Stationary Kirchhoff equations with powers

We discuss existence of solutions, compactness and stability properties in closed manifolds for the critical Kirchhoff equations (a + b integral(M) vertical bar del u vertical bar(2) dv(g))(theta 0) Delta(g)u + hu = u(p-1), where Delta(g) is the Laplace-Beltrami operator, h is a C-1-function in M, p epsilon (2, 2(star)], a, b, theta(0) > 0 are positive real numbers, and 2(star) is the critical Sobolev exponent. A fractional critical dimension d(0) = 2 (1+theta(0))/theta(0) appears in the critical case p = 2(star).