Soliton Solutions for Quasilinear Schrödinger Equations

By using a change of variables, we get new equations, whose respective associated functionals are well defined in and satisfy the geometric hypotheses of the mountain pass theorem. Using this fact, we obtain a nontrivial solution.


Introduction
We study the existence of solutions for the following quasilinear Schrödinger equations: where  ∈ (R  , R + ) is bounded and periodic in each variable of   , 1 ≤  ≤ ,  ≥ 3, () <  + 1 <  + 1 < 2 * := 2/( − 2),  ≥ 1, and here where  0 is defined in Lemma 2. These equations are related to existence of standing wave solutions for quasilinear Schrödinger equations of the form where  is a given potential and  and ℎ are real functions.Quasilinear equations such as (3) have been accepted as models of several physical phenomena corresponding to various types of .The case of () =   was used for the superfluid film equation in plasma physics [1].Besides, (3) also appears in plasma physics and fluid mechanics [2], in dissipative quantum mechanics [3], and in the theory of Heisenberg ferromagnetism and magnons [4,5].See also [6,7] for more physical backgrounds.Equations (3) with  = 1 have been studied extensively recently; see [8,9].When () = (1+) /2 , then (3) turn into our equations (1) with ℎ() =   +   .In particular if we let  = 1, that is, () = (1 + ) 1/2 , (3) models the self-channeling of a high-power ultrashort laser in matter [10].In this case, few results are known.In [11], the authors proved global existence and uniqueness of small solutions in transverse space dimensions 2 and 3 and local existence without any smallness condition in transverse space dimension 1.In [12], the authors proved the existence of nontrivial solution.When  > 1, although we do not know the physical background of (3), in a mathematical sense, we give the proof of the existence of nontrivial solution.
For (1), the main difficulty is that the energy functional associated to (1) is not well defined in  1 (R  ).To overcome this difficulty, enlightened by [8,9], we give a new change of variables.Then we reduce the quasilinear problem (1) to a semilinear one, which we will prove has a nontrivial solutions.
Our main result is the following.
In this paper,  denotes positive (possibly different) constant,   (R  ) denotes the usual Lebesgue space with norm

The Change of Variables
We note that the solutions of (1) are the critical points of the following functional: Since the functional () may not be well defined in the usual Sobolev spaces  1 (R  ), we make a change of variables as where Then after the change of variables, () can be written by By Lemma 2 listed below, we have lim  → 0  −1 ()/ = 1 and lim  → ∞ | −1 ()|  / = √ 2 ( > 1) or √2/3 ( = 1), so (V) We show that (7) is equivalent to Indeed, if we choose  = (1/()) in (7), then we get (8).On the other hand, since  =  −1 (V), if we let  = () in (8), we get (7).Therefore, in order to find the nontrivial solutions of (1), it suffices to study the existence of the nontrivial solutions of the following equations: Before we close this section, we give some properties of the change of variables.
Lemma 2. For all  > 0, one has the following: When  = 1, the result is obvious since () is an increasing bounded function.

Mountain Pass Geometry
In this section, we establish the geometric hypotheses of the mountain pass theorem.
Now we give the completion of the proof of Theorem 1.