Multiplicity and Limiting Profiles of Solutions for Schrödinger Equations with Asymptotically Linear Nonlinearities

In this paper, we are concerned with singularly perturbed Schr & ouml;dinger equation -is an element of Au-2 + V(x)u = f(u), x is an element of R-N , where is an element of > 0 is a small parameter, V possesses global minimum points, and f is asymptotically linear at infinity. Using variational methods and construction methods, we reveal the relationship between the number of solutions and the profile of the potential V. In particular, some new tricks and the method of Nehari manifold dependent on a suitable restricted set are introduced to overcome the difficulty resulting from the appearance of asymptotically linear nonlinearity.