Multi-peak positive solutions to a class of Kirchhoff equations

In the present paper, we consider the nonlocal Kirchhoff problem -(epsilon(2)a + epsilon b integral(R3) vertical bar del u vertical bar(2) Delta u + V(x)u(p) = u> 0 in R-3, where a, b > 0, 1 < p < 5 are constants, epsilon > 0 is a parameter. Under some mild assumptions on the function V, we obtain multi-peak solutions for epsilon sufficiently small by Lyapunov-Schmidt reduction method. Even though many results on single peak solutions to singularly perturbed Kirchhoff problems have been derived in the literature by various methods, there exist no results on multi-peak solutions before this paper, due to some difficulties caused by the nonlocal term (integral(R3)vertical bar del u vertical bar(2)).u. A remarkable new feature of this problem is that the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single Kirchhoff equation, which is quite different from most of the elliptic singular perturbation problems.