A new type of solutions for a nonlinear Schrödinger system with χ (2) nonlinearities

We are concerned with the question of constructing a new type of solution to the problem with χ (2) nonlinearities where P ( x ) = P ( | x | ) and Q ( x ) = Q ( | x | ) are positive bounded radial potentials, 3 ≤ N < 6, α > 0 and α > β . Assuming that the potentials P ( x ) and Q ( x ) satisfy certain conditions, the existence of a new type of solutions is proved. MSC: 35B99; 35J10; 35J60


Introduction
This paper deals with the questions of the existence of a new type of solution of the following systems of coupled elliptic equations with χ (2) nonlinearities where P(x) = P(|x|) and Q(x) = Q(|x|) are positive bounded radial potentials, 3 ≤ N < 6, α > 0 and α > β.
The system (1.1) has strongly attracted researchers' attention and has been extensively studied because it arises from nonlinear optical theory. In the nonlinear optical theory, the following cubic nonlinear Schrödinger equation is the basic equation that can be used to describe the formation and propagation of optical solutions in Kerr-type materials [6,15]. Here, the slowly varying envelope of electric field φ represents the relative strength, the real-valued parameter r is concerned with the sign of dispersion/diffraction, and χ represents the nonlinearity. z describes the propagation distance coordinate. The Laplacian operator ∇ 2 can either be ∂ 2 ∂τ 2 for temporal solitons, where τ is the normalized retarded time, or , where x = (x 1 , . . . , x N ) is in the direction orthogonal to z. Solitary wave solutions to (1.2) and its generations have been proved in [2,14].
As the optical material has a χ (2) (i.e., quadratic) nonlinear response instead of a conventional χ (3) material for which the problem (1.2) is based on (see [3,4]), the nonlinear optical effects such as Second Harmonic Generation were discovered. As is known, the χ (3) nonlinear Schrödinger system is well studied and has been explored by many authors in recent years, one can refer to [1,7,9,10,12,13,18] and the references therein. For the χ (2) nonlinear Schrödinger system, in the case α = 1, β = 0, N = 1, Zhao, Zhao and Shi in [22] proved the existence of a ground-state solution for (1.1). Very recently, using the finitely dimensional reduction method, Wang and Zhou in [17] studied the existence of infinitely many nonradial positive synchronized solutions of the system (1.1) under radial potentials satisfying some algebraic decay. Using the same method in [19], Yang and Zhou proved the existence of a single peak solution for (1.1). For more results, we refer the readers to [16,20,21] and the references therein.
In this paper, we construct a new type of solutions for the χ (2) nonlinear Schrödinger system (1.1). The new type of solutions that were first introduced by Duan and Musso recently in [8] have polygonal symmetry in the (x 1 , x 2 )-plane, even symmetry in the x 3 direction, and radial symmetry in other variables. Due to the χ (2) nonlinearity that appears in our paper, we need to improve some estimates and give precise computing techniques. To the best of our knowledge, there is no result on such a question in the current the literature.
First, we will give some notations. Let where k is an integer and 0 is the zero vector in R N-3 , r ∈ [r 1 m ln m, r 2 m ln m] for some It is well known that the following equation has a unique positive ground state W that satisfies: By direct computation, we know that (U, V ) = (μW , γ W ) solves the following limit system for (1.1): For any function P(x) > 0, we define the norm of H 1 P (R N ) as follows with the inner product We define the product space where j = 1, 2, . . . , m and , R 2 denote the dot product in R 2 . We see that R N can be divided into k parts: 1 , . . . , k . For j , we divided it into two parts: We know the interior of j ∩ i , + j ∩j are empty sets for i = j. Define H P,s = u : u ∈ H 1 P R N , u is even in x l , l = 2, 4, 5, . . . , N, In what follows, we make some assumptions: (P) There exist constants a > 0, s > 1 and θ > 0, such that as r → +∞, (Q) There exist constants b > 0, t > 1 and ε > 0, such that as r → +∞, In this paper, we always assume for some α, β > 0 small. Our main result can be stated as follows: The results in [22] show that N = 6 is a critical dimension, that is to say, the positive solutions do not exist when N ≥ 6. At the same time, by the structure of the new type of solution in our paper, the dimension N must be greater than or equal to 3.
Remark 1.5 The solutions of the system (1.1) has a large number of bumps near infinity, which causes the energy to become very large.

Finite-dimensional reduction
and and It is standard to see that I ∈ C 2 (H, R).
We expand the functional K(ϕ, ψ) as follows and Here, L is a linear operator from E to E, that satisfies We have the following important results that have been proved in [8].

Lemma 2.1 (Lemma 3.1, [8]) For r, h as parameters in m and any
By the same argument as that of Lemma 2.2 in [8], we can prove: Lemma 2.2 There exists a constant C > 0, independent of m, such that for any (r, h) ∈ m , The following important Proposition can be found in [17], we only need to use the contraction theorem to prove it. Meanwhile, the new type of solution is not weighing on the proof of this Proposition. Hence, we omit it for conciseness.

Proposition 2.3
There exists an integer m 0 > 0, such that for any m ≥ m 0 , there is a unique Moreover, there is a positive constant C, such that where τ > 0 is small enough.
Next, we will give the estimate for l m . Proof Recall that By symmetry, we have By using the estimates (3.14) and (3.15) in [8], we obtain e -(2-τ )|y i -y i | where τ > 0 is sufficiently small.