the Laplace–Beltrami operator on S N . By using a finite reduction argument and local Pohozaev-type identities for polyharmonic operator, we prove that if N ≥ 4 m + 1 and Q (r , y ′ ′) has a stable critical point (r 0 , y 0 ′ ′) , then the above problem has infinitely many solutions, whose energy can be arbitrarily large. [ABSTRACT FROM AUTHOR]

Abstract: Let be a 4-dimensional compact Riemannian manifold and let be positive smooth functions on M. In this note, we prove that the problem always admits a positive solution, up to a conformal deformation of g. This leads to a geometric obstruction result for the prescribed scalar curvature problem. [Copyright &y& Elsevier]