Next Article in Journal
Controllability of a Class of Impulsive ψ-Caputo Fractional Evolution Equations of Sobolev Type
Next Article in Special Issue
On Implicit Time–Fractal–Fractional Differential Equation

Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

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

by
Guotao Wang
1,*,
Zhuobin Zhang
1 and
Zedong Yang
2
1
School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan 030031, China
2
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China
*
Author to whom correspondence should be addressed.
Axioms 2022, 11(6), 282; https://doi.org/10.3390/axioms11060282
Submission received: 20 May 2022 / Revised: 4 June 2022 / Accepted: 6 June 2022 / Published: 10 June 2022
(This article belongs to the Special Issue Fractional Calculus and Differential Equations)

## Abstract

:
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.
MSC:
35B08; 35B09; 35J10

## 1. 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
$∫ 0 U d s ψ ( s ) = ∞ , ∀ U > 0 ,$
Osgood [1] presented the existence result of solutions for the following equation without the Cauchy–Lipschitz condition
$d y d χ = ψ ( χ , y ) ,$
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 2017, by employing the analysis technique and weighted norm method, Sun [17] established the existence result of solutions to the following Schrödinger equation
$Δ μ + ψ ( | χ | ) b ( μ ) = 0 ,$
where $| χ | ∈ E D$, $ψ ( | χ | ) ∈ C l o c λ ( E D , R )$, $λ ∈ ( 0 , 1 )$, $b ( μ ) ∈ C l o c λ ( R , R )$ (locally Hölder continuous), $E D = { χ ∈ R 2 : | χ | > D }$, $S D = { χ ∈ R 2 : | χ | = D }$, for $D > 0$.
In 2018, by introducing a growth condition and employing the iterative technique, Zhang, Wu and Cui [18] established the nonexistence and existence results of the entire blow-up solutions to the following Schrödinger equation
$d i v ( Λ ( | ∇ μ | ) ∇ μ ) = b ( | χ | ) ψ ( μ ) , χ ∈ R n ,$
where $n ≥ 2$, $Λ$ is a nonlinear operator belonging to the set ${ Λ ∈ C 2 ( [ 0 , ∞ ) , ( 0 , ∞ ) ) | ∃ β ∈ ( 0 , ∞ ) : Λ ( m s ) ≤ m β Λ ( s ) , 0 < m < 1 }$.
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
$d i v Λ ( | ∇ μ | p − 2 ) ∇ μ = b ( | χ | ) ψ ν , χ ∈ R n , d i v Λ ( | ∇ ν | p − 2 ) ∇ ν = h ( | χ | ) φ μ , χ ∈ R n ,$
where $n ≥ 3$, $b , h , ψ , φ ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) )$ and $Λ$ is a nonlinear operator belonging to $θ = { Λ ∈ C 2 ( [ 0 , ∞ ) , ( 0 , ∞ ) ) | ∃ p ∈ ( 2 , ∞ ) : Λ ( m s ) ≤ m p − 2 Λ ( s ) , 0 < m < 1 }$.
Motivated by the above work, we studied the existence of entire positive radial solutions to the following Schrödinger system
$d i v Λ ( | ∇ μ | p − 2 ) ∇ μ = b ( | χ | ) ψ μ , ν , χ ∈ R n , d i v Λ ( | ∇ ν | p − 2 ) ∇ ν = h ( | χ | ) φ μ , ν , χ ∈ R n ,$
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
$∫ i ∞ 1 ψ t , ( φ ( t , t ) ) 1 p − 1 + 1 1 p − 1 d t = ∞ , ∀ i > 0$
and
$∫ j ∞ 1 φ ( ψ ( t , t ) ) 1 p − 1 , t + 1 1 p − 1 d t = ∞ , ∀ j > 0 .$
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].

