Existence and properties of soliton solution for the quasilinear Schrödinger system

: In this article, we consider the following quasilinear Schrödinger system:


Introduction
We consider the following quasilinear Schrödinger system: , (1.1) where ≥ N 3, > < ε k 0, 0 are the real constants.In recent years, much more attention has been devoted to the quasilinear Schrödinger system Li proved the existence of nontrivial solution by using a change of variable and the Mountain Pass Theorem when > k 0 is large enough [1].Let ( ) ( ) ( ) 2 , Guo and Tang proved the existence of ground state solution by using the Nehari manifold method and concentration compactness principle in [2], which is localized near the potential well for λ large enough.In [3], minimax methods in a suitable Orlicz space were employed to establish the existence of standing wave solution for a quasilinear Schrödinger system involving subcritical nonlinearities.
For some systems similar to (1.2), Chen and Zhang have done several contributions.In [4], they obtained the existence of positive ground state solution by using Morse iteration to define a Pohožaev manifold.They obtained the existence of positive solution by using monotonicity trick and the Morse iteration in [5].They proved the existence of ground state solution by minimization under a convenient constraint and concentration compactness lemma in [6].They found the existence of ground state solution by establishing a suitable constraint set and studying related minimization problem in [7].
By establishing a suitable Nehari-Pohožaev-type constraint set and considering related minimization problem, the existence of ground state solution for a class of systems was proved in [8].The symmetric Mountain Pass Theorem was employed to establish the existence of infinitely many solutions for the quasilinear Schrödinger system in N in [9], which involves a parameter α and subcritical nonlinearities.By developing a new iterative technique and suitable estimation, the existence of the entire radial large solution was established for the modified quasilinear Schrödinger elliptic system in [10].
The study of System (1.1) was in part motivated by the nonlinear Schrödinger equation: where ( ) W x is a given potential, k is a real constant, and l and h are real functions that are essentially pure power forms.The quasilinear Schrödinger equation (1.3) describes several physical phenomena with different h, see [11][12][13] and references therein.We consider the case ( ) , one obtains a corresponding equation of elliptic type which has the formal variational structure (1.4) where ( ) F is the new potential function.Problem (1.4) has caused a heated discussion.When > k 0 is small enough, the existence result of multiple solutions was studied via dual approach techniques and variational methods [14].The paper [15] established the existence of soliton solution by a minimization argument.The minimax principles for lower semicontinuous functionals, developed by Szulkin [16], were used to find solutions in [17].The Mountain Pass Theorem combined with the principle of symmetric criticality to establish multiplicity of solutions in [18].Moameni [19] changed variables to remove nonconvex term and created a suitable Orlicz space to meet the Mountain Pass Theorem, and proved the existence of soliton solution for a quasilinear Schrödinger equation involving critical exponent in N .
Inspired by the above results, we apply the methods in [19] to solve system (1.1) and try to find nonnegative soliton solution.During this process, it is required that α β , are integers multiple of constant 2. The main result of this article is the following: The article is organized as follows.In Section 2, we reformulate this problem in an appropriate Orlicz space.In Section 3, we prove the existence of a solution for a special deformation of problem (1.1).Theorem 1.1 is proved in Section 4.

Reformulation of the problem and preliminaries
The energy functional associated with (1.1) is By changing variables, we treated this problem in an Orlicz space.From [20] and [19], we changed variables as follows: Since h is strictly monotone and has a well-defined inverse function Also, for some > C 0 0 it holds ( ) i 1, 2. Now we introduce the Orlicz space (see [21]): Existence and properties of soliton solution for quasilinear Schrödinger system  3 equipped with the norm: Using this change of variable, we can rewrite the functional ( ) defined in the space From the definition of G, only radially symmetric functions are in this space, and equipped with the norm: where L r is the Lebesgue function space with the norm Here are some related facts: (iv) The dual space ( ) , where C 1 is a constant independent of μ.
from H G 1 into ( ) 2 2 * and is compact for < < q 2 2 2 *. (vii) Suppose B k is the ball with center at the coordinate origin and radius Proof.See [20] for the proof of parts (i)-(vi).The proof of part (vii) is similar to the proof of [19].□ Denote by H r 1 the space of radially symmetric functions in Throughout this article, we use the standard notations.∫, H 1 , E G , L t , and ∥ ∥ ⋅ stand for ∫ N , H 1,2 , ( ) , respectively.We use C to denote any constant that is independent of the sequences considered, and the operation: ⁎ , ⁎ , where ⁎ represents any operation.

