This article is concerned with the following quasilinear Schrödinger system: − Δ u + u + k 2 Δ | u | 2 u = 2 α α + β | u | α − 2 u | v | β , x ∈ R N , − Δ v + v + k 2 Δ | v | 2 v = 2 β α + β | u | α | v | β − 2 v , x ∈ R N , (1.1) where k < 0 is a real constant.

Many scholars have made significant contributions to the study of the quasilinear Schrödinger system. Wang and Huang proved the existence of ground-state solutions for a class of systems by establishing a suitable Nehari–Pohožaev-type constraint set and considering related minimization problems in [2]. The existence of infinitely many solutions was established for the quasilinear Schrödinger system by the symmetric Mountain Pass Theorem; see [3]. The existence of positive solutions was obtained by using the monotonicity trick and Morse iteration in [4]. Chen and Zhang proved the existence of ground-state solutions by minimization under a convenient constraint and concentration compactness lemma in [5].

The quasilinear Schrödinger system (1.1) is in part motivated by the following quasilinear Schrödinger equation: i ϵ ∂ z = − ϵ Δ z + W x z − l | z | 2 z − k ϵ Δ h | z | 2 h ′ | z | 2 z , for x ∈ R N , N > 2 , (1.2) 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 1.2 describes several physical phenomena with different h; see [6–8].

Let the case h ( s ) = s , l ( s ) = μ s p − 1 2 and k > 0. Setting z(t, x) = exp(−iFt)u(x), one can obtain a corresponding equation of elliptic type which has the formal variational structure: ϵ Δ u + V x u − ϵ k Δ | u | 2 u = μ | u | p − 1 u , u > 0 x ∈ R N , N > 2 , (1.3) where V(x) = W(x) – F is the new potential function. The problem (1.3) has been studied by many academics. In [9], the existence results of multiple solutions were studied via dual approach techniques and variational methods when k > 0 was small enough. The existence of soliton solutions was established by a minimization argument; see [10]. The Mountain Pass Theorem is combined with the principle of symmetric criticality to establish the multiplicity of solutions in [11]. In [12], the author proved the existence of soliton solutions via making a change in variables and creating a suitable Orlicz space. The minimax principles for lower semicontinuous functionals were used to find solutions in [13].

In [14], the authors used the method developed by [1, 15] to divide the energy functional into two parts and established the existence of ground-state solutions for a type of quasilinear Schrödinger equation like 1.3. Inspired by [14], we try to find the existence of ground-state solutions for system 1.1. This achievement can enrich the relatively few existing results about this system.

The main result of this paper is the following:

Theorem 1.1

When k < 0, α > 1, β > 1, and α + β < 2*, then (1.1) has a ground-state solution.

This paper is organized as follows. In Section 2, preparation work is completed. In Section 3, we reformulate this problem and prove Theorem 1.1. In this paper, C is defined as different constants.

First, we explain that L q ( R N ) denotes the Lebesgue space with the norm u p = ∫ R N | u | p d x 1 p , where 1 ≤ p < ∞. L q = L q ( R N ) × L q ( R N ) with the norm u , v p = ∫ R N | u | p d x 1 p + ∫ R N | v | p d x 1 p , where 1 ≤ p < ∞. H 1 = u , v : u , v ∈ L 2 R N , ∇ u , ∇ v ∈ L 2 R N with norms u , v = u + v = ∫ R N | ∇ u | 2 + u 2 d x 1 2 + ∫ R N | ∇ v | 2 + v 2 d x 1 2 and u , v 2 = u 2 + v 2 .

The embedding H ¹↪L ^q is continuous and compact for q ∈ (2, 2*).

In (1.1), the Euler–Lagrange functional associated with Equation 1.1 is given by I u , v = 1 2 ∫ R N 1 − k u 2 | ∇ u | 2 d x + 1 2 ∫ R N | u | 2 d x + 1 2 ∫ R N 1 − k v 2 | ∇ v | 2 d x + 1 2 ∫ R N | v | 2 d x − 2 α + β ∫ R N | u | α | v | β d x .

For (u, v), constructing the variable like [16, 17], we have d z = − k 1 − k u 2 d u , z = h u = 1 2 − k u 1 − k u 2 + 1 2 l n − k u + 1 − k u 2 , d w = − k 1 − k v 2 d v , w = h v = 1 2 − k v 1 − k v 2 + 1 2 l n − k v + 1 − k v 2 .

