Multiple nonsymmetric nodal solutions for quasilinear Schr\"{o}dinger system

In this paper, we consider the quasilinear Schr\"{o}dinger system in $\mathbb R^{N}$($N\geq3$): $$\left\{\begin{align}&-\Delta u+ A(x)u-\frac{1}{2}\triangle(u^{2})u=\frac{2\alpha }{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\\&-\Delta v+ Bv-\frac{1}{2}\triangle(v^{2})v=\frac{2\beta }{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\end{align}\right. $$ where $\alpha,\beta>1$, $2<\alpha+\beta<\frac{4N}{N-2}$,$B>0$ is a constant. By using a constrained minimization on Nehari-Poho\v{z}aev set, for any given integer $s\geq2$, we construct a non-radially symmetrical nodal solution with its $2s$ nodal domains.


Introduction
We study the following quasilinear Schrödinger system where u(x) → 0, v(x) → 0 as |x| → ∞, N ≥ 3, u := u(x), v := v(x) be real valued functions on R N , α, β > 1, 2 < α + β < 4N N −2 , B > 0 is a constant. In the last two decades, much attention has been devoted to the quasilinear Schrödinger equation of the form The equation (1.2) is related to the existence of standing waves of the following quasilinear Schrödinger equation where V is a given potential, l and g are real functions. The equation (1.3) has been used as models in several areas of physics corresponding to various types of g. The superfluid film equation in plasma physics has this structure for g(s) = s [10]. In the case g(s) = (1+s) 1 2 , the equation (1.3) models the self-channeling of a high-power ultra short laser in matter [20]. The equation (1.3) also appears in fluid mechanics [10,11], in the theory of Heidelberg ferromagnetism and magnus [12], in dissipative quantum mechanics and in condensed matter theory [15]. When considering the case g(s) = s, one obtains a corresponding equation of elliptic type like (1.2). For more detailed mathematical and physical interpretation of equations like (1.2), we refer to [1,3,5,13,19,22] and the references therein.
In recent years, there has been increasing interest in studying problem (1.2), see for examples, [6,7,9,16,17,25,26] and the references therein. More precisely, by Mountain-Pass theorem and the principle of symmetric criticality, Severo [23] obtained symmetric and nonsymmetric solutions for quasilinear Schrödinger equation (1.2). In [14], when 4 ≤ p < 4N N −2 , Liu, Wang and Wang established the existence results of a positive ground state solution and a sign-changing ground state solution were given by using the Nehari method for (1.2). Based on the method of perturbation and invariant sets of descending flow, Zhang and Liu [28] studied the nonautonomous case of (1.2), they obtained the existence of infinitely many sign-changing solutions for 4 < p < 4N N −2 . With the help of Nehari method and change of variables, Deng, Peng and Wang [8] considered (1.4) and proved that (1.4) has at least one pair of k-node solutions if either N ≥ 6 and 4 < p < 4N N −2 or 3 ≤ N < 6 and 2(N +2) In addition, problem (1.4) still has at least one pair of k-node solutions if 3 ≤ N < 6 , 4 < q ≤ 2(N +2) N −2 and λ sufficiently large. Note that all sign-changing solutions obtained in [8,14,28] are only valid for 4 < p < 4N N −2 . When 2 < p < 4N N −2 , Ruiz and Siciliano [21] showed equation (1.2) has a ground states solution via Nehari-Pohožaev type constraint and concentration-compactness lemma, Wu and Wu [27] obtained the existence of radial solutions for (1.2) by using change of variables.
It is natural to pose a series of interesting questions: whether we can find an unified approach to obtain sign-changing solutions for the full subcritical range of 2 < α + β < 4N N −2 ? Further, whether we can extend these results to quasilinear Schrödinger system? To answer these two questions, we adopt an action of finite subgroup G of O(2) from Szulkin and Waliullah [24] and look for the existence of non-radially symmetrical nodal solutions for quasilinear Schrödinger system (1.1).
Before stating our main results, we make the following assumptions: is radially symmetric with respect to the first two coordinates, that is to say, if (x 1 , x 2 , x 3 , · · · , x N ), (y 1 , y 2 , y 3 , · · · , y N ) ∈ R N and x 2 1 +x 2 2 = y 2 1 +y 2 2 , then A(x 1 , x 2 , z 3 , · · · , z N ) = A(y 1 , y 2 , z 3 , · · · , z N ). It is worth noting that (A 1 ) is used to derive the existence of a strongly convergent subsequence, while for the system, we only need one such kind of condition in our system, which seems to be a different phenomenon due to the coupling of u and v. (A 2 )-(A 3 ) once appeared in [21,27] to obtain the existence of ground states solutions for the quasilinear Schrödinger equation. (A 4 ) once appeared in [4] to prove the existence of nodal solutions of single p-Laplacian equation.
Our main results reads as follows.
For any given integer s ≥ 2, the problem (1.1) possesses a non-radially symmetrical nodal solution with its 2s nodal domains.
is a positive constant, one can still obtain the same results as Theorem 1.1 for system (1.1).

