Existence Of Solutions For Kirchhoff Type Equations With Critical Exponents

In this paper, we mainly consider two problems. Firstly, we investigate the following Kirchhoff type boundary value problem involving a critical nonlinearity: where Ω (?) R3 is a bounded domain with smooth boundary (?)Ω, μ, is positive parameter, a,b> 0 are constants.6 is the critical Sobolev exponent, V is a positive continuous potential satisfying some conditions, and f ∈ C(Ω × R, R) is a subcritical nonlinear term. We use the variational methods to prove the existence of a positive solution.Secondly, we consider the following Kirchhoff type equation: where a, b> 0 are constants, V, Q and f are asymptotically periodic in the variable x. For a class of asymptotically periodic Kirchhoff systems with critical growth, the existence of ground states is established. The proof is based on the method of Nehari manifold and concentration compactness principle.