Remarks on a Class of Nonlinear Schrödinger Equations with Potential Vanishing at Infinity

We study the following nonlinear Schrödinger equation −Δu + V(x)u = K(x)f(u), x ∈ RN, u ∈ H(R), where the potential V(x) vanishes at infinity. Working in weighted Sobolev space, we obtain the ground states of problem (P) under a Nahari type condition. Furthermore, if V(x), K(x) are radically symmetric with respect to x ∈ R, it is shown that problem (P) has a positive solution with some more general growth conditions of the nonlinearity. Particularly, if f(u) = u, then the growth restriction σ ≤ p ≤ N + 2/N − 2 in Ambrosetti et al. (2005) can be relaxed to ̃ σ ≤ p ≤ N + 2/N − 2, where ̃ σ < σ if 0 < β < α < 2.


Introduction
The motivation of the paper is concerned with the existence of standing waves of the following nonlinear Schrödinger equation: where  is the imaginary unit, Ṽ is a real function on R  , (, ) : R  ×[0, +∞) → C, and  is supposed to satisfy that (,   ) = (, )  for all ,  ∈ R. 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.
If the potential () decays to zero at infinity, the methods used in the proceeding papers cannot be employed because the variational theory in  1 (R  ) 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 − 2 Δ + () = ()  ,  ∈ R  ,  > 0, for all  satisfying 1 <  < ( + 2)/( − 2), 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][13][14][15][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 is, for some  > 0, and simultaneously () decays to zero at infinity.The main aim of the paper is to extend the result of [9] to problem (5) with much more general classes of ().Moreover, if (), () are radically symmetric with respect to  ∈ R  , problem (5) will be proved to also have a positive solution with some more general growth conditions of the nonlinearity.Particularly, the result can exactly extend the growth restriction of the special case of () =   to a new one, and the range of which is bigger than the usual  ≤  ≤ ( + 2)/( − 2) in [9]; for more details please see Theorem 6.
Throughout the paper, we make the following assumptions on (), (), and ().
Furthermore, we define the energy functional associated with problem (5) by By Proposition 2,  is well defined on  and  ∈  1 (, R) with Definition 4. A function  ∈  is said to be a solution of problem (5) provided that  ̸ ≡ 0 satisfies Notation.Hereafter we use the following notation.
(iii) For any  > 0 and for any  ∈ R  ,   () denotes the ball of radius  centered at .
Our main results are the following.

Variational Setting and Some Preliminaries
In this section, we describe the variational framework for the study of the critical points of the functional defined in (14).Set where N = { ∈  \ {0} : () ≜ ⟨  (), ⟩ = 0}.First, it is necessary to show that  is a positive number.Now we give the following two lemmas.
Proof.The proof of this lemma is similar to the case of assuming () condition, which can be found in [18], so we omit it here.

Proof of the Main Results
The aim of this section is to prove Theorems 5 and 6.For Theorem 5, we will take two steps; the first is to show the existence of nonzero critical point  ∈  of , and the second is to prove the critical point  is a bound state; that is  ∈  2 (R  ).
To prove Theorem 6, we use a weighted Sobolev embedding theorem, which is based upon the results discussed in [19].

Proof of Theorem 5
Step 1.Let {  } ⊂ N be the sequence minimizing for  given in (18).By Lemma 10, {  } is bounded in , and there is some 0 ̸ ≡  ∈  such that Since N is smooth, the minimizer is a critical point of .
Proof of Theorem 6.Based upon Proposition 14, the proof of Theorem 6 can be followed from some standard techniques; we leave the details to the readers.