Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities

We study the following Kirchhoff type equation: -(a+b integral(RN) vertical bar del u vertical bar(2)dx)Delta u + u = k(x)vertical bar u vertical bar(p-2)u + m(x)vertical bar u vertical bar(q-2)u in R-N, where N >= 3, a, b > 0, 1 < q < 2 < p < min{ 4, 2*}, 2* = 2N/(N - 2), k is an element of C(R-N) is bounded and m is an element of Lp/(p-q)(R-N). By imposing some suitable conditions on functions k(x) and m(x), we firstly introduce some novel techniques to recover the compactness of the Sobolev embedding H-1(R-N) -> L-r(R-N)(2 <= r < 2*); then the Ekeland variational principle and an innovative constraint method of the Nehari manifold are adopted to get three positive solutions for the above problem.