Stationary Kirchhoff equations with powers

We discuss existence of solutions, compactness and stability properties in closed manifolds for the critical Kirchhoff equations ( a + b ∫ M | ∇ u | 2 d v g ) θ 0 Δ g u + h u = u p - 1 , \Bigg{(}a+b\int_{M}\lvert\nabla u|^{2}\,dv_{g}\Bigg{)}^{\theta_{0}}\Delta_{g}u% +hu=u^{p-1}, where Δ g {\Delta_{g}} is the Laplace–Beltrami operator, h is a C 1 {C^{1}} -function in M, p ∈ ( 2 , 2 ⋆ ] {p\in(2,2^{\star}]} , a , b , θ 0 > 0 {a,b,\theta_{0}>0} are positive real numbers, and 2 ⋆ {2^{\star}} is the critical Sobolev exponent. A fractional critical dimension d 0 = 2 ⁢ ( 1 + θ 0 ) θ 0 {d_{0}=\frac{2(1+\theta_{0})}{\theta_{0}}} appears in the critical case p = 2 ⋆ {p=2^{\star}} .