Solutions for a Kirchhoff type problem with critical exponent in RN

In this paper, we consider the existence of solutions of the Kirchhoff problem ( K ) − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + u = k f ( u ) + | u | 2 ⁎ − 2 u , in R N , where a , b > 0 , k > 0 and N ≥ 3 . We transform it into an equivalent system with respect to ( u , λ ) , which is easier to solve, ( S ) { − Δ u + u = k f ( u ) + | u | 2 ⁎ − 2 u , λ − a − b λ N − 2 2 ∫ R N | ∇ u | 2 d x = 0 , in R N × R + . With the equivalence of ( K ) to ( S ) , we obtain the existence of solutions for ( K ) by solving ( S ) . Existence of solutions of ( S ) is verified mainly by mountain pass theorem without (PS) condition. Our method works without the restriction of the (AR) condition in spaces of three or more dimensions.