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Existence of positive solutions for quasilinear elliptic equation on Riemannian manifolds

Let (M ,g) be a smooth compact Riemannian manifold of dimension n 3 . We study the existence of positive weak solutions for the following quasilinear elliptic equation −(Δp)gu+up−1 = f (x,u,∇gu) in M , where (Δp)gu = divg(|∇u|p−2 g ∇u) is the p -Laplacian operator on Riemannian manifold (M ,g) with 1 < p < n .