Multiple positive solutions to a Kirchhoff type problem involving a critical nonlinearity

In this paper, we investigate a class of Kirchhoff type problem involving a critical nonlinearity -(1+b integral(Omega) vertical bar del u vertical bar(2)dx) Delta u = lambda u + vertical bar u vertical bar(4)u, u is an element of H-0(1)(Omega), where b > 0, lambda(1) , lambda(1) is the principal eigenvalue of (- Delta, H-0(1)(Omega). With the help of the Nehari manifold, we obtain the multiplicity of positive solutions for A in a small right neighborhood of lambda(1) and prove that one of the solutions is a positive ground state solution, which is different from the result of Brezis-Nirenberg in 1983. This paper can be regarded as the complementary work of Naimen (2015). (C) 2016 Elsevier Ltd. All rights reserved.