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In this paper, we consider the following quasilinear equation:where  M  is a compact Riemannian manifold with dimensionwithout boundary, and. Here,andare continuous functions on  M  satisfying some further conditions. The operatoris the  p -Laplace–Beltrami operator on  M  associated with the metric  g , andis the Riemannian distance on. Moreover, we assume,, andwith. The notionis the critical Hardy–Sobolev  exponent . With the help of Mountain Pass Theorem, we get the existence results under different assumptions.

Mountain pass solution to a perturbated Hardy–Sobolev equation involving p -Laplacian on compact Riemannian manifolds