Multi-bumps solutions for the nonlinear Schrodinger equation under a slowing decaying potential

In this paper, we consider the following nonlinear Schrodinger equation: -Delta u + V(vertical bar x vertical bar)u = Q(vertical bar x vertical bar)u(p), u > 0 in R-N, u is an element of H-1 (R-N), (0.1) where N >= 3, p is an element of (1, N+2/N-2), and V, Q have the algebraical decay that V(vertical bar x vertical bar) = V-0 + a/vertical bar x vertical bar(m) + O (1/vertical bar x vertical bar(m+kappa)), Q(vertical bar x vertical bar) = Q(0) + b/vertical bar x vertical bar n + O (1/vertical bar x vertical bar(n+theta)), as vertical bar x vertical bar -> infinity, where V-0, Q(0,) kappa, theta, a > 0, b is an element of R, and m, n > 1. By introducing the Miranda theorem, via the Lyapunov-Schmidt finite-dimensional reduction method, we construct infinitely many multi-bumps solutions of (0.1), whose maximum points of bumps lie on the top and bottom circles of a cylinder provided m < n, b > 0, or b <= 0. This result complements and extends the ones in [Duan and Musso, JDE, 2022] for a slow decaying rate of the potential function at infinity from m > max {2, 4/p-1} to m > 1.