Since h is strictly monotone, it has a well-defined inverse function f and u = f(z), v = f(w). Note that h u ∼ − k u , | u | ≪ 1 − k − k 2 u | u | , | u | ≫ 1 − k , h ′ u = − k 1 − k u 2 and f z ∼ 1 − k z , | z | ≪ 1 − k 2 − k | z | z , | z | ≫ 1 − k , f ′ z = 1 h ′ u = 1 − k 1 − k v 2 = 1 − k 1 − k f z 2 .

Similarly, the same operation holds true for v = f(w).

Using the variable, (1.1) will become − 1 k Δ z + f z f ′ z = 2 α α + β | f z | α − 2 f z | f w | β , x ∈ R N , − 1 k Δ w + f w f ′ w = 2 β α + β | f z | α | f w | β − 2 f w , x ∈ R N , (2.1) where f : [ 0 , ∞ ) → R and f ′ = 1 − k 1 − k f 2 on [0, ∞), f(0) = 0, and f(−t) = f(t) on [0, ∞). From the above facts, if (z, w) is a weak solution for (2.1), then ( u , v ) = f ( z ) , f ( w ) is a weak solution for (1.1). The energy functional I(u, v) reduces to the following functional: ϕ z , w = 1 2 ∫ R N 1 − k | ∇ z | 2 d x + 1 2 ∫ R N f 2 z d x + 1 2 ∫ R N 1 − k | ∇ w | 2 d x + 1 2 ∫ R N f 2 w d x − 2 α + β ∫ R N | f z | α | f w | β d x . (2.2)

There are some properties of f : R → R as follows, which are proved in [16, 17].

Lemma 2.1

The function f(t) and its derivative satisfy the following properties:

(i) f ( t ) t → 1 as t → 0;

In this section, we will complete the proof of Theorem 1.1. First, we will recall the critical point theorem in [1], which is crucial for proving Theorem 1.1.

Theorem 3.1

Let X , ( ⋅ , ⋅ ) be a Banach space and L ⊂ R + an interval. Consider the following family of C ¹ -functionals on X : Φ λ z , w = A z , w + λ B z , w , λ ∈ L , with B being non-negative and either A(z, w) → +∞ or B(z, w) → +∞ as ( z , w ) → ∞ . Assume that there are two points (z ₁, w ₁), (z ₂, w ₂) ⊂ X such that c λ = inf γ ∈ Γ λ max t 1 , t 2 ∈ 0,1 × 0,1 Φ λ γ t 1 , t 2 > max Φ λ z 1 , w 1 , Φ λ z 2 , w 2 for all λ ∈ L , where Γ _λ = {γ ∈ C([0, 1] × [0, 1], X): γ(0, 0) = (z ₁, w ₁), γ(1, 1) = (z ₂, w ₂)}. Then, for almost every λ ∈ L, there is a sequence {(z _n , w _n )} ⊂ X such that

(i) (z _n , w _n ) is bounded;

Moreover, the map λ → c _λ is non-increasing and continuous from the left.

Let λ ∈ L be an arbitrary but fixed value where c λ ′ exists, where c λ ′ is the derivative of c _λ with respect to λ. Let {λ _n } ⊂ L be a strictly increasing sequence such that λ _n → λ. To prove Theorem 3.1, we will show the following lemmas:

Lemma 3.1. There exists a sequence of path {γ _n } ⊂ Γ and K = K ( c λ ′ ) > 0 such that

(ii) max ( t 1 , t 2 ) ∈ [ 0,1 ] Φ λ ( γ n ( t 1 , t 2 ) ) ≤ c λ + ( − c λ ′ + 2 ) ( λ − λ n ) .

Proof. The proof is standard; see [1].

Lemma 3.1

means that there exists a sequence of paths {γ _n } ⊂ Γ such that max t 1 , t 2 ∈ 0,1 × 0,1 Φ λ γ n t 1 , t 2 → c λ , for all n ∈ N sufficiently large; starting from a level strictly below c _λ , all the “top” of the path is contained in the ball centered at the origin of fixed radius K = K ( c λ ′ ) > 0 . Now, for α > 0, we define F α = z , w ∈ X : z , w ≤ K + 1  and  | Φ λ z , w − c λ | ≤ α , where K is given in lemma 3.1.