Remark 1.3.
Since s ∈ N is arbitrary, the solution we obtained in Theorem 1.1 is actually a result of multiplicity. Remark 1.4. As a main novelty with respect to some results in [8,14,28], we are able to deal with exponents α + β ∈ (2, 4N N −2 ) and obtain the existence and multiplicity of nodal solution.
The rest of the paper is organized as follows. In Section 2, we establish some preliminary results. Theorem 1.1 is proved in Section 3.

Preliminaries
Throughout this paper, u H 1 and |u| r denote the usual norms of H 1 (R N ) and L r (R N ) for r > 1, respectively. C and C i (i = 1, 2, . . .) denote (possibly different) positive constants and R N g denotes the integral R N g(z)dz. The → and ⇀ denote strong convergence and weak convergence, respectively. Let The term R N u 2 |∇u| 2 is not convex and H is not even a vector space. So, the usual min-max techniques cannot be directly applied, nevertheless H is a complete metric space with distance Then we call (u, v) ∈ X is a weak solution of (1.1) if for any ϕ 1 , Hence there is a one-to-one correspondence between solutions of (1.1) and critical points of the following functional I : X → R defined by Then, (u, v) ∈ X is a solution of (1.1) if and only if . Motivated by [24], we recall that a subset U of a Banach space E is called invariant with respect to an action of a group G (or Ginvariant) if gU ⊂ U for all g ∈ G, and a functional I : Let x = (y, z) = (y 1 , y 2 , z 1 , · · · , z N ) ∈ R N and let O(2) be the group of orthogonal transformations acting on R 2 by (g, y) → gy. For any positive integer s we define G s to be the finite subgroup of O(2) generated by the two elements α and β in O(2), where α is the rotation in the y-plane by the angle 2π s and β is the reflection in the line y 1 = 0 if s = 2, and in the line y 2 = tan π s y 1 for other s (so in complex notation w = y 1 + iy 2 , αw = we 2πi s and βw = we Then our aim is to prove that m is achieved. In the rest of this section, we will give some properties of the set M. For any u ∈ H 1 (R N ), we define u t : Let t ∈ R + and (u, v) ∈ X. We have that Denote h uv (t) := I(u t , v t ). Since α + β > 2, we see that h uv (t) > 0 for t > 0 small enough and h uv (t) → −∞ as t → +∞, this implies that h uv (t) attains its maximum. Moreover, h uv (t) : Lemma 2.1. If (u, v) ∈ X is a weak solution of (1.1), then (u, v) satisfies the following P (u, v) = 0, where Proof. The proof is standard, so we omit it here.
The lemma below shows (2.2) is well defined.
Lemma 2.2. For any (u, v) ∈ X and u, v = 0, the map h uv attains its maximum at exactly one pointt. Moreover, h uv is positive and increasing for t ∈ [0,t] and decreasing for t >t. Finally Proof. For any t > 0, set s = t N +α+β , we obtain This is a concave function by condition (A 3 ) and we already know that it attains its maximum, lett be the unique point at which this maximum is achieved. Notice that G(u t , v t ) = th ′ uv (t), thent is the unique critical point of h uv and h uv is positive and increasing for 0 < t <t and decreasing for t >t. In particular,t ∈ R is the unique value such that ut ∈ M, and I(ut, vt) reaches a global maximum for t =t. This finishes the proof. Proof. For every (u, v) ∈ M, it follows from (A 2 ) that The proof is complete.

Proof of Theorem 1.1
We need the following variant of Lions lemma.
Observe that in the family {B(gz n , 1)} g∈O(2) , we find an increasing number of disjoint balls provided that |(z 1 N −2 ), by (3.2), |(z 1 n , z 2 n )| must be bounded. Then for sufficiently large r ≥ r 0 , one obtains and we get a contradiction with (3.1).
The following Lemma is due to Poppenberg, Schmitt and Wang from [19, Lemma 2].
Proof. The proof is analogous to that of [19,Lemma 2], so we omit it here.
Using (u n , v n ) ⊂ M and (A 2 ), we may obtain which implies that {u n }, {v n }, {u 2 n } and {v 2 n } are bounded in H 1 (R N ), then, there exists a subsequence of (u n , v n ), still denoted by (u n , v n ) such that (u n , v n ) ⇀ (u, v) in X. Then {u n } and {v n } are bounded in L α+β (R N ). The proof includes the following three steps.