## 2. Preliminaries

In this section, we give a definition, some notations, assumptions and Lemmas that are subsequently needed in the proof.
Firstly, we present the definition about the classification of solutions.
Definition 1
([34]). A solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$ of system (1) is called an entire bounded solution if condition (2) is established; it is called an entire blow-up solution if condition (3) is established; it is called a semifinite entire blow-up solution if condition (4) or (5) is established.
Finite case: both components μ and ν are bounded, that is
$lim | χ | → ∞ μ ( | χ | ) < ∞ a n d lim | χ | → ∞ ν ( | χ | ) < ∞ .$
Infinite case: both components μ, ν are blow-up, that is
$lim | χ | → ∞ μ ( | χ | ) = ∞ a n d lim | χ | → ∞ ν ( | χ | ) = ∞ .$
Semifinite Case: one of the components is bounded, while the other is blow-up, that is
$lim | χ | → ∞ μ ( | χ | ) < ∞ a n d lim | χ | → ∞ ν ( | χ | ) = ∞$
or
$lim | χ | → ∞ μ ( | χ | ) = ∞ a n d lim | χ | → ∞ ν ( | χ | ) < ∞ .$
We then present the notations as follows: $τ = | χ |$, $i , j , c 1 , c 2 ∈ ( 0 , ∞ )$ are suitably chosen,
$G 1 ( τ ) = ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 b ( s ) d s d t , G 2 ( τ ) = ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 h ( s ) d s d t , L ( τ ) = ∫ i + j τ d t [ ( ψ + φ ) ( t , t ) + 1 ] 1 p − 1 , L ( ∞ ) : = lim τ → ∞ L ( τ ) , G ( τ ) = G 1 ( τ ) + G 2 ( τ ) , G k ( ∞ ) : = lim τ → ∞ G k ( τ ) , k = 1 , 2 , ω 1 ( τ ) = ψ 1 , j φ ( i , i ) p − 1 + c 2 φ ( 1 , 1 + L − 1 ( G ( τ ) ) i ) + 1 φ ( i , i ) 1 p − 1 G 2 ( τ ) , ω 2 ( τ ) = φ i ψ ( j , j ) p − 1 + c 1 ψ ( 1 + L − 1 ( G ( τ ) ) j , 1 ) + 1 ψ ( j , j ) 1 p − 1 G 1 ( τ ) , 1 , U 1 ( τ ) = ∫ 0 τ ℑ − 1 1 ϱ n − 1 ∫ 0 ϱ t n − 1 b ( t ) ψ i , j + ( 1 φ ( i , j ) + 1 ) 1 p − 1 G 2 ( t ) d t d ϱ , V 1 ( τ ) = ∫ 0 τ ℑ − 1 1 ϱ n − 1 ∫ 0 ϱ t n − 1 h ( t ) φ i + ( 1 ψ ( i , j ) + 1 ) 1 p − 1 G 1 ( t ) , j d t d ϱ , U 2 ( τ ) = ∫ 0 τ c 1 ω 1 ( t ) + 1 1 p − 1 ℑ − 1 ∫ 0 t b ( s ) d s d t , V 2 ( τ ) = ∫ 0 τ c 2 ω 2 ( t ) + 1 1 p − 1 ℑ − 1 ∫ 0 t h ( s ) d s d t , U k ( ∞ ) : = lim τ → ∞ U k ( τ ) , V k ( ∞ ) : = lim τ → ∞ V k ( τ ) , f o r k = 1 , 2 , F 1 ( τ ) = ∫ i τ 1 ψ t , ( φ ( t , t ) ) 1 p − 1 + 1 1 p − 1 d t , F 1 ( ∞ ) : = lim r → ∞ F 1 ( τ ) , F 2 ( τ ) = ∫ j τ 1 φ ( ψ ( t , t ) ) 1 p − 1 , t + 1 1 p − 1 d t , F 2 ( ∞ ) : = lim r → ∞ F 2 ( τ ) .$
Assume that $ψ$ and $φ$ satisfy the following assumptions.
$( N 1 )$$ψ , φ ∈ C [ 0 , ∞ ) × [ 0 , ∞ ) , [ 0 , ∞ )$ are increasing for every variable and $ψ ( μ , ν ) > 0 , φ ( μ , ν ) > 0$ for all $μ , ν > 0$;
$( N 2 )$ for fixed constants $i , j ∈ ( 0 , ∞ )$, there exist $c 1 , c 2 ∈ ( 0 , ∞ )$ such that
$ψ ( t 1 s 1 , t 2 s 2 ) ≤ c 1 ψ ( t 1 , t 2 ) ψ ( s 1 , s 2 ) ,$
$φ ( t 1 s 1 , t 3 s 3 ) ≤ c 2 φ ( t 1 , t 3 ) φ ( s 1 , s 3 ) ,$
$ψ ( i , j ) ≥ 5 − 1 2 a n d φ ( i , j ) ≥ 5 − 1 2 ,$
where $t 1 ≥ min i , j , ψ 1 p − 1 ( j , j )$, $s 1 ≥ min 1 , i ψ 1 1 − p ( j , j )$, $t 2 ≥ min j , φ 1 p − 1 ( i , i )$, $s 2 ≥ min 1 , j φ 1 1 − p ( i , i )$, $t 3 ≥ min i , j$, $s 3 ≥ 1$;
$( S 1 )$$U 2 ( ∞ ) < F 1 ( ∞ ) < ∞ , V 2 ( ∞ ) < F 2 ( ∞ ) < ∞$;
$( S 2 )$$U 1 ( ∞ ) < ∞ , V 1 ( ∞ ) < ∞$;
$( S 3 )$$F 1 ( ∞ ) = F 2 ( ∞ ) = ∞ , U 2 ( ∞ ) = V 2 ( ∞ ) = ∞$;
$( S 4 )$$U 1 ( ∞ ) = V 1 ( ∞ ) = ∞$;
$( S 5 )$$F 1 ( ∞ ) = ∞ , U 1 ( ∞ ) = ∞ , U 2 ( ∞ ) = ∞$;
$( S 6 )$$V 1 ( ∞ ) < ∞ , V 2 ( ∞ ) < F 2 ( ∞ ) < ∞$;
$( S 7 )$$F 2 ( ∞ ) = ∞ , V 1 ( ∞ ) = ∞ , V 2 ( ∞ ) = ∞$;
$( S 8 )$$U 1 ( ∞ ) < ∞ , U 2 ( ∞ ) < F 1 ( ∞ ) < ∞$.
Lemma 1
([18]). If $Λ ∈ θ$, let $ℑ ( s ) = s Λ ( s p − 2 )$. We have
$( 1 ) :$$ℑ ( s )$ has a nonnegative increasing inverse mapping $ℑ − 1 ( s )$;
$( 2 ) :$ If $0 < q < 1$, we have
$ℑ − 1 ( q s ) ≥ q 1 p − 1 ℑ − 1 ( s ) ;$
$( 3 ) :$ If $q ≥ 1$, we have
$ℑ − 1 ( q s ) ≤ q 1 p − 1 ℑ − 1 ( s ) .$
Through the similar proof as in [19], we can obtain the following Lemma.
Lemma 2.
$( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$ is a radial solution of the Schrödinger system (1) if and only if it is a solution of the following ordinary differential system
$Λ ( | μ ′ | p − 2 ) μ ′ ′ + n − 1 τ Λ ( | μ ′ | p − 2 ) μ ′ = b ( τ ) ψ μ , ν , τ > 0 , Λ ( | ν ′ | p − 2 ) ν ′ ′ + n − 1 τ Λ ( | ν ′ | p − 2 ) ν ′ = h ( τ ) φ μ , ν , τ > 0 .$

