Construct new type solutions for the fractional Schrödinger equation

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                </mml:math></jats:alternatives></jats:inline-formula> is a positive radical function. We constructed infinitely many non-radial solutions of the new type which have a more complex concentration structure for (0.1).</jats:p>


Introduction and main result
In this paper, we focus on the following fractional problem: ⎧ ⎨ ⎩ (-) s u + u = K(|x|)u p , u > 0, x ∈ R N , u(x) ∈ H s (R N ), (1.1) where N ≥ 3, 0 < s < 1, 1 < p < N+2s N-2s , K(|x|) is a positive radical function. Problem (1.1) arises from the study of time-independent waves ψ(x, t) = e -iEt u(x) of the following nonlinear fractional Schrödinger equation: In (1.1), the fractional Laplacian operator (-) s is defined as |x -y| N+2s dy, (1.3) where C N,s is some normalization constant and P.V. stands for the Cauchy principle value. The fractional Schrödinger equation was discovered by Laskin [23,24] as a result of ex-tending the Feynman path integral from the Brownian-like paths to Lévy-like quantum mechanical paths. This problem has a strong physical background and it has frequently been studied by a lot of researchers. Problem (1.1) is a typical case of the equation For a subcritical power, under suitable conditions on V and f , Bieganowski and Secchi in [6] showed that the ground state solutions to (1.4) converge in L 2 loc (R N ). Using the penalized technique and the variational methods, An et al. in [3] proved the existence of a positive solution to (1.4) with the fast decaying potential V . In [10], Dávila, del Pino, and Wei generalized some previous results on the Schrödinger equation to fractional problem (1.4) by using the Lyapunov-Schmidt variational reduction. In [27], Long, Peng, and Yang obtained infinitely many non-radial positive solutions for (1.1) whose functional energy is very large. For the general nonlinearity, in [32], Secchi constructed solutions to (1.4), and the approach based on minimization on the Nehari manifold. For the critical case, He and Zou in [21] considered the following problem with critical growth: where the parameter λ > 0, 2 * s is the critical Sobolev exponent, f and h satisfy certain conditions. They showed the bifurcation and multiplicity of positive solutions for (1.5) in their paper. In [19], Guo and He proved the existence and concentration of positive solutions to the following fractional nonlinear Schrödinger equation: For the supercritical case, Ao et al. in [4] proved the existence of bound state solutions for (1.4). When the potential and nonlinearity satisfy certain conditions, Bisci and Rădulescu in [30] studied the existence of multiple ground state solutions for the following problem: For other existence results, we refer to [1, 2, 5, 7-9, 12, 14, 17, 18, 20, 22, 25, 26, 28, 29, 31, 35-37] and the references therein. After the above bibliography review, we want to specifically mention two papers [13] and [33]. In [33], Wei and Yan used a constructive method to produce infinitely many non-radial solutions to (1.1) when s = 1 with high energy. Using the same method, Duan and Musso in [13] got other type of building blocks for the same problem in [33]. It is worth mentioning here that the structure of solutions in [13,33] is dissimilar. Compared to the results in [33], the solutions in [13] have a more complex concentration structure. The solutions in [33] show the polygonal symmetry in the (x 1 , x 2 )-plane and are radially symmetric in other variables. However, the solutions in [13] have the polygonal symmetry in the (x 1 , x 2 )-plane, the even symmetry in the x 3 direction, and the radial symmetry in other variables. Inspired by the two works mentioned above, our aim is to construct new type solutions like the solutions in [13] which have a more complex concentration structure for problem (1.1). Since the fractional Laplacian operator is a nonlocal one, it is difficult to use the methods for local operator directly. For instance, the ground state for -decays exponentially at infinity. In contrast, the ground state for (1.7) decays algebraically at infinity. So, the research of problem (1.1) becomes more complicated. At the same time, we need to redetermine the range for the parameters h and r which will be used to determine the position of the locations y j , y j of the bumps in W r,h (x). Next, we give some notation and definitions which are used in our paper. Let N ≥ 3, m be an integer and introduce the points , h, 0), j = 1, 2, . . . , k, where 0 is the zero vector in R N-3 . Throughout this paper, we assume that (r, h) ∈ k and define k : = where α, β > 0 are small constants,Ã 1 , A 3 , and E are defined in Proposition 4.1. Set where θ = arctan x 2 x 1 . We choose the unique positive solution W (up to translations and dilations) of the following equation: to construct the approximate solution for (1.1). Denote For j = 1, . . . , k, we divide R N into k points: For j , we divide it into two points: We see that and empty for j = l. Before we give the main theorem, we first give the following conditions on the potential function K : (K) for some a > 0, θ > 0 and N+2s N+2s+1 < m < N + 2s. Our main result is summarized as follows.
where v r,h ∈ ℵ s , (r, h) ∈ k , and as k → +∞, In this paper, we first give some preliminaries in Sect. 2, and then we study the reduced finite dimensional problem in Sect. 3. In Sect. 4, we prove Theorem 1.1. Some useful lemmas are left in the Appendix.

Some preliminaries
In this section, we list a couple of important properties and basic theory of fractional Sobolev spaces which are used in our paper. For more technical details, we refer the reader to [11].
For any s ∈ (0, 1), the space The norm of H s (R N ) is written as The following identity comes from Proposition 3.6 in [11]: where is the Fourier transform, C is a suitable positive constant depending only on s, and The following results play a key role in proving Theorem 1.1.
(2) (Symmetry, regularity, and decay) W is radially symmetric and strictly decreasing in |x|. Moreover, the solution W satisfies with some constants C 2 ≥ C 1 > 0.

Variational reduction
The energy functional corresponding to (1.1) is defined as Letting we can expand J(φ) as follows: and The following result implies that L is invertible in H s .

Lemma 3.1
There exists an integer k 0 > 0 such that, for k ≥ k 0 , there is a constant C > 0 independent of k, satisfying that, for any (r, h) ∈ k , Proof We argue by contradiction. Suppose that there are n → +∞, u n ∈ H s , (r k , h k ) ∈ k such that By symmetry, we have In particular Letũ n =ũ n (x + y 1 ), we can choose R > 0 such that B R (y 1 ) ⊂ 1 . Thus So, we can conclude u n u, weakly in H s R N and u n → u, strongly in L 2 loc R N .
At the same time, we can obtain that u is even in x j , j = 2, . . . , N . From the orthogonal conditions for functions of H s Letting k → +∞, we obtain Now, we claim that u satisfies For any R > 0, let ψ ∈ C ∞ 0 (B R (0)) ∩ H s be any function satisfying that ψ is even in x j , j = 2, . . . , N . Then ψ 1 (x) = ψ(xy 1 ) ∈ B R (y 1 ). By Lemma 5.3, and inserting ψ 1 (x) into (3.2), we have Thus, we can take R > 0 large enough, then which is a contradiction to (3.3).
Moreover, we give the estimate for l. (3.6) Using the fact N+2s N+2s+1 < m < N + 2s, we can check that for some small constant τ > 0. Inserting the above estimates into (3.6), we obtain the desired results.
The proof of the following significant Proposition 3.3 is the same as in [27], we only describe the content of it briefly. where τ > 0 is small enough.

Proof of our main result
In this part, we mainly give the estimate in Proposition 4.1 and show Theorem 1.1.
Using the fact that W y j ≥ W y j for x ∈ + 1 , (4.2) and (4.3), we can check that
Moreover, we find The proofs of the following two lemmas are similar to Lemma A. 2 and (28) in [20] respectively.