ORBITAL STABILITY OF STANDING WAVES FOR A CLASS OF SCHRÖDINGER EQUATIONS WITH UNBOUNDED POTENTIAL

This paper is concerned with the nonlinear Schr¨odinger equation with an unbounded potential iϕ t = − (cid:2) ϕ + V ( x ) ϕ − μ | ϕ | p − 1 ϕ − λ | ϕ | q − 1 ϕ , x ∈ R N , t ≥ 0, where μ > 0, λ > 0, and 1 < p < q < 1 + 4 /N . The potential V ( x ) is bounded from below and satisﬁes V ( x ) → ∞ as | x | → ∞ . From variational calculus and a compactness lemma, the existence of standing waves and their orbital stability are obtained.


Introduction
In this paper, we consider the nonlinear Schrödinger equation with an unbounded potential where μ > 0, λ > 0, and 1 < p < q < 1 + 4/N. The potential V (x) is bounded from below and satisfies V (x) → ∞ as |x| → ∞. Equation (1.1) has its physical background. For example, when V (x) = |x| 2 , it models the Bose-Einstein condensate with attractive interparticle interactions under magnetic trap [2,7,11,17,20]. When |D α V | is bounded for all |α| ≥ 2, in terms of the smoothness of the time 0 of Schrödinger kernel for potentials of quadratic growth provided by Fujiwara [9], Oh [13] established the local well-posedness of (1.1) in the corresponding energy space. Since Yajima [19] showed that for superquadratic potentials, the Schrödinger kernel is nowhere C 1 , we see that quadratic potentials are the highest-order potential for local well-posedness of (1.1). Thus the result of Oh [13], the local well-posedness of nonlinear Schrödinger equation with the potential function V (x), is indeed sharp.
We are interested in the following standing waves of (1.1): 2 Orbital stability where w ∈ R is a parameter and u(x) is the solution of the nonlinear elliptic equation The interesting topics to investigate standing waves are pursued strongly by many physicians and mathematicians [4,3,12,14,16]. For (1.3), Ding and Ni [8] by using "mountain pass" and comparison arguments got the existence of positive solutions. Rabinowitz [15] and Zhang [20,21] also studied the existence of the solutions for (1.3) by the method of variation. Hirose and Ohta [10] studied the uniqueness of the solution for (1.3).
This paper is organized as follows. In the second section, we give some necessary preliminaries which include the compactness lemma. In the third section, we prove the existence of the standing waves. And in the last section, we obtain their orbital stability.

Preliminaries
For (1.1), we impose the initial value as follows: In the course of nature, we set Here and hereafter, for simplicity, we denote R N dx by dx. H becomes a Hilbert space, continuously embedded in H 1 (R N ), when endowed with the inner product whose associated norm is denoted by · H .
and energy for all t ∈ [0,∞). Then when n ≥ m, we get (2.10) Here and hereafter C denotes various positive constant. Thus we get that It follows that the embedding H L 2 (R N ) is compact. For p > 1, using the conclusion of p = 1 and the Gagliardo-Nirenberg inequality, we can get the conclusion immediately.

The existence of standing waves
Firstly, we define a variational problem as follows: By the Gagliardo-Nirenberg inequality and (3.4), for 1 < p < q < 1 + 4/N, one has where 0 < θ 1 < θ 2 < 1. Hence, from (3.3) and (3.5), we have where ξ is between x 0 and x, and choosing Therefore, it follows from (3.4) and (3.10) that (3.11) which implies that E(u) ≥ d ρ . From (3.10) and with being coercive and convex, one has For any ρ > 0, let Ω ρ denote the set of the minimizers of the variational problem (3.2). Then for any u ∈ Ω ρ , by Theorem 3.1, there must exist a Lagrange multiplier w such that (3.14) It follows that ϕ(t,x) = e iwt u(x) is the standing wave solution of (1.1), which also called ground state since u is a minimizer of (3.2). Thus e iwt u(x) is the orbit of u. It is obvious that for any t ≥ 0, if u is a solution of (3.2), then e iwt u is also a solution of (3.2), which yields e iwt u ∈ Ω ρ .

Orbital stability of standing waves
Now in terms of Cazenave and Lion's argument [6], we have the following orbital stability.   It follows from (4.5) and the conservation laws in Lemma 2.1 that {ϕ n (t,·)} n∈N is a minimizing sequence for the problem (3.2). Therefore, there exists a u ∈ Ω ρ such that ϕ n t n ,· − u H −→ 0 as n −→ ∞. (4.6) This is contradictory with (4.4). The proof is complete.