The Existence of Radial Solutions to the Schrödinger System Containing a Nonlinear Operator

In this paper, we investigate a class of nonlinear Schrödinger systems containing a nonlinear operator under Osgood-type conditions. By employing the iterative technique, the existence conditions for entire positive radial solutions of the above problem are given under the cases where components μ and ν are bounded, μ and ν are blow-up, and one of the components is bounded, while the other is blow-up. Finally, we present two examples to verify our results.


Introduction
Osgood-type condition is of great significance in the field of mathematics and has been widely applied to different equations or systems by many authors. In 1898, under the Osgood type condition Osgood [1] presented the existence result of solutions for the following equation without the Cauchy-Lipschitz condition where ψ(s) is a continuous function satisfying |ψ(χ, y) − ψ(χ, y )| ≤ ϕ(|y − y |). Then, lots of authors began to consider applying the Osgood-type condition to other problems and gained many excellent results such as the comparison result of viscosity upper and lower solutions for fully nonlinear parabolic equations [2], the existence result of solutions for backward stochastic differential equations (BSDEs) [3], and the nonexistence result of the local solution for semilinear fractional heat Equation [4]. For more results, see [5][6][7][8][9]. The Schrödinger equation was derived from mathematical physics and closely related to several physical phenomena. In [10], Kurihura used it to model the superfluid film equation in plasma physics. In [11,12], it was used to model the phenomena of the self-channeling of a high-power ultrashort laser in matter. More examples and details of applications can be found in [13][14][15][16].
In 2020, by employing the iterative technique, Wang et al. [19] established the existence result of the entire radial solutions for the following Schrödinger system Motivated by the above work, we studied the existence of entire positive radial solutions to the following Schrödinger system where n ≥ 3, b, h are continuous functions, Λ is a nonlinear operator belonging to θ and ψ, ϕ are continuous functions satisfying Osgood-type conditions By employing the monotone iterative method, we give the existence results of positive entire radial solutions to the Schrödinger system (1) under the cases where the components µ and ν are bounded, µ and ν are blow-up, and one of the components is bounded while the other is blow-up. The monotone iterative method plays a significant role in the study of nonlinear problem, as can be seen in [18][19][20][21][22][23][24][25][26][27][28] and the references therein. To the best of our knowledge, there is no work about the existence of the positive radial solutions of the Schrödinger system (1) under the Osgood-type conditions. In addition, our results extended the work of authors in [18,[28][29][30][31][32][33].

Preliminaries
In this section, we give a definition, some notations, assumptions and Lemmas that are subsequently needed in the proof.
is a radial solution of the Schrödinger system (1) if and only if it is a solution of the following ordinary differential system

The Entire Positive Bounded Radial Solutions
In this section, we prove Theorems 1 and 2.

The Entire Positive Blow-Up Radial Solutions
In this section, we prove Theorem 3.

The Semifinite Entire Positive Blow-Up Radial Solutions
In this section, we prove Theorems 4 and 5.

Example
Example 1. Consider the following Schrödinger system meaning that (N2) is established. From Theorem 1, the Schrödinger system (29) has an entire positive radial solution (µ, ν)