Multiple solutions to the nonhomogeneous p-Kirchhoff elliptic equation with concave-convex nonlinearities

In this paper, we study the multiplicity of solutions for the nonhomogeneous p-Kirchhoff elliptic equation -M(parallel to del u parallel to(p)(p))Delta(p)u = lambda h(1)(x)vertical bar u vertical bar(q-2)u + h(2)(x)vertical bar u vertical bar(r-2)u + h(3)(x), x is an element of Omega, (0.1) with zero Dirichlet boundary condition on partial derivative Omega, where Omega is the complement of a smooth bounded domain D in R-N (N >= 3). lambda > 0, M(s) = a + bs(k), a, b > 0, k >= 0, h(1)(x), h(2)(x) and h(3)(x) are continuous functions which may change sign on Omega. The parameters p, q, r satisfy 1 < q < p(k + 1) < r < p* = Np/N-p. A new existence result for multiple solutions is obtained by the Mountain Pass Theorem and Ekeland's variational principle. (C) 2013 Elsevier Ltd. All rights reserved.