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

In this paper, we consider the following quasilinear equation: Delta(p,g)u + a(x)vertical bar u vertical bar(p-2)u = K(x)vertical bar u vertical bar(p)* ((s)-2)u/d(g)(x, x(0))(s) + h(x)vertical bar u vertical bar(r-2)u, x is an element of M, where M is a compact Riemannian manifold with dimension n >= 3 without boundary, and x(0) is an element of M. Here a(x), K(x) and h(x) are continuous functions on M satisfying some further conditions. The operator Delta(p,g) is the p-Laplace- Beltrami operator on M associated with the metric g, and d(g) is the Riemannian distance on (M, g). Moreover, we assume p is an element of (1, n), s is an element of [0, p), and r is an element of (p, p*) with p* = np/n-p. The notion p* (s) = (n-s)p/n-p is the critical Hardy-Sobolev exponent. With the help of Mountain Pass Theorem, we get the existence results under different assumptions.