Auxiliary problem
In this section, we show some results needed to prove Theorem 1.1.Indeed, we consider a special deformation ( ) H z w ¯, ε (see (3.1) in the following) of ( ) satisfies the PS condition and to which we can apply the Mountain Pass Theorem.Consequently, ( ) H z w ¯, ε has a critical point for each > ε 0. We can use it to prove Theorem 1.1 in the next section.In fact, we will see that the functionals ( ) I z w ¯, ε and ( ) H z w ¯, ε will coincide for the small values of ε.This idea was explored in [19,22].
To do this, we shall consider constants θ and l satisfying Let > a 0 i be the value at which ( ) = ξ a a l where χ Λ denotes the characteristic function of the set Λ, which is a bounded domain.For convenience of later calculation, let Λ be a ring domain.Set ( ) ( ) 2 .According to [19], the functions y, Y satisfy the following conditions: Using the modification, the energy functional ( ) As in Section 2, we can rewrite the functional ( ) as follows: . Some properties of the functional H ¯are stated as follows: for every We will use the Mountain Pass Theorem like [23,24].First, let us define the Mountain Pass value where Next, we will prove Theorem 3.1 by the following lemmas.
Lemma 3.1.The functional H ¯satisfies the Mountain Pass geometry.
Proof.From the definition of Y , we have ( ) = H ¯0, 0 0. Clearly, there exists with ≢ e 0 and for the large values of t.Consequently, there exists ( ) ( ) . Therefore, the functional H ¯satisfies the Mountain Pass Geometry.□ This lemma guaranties the existence of a ( ) where < ≪ ρ 0 1.
For ( ) ∈ z w S , ρ , we have ( ) . By Hölder inequality and continuity of f , we obtain Also, it follows from ( ) y 1 and ( ) From (3.3), (3.4), and (3.5), we obtain . Hence, for ( which combines the definition of C 0 , and we have The following statements hold: (ii) For each > δ 0, there exists > R 0, such that Existence and properties of soliton solution for quasilinear Schrödinger system  7 and ε For part (i), pick .
For part (ii), let Hence, from (3.8) we have which stands for By ( ) y 2 , we obtain . Therefore, is bounded.Therefore, it follows from (3.11) that It proves part (ii).
For part (iii), from part (ii) we know that for each > δ 0, there exists > R 0, such that We might as well make , and it follows from ( ) Existence and properties of soliton solution for quasilinear Schrödinger system  9 and from (3.12), (3.13), we obtain and .
Then, by the compact embedding theorem and Lebesgue theorem, we obtain a subsequence still denoted by (3.17) It follows from part (vii) of Proposition 2.1 that the map: from H G 1 into ( ) Considering (3.17) and (3.18), it follows from (3.16) that for every > δ 0. Consequently, as → ∞ n .It proves part (iii).For part (iv), we know f is increasing and = f 0, hence ( ) For the second term and the fourth term on the right-hand side of (3.8), we have weakly in H G 1 , for the right-hand side of the above two inequalities we have n n By the dominated convergence theorem and the fact that ( a.e. in N , we obtain and For the fifth term on the right-hand side of (3.8), we have and similarly by the dominated convergence theorem, we obtain Also, we know and follows from part (vii) of Proposition 2.1 that the map: hence it follows from the dominated convergence theorem that a.e. in N .We may replace and Existence and properties of soliton solution for quasilinear Schrödinger system  11 It follows from part (iv) of Lemma 3.3 and (3.

Proof of Theorem 1.1
To prove Theorem 1.1, we need to find a critical point for the functional I ¯ε.It follows from [19] that the functionals I ¯ε and H ¯ε will coincide for the small values of ε, every critical point of H ¯ε will be a critical point of I ¯ε.Without loss of generality, assume ε 2 instead of ε in the functionals H ¯ε and I ¯ε, i.e., Existence and properties of soliton solution for quasilinear Schrödinger system  13 It follows from Theorem 3.1 that there exists a critical point ( The following lemmas are crucial for the proof of Theorem 1.1.
2 .Hence, we have It follows from the definition of the Mountain Pass value that And following the maximum property of function, we obtain Combining (4.1) and (4.3), we obtain Therefore, This lemma is liking [25].from which we obtain [19], it follows from Proposition 2.1(vi) that { ( )} f v n n is a bounded sequence in ∩ into account (3.35)-(3.40), the above limit implies = = = ′ = ′ = ′ = δ

Proof. For each > ε 0 ,2
it follows fromLemma 4Existence and properties of soliton solution for quasilinear Schrödinger system  15