## 3. The Entire Positive Bounded Radial Solutions

In this section, we prove Theorems 1 and 2.
Theorem 1.
Assume that $( N 1 )$, $( N 2 )$ hold, then the system (1) has an entire positive radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$.
Proof.
Through an operation on system (6), we obtain
$ℑ ( μ ′ ) ′ + n − 1 τ ℑ ( μ ′ ) = b ( τ ) ψ μ ( τ ) , ν ( τ ) , τ > 0 , ℑ ( ν ′ ) ′ + n − 1 τ ℑ ( ν ′ ) = h ( τ ) φ μ ( τ ) , ν ( τ ) , τ > 0 .$
Obviously, the above system can be transformed into the following system
$μ ( τ ) = μ ( 0 ) + ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 b ( s ) ψ μ ( s ) , ν ( s ) d s d t , τ ≥ 0 , ν ( τ ) = ν ( 0 ) + ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 h ( s ) φ μ ( s ) , ν ( s ) d s d t , τ ≥ 0 .$
Define the sequences ${ μ m ( τ ) } m ≥ 0$ and ${ ν m ( τ ) } m ≥ 0$ on $[ 0 , ∞ )$ by
$μ 0 ( τ ) = μ ( 0 ) = i , ν 0 ( τ ) = ν ( 0 ) = j , τ ≥ 0 , μ m ( τ ) = μ ( 0 ) + ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 b ( s ) ψ μ m − 1 ( s ) , ν m − 1 ( s ) d s d t , τ ≥ 0 , ν m ( τ ) = ν ( 0 ) + ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 h ( s ) φ μ m − 1 ( s ) , ν m − 1 ( s ) d s d t , τ ≥ 0 .$
Using the similar arguments as in [19], we obtain the sequences ${ μ m ( τ ) } m ≥ 0$ and ${ ν m ( τ ) } m ≥ 0$ are increasing and
$( μ m ( τ ) + ν m ( τ ) ) ′ [ ( ψ + φ ) ( μ m ( τ ) + ν m ( τ ) , μ m ( τ ) + ν m ( τ ) ) + 1 ] 1 p − 1 ≤ G 1 ′ ( τ ) + G 2 ′ ( τ ) .$
We then arrive at
$∫ i + j μ m ( τ ) + ν m ( τ ) d t [ ( ψ + φ ) ( t , t ) + 1 ] 1 p − 1 ≤ G ( τ ) .$
Therefore,
$L ( μ m ( τ ) + ν m ( τ ) ) ≤ G ( τ ) .$
By $( N 1 )$, we can obtain that $L ′ ( τ ) > 0$ and $L ( τ )$ is a bijection. Clearly, the inverse function $L − 1$ is strictly increasing on $[ 0 , L ( ∞ ) )$ and
$μ m ( τ ) + ν m ( τ ) ≤ L − 1 ( G ( τ ) ) .$
By Lemma 1, $( N 1 )$, $( N 2 )$, (7) and (8), the monotonicity of ${ μ m ( τ ) } m ≥ 0$ and ${ ν m ( τ ) } m ≥ 0$, we obtain
$μ m ( τ ) ≤ i + ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 b ( s ) ψ μ m ( s ) , ν m ( s ) d s d t ≤ i + ψ μ m ( τ ) , ν m ( τ ) + 1 1 p − 1 ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 b ( s ) d s d t ≤ i + ψ ν m ( τ ) + L − 1 ( G ( τ ) ) , ν m ( τ ) + 1 1 p − 1 G 1 ( τ ) = i + ψ ν m ( τ ) ( 1 + L − 1 ( G ( τ ) ) ν m ( τ ) ) , ν m ( τ ) + 1 1 p − 1 G 1 ( τ ) ≤ i + ψ ν m ( τ ) ( 1 + L − 1 ( G ( τ ) ) j ) , ν m ( τ ) + 1 1 p − 1 G 1 ( τ ) ≤ i + c 1 ψ ν m ( τ ) , ν m ( τ ) ψ 1 + L − 1 ( G ( τ ) ) j , 1 + 1 1 p − 1 G 1 ( τ ) = ψ ν m ( τ ) , ν m ( τ ) 1 p − 1 ( i ψ ( ν m ( τ ) , ν m ( τ ) ) 1 p − 1 + ( c 1 ψ ( 1 + L − 1 ( G ( τ ) ) j , 1 ) + 1 ψ ( ν m ( τ ) , ν m ( τ ) ) ) 1 p − 1 G 1 ( τ ) ) ≤ ψ ν m ( τ ) , ν m ( τ ) 1 p − 1 i ψ ( j , j ) 1 p − 1 + c 1 ψ ( 1 + L − 1 ( G ( τ ) ) j , 1 ) + 1 ψ ( j , j ) 1 p − 1 G 1 ( τ )$
and
$ν m ( τ ) ≤ j + ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 h ( s ) φ μ m ( s ) , ν m ( s ) d s d t ≤ j + φ μ m ( τ ) , ν m ( τ ) + 1 1 p − 1 ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 h ( s ) d s d t ≤ j + φ μ m ( τ ) , μ m ( τ ) + L − 1 ( G ( τ ) ) + 1 1 p − 1 G 2 ( τ ) = j + φ μ m ( τ ) , μ m ( τ ) ( 1 + L − 1 ( G ( τ ) ) μ m ( τ ) ) + 1 1 p − 1 G 2 ( τ ) ≤ j + φ μ m ( τ ) , μ m ( τ ) ( 1 + L − 1 ( G ( τ ) ) i ) + 1 1 p − 1 G 2 ( τ ) ≤ j + c 2 φ μ m ( τ ) , μ m ( τ ) φ 1 , 1 + L − 1 ( G ( τ ) ) i + 1 1 p − 1 G 2 ( τ ) = φ μ m ( τ ) , μ m ( τ ) 1 p − 1 ( j φ ( μ m ( τ ) , μ m ( τ ) ) 1 p − 1 + ( c 2 φ ( 1 , 1 + L − 1 ( G ( τ ) ) i ) + 1 φ ( μ m ( τ ) , μ m ( τ ) ) ) 1 p − 1 G 2 ( τ ) ) ≤ φ μ m ( τ ) , μ m ( τ ) 1 p − 1 j φ ( i , i ) 1 p − 1 + c 2 φ ( 1 , 1 + L − 1 ( G ( τ ) ) i ) + 1 φ ( i , i ) 1 p − 1 G 2 ( τ ) .$
By $( N 1 )$, $( N 2 )$, (9) and (10) and the monotonicity of ${ μ m ( τ ) } m ≥ 0$ and ${ ν m ( τ ) } m ≥ 0$, we obtain
$ℑ ( μ m ( τ ) ) ′ ′ + n − 1 τ ℑ μ m ( τ ) ′ = b ( τ ) ψ μ m − 1 ( τ ) , ν m − 1 ( τ ) ≤ b ( τ ) ψ μ m ( τ ) , ν m ( τ ) ≤ b ( τ ) ψ ( μ m ( τ ) , φ μ m ( τ ) , μ m ( τ ) 1 p − 1 ( j φ ( i , i ) 1 p − 1 + c 2 φ ( 1 , 1 + L − 1 ( G ( τ ) ) i ) + 1 φ ( i , i ) 1 p − 1 G 2 ( τ ) ) ) ≤ b ( τ ) c 1 ψ μ m ( τ ) , φ μ m ( τ ) , μ m ( τ ) 1 p − 1 ω 1 ( τ )$
and
$ℑ ( ν m ( τ ) ) ′ ′ + n − 1 τ ℑ ν m ( τ ) ′ = h ( τ ) φ μ m − 1 ( τ ) , ν m − 1 ( τ ) ≤ h ( τ ) φ μ m ( τ ) , ν m ( τ ) ≤ h ( τ ) φ ( ψ ν m ( τ ) , ν m ( τ ) 1 p − 1 ( i ψ ( j , j ) 1 p − 1 + c 1 ψ ( 1 + L − 1 ( G ( τ ) ) j , 1 ) + 1 ψ ( j , j ) 1 p − 1 G 1 ( τ ) ) , ν m ( τ ) ) ≤ h ( τ ) c 2 φ ψ ν m ( τ ) , ν m ( τ ) 1 p − 1 , ν m ( τ ) ω 2 ( τ ) .$
From the above inequalities, we obtain
$ℑ ( μ m ( τ ) ) ′ ′ ≤ ℑ ( μ m ( τ ) ) ′ ′ + n − 1 τ ℑ − 1 μ m ( τ ) ′ ≤ b ( τ ) c 1 ψ μ m ( τ ) , φ μ m ( τ ) , μ m ( τ ) 1 p − 1 ω 1 ( τ )$
and
$ℑ ( ν m ( τ ) ) ′ ′ ≤ ℑ ( ν m ( τ ) ) ′ ′ + n − 1 τ ℑ − 1 ν m ( τ ) ′ ≤ h ( τ ) c 2 φ ψ ν m ( τ ) , ν m ( τ ) 1 p − 1 , ν m ( τ ) ω 2 ( τ ) .$
We then arrive at
$ℑ μ m ( τ ) ′ ≤ ∫ 0 τ b ( s ) c 1 ψ μ m ( s ) , φ μ m ( s ) , μ m ( s ) 1 p − 1 ω 1 ( s ) d s$
