Solutions for a Kirchhoff type problem with critical exponent in RN

In this paper, we consider the existence of solutions of the Kirchhoff problem -(a +b integral(RN) vertical bar del u vertical bar(2)dx)Delta u + u = kf(u) + vertical bar u vertical bar(2*-2) u, in R-N (kappa) where a, b > 0, k > 0 and N >= 3. We transform it into an equivalent system with respect to (u, lambda), which is easier to solve, { -Delta u + u = kf(u) + vertical bar u vertical bar(2*-2) u, in R-N x R+ lambda - a - b lambda N-2/2 integral (RN) vertical bar del u vertical bar(2)dx = 0, With the equivalence of (kappa) to (S), we obtain the existence of solutions for (kappa) 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. (c) 2020 Elsevier Inc. All rights reserved.