LAUSR

Abstract In [3] , J.S. Case considered a Yamabe type problem in the setting of smooth measure space for a Riemannian manifold and a parameter m, which generalize the original Yamabe problem when m = 0 and Perelman's ν-entropy when m = ∞ . In the context of Euclidean space this generalization consists to find the functions that satisfy the sharp Gagliardo, Nirenberg, Sobolev inequalities. We point out that J.S. Case also solved this problem when the parameter m is natural number. In this work, by using the Schoen's argument [12] , we prove the result of J.S. Case for a special class of compact n-dimensional smooth metric measure space when the parameter m belongs to I = ⋃ i = 0 ∞ [ i , 1 4 ( 4 − 3 n + n 2 + 16 n + 16 ) + i ) .