Lemma 3.2

For all α > 0, inf Φ λ ′ z , w : z , w ∈ F α = 0 . (3.2)

Proof. We assume that (3.2) does not hold. Then, there exists α > 0 such that for any (z, w) ∈ F _α , we obtain Φ λ ′ z , w ≥ α . (3.3)

Without loss of generality, we can assume that 0 < α < 1 2 c λ − max Φ λ z 1 , w 1 , Φ λ z 2 , w 2 .

A classical deformation argument then says that there exists ϵ ∈ [0, α] and a homeomorphism η: X → X such that η u = u , if | Φ λ z , w − c λ | ≥ α , (3.4) Φ λ η z , w ≤ Φ λ z , w , ∀ z , w ∈ X , (3.5) Φ λ η z , w ≤ c λ − ϵ , ∀ z , w ∈ X ,  satisfying  z , w ≤ K  and  Φ λ z , w ≤ c λ + ϵ . (3.6)

Let {γ _n } ⊂ Γ be the sequence obtained in lemma 3.1. We choose and fix m ∈ N sufficiently large in order that − c λ ′ + 2 λ − λ m ≤ ϵ . (3.7)

By lemma 3.1 and (3.4), η(γ _m ) ∈ Γ. Now if (z, w) = γ _m (t ₁, t ₂) satisfies Φ λ z , w ≤ c λ − λ − λ m , then (3.5) implies that Φ λ η z , w ≤ c λ − λ − λ m . (3.8)

If (z, w) = γ _m (t ₁, t ₂) satisfies Φ λ z , w > c λ − λ − λ m , by lemma 3.1 and (3.7), we obtain (z, w) such that ( z , w ) ≤ K with Φ _λ (z, w) ≤ c _λ + ϵ. From (3.6), we obtain Φ λ η z , w ≤ c λ − ϵ ≤ c λ − λ − λ m . (3.9)

Combining (3.8) with (3.9), we obtain max t 1 , t 2 ∈ 0,1 × 0,1 Φ λ η γ m t 1 , t 2 ≤ c λ − λ − λ m , which contradicts the variational characterization of c _λ .

Next, we prove theorem 3.1.

Proof. Since lemma 3.2 is true, there exists a PS sequence for Φ _λ at the level c λ ∈ R , which is contained in the ball of radius K + 1 centered at the origin. Hence, this is proved.

Let L = 1 2 , 1 , we define the following energy functional: Φ λ z , w = 1 2 ∫ R N 1 − k | ∇ z | 2 + z 2 + 1 − k | ∇ w | 2 + w 2 d x − λ ∫ R N 1 2 z 2 − f 2 z + w 2 − f 2 w + 2 α + β | f z | α | f w | β d x , (3.10) where λ ∈ L. Moreover, let A z , w = 1 2 ∫ R N 1 − k | ∇ z | 2 + z 2 + 1 − k | ∇ w | 2 + w 2 d x and B z , w = λ ∫ R N 1 2 z 2 − f 2 z + w 2 − f 2 w + 2 α + β | f z | α | f w | β d x .

Letting ( z , w ) → + ∞ , then A(z, w) → +∞ and B(z, w) ≥ 0.

By a standard argument in [18, 19], we have the following Pohožaev-type identity:

Lemma 3.3

If (z, w) ∈ H ¹ is a critical point of (3.10), then (z,w) satisfies P _λ (z, w) = 0, where P λ z , w ≔ N − 2 2 ∫ R N 1 − k | ∇ z | 2 + | ∇ w | 2 d x + N 2 ∫ R N f 2 z + f 2 w d x − 2 N λ α + β ∫ R N | f z | α | f w | β d x . (3.11)

Similar to [9], we obtain the following lemma:

Lemma 3.4

Φ _λ (z, w) meet the conditions as follows:

(i) there exists (z, w) ∈ H ¹ \{(0, 0)} such that Φ _λ (z, w) < 0 for all λ ∈ L ;

Proof. (i) Let (z, w) ∈ H ¹ \{(0, 0)} be fixed. For any λ ∈ L = 1 2 , 1 , we obtain Φ λ z , w ≤ Φ 1 2 z , w = 1 2 ∫ R N 1 − k | ∇ z | 2 + | ∇ w | 2 d x + 1 4 ∫ R N z 2 + f 2 z + w 2 + f 2 w d x − 1 α + β ∫ R N | f z | α | f w | β d x .

