Existence of positive solutions for nonlinear Kirchhoff type problems in R 3 with critical Sobolev exponent

In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent:− a + b ∫R 3| Du | 2Δu + u = f ( x , u ) + u 5 , x ∈ R 3 ,u ∈ H 1(R 3) , u > 0 , x ∈ R 3 ,where a , b  > 0 are constants. Under certain assumptions on the sign‐changing function f ( x , u ), we prove the existence of positive solutions by variational methods. Our main results can be viewed as a partial extension of a recent result of He and Zou in [Journal of Differential Equations, 2012] concerning the existence of positive solutions to the nonlinear Kirchhoff problem−ϵ 2 a + ϵb ∫R 3| Du | 2Δu + V ( x ) u = f ( u ) , x ∈ R 3 ,u ∈ H 1(R 3) , u > 0 , x ∈ R 3 ,where ϵ  > 0 is a parameter, V ( x ) is a positive continuous potential, and f ( u ) ∽ | u | p − 2 u with 4 <  p  < 6 and satisfies the Ambrosetti–Rabinowitz type condition. Copyright © 2013 John Wiley & Sons, Ltd.