and
$ℑ ν m ( τ ) ′ ≤ ∫ 0 τ h ( s ) c 2 φ ψ ν m ( s ) , ν m ( s ) 1 p − 1 , ν m ( s ) ω 2 ( s ) d s .$
By Lemma 1, $( N 1 )$, (13) and (14), we obtain
$μ m ( τ ) ′ ≤ ℑ − 1 ∫ 0 τ b ( s ) c 1 ψ μ m ( s ) , φ ( μ m ( s ) , μ m ( s ) ) 1 p − 1 ω 1 ( s ) d s ≤ ℑ − 1 c 1 ω 1 ( τ ) ∫ 0 τ b ( s ) ψ μ m ( s ) , φ ( μ m ( s ) , μ m ( s ) ) 1 p − 1 d s ≤ c 1 ω 1 ( τ ) + 1 1 p − 1 ℑ − 1 ∫ 0 τ b ( s ) ψ μ m ( s ) , φ ( μ m ( s ) , μ m ( s ) ) 1 p − 1 d s ≤ c 1 ω 1 ( τ ) + 1 1 p − 1 ℑ − 1 ψ μ m ( τ ) , φ ( μ m ( τ ) , μ m ( τ ) ) 1 p − 1 ∫ 0 τ b ( s ) d s ≤ c 1 ω 1 ( τ ) + 1 1 p − 1 ψ μ m ( τ ) , ( φ ( μ m ( τ ) , μ m ( τ ) ) ) 1 p − 1 + 1 1 p − 1 ℑ − 1 ∫ 0 τ b ( s ) d s$
and
$ν m ( τ ) ′ ≤ ℑ − 1 ∫ 0 τ h ( s ) c 2 φ ψ ( ν m ( s ) , ν m ( s ) ) 1 p − 1 , ν m ( s ) ω 2 ( s ) d s ≤ ℑ − 1 c 2 ω 2 ( τ ) ∫ 0 τ h ( s ) φ ψ ( ν m ( s ) , ν m ( s ) ) 1 p − 1 , ν m ( s ) d s ≤ c 2 ω 2 ( τ ) + 1 1 p − 1 ℑ − 1 ∫ 0 τ h ( s ) φ ψ ( ν m ( s ) , ν m ( s ) ) 1 p − 1 , ν m ( s ) d s ≤ c 2 ω 2 ( τ ) + 1 1 p − 1 ℑ − 1 φ ψ ( ν m ( τ ) , ν m ( τ ) ) 1 p − 1 , ν m ( τ ) ∫ 0 τ h ( s ) d s ≤ c 2 ω 2 ( τ ) + 1 1 p − 1 φ ( ψ ( ν m ( τ ) , ν m ( τ ) ) ) 1 p − 1 , ν m ( τ ) + 1 1 p − 1 ℑ − 1 ∫ 0 τ h ( s ) d s .$
From the above two inequalities, we easily deduce that
$μ m ( τ ) ′ ψ μ m ( τ ) , ( φ ( μ m ( τ ) , μ m ( τ ) ) ) 1 p − 1 + 1 1 p − 1 ≤ c 1 ω 1 ( τ ) + 1 1 p − 1 ℑ − 1 ∫ 0 τ b ( s ) d s$
and
$ν m ( τ ) ′ φ ( ψ ( ν m ( τ ) , ν m ( τ ) ) ) 1 p − 1 , ν m ( τ ) + 1 1 p − 1 ≤ c 2 ω 2 ( τ ) + 1 1 p − 1 ℑ − 1 ∫ 0 τ h ( s ) d s .$
We then arrive at
$∫ i μ m ( τ ) 1 ψ t , ( φ ( t , t ) ) 1 p − 1 + 1 1 p − 1 d t ≤ ∫ 0 τ c 1 ω 1 ( t ) + 1 1 p − 1 ℑ − 1 ∫ 0 t b ( s ) d s d t$
and
$∫ j ν m ( τ ) 1 φ ( ψ ( t , t ) ) 1 p − 1 , t + 1 1 p − 1 d t ≤ ∫ 0 τ c 2 ω 2 ( t ) + 1 1 p − 1 ℑ − 1 ∫ 0 t h ( s ) d s d t .$
Now the above two inequalities can be expressed as
$F 1 ( μ m ( τ ) ) ≤ U 2 ( τ ) , ∀ τ ≥ 0$
and
$F 2 ( ν m ( τ ) ) ≤ V 2 ( τ ) , ∀ τ ≥ 0 .$
It follows from the $( N 1 )$ that $F 1 − 1$ and $F 2 − 1$ are strictly increasing on $[ 0 , F 1 ( ∞ ) )$ and $[ 0 , F 2 ( ∞ ) )$ separately, we obtain
$μ m ( τ ) ≤ F 1 − 1 U 2 ( τ ) , ∀ τ ≥ 0$
and
$ν m ( τ ) ≤ F 2 − 1 V 2 ( τ ) , ∀ τ ≥ 0 .$
Since
$( μ m ( τ ) ) ′ ≥ 0 a n d ( ν m ( τ ) ) ′ ≥ 0 , ∀ τ ≥ 0 ,$
we obtain
$μ m ( τ ) ≤ μ m ( c 0 ) ≤ W 1 a n d ν m ( τ ) ≤ ν m ( c 0 ) ≤ W 2 , o n [ 0 , c 0 ] ,$
where $W 1 = F 1 − 1 ( U 2 ( c 0 ) )$ and $W 2 = F 2 − 1 ( V 2 ( c 0 ) )$ are positive constants. Moreover, from (15) and (16), we can deduce that ${ ( μ m ( τ ) ) ′ }$ and ${ ( ν m ( τ ) ) ′ }$ are bounded on $[ 0 , c 0 ]$ for arbitrary $c 0 > 0$. Therefore, the monotone sequences ${ μ m ( τ ) }$ and ${ ν m ( τ ) }$ are bounded and equicontinuous on $[ 0 , c 0 ]$. By employing the Arzela–Ascoli theorem, we obtain the subsequences of ${ μ m ( τ ) }$ and ${ ν m ( τ ) }$ uniformly converging towards $μ ( r )$ and $ν ( r )$ on $[ 0 , c 0 ]$. According to the arbitrariness of $c 0$, we obtain that $( μ , ν )$ is an entire positive solution of the system (6). Thus, from Lemma 2, we obtain that $( μ , ν )$ is an entire positive radial solution of the system (1). □
Theorem 2.
Assuming that $( N 1 )$, $( N 2 )$, $( S 1 )$ and $( S 2 )$ hold, then the system (1) has an entire positive bounded radial solution $( μ , ν )$ such that
$i + U 1 ( τ ) ≤ μ ( τ ) ≤ F 1 − 1 ( U 2 ( τ ) ) , j + V 1 ( τ ) ≤ ν ( τ ) ≤ F 2 − 1 ( V 2 ( τ ) ) .$
Proof.
On the basis of $( N 1 )$ and $( N 2 )$, by Theorem 1, we see that the system (1) has an entire positive radial solution $( μ , ν )$. Moreover, it follows from (19), (20) and $( S 1 )$ that
$F 1 ( μ m ( τ ) ) ≤ U 2 ( ∞ ) < F 1 ( ∞ ) < ∞ , ∀ τ ≥ 0$
and
$F 2 ( ν m ( τ ) ) ≤ V 2 ( ∞ ) < F 2 ( ∞ ) < ∞ , ∀ τ ≥ 0 .$
Since $F 1 − 1$ and $F 2 − 1$ are strictly increasing on $[ 0 , F 1 ( ∞ ) )$ and $[ 0 , F 2 ( ∞ ) )$ separately, we obtain
$μ m ( τ ) ≤ F 1 − 1 U 2 ( ∞ ) < ∞ , ∀ τ ≥ 0$
and
$ν m ( τ ) ≤ F 2 − 1 V 2 ( ∞ ) < ∞ , ∀ τ ≥ 0 .$
Letting $m → ∞$ into the above two inequalities, we obtain
$μ ( τ ) ≤ F 1 − 1 U 2 ( ∞ ) < ∞ , ∀ τ ≥ 0$
and
$ν ( τ ) ≤ F 2 − 1 V 2 ( ∞ ) < ∞ , ∀ τ ≥ 0 .$
Letting $m → ∞$ in (7), we obtain
$μ ( τ ) = i + ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 b ( s ) ψ μ ( s ) , ν ( s ) d s d t$
and
$ν ( τ ) = j + ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 h ( s ) φ μ ( s ) , ν ( s ) d s d t .$
Then, it follows from Lemma 1, $( N 1 )$, $( N 2 )$ and $( S 2 )$ that
$μ ( r ) = i + ∫ 0 τ ℑ − 1 1 t n − 1 ∫ 0 t s n − 1 b ( s ) ψ μ ( s ) , ν ( s ) d s d t ≥ i + ∫ 0 τ ℑ − 1 ( 1 ϱ n − 1 ∫ 0 ϱ t n − 1 b ( t ) ψ ( i , j + ∫ 0 t ℑ − 1 1 σ n − 1 ∫ 0 σ s n − 1 h ( s ) φ ( μ ( s ) , ν ( s ) ) d s d σ ) d t ) d ϱ ≥ i + ∫ 0 τ ℑ − 1 1 ϱ n − 1 ∫ 0 ϱ t n − 1 b ( t ) ψ i , j + ( 1 φ ( i , j ) + 1 ) 1 p − 1 G 2 ( t ) d t d ϱ = i + U 1 ( τ ) .$
As with the above proof, we can prove that
$ν ( τ ) ≥ j + V 1 ( τ ) .$

