The Existence and Uniqueness Result for a Relativistic Nonlinear Schrödinger Equation

and Applied Analysis 3 For (4), the result is obvious since g(t) is an increasing bounded function. Since

The superfluid film equation in plasma physics has this structure for () =  (see [6]).Putting (, ) = exp(−)(), where  ∈ R and  > 0 is a real function, (2) turns into the following equation: where () = () −  is the new potential function and  is the new nonlinearity.In this case, the first existence results are due to [7].In [7], the main existence results are obtained through a constrained minimization argument.Subsequently, a general existence result was derived in [8].
The idea in [8] is to make a change of variables and reduce the quasilinear problem to semilinear one and Orlicz space framework was used to prove the existence of positive solutions via the Mountain pass theorem.The same method of changing of variables was also used in [9] but the usual Sobolev space  1 (R  ) framework was used as the working space.Precisely, since the energy functional associated (3) is not well defined in  1 (R  ), they first make the changing of unknown variables V =  −1 (), where  is defined by ODE as follows: and () = −(−),  ∈ (−∞, 0].Then, after the changing of variable, to find the solutions of (2), it suffices to study the existence of solutions for the following semilinear equation: By using the classical results given by [10], they proved the existence of a spherically symmetric solution.In [11], the authors give a sufficient condition for uniqueness of the ground state solutions by using the same change of variables as [9].
In the case () = (1+) 1/2 , (2) models the self-channeling of a high-power ultrashort laser in matter (see [12]).In this case, few results are known.In [13], 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.But they did not study the existence of standing waves.But we have to point out that the method of change of variables as (4) cannot be generalized to treat the case () = (1 + ) 1/2 .In [14], the authors made the changing of known variable (see also [15]) and proved the existence of nontrivial solution with  ≥ 3 and  = 1.In this paper, for () = (1 + ) 1/2 and  > 0, we will show the existence and uniqueness result for (1) by using a change of variables due to [14,15].One main difficulty in dealing with this problem seems to be that of obtaining the boundedness of a (PS) sequence for the corresponding functional.We overcome this difficulty by using Jeanjean's result [16].
Our main result is the following.
In this paper, C denotes positive (possibly different) constant,   (R  ) denotes the usual Lebesgue space with norm

Preliminaries
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 () = √1 + ( 2 /2(1 +  2 )).Since () is monotonous with ||, the inverse function  −1 () of () exists.Then after the change of variables, () can be written as By Lemma 2 listed below, we have lim We show that ( 11) is equivalent to Indeed, if we choose  = (1/()) in (11), then we get (12).On the other hand, since  =  −1 (V), if we let  = () in (12), we get (11).Therefore, in order to find the nontrivial solutions of (1), it suffices to study the existence of the nontrivial solutions of the following equation: Before we close this section, we give some properties of the change of variables.

Existence
At first, we give two Lemmas.
Lemma 4.There exists ) with supp  :=  1 , we will prove that () → −∞ as  → ∞, which will prove the result if we take V =  with  large enough.By Lemma 2, we have  −1 () ≥  as  ≥ 1, so as  → ∞.Thus, we get the result.
We will use the following Theorem which is due to Jeanjean [16].
Theorem 5. Let  be a Banach space equipped with the norm ‖ ⋅ ‖ and let  ⊂ R + be an interval.One considers a family (  ) ∈ of  1 -functionals on  of the form where () ≥ 0, for all  ∈ , and such that either () → +∞ or () → +∞ as ‖  ‖ → ∞.One assumes that there are two points there hold, for all  ∈ , Then, for almost every  ∈ , there is a subsequence {V  ()} ⊂  such that We consider the functional where since and by Lemma 2 (6), we have so (V) → +∞.
Proof.We first note that {V  ()} ⊂  1 (R  ) satisfies and, for any Since {V  ()} is a bounded Palais-Smale sequence, there exists By the Lebesgue Dominated Theorem, we have Hence, V is a weak solution of (1).If V() ̸ ≡ 0, then we get the result.
From Lemma 6, we see that, for almost all  ∈ [1/2, 1], there exists a solution V() to the following Schrödinger equation: where Therefore, we can choose {  } ⊂ [1/2, 1] such that   → 1. Setting V  := V(  ), we have     (V  ) = 0. We can deduce that V is a solution to (13) if we show that   (V) = 0. To prove this, in view of Lemma 6, we first check that {V  } is bounded in  1 (R  ).
Notice that the Pohozaev identity implies that the solutions of (45) satisfy Lemma 7. The sequence {V  } is bounded.
In the second case, we define the sequence (if for a  ∈ N,   defined by ( 56) is not unique, we choose the smaller possible value).By construction {  } ⊂  1 (R  ) is bounded.Moreover by the definition of (56), we have so     (  )  = 0. Then following the proof above, we have   (  ) → 0 and lim inf  → ∞ (  ) = lim inf  → ∞    (  ).On the other hand, by the proof of Lemmas 3 and 2 (6), there exists a constant  > 0 such that     (V)V ≥  ‖ V‖ 2 + (‖ V‖ 2 ) as V → 0, uniformly in  ∈ N. Thus, since     (V  ) = 0, there is  > 0 such that ‖ V  ‖≥ , for all  ∈ N. Similarly, following the proof of Lemma 3, we have Using Lemma 6 again, we complete the proof of Lemma 8 which implies that  =  −1 (V) is a solution for (1).
Remark 9.In [14], the authors considered the existence of solutions for the following quasilinear Schrödinger equation: where the nonlinearity ℎ is Hölder continuous and satisfies the following conditions: there exists  > 2 such that for any  > 0, there holds 0 < ()() ≤ ()ℎ().

Uniqueness
In this section, we study the uniqueness of the positive radial solution of (13).We put (65) We apply the following uniqueness result due to Serrin and Tang [19].
Then the semilinear problem has at most one positive radial solution.
By Lemma 8, we can apply Theorem 11, Hence we obtain the uniqueness of positive radial solutions of (13).