New multiplicity results for a class of nonlocal equation with steep potential well

In this paper, we consider a class of Kirchhoff type equation with Hartree nonlinearity: - (1 + a integral(RN)vertical bar del u vertical bar(2) dx) Delta u + lambda V(x)u = (l(alpha) * Q vertical bar u vertical bar(p)) Q vertical bar u vertical bar(p-2) u in R-N, where N >= 3, a, lambda > 0 parameters, V is an element of C(R-N, R+), l(alpha) is the Riesz potential, Q(x) >= 0 in R-N, and 1 + alpha/N < p < N+alpha/N-2. Under some suit- able assumptions for potential V(x) and spatial dimensions N, by using the filtration of Nehari manifold and Ekeland variational principle, we prove that above nonlocal equation admits at least two positive solutions. The effect of the steep potential well and spatial dimensions on the number of positive solutions are completely showed.