As [20, 21], we consider ϕ , φ ∈ C 0 ∞ ( R ) such that 0 ≤ ϕ(x) ≤ 1, 0 ≤ φ(x) ≤ 1 and ϕ x = 1 , if | x | ≤ 1 , 0 , if | x | ≥ 1 , φ x = 1 , if | x | ≤ 1 , 0 , if | x | ≥ 1 .

By Lemma 2.1 (ii) and (v), we obtain | f t ϕ | ≥ C | t ϕ | ≥ C f t ϕ .

By Lemma 2.1 (ii), Φ λ t 1 ϕ , t 2 φ ≤ 1 2 ∫ R N 1 − k | ∇ t 1 ϕ | 2 + t 1 2 ϕ 2 d x + 1 2 ∫ R N 1 − k | ∇ t 2 φ | 2 + t 2 2 φ 2 d x − 1 α + β ∫ R N | f t 1 ϕ | α | f t 2 φ | β d x ≤ t 1 2 2 ∫ R N 1 − k | ∇ ϕ | 2 + ϕ 2 d x + t 2 2 2 ∫ R N 1 − k | ∇ φ | 2 + φ 2 d x − C | f t 1 | α + | f t 2 | β α + β ∫ R N | ϕ | α | φ | β d x .

It follows that Φ _λ (t ₁ ϕ, t ₂ φ) → −∞ as (t ₁, t ₂) → (+∞, + ∞). Thus, there exists (t ₃, t ₄) > 0 such that Φ _λ (t ₃ ϕ, t ₄ φ) < 0. Thus, taking (z, w) = (t ₃ ϕ, t ₄ φ), we obtain Φ _λ (z, w) < 0 for all λ ∈ L.

(ii) As [20, 22], there exists C > 0 and ρ ₁ > 0 small enough such that ∫ R N 1 − k | ∇ z | 2 + f 2 z + 1 − k | ∇ w | 2 + f 2 w d x ≥ C z , w , for ( z , w ) ≤ ρ 1 . From Lemma 2.1 (iii) and Hölder inequality, we obtain Φ λ z , w ≥ 1 2 ∫ R N 1 − k | ∇ z | 2 + f 2 z d x + 1 2 ∫ R N 1 − k | ∇ w | 2 + f 2 w d x − 1 α + β ∫ R N | f z | α | f w | β d x ≥ C z , w − 1 α + β ∫ R N | f z | α | f w | β d x ≥ C z , w − C z α 1 p w β 1 p ′ for all z , w ≤ ρ 1 , where α ₁ = α or α 2 , β ₁ = β or β 2 , and ( 1 p + 1 p ′ ) = 1 . It can conclude that Φ _λ has a strict local minimum at 0, and hence, c _λ > 0.

By Theorem 3.1, it is easy to know that for every λ ∈ 1 2 , 1 , there exists a bounded sequence (z _n , w _n ) ⊂ H ¹ such that Φ _λ (z _n , w _n ) → c _λ and Φ λ ′ ( z n , w n ) → 0 .

Lemma 3.5

If (z _n , w _n ) ⊂ H ¹ is the sequence obtained above, then for almost every λ ∈ L = [ 1 2 , 1 ] , there exists (z _λ , w _λ ) ∈ H ¹ \{(0, 0)} such that Φ _λ (z _λ , w _λ ) → c _λ and Φ λ ′ ( z λ , w λ ) → 0 .

Proof. Since (z _n , w _n ) is bounded in H ¹, up to a subsequence, there exists (z _λ , w _λ ) ∈ H ¹ such that z n , w n ⇀ z λ , w λ in H 1 , z n , w n → z λ , w λ  in  L s  for all  2 < s < 2 * , z n x , w n x → z λ x , w λ x  a. e. in  R N .

Since Φ λ ′ ( z n , w n ) → 0 , by the Lebesgue dominated convergence theorem, it is easy to get Φ λ ′ ( z n , w n ) → Φ λ ′ ( z λ , w λ ) , that is, Φ λ ′ ( z λ , w λ ) = 0 , as shown in [23]. Similar to [22, 24, 25], there exists C > 0 such that ∫ R N 1 − k | ∇ z n − z λ | 2 + f z n f ′ z n − f z λ f ′ z λ z n − z λ d x ≥ C z n − z λ 2 , (3.12) ∫ R N 1 − k | ∇ w n − w λ | 2 + f w n f ′ w n − f w λ f ′ w λ w n − w λ d x ≥ C w n − w λ 2 . (3.13)