## 4. The Entire Positive Blow-Up Radial Solutions

In this section, we prove Theorem 3.
Theorem 3.
Assume that $( N 1 )$, $( N 2 )$, $( S 3 )$ and $( S 4 )$ hold, then the system (1) has an entire positive blow-up radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$.
Proof.
On the basis of $( N 1 )$, $( N 2 )$, by Theorem 1, we see that the system (1) has an entire positive radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$. Moreover, it follows from (19) and (20) that
$F 1 ( μ m ( τ ) ) ≤ U 2 ( ∞ ) , ∀ τ ≥ 0$
and
$F 2 ( ν m ( τ ) ) ≤ V 2 ( ∞ ) , ∀ τ ≥ 0 .$
Since $F 1 − 1$ and $F 2 − 1$ are strictly increasing on $[ 0 , F 1 ( ∞ ) )$ and $[ 0 , F 2 ( ∞ ) )$ separately, we arrive at
$μ m ( τ ) ≤ F 1 − 1 U 2 ( ∞ ) , ∀ τ ≥ 0$
and
$ν m ( τ ) ≤ F 2 − 1 V 2 ( ∞ ) , ∀ τ ≥ 0 .$
When $( S 3 )$ holds, we see that $F 1 − 1 ( ∞ ) = F 2 − 1 ( ∞ ) = ∞$. Letting $m → ∞$ into the above two inequalities, we have
$μ ( τ ) ≤ F 1 − 1 U 2 ( ∞ ) , ∀ τ ≥ 0$
and
$ν ( τ ) ≤ F 2 − 1 V 2 ( ∞ ) , ∀ τ ≥ 0 .$
By condition $( S 3 )$, letting $τ → ∞$ into the above two inequalities, we obtain
$lim τ → ∞ μ ( τ ) ≤ F 1 − 1 U 2 ( ∞ ) = ∞ , ∀ τ ≥ 0$
and
$lim τ → ∞ ν ( τ ) ≤ F 2 − 1 V 2 ( ∞ ) = ∞ , ∀ τ ≥ 0 .$
Then, it follows from $( S 4 )$, (23) and (24) that
$lim τ → ∞ μ ( τ ) ≥ i + lim τ → ∞ U 1 ( τ ) > U 1 ( ∞ ) = ∞$
and
$lim τ → ∞ ν ( τ ) ≥ j + lim τ → ∞ V 1 ( τ ) > V 1 ( ∞ ) = ∞ .$
Consequently,
$lim τ → ∞ μ ( τ ) = ∞ a n d lim τ → ∞ ν ( τ ) = ∞ ,$
which imply that the system (6) has an entire positive blow-up solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$. From Lemma 2, the system (1) has an entire positive blow-up radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$. □

## 5. The Semifinite Entire Positive Blow-Up Radial Solutions

In this section, we prove Theorems 4 and 5.
Theorem 4.
Assuming that $( N 1 )$, $( N 2 )$, $( S 5 )$ and $( S 6 )$ hold, the system (1) then has a semifinite entire positive blow-up radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$.
Proof.
In view of $( N 1 )$, $( N 2 )$, by Theorem 1, we see that system (1) has an entire positive radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$. By $( S 5 )$, (25) and (27), we obtain
$lim τ → ∞ μ ( τ ) ≤ F 1 − 1 U 2 ( ∞ ) = ∞$
and
$lim τ → ∞ μ ( τ ) ≥ i + lim τ → ∞ U 1 ( τ ) > U 1 ( ∞ ) = ∞ ,$
which imply that
$lim τ → ∞ μ ( τ ) = ∞ .$
Moreover, by $( S 6 )$, (22) and (24), we obtain
$lim τ → ∞ ν ( τ ) ≤ F 2 − 1 V 2 ( ∞ ) < ∞$
and
$lim τ → ∞ ν ( τ ) ≥ j + lim τ → ∞ V 1 ( τ ) > V 1 ( ∞ ) , V 1 ( ∞ ) < ∞ ,$
which imply that
$lim ν → ∞ ν ( τ ) < ∞ .$
Therefore, system (6) has a semifinite entire positive blow-up solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$. From Lemma 2, the system (1) has a semifinite entire positive blow-up radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$. □
Theorem 5.
Assume that $( N 1 )$, $( N 2 )$, $( S 7 )$ and $( S 8 )$ hold, then the system (1) has a semifinite entire positive blow-up radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$.
Proof.
In view of $( N 1 )$, $( N 2 )$, by Theorem 1, we see that system (1) has an entire positive radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$. By $( S 7 )$, (26) and (28), we obtain
$lim τ → ∞ ν ( τ ) ≤ F 2 − 1 V 2 ( ∞ ) = ∞$
and
$lim τ → ∞ ν ( τ ) ≥ j + lim τ → ∞ V 1 ( τ ) > V 1 ( ∞ ) = ∞ ,$
which imply that
$lim τ → ∞ ν ( τ ) = ∞ .$
Moreover, by $( S 8 )$, (21) and (23), we obtain
$lim τ → ∞ μ ( τ ) ≤ F 1 − 1 U 2 ( ∞ ) < ∞$
and
$lim τ → ∞ μ ( τ ) ≥ i + lim τ → ∞ U 1 ( τ ) > U 1 ( ∞ ) , U 1 ( ∞ ) < ∞ ,$
which imply that
$lim τ → ∞ μ ( τ ) < ∞ .$
Therefore, system (6) has a semifinite entire positive blow-up solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$. From Lemma 2, system (1) has a semifinite entire positive blow-up radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$. □

