The prescribed Gauduchon scalar curvature problem in almost Hermitian geometry

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.