RETRACTED ARTICLE: Existence results for the general Schrödinger equations with a superlinear Neumann boundary value problem

The main goal of this paper is to study the general Schrödinger equations with a superlinear Neumann boundary value problem in domains with conical points on the boundary of the bases. First the formulation and the complex form of the problem for the equations are given, and then the existence result of solutions for the above problem is proved by the complex analytic method and the fixed point index theory, where we absorb the advantages of the methods in recent works and give some improvement and development. Finally, we are also interested in the asymptotic behavior of solutions of the mentioned equation. These results generalize some previous results concerning the asymptotic behavior of solutions of non-delay systems of Schrödinger equations or of delay Schrödinger equations.


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The general theory of PDEs like (1) with variable exponent has gained the interest of many mathematicians in recent years. We refer to the surveys [1,8,14,15,23,27]. From a physical point of view, such Schrödinger equations with a superlinear Neumann boundary value problem have gained a lot of interest in recent years, in particular in the context of systems for the mean field dynamics of Bose-Einstein condensates [2,5] and in applications to fields like nonlinear and fibers optics [25].
To define the solution of (1), we introduce a class of functions related to the exponent (x) (see [30]) This set is a Sobolev space of functions, locally summable on S together with their first order generalized derivatives. It follows that there exists a good approximation of g based on a set of independent and identically distributed random To the best of our knowledge, this notion of indirect observability was introduced for the first time in the context of coupled elliptic equations in [7], to obtain an exact indirect controllability result, in which one wants to drive back the fully coupled system to equilibrium by controlling only one component of the system. In 2017, Lai, Sun and Li (see [17]) used a two level energy method to estimate the solution of (1). In the case when ω ε (x) and (x) are fixed constants, there have been many results about the existence, uniqueness, blowing-up and so on; we refer to the bibliography (see [19,29]). It follows that the hypothesis space is a Hilbert space H E induced by a Mercer kernel K which is a continuous, symmetric, and positive semi-definite function on S × S (see [24]). Space H E is the completion of the linear span of the set of functions {E s := E(s, ·) : s ∈ S} with respect to the inner product The reproducing property in H E is (see [3]) where g ∈ H E and s ∈ S. Then by (2), we have (see [4]) It implies that H E ⊆ C(S).
Scheme (3) shows that regularization not only ensures computational stability but also preserves localization property for the algorithm. In this paper, we further study the asymptotic behaviors of solutions of (1).
We adopt the coefficient-based regularization and the data-dependent hypothesis space (see [6,20,26]) g w,ς (s) = g w,ζ ,ς,s (s) = g w,ζ ,ς,s (u)| u=s , g w,ζ ,ς,s = arg min where 1 ≤ q ≤ 2, and Compared with scheme (3), the first advantage of (4) is the efficacy of computations without any optimization processes. Another advantage is that we can choose a suitable parameter q according to the research interest, e.g., smoothness and sparsity.
To study the approximation quality of g w,ς , we derive an upper bound of the error g w,ςg S with g(·) S := S g(·) 2 d S 1 2 and establish its convergence rate as m → ∞ (see [10]). The remainder of this paper is organized as follows. In Sect. 2, we will provide the main results. In Sect. 3, some basic, but important estimates and properties are summarized.

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The proofs of main results will be given in Sect. 4. Section 5 contains the conclusions of the paper.

Main results
We first formulate some basic notations and assumptions.
Let S be the marginal distribution of on S and L 2 S (S) be the Hilbert space of functions from S to T, which are square-integrable with respect to S with the norm denoted by · S . The integral operator L E : where s ∈ S. Let {μ i } be the eigenvalues of L E and {e i } be the corresponding eigenfunctions. Then for g ∈ L 2 S (S), see [9]. We assume that g satisfies the regularity condition L -r E g ∈ L 2 S , where r > 0. We show the following useful feature of the capacity of H E,w when the l 2 -empirical covering number is used (see [11]), namely where > 0, B 1 = {f ∈ H E,w : g E ≤ 1}, 0 < p < 2 and c p > 0 (see [22]). We use the projection operator to obtain a faster learning rate under the condition |y| ≤ M almost surely (see [18,21]).

Definition 2.1
Let A > 0. Then the projection operator γ A on the space of solutions g : S → R is defined as We assume all the constants are positive and independent of δ, m, χ , ς and ζ . Now we state our main results.
where u, v ∈ S and c 0 is a positive constant. Then Then we need an upper bound of the integral in (9). In order to get it, we only need to give its decomposition by using g w,χ which provides a crucial connection between g w,ς and the regularization function g χ , while different regularization parameters χ and ς are adopted.
Theorem 2 Let g w,ζ ,ς,s be defined as in (4) and let be a solution of (1). Then we have

Lemmas
Some basic, but important estimates and properties of solutions γ A (g) are summarized in the following lemma.

Lemma 1 Under the assumptions of Theorem
Proof We will split the proof into four steps.
Step 1. Obtaining estimates of the terms: We take the sum of the inner products with g χ (t) and -g (t), respectively, and obtain where On the other hand, In view of the latter two inequalities, we have Using (13) and (14), we have for each ε 1 > 0. So Integrating by parts, we have where However, for this term we have Moreover, Inserting the latter inequality into (17), we have On the other hand, Using (16), (18), (19) and (15), we have Next, we estimate E g (g χ ; τ ) + γ A (g χ ; 0). For this purpose, we take the inner product with (-∂ 2 g ) -1 g χ (t) in the space (R N , · R N ,g ) to obtain It follows that We now estimate the second term of the right-hand side of the above equation as

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Moreover, by (20) and having in mind (21), we can write Step 2. Improving estimates (15) and (20).
Inserting (22) into the latter inequality, we have On the other hand, equation (20) implies that from (22).

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Step 4. Estimating S γ A (g ; t) dt.
From (25), we have It follows that for all ε 3 > 0.
Integrating the latter inequality between 0 and τ , we obtain and having in mind equation (27), we can improve the last estimate as follows: In other words, Since δ ≤ √ δ 0 /2, it follows that (12) holds. This completes the proof.
The following result provides a uniform observability inequality.
Proof We first have the discrete identity by Lemma 1, where We now estimate separately A, X g and B.
Estimate for A. We have Estimate for X g . Notice that Estimate for B. We have Next we obtain due to (32) and (33)-(35).

Proofs of main results
Now we derive the learning rates.
We complete the proof of Theorem 1.

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which yields This completes the proof of Theorem 2.

Conclusions
In this paper, we studied a class of Schrödinger equations with Neumann boundary condition L ε g = div(ω ε (x)|∇g| (x)-2 ∇g) = 0 on a compact metric space S ⊂ R n , n ≥ 2, with a positive weight ω ε (x). We were interested in the asymptotic behavior of solutions of the mentioned equation. More precisely, we formulated conditions on a function g, which guarantee that the graph of at least one solution for the above-mentioned equation stays in the prescribed domain. These results generalized some previous results concerning the asymptotic behavior of solutions of non-delay systems of Schrödinger equations or of delay Schrödinger equations.

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