Existence of positive solutions for nonlinear Kirchhoff type problems in R^3 with critical Sobolev exponent and sign-changing nonlinearities

Abstract: In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent: -\left(a+b\ds\int_{\R^3}|D u|^2\right)\Delta u+u=f(x,u)+u^{5}, u\in H^1(\R^3), u>0, $x\in \R^3$ 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 [17] concerning the existence of positive solutions to the nonlinear Kirchhoff problem \left(\varepsilon^2a+\varepsilon b\ds\int_{\R^3}|D u|^2\right)\Delta u+V(x)u=f(u), u\in H^1(\R^3), u>0, $x\in \R^3$, where $\varepsilon>0$ is a parameter, $V(x)$ is a positive continuous potential and $f(u)\thicksim |u|^{p-2}u$ with $4<p<6$ and satisfies the Ambrosetti-Rabinowitz type condition.