The estimation and profile of the critical value for a Schrödinger equation

In this paper, we are concerned with the following Schrodinger problem: − Δ u + V ( x ) u = f ( u ) Open image in new window, u > 0 Open image in new window, in R N Open image in new window, where f : R → R Open image in new window is of class C 1 Open image in new window. The estimation and profile of the critical value of the corresponding functional is proved, which entails the relationship between the critical value on the balls and the least-energy value on the whole space. Our results are also true for three cases of the potential function V ( x ) Open image in new window.