## 6. Example

Example 1.
Consider the following Schrödinger system
$d i v Λ ( | ∇ μ | 5 ) ∇ μ = 3 4 3 − | χ | | χ | e | χ | μ 1 2 ν 1 2 , χ ∈ R 6 , d i v Λ ( | ∇ ν | 5 ) ∇ ν = 1 4 3 − 2 | χ | | χ | e 2 | χ | μ 1 3 ν 2 3 , χ ∈ R 6 .$
Let $Λ ( s ) = s 5$, $p = 7$, then $Λ ∈ θ$. Here $b ( s ) = 3 − s s e s$, $h ( s ) = 3 − 2 s s e 2 s$, $ψ ( μ , ν ) = 3 4 μ 1 2 ν 1 2$, $φ ( μ , ν ) = 1 4 μ 1 3 ν 2 3$, then $ψ$ and $φ$ are increasing for each variable and $( N 1 )$ holds. Obviously, when $i = j = 4$, we have $t 1 ≥ 3 6$, $s 1 ≥ 1$, $t 2 ≥ 1$, $s 2 ≥ 1$, $t 3 ≥ 4$, $s 3 ≥ 1$,
$ψ ( t 1 s 1 , t 2 s 2 ) = 3 4 t 1 1 2 s 1 1 2 t 2 1 2 s 2 1 2 ≤ c 1 3 4 t 1 1 2 t 2 1 2 3 4 s 1 1 2 s 2 1 2 = c 1 ψ ( t 1 , t 2 ) ψ ( s 1 , s 2 ) , ∀ c 1 ≥ 4 3 ,$
$φ ( t 1 s 1 , t 3 s 3 ) = 1 4 t 1 1 3 s 1 1 3 t 3 2 3 s 3 2 3 ≤ c 2 1 4 t 1 1 3 t 3 2 3 1 4 s 1 1 3 s 3 2 3 = c 2 φ ( t 1 , t 3 ) φ ( s 1 , s 3 ) , ∀ c 2 ≥ 4 ,$
$ψ ( i , j ) ≥ 5 − 1 2 a n d φ ( i , j ) ≥ 5 − 1 2 ,$
meaning that $( N 2 )$ is established. From Theorem 1, the Schrödinger system (29) has an entire positive radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$.
Example 2.
Consider the following Schrödinger system
$d i v Λ ( | ∇ μ | 3 ) ∇ μ = | χ | 3 ( μ 4 + ν 3 ) , χ ∈ R 4 , d i v Λ ( | ∇ ν | 3 ) ∇ ν = ( 3 | χ | − 1 e | χ | + e | χ | ) μ ν 3 , χ ∈ R 4 .$
Let $Λ ( s ) = s 3$, $p = 5$, then $Λ ∈ θ$. Here, $b ( s ) = s 3$, $h ( s ) = 3 s − 1 e s + e s$, $ψ ( μ , ν ) = μ 4 + ν 3$, $φ ( μ , ν ) = μ ν 3$, then $φ$ and $ψ$ are increasing for each variable and $( N 1 )$ holds. Obviously, when $i = j = 1$, we have $t 1 ≥ 1$, $s 1 ≥ 1 2 4$, $t 2 ≥ 1$, $s 2 ≥ 1$, $t 3 ≥ 1$, $s 3 ≥ 1$,
$ψ ( t 1 s 1 , t 2 s 2 ) = t 1 4 s 1 4 + t 2 3 s 2 3 ≤ c 1 ( t 1 4 + t 2 3 ) ( s 1 4 + s 2 3 ) = c 1 ψ ( t 1 , t 2 ) ψ ( s 1 , s 2 ) , ∀ c 1 ≥ 1 ,$
$φ ( t 1 s 1 , t 3 s 3 ) = t 1 s 1 t 3 3 s 3 3 ≤ c 2 t 1 t 3 3 s 1 s 3 3 = c 2 φ ( t 1 , t 3 ) φ ( s 1 , s 3 ) , ∀ c 2 ≥ 1 ,$
$ψ ( i , j ) ≥ 5 − 1 2 a n d φ ( i , j ) ≥ 5 − 1 2 ,$
meaning that $( N 2 )$ is established. After a simple calculation, one has
$U 2 ( ∞ ) = ∫ 0 ∞ c 1 ω 1 ( t ) + 1 1 4 ℑ − 1 ∫ 0 t b ( s ) d s d t > ∫ 0 ∞ 1 4 t 4 10 d t = 1 4 10 ∫ 0 ∞ t 2 5 d t = ∞ ,$
$V 2 ( ∞ ) = ∫ 0 ∞ c 2 ω 2 ( t ) + 1 1 4 ℑ − 1 ∫ 0 t h ( s ) d s d t > ∫ 0 ∞ e t 10 d t = ∫ 0 ∞ e t 10 d t = ∞ ,$
$F 1 ( ∞ ) = ∫ i ∞ 1 ψ t , ( φ ( t , t ) ) 1 4 + 1 1 4 d t = ∫ i ∞ 1 t 4 + t 3 + 1 4 d t = ∞$
and
$F 2 ( ∞ ) = ∫ j ∞ 1 φ ( ψ ( t , t ) ) 1 4 , t + 1 1 4 d t = ∫ j ∞ 1 ( t 4 + t 3 ) 1 4 t 3 + 1 4 d t = ∞ ,$
meaning that $( S 3 )$ is established. We then have
$G 1 ( τ ) = ∫ 0 τ ℑ − 1 1 t 3 ∫ 0 t s 3 b ( s ) d s d t = ∫ 0 τ 1 t 3 ∫ 0 t s 6 d s 1 10 d t = 1 7 10 ∫ 0 τ t 2 5 d t = 5 7 1 7 10 τ 7 5 ,$
$G 2 ( τ ) = ∫ 0 τ ℑ − 1 1 t 3 ∫ 0 t s 3 h ( s ) d s d t = ∫ 0 τ 1 t 3 ∫ 0 t s 2 e s ( 3 + s ) d s 1 10 d t = ∫ 0 τ e t 10 d t = 10 e τ 10 ,$
$U 1 ( ∞ ) = ∫ 0 ∞ ℑ − 1 1 ϱ 3 ∫ 0 ϱ t 3 b ( t ) ψ i , j + ( 1 φ ( i , j ) + 1 ) 1 p − 1 G 2 ( t ) d t d ϱ > ∫ 0 ∞ ℑ − 1 1 ϱ 3 ∫ 0 ϱ t 3 b ( t ) ψ ( i , j ) d t d ϱ > ∫ 0 ∞ ℑ − 1 1 ϱ 3 ∫ 0 ϱ t 3 b ( t ) 1 ψ ( i , j ) + 1 d t d ϱ > 1 ψ i , j + 1 1 p − 1 ∫ 0 ∞ ℑ − 1 1 ϱ 3 ∫ 0 ϱ t 3 b ( t ) d t d ϱ = 1 2 + 1 1 4 G 1 ( ∞ ) = ∞$
