Remarks on a Class of Nonlinear Schr？dinger Equations with Potential Vanishing at Infinity

We study the following nonlinear Schr？dinger equation , where the potential vanishes at infinity. Working in weighted Sobolev space, we obtain the ground states of problem under a Nahari type condition. Furthermore, if are radically symmetric with respect to , it is shown that problem has a positive solution with some more general growth conditions of the nonlinearity. Particularly, if , then the growth restriction in Ambrosetti et al. (2005) can be relaxed to , where if . 1. Introduction The motivation of the paper is concerned with the existence of standing waves of the following nonlinear Schr？dinger equation: where is the imaginary unit, is a real function on , , and is supposed to satisfy that for all . Problem (1) arises in many applications. For example, in some problems arising in nonlinear optics, in plasma physics, and in condensed matter physics, the presence of many particles leads one to consider nonlinear terms which simulates the interaction effect among them. For problem (1), we are interested in looking for a stationary solution; that is, with in and (the frequency); then it is not difficult to see that must satisfy Here and below, . Variational approach to (2) was initiated by Rabinowtiz [1], and since then several authors have studied (2) under different assumptions on and the nonlinearity. If is positive and bounded away from zero, then, by the well-known concentration compactness principle [2, 3], it is shown that there is bound states for problem (2); we mention here the work by Jeanjean and Tanaka [4, 5], Liu and Wang [6], Li et al. [7], Zhu [8], and the references therein. If the potential decays to zero at infinity, the methods used in the proceeding papers cannot be employed because the variational theory in cannot be used here. The earlier work on (2) we know of where decays at infinity, is that by Ambrosetti et al. [9]; the authors proved that problem (2) has bounded states for with where Following [9], by requiring some further assumptions on , in [10], the authors showed that there exist bound states of equation , , , for all satisfying , provided that is sufficiently small. Motivated by the works [9, 10], in paper [11], the authors extended the results to potentials that can both vanish and decay to zero at infinity. And since then, there are many papers on problem (2) with potential vanishing at infinity; see, for example, [12–16]. In this paper, more precisely we will focus on the following model equation: To our best knowledge, it seems that there are few results on problem (5), where does not satisfy condition; that