Picture below is from the 20th page of Oh, Yong-Geun, Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, Commun. Math. Phys. 121, No. 1, 11-33 (1989). ZBL0693.35132.
First, how to put off the cut-off function $\chi_h$ in the first red ?
Besides, I can't understand why $\int V''(hy)\chi _h^2(y)u_0^2(y)dy$ is smooth at $h=0$ ?
Third, how to get the second red line ?