and
$V 1 ( ∞ ) = ∫ 0 ∞ ℑ − 1 1 ϱ 3 ∫ 0 ϱ t 3 h ( t ) φ i + ( 1 ψ ( i , j ) + 1 ) 1 p − 1 G 1 ( t ) , j d t d ϱ > ∫ 0 ∞ ℑ − 1 1 ϱ 3 ∫ 0 ϱ t 3 h ( t ) φ ( i , j ) d t d ϱ > ∫ 0 ∞ ℑ − 1 1 ϱ 3 ∫ 0 ϱ t 3 h ( t ) 1 φ ( i , j ) + 1 d t d ϱ > 1 φ ( i , j ) + 1 1 p − 1 ∫ 0 ∞ ℑ − 1 1 ϱ 3 ∫ 0 ϱ t 3 h ( t ) d t d ϱ = 1 1 + 1 1 4 G 2 ( ∞ ) = ∞ ,$
meaning that $( S 4 )$ is established. From Theorem 3, the Schrödinger system (30) has an entire positive blow-up radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$.
Example 3.
Consider the following Schrödinger system
$d i v Λ ( | ∇ μ | 4 ) ∇ μ = | χ | 5 ( μ 4 + ν ) , χ ∈ R 5 , d i v Λ ( | ∇ ν | 4 ) ∇ ν = ( 6 | χ | − 1 e | χ | + 2 e | χ | ) μ 2 ν 2 , χ ∈ R 5 .$
Let $Λ ( s ) = s 4$, $p = 6$, then $Λ ∈ θ$. Here, $b ( s ) = s 5$, $h ( s ) = 6 s − 1 e s + 2 e s$, $ψ ( μ , ν ) = μ 4 + ν$, $φ ( μ , ν ) = μ 2 ν 2$, then $φ$ and $ψ$ are increasing for each variable and $( N 1 )$ holds. Obviously, when $i = j = 1$, we have $t 1 ≥ 1$, $s 1 ≥ 1 2 5$, $t 2 ≥ 1$, $s 2 ≥ 1$, $t 3 ≥ 1$, $s 3 ≥ 1$,
$ψ ( t 1 s 1 , t 2 s 2 ) = t 1 4 s 1 4 + t 2 s 2 ≤ c 1 ( t 1 4 + t 2 ) ( s 1 4 + s 2 ) = c 1 ψ ( t 1 , t 2 ) ψ ( s 1 , s 2 ) , ∀ c 1 ≥ 1 ,$
$φ ( t 1 s 1 , t 3 s 3 ) = t 1 2 s 1 2 t 3 2 s 3 2 ≤ c 2 t 1 2 t 3 2 s 1 2 s 3 2 = c 2 φ ( t 1 , t 3 ) φ ( s 1 , s 3 ) , ∀ c 2 ≥ 1 ,$
$ψ ( i , j ) ≥ 5 − 1 2 a n d φ ( i , j ) ≥ 5 − 1 2 ,$
meaning that $( N 2 )$ is established. After a simple calculation, one has
$U 2 ( ∞ ) = ∫ 0 ∞ c 1 ω 1 ( t ) + 1 1 4 ℑ − 1 ∫ 0 t b ( s ) d s d t > ∫ 0 ∞ 1 6 t 6 17 d t = 1 6 17 ∫ 0 ∞ t 6 17 d t = ∞ ,$
$V 2 ( ∞ ) = ∫ 0 ∞ c 2 ω 2 ( t ) + 1 1 4 ℑ − 1 ∫ 0 t h ( s ) d s d t > ∫ 0 ∞ e t 17 d t = ∫ 0 ∞ e t 17 d t = ∞ ,$
$F 1 ( ∞ ) = ∫ i ∞ 1 ψ t , ( φ ( t , t ) ) 1 4 + 1 1 4 d t = ∫ i ∞ 1 t 4 + t + 1 4 d t = ∞$
and
$F 2 ( ∞ ) = ∫ j ∞ 1 φ ( ψ ( t , t ) ) 1 4 , t + 1 1 4 d t = ∫ j ∞ 1 t 2 ( t 4 + t ) 1 2 + 1 4 d t = ∞ ,$
meaning that $( S 3 )$ is established. We then have
$G 1 ( τ ) = ∫ 0 τ ℑ − 1 1 t 3 ∫ 0 t s 3 b ( s ) d s d t = ∫ 0 τ 1 t 3 ∫ 0 t s 8 d s 1 17 d t = 1 9 17 ∫ 0 τ t 6 17 d t = 17 23 1 9 10 τ 23 17 ,$
$G 2 ( τ ) = ∫ 0 τ ℑ − 1 1 t 3 ∫ 0 t s 3 h ( s ) d s d t = ∫ 0 τ 1 t 3 ∫ 0 t s 2 e s ( 6 + 2 s ) d s 1 17 d t = 2 17 ∫ 0 τ e t 17 d t = 17 2 17 e τ 17 ,$
$U 1 ( ∞ ) = ∫ 0 ∞ ℑ − 1 1 ϱ 4 ∫ 0 ϱ t 4 b ( t ) ψ i , j + ( 1 φ ( i , j ) + 1 ) 1 p − 1 G 2 ( t ) d t d ϱ > ∫ 0 ∞ ℑ − 1 1 ϱ 4 ∫ 0 ϱ t 4 b ( t ) ψ ( i , j ) d t d ϱ > ∫ 0 ∞ ℑ − 1 1 ϱ 4 ∫ 0 ϱ t 4 b ( t ) 1 ψ ( i , j ) + 1 d t d ϱ > 1 ψ i , j + 1 1 p − 1 ∫ 0 ∞ ℑ − 1 1 ϱ 4 ∫ 0 ϱ t 4 b ( t ) d t d ϱ = 1 2 + 1 1 5 G 1 ( ∞ ) = ∞$
and
$V 1 ( ∞ ) = ∫ 0 ∞ ℑ − 1 1 ϱ 4 ∫ 0 ϱ t 4 h ( t ) φ i + ( 1 ψ ( i , j ) + 1 ) 1 p − 1 G 1 ( t ) , j d t d ϱ > ∫ 0 ∞ ℑ − 1 1 ϱ 4 ∫ 0 ϱ t 4 h ( t ) φ ( i , j ) d t d ϱ > ∫ 0 ∞ ℑ − 1 1 ϱ 4 ∫ 0 ϱ t 4 h ( t ) 1 φ ( i , j ) + 1 d t d ϱ > 1 φ ( i , j ) + 1 1 p − 1 ∫ 0 ∞ ℑ − 1 1 ϱ 4 ∫ 0 ϱ t 4 h ( t ) d t d ϱ = 1 1 + 1 1 5 G 2 ( ∞ ) = ∞ ,$
which mean that $( S 4 )$ is established. From Theorem 3, the Schrödinger system (31) has an entire positive blow-up radial solution $( μ , ν ) ∈ C 2 [ 0 , ∞ ) × C 2 [ 0 , ∞ )$.

