On interacting bumps of semi-classical states of nonlinear Schrödinger equations

We study concentrated positive bound states of the following nonlinear Schr\"odinger equation: \[ h^2 Δ u - V(x) u + u^p=0,\ \ \ u>0, \ \ x ∈ R^N , \] where p is subcritical. We prove that, at a local maximum point x₀ of the potential function $V(x)$ and for arbitrary positive integer $K (K>1)$, there always exist solutions with K interacting bumps concentrating near x₀. We also prove that at a nondegenerate local minimum point of $V(x) $ such solutions do not exist.