We study the existence and
concentration of bound states to N-dimensional nonlinear Schrödinger equation
−𝜀2△u𝜀+ V (x)u𝜀= K(x)f(u𝜀), where N ≥ 3, 𝜀 > 0 is sufficiently small, and the
function f(s) is nonnegative and asymptotically linear at infinity. More concretely,
when f(s) ∼ O(s) as s → +∞, the potential function V (x) lies in C01(ℝN) with
V (x) ≥ 0 and V (x)≢0, and K(x) ≥ 0 is permitted to be unbounded under some
other necessary restrictions, we can show that a positive H1(ℝN)-solution u𝜀(x)
exists and concentrates around the local maximum point of the corresponding ground
energy function.