## Author Contributions

G.W., Z.Z. and Z.Y. equally contributed this manuscript and approved the final version. All authors have read and agreed to the published version of this manuscript.

## Funding

The work was supported by NSF of Shanxi Province, China (No. 20210302123339) and the Graduate Education and Teaching Innovation Project of Shanxi, China (No. 2021YJJG142).

Not applicable.

Not applicable.

Not applicable.

## Acknowledgments

We thank the referees for their useful comments on our work that led to its improvement.

## Conflicts of Interest

The authors declare that they have no competing interests.

## References

1. Osgood, W.F. Beweis der Existenz einer LÖsung der Differential gleichung $d y d x$ = f (x, y) ohne Hinzunahme der Cauchy-Lipschitz’schen Bedingung. Monatsh. Math. Phys. 1898, 9, 331–345. [Google Scholar] [CrossRef] [Green Version]
2. Papi, M. A generalized Osgood condition for viscosity solutions to fully nonlinear parabolic degenerate equations. Adv. Differ. Equ. 2002, 7, 1125–1151. [Google Scholar]
3. Fan, S.; Jiang, L.; Davison, M. Existence and uniqueness result for multidimensional BSDEs with generators of Osgood type. Front. Math. China 2013, 8, 811–824. [Google Scholar] [CrossRef]
4. Li, K. No local L1 solutions for semilinear fractional heat equations. Fract. Calc. Appl. Anal. 2017, 20, 1328–1337. [Google Scholar] [CrossRef] [Green Version]
5. Villa-Morales, J. An Osgood condition for a semilinear reaction-diffusion equation with time-dependent generator. Arab J. Math. Sci. 2016, 22, 86–95. [Google Scholar] [CrossRef] [Green Version]
6. Zhang, L.; Hou, W. Standing waves of nonlinear fractional p-Laplacian Schrödinger equation involving logarithmic nonlinearity. Appl. Math. Lett. 2020, 102, 106149. [Google Scholar] [CrossRef]
7. Laister, R.; Robinson, J.C.; Sierżȩga, M. Non-existence of local solutions of semilinear heat equations of Osgood type in bounded domains. Comptes Rendus Math. 2014, 352, 621–626. [Google Scholar] [CrossRef] [Green Version]
8. Laister, R.; Robinson, J.C.; Sierżȩga, M. A necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations. J. Differ. Equ. 2017, 262, 4979–4987. [Google Scholar] [CrossRef] [Green Version]
9. Fan, S. Existence, uniqueness and approximation for Lp solutions of reflected BSDEs with generators of one-sided Osgood type. Acta Math. Sin. 2017, 33, 1–32. [Google Scholar] [CrossRef]
10. Kurihura, S. Large-amplitude quasi-solitons in superfluid films. J. Phys. Soc. 1981, 50, 3262–3267. [Google Scholar] [CrossRef]
11. Kosevich, A.; Ivanov, B.; Kovalev, A. Magnetic solitons. Phys. Rep. 1990, 194, 117–238. [Google Scholar] [CrossRef]
12. Quispel, G.; Capel, H. Equation of motion for the Heisenberg spin chain. Phys. Lett. A 1982, 110, 41–80. [Google Scholar] [CrossRef]
13. Severo, J. Quasilinear Schrödinger equations involving concave and convex nonlinearities. Commun. Pure Appl. Anal. 2009, 8, 621–644. [Google Scholar]
14. Miyagaki, O.H.; Soares, S.H. Soliton solutions for quasilinear Schrödinger equations with critical growth. J. Differ. Equ. 2010, 248, 722–744. [Google Scholar]
15. Floer, A.; Weinstein, A. Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 1986, 69, 397–408. [Google Scholar] [CrossRef] [Green Version]
16. Litvak, A.; Sergeev, A. One-dimensional collapse of plasma waves. JETP Lett. 1978, 27, 517–520. [Google Scholar]
17. Sun, Y.; Liu, L.; Wu, Y. The existence and uniqueness of positive monotone solutions for a class of nonlinear Schrödinger equations on infinite domains. J. Comput. Appl. Math. 2017, 321, 478–486. [Google Scholar] [CrossRef]
18. Zhang, X.; Wu, Y.; Cui, Y. Existence and nonexistence of blow-up solutions for a Schrödinger equation involving a nonlinear operator. Appl. Math. Lett. 2018, 82, 85–91. [Google Scholar] [CrossRef]
19. Wang, G.; Yang, Z.; Agarwal, R.P.; Zhang, L. Study on a class of Schrödinger elliptic system involving a nonlinear operator. Nonlinear Anal. Model. Control 2020, 25, 846–859. [Google Scholar] [CrossRef]
20. Zhang, Z.; Zhou, S. Existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Appl. Math. Lett. 2015, 50, 48–55. [Google Scholar] [CrossRef]
21. Zhang, Z. Optimal global and boundary asymptotic behavior of large solutions to the Monge-Ampère equation. J. Funct. Anal. 2020, 278, 108512. [Google Scholar] [CrossRef]
22. Qin, J.; Wang, G.; Zhang, L.; Ahmad, B. Monotone iterative method for a p-Laplacian boundary value problem with fractional conformable derivatives. Bound. Value Probl. 2019, 2019, 145. [Google Scholar] [CrossRef] [Green Version]
23. Wang, G. Twin iterative positive solutions of fractional q-difference Schrödinger equations. Appl. Math. Lett. 2018, 76, 103–109. [Google Scholar] [CrossRef]
24. Wang, G.; Ren, X.; Zhang, L.; Ahmad, B. Explicit iteration and unique positive solution for a Caputo-Hadamard fractional turbulent flow model. IEEE Access 2019, 7, 109833–109839. [Google Scholar] [CrossRef]
25. Pei, K.; Wang, G.; Sun, Y. Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain. Appl. Math. Comput. 2017, 312, 158–168. [Google Scholar] [CrossRef]
26. Zhang, L.; Ahmad, B.; Wang, G. Explicit iterations and extremal solutions for fractional differential equations with nonlinear integral boundary conditions. Appl. Math. Comput. 2015, 268, 388–392. [Google Scholar] [CrossRef]
27. Peterson, J.D.; Wood, A.W. Large solutions to non-monotone semilinear elliptic systems. J. Math. Anal. Appl. 2011, 384, 284–292. [Google Scholar] [CrossRef] [Green Version]
28. Zhang, X.; Liu, L. The existence and nonexistence of entire positive solutions of semilinear elliptic systems with gradient term. J. Math. Anal. Appl. 2010, 371, 300–308. [Google Scholar] [CrossRef] [Green Version]
29. Covei, D.-P. Radial and nonradial solutions for a semilinear elliptic system of Schrödinger type. Funkcial. Ekvac. 2011, 54, