Existence of global solutions to a quasilinear Schrödinger equation with general nonlinear optimal control conditions

In this article, we study a modiﬁed maximum principle approach under condition on the weight of the delay term in the feedback and the weight of the term without delay. On that basis, we prove the existence of global solutions for a quasilinear Schrödinger equation in an unbounded domain with a general nonlinear nonlinear optimal control condition in the weakly nonlinear internal feedback. The equation includes many special cases such as classical Schrödinger equations, fractional Schrödinger equations, and relativistic Schrödinger equations, etc. Our results are established by means of the ﬁxed point theory associated with the Schrödinger operator in suitable b -metric spaces. Moreover, we establish general stability estimates by using some properties of Schrödinger convex functions.


R E T R A C T E D A R T I C L E
Problem (1.1), which is driven by the infinitesimal generator of a Brownian motion, has been studied by many authors. There are many references to equation (1.1); see, for example, [1][2][3]. It arises when one looks for standing waves to the fourth-order Schrödinger equation (see [4]) where γ ∈ R, p ≥ 1, and the space dimension is no more than three. Problem (1.3) describes a stable soliton; in particular, there are solitons in magnetic materials for p = 1 in 3D space. There has been a lot of interest in fourth-order semilinear Schrödinger equations because of their strong physical background. Because of both physical and mathematical reasons, the ground states are the most important solutions. At the same time, the existence, uniqueness, and multiplicity of solutions are important characteristics. Therefore, we pay attention to the existence, uniqueness, and multiplicity of the ground states. Researchers studied the existence, nonexistence, and uniqueness of ground states to the scalar equation in [5][6][7][8][9] and the references therein. Results about ground states for 2 and 3 coupled systems can be found in [10][11][12][13]. Recently, Sun [14] studied the Cauchy problem of the equation where λ ∈ R and μ = 0. The classical maximum principle approach associated with Schrödinger operator (see [15]) was introduced by the Schrödinger principal value of the Schrödinger integral. Liu studied the general Schrödinger equations with a superlinear Neumann boundary value problem in domains with conical points on the boundary of the bases in [4]. Sun also obtained more general sufficient conditions for maximum principle approach associated with a class of linear Schrödinger equations with mixed boundary conditions in [14]. Based on variational methods, the existence of infinitely many solutions for a fractional Kirchhoff-Schrödinger-Poisson system was studied in [16]. Coveri considered the existence and symmetry of positive solutions for a modified Schrödinger system under the Keller-Osserman type conditions in [17]. Chaharlang and Razania considered the fourthorder singular elliptic problem involving p-biharmonic operator with Dirichlet boundary condition in [18]. The existence of at least one weak solution was proved in two different cases of the nonlinear term at the origin. Some nonlocal problems of Kirchhoff type with Dirichlet boundary condition in Orlicz-Sobolev spaces were also considered in [19]. The existence and multiplicity of solutions for the Schrödinger-Kirchhoff type problems involving the fractional p-Laplacian and critical exponent were considered in [20]. The authors in [21] were concerned with the existence of nonnegative solutions of a Schrödinger-Choquard-Kirchhoff-type fractional p-equation. The existence of solutions for a class of fractional Kirchhoff-type problems with Trudinger-Moser nonlinearity was studied in [22].
In this paper, we shall use this method to study the quasilinear Schrödinger equation with a general nonlinear nonlinear optimal control condition (1.1), which contains spatial heterogeneities with arbitrary sign along the boundary. This result is new in the general framework of a heterogeneous and bounded rectangular domain. Regarding the existence of a solution of the quasilinear Schrödinger equation with a general nonlinear nonlinear optimal control condition (1.1), it was obtained in [1] by applying the maximum principle approach with respect to the Schrödinger operator. For the evolution-free Schrödinger boundary problem (1.1) in different spaces, we refer to [19,23,24].

A modified maximum principle approach
First we define the space the auxiliary functionals and where P(f ) is composed of both mass and energy and I(f ) is considered as a Nehari functional. The Nehari manifold is defined by The stable set G and unstable set B are defined as follows: and Equation (1.1) also stems from looking for the standing wave J (t, s) = e it f (s) of the equation where A is the infinitesimal generator of a rotationally invariant Lévy process (see [18]). Now we present a modified maximum principle approach with respect to the Schrödinger operator for the quasilinear Schrödinger equation with a general nonlinear nonlinear optimal control condition (1.1), which plays an important role in our discussions.
Then J has the expression, J (s) = R n |t| 2 |f | 2 dt, and the computation of the modified modified maximum principle approach associated with (1.1) is given by Proof It follows that which yields where We present the estimates of K 1 and K 2 as follows: which together with (2.6) gives On the one hand, we have On the other hand, we have which together with (2.6) gives Substituting (2.7) and (2.8) into (2.6), we have Further, we derive which yields Further, we have which yields So which yields Substituting the above equalities for I 1 , I 2 , and I 3 into (2.9), we have which yields The proof is complete.
After clarifying the required assumptions on the infinitesimal generator A, we provide some examples satisfying the assumptions of Eq. (2.3). We assume the following: (H1) There exist positive constants t, c and K such that -J (s) ≥ c|s| 2t for all s ∈ R n with |s| ≥ K.

Remark 2.2
(1) Since A is the infinitesimal generator of a rotationally invariant Lévy process (see [25][26][27]), we have where a ≥ 0 and ν is an invariant Lévy measure. Thus s ≤ 1, J (s) ≤ 0 for all s ∈ R n , and sup s : there exist constants c and K such that -J (s) ≥ c|s| 2t for all s ∈ R n with |s| ≥ K ≥ 0.

Main results
In this section, we shall state and prove our main result. Now we state the local existence theory of solution for the Cauchy problem (1.1).  Proof We only prove that G is invariant, B can be proved similarly. Suppose that f 0 ∈ G, we claim that f (s) ∈ G for every t ∈ (0, L).  Then according to Lemma 3.1, the existence time of a local solution of (1.1) can be extended to infinity, thus the solution of problem (1.1) is global.

Conclusions
In this article, we studied a modified maximum principle approach under a condition on the weight of the delay term in the feedback and the weight of the term without delay. On that basis, we proved the existence of global solutions for a quasilinear Schrödinger