By Hölder inequality and Lemma 2.1(ii) and (iv), we deduce that 2 α α + β ∫ R N | f z n | α − 2 f z n f ′ z n | f w n | β z n − z λ d x + 2 β α + β ∫ R N | f z n | α | f w n | β − 2 f w n f ′ w n w n − w λ d x ≤ 2 α α + β ∫ R N | z n | α − 1 | w n | β z n − z λ d x + 2 β α + β ∫ R N | z n | α | w n | β − 1 w n − w λ d x ≤ 2 α α + β ∫ R N | z n | 1 β | w n | 1 α − 1 d x α − 1 β z n − z λ p 1 + 2 α α + β ∫ R N | z n | 1 β − 1 | w n | 1 α d x β − 1 α w n − w λ p 2 → 0 , (3.14) where p 1 = 1 ( α − 1 ) β and p 2 = 1 ( β − 1 ) α . Similarly, we obtain 2 α α + β ∫ R N | f z λ | α − 2 f z λ f ′ z λ | f w λ | β z n − z λ d x + 2 β α + β ∫ R N | f z λ | α | f w λ | β − 2 f w λ f ′ w λ w n − w λ d x → 0 . (3.15)

Following (3.12), 3.13, 3.14, and .3.15, we obtain 0 ← ⟨ Φ λ ′ z n , w n − Φ λ ′ z λ , w λ , z n − z λ , w n − w λ ⟩ = ∫ R N 1 − k | ∇ z n − z λ | 2 + f z n − z λ f ′ z n − z λ z n − z λ d x + ∫ R N 1 − k | ∇ w n − w λ | 2 + f w n − w λ f ′ w n − w λ w n − w λ d x − 2 α α + β ∫ R N | f z n | α − 2 f z n f ′ z n | f w n | β − | f z λ | α − 2 f z λ f ′ z λ | f w λ | β z n − z λ d x − 2 β α + β ∫ R N | f z n | α | f w n | β − 2 f w n f ′ w n − | f z λ | α | f w λ | β − 2 f w λ f ′ w λ w n − w λ d x ≥ C z n − z λ 2 + C w n − w λ 2 + o n 1 , (3.16) which implies that (z _n , w _n ) → (z _λ , w _λ ) in H ¹. Thus, (z _λ , w _λ ) is a non-trivial critical point of Φ _λ (z, w) with Φ _λ (z _λ , w _λ ) = c _λ .

Next, we prove Theorem 1.1.

Proof. At first, using Theorem 3.1, for arbitrary λ ∈ L = 1 2 , 1 , there is a (z _λ , w _λ ) ∈ H ¹ such that z n , w n ⇀ z λ , w λ ≠ 0,0  in  H 1 , Φ λ z n , w n → c λ  and  Φ λ ′ z n , w n → 0 .

By Lemma 3.5, we obtain Φ λ z λ , w λ → c λ  and  Φ λ ′ z λ , w λ = 0 .

Thus, there exists λ n ⊂ 1 2 , 1 such that λ n → 1 , z λ n , w λ n ∈ H 1 , Φ λ n ′ z λ n , w λ n = 0  and  Φ λ n z λ n , w λ n = c λ n .

Next, we prove that { ( z λ n , w λ n ) } is bounded in H ¹. From Lemma 3.4 Φ λ n z λ n , w λ n = c 1 2 , Φ λ n ′ z λ n , w λ n = 0 , it follows that c 1 2 ≥ Φ λ n z λ n , w λ n = Φ λ n z λ n , w λ n − 1 N P λ n z λ n , w λ n = N − 2 2 N ∫ R N 2 N − 2 1 − k | ∇ z λ n | 2 + | ∇ w λ n | 2 + f 2 z λ n + f 2 w λ n d x . (3.17)

By Lemma 2.1 (v) and Sobolev inequality, it follows that ∫ | z λ n | ≤ 1 z λ n 2 d x ≤ C ∫ R N f 2 z λ n d x , ∫ | w λ n | ≤ 1 w λ n 2 d x ≤ C ∫ R N f 2 w λ w d x and ∫ | z λ n | > 1 z λ n 2 d x ≤ ∫ | z λ n | > 1 z λ n 2 * d x ≤ C ∫ R N | ∇ z λ n | 2 d x 2 * 2 , ∫ | w λ n | > 1 w λ n 2 d x ≤ ∫ | w λ n | > 1 w λ n 2 * d x ≤ C ∫ R N | ∇ w λ n | 2 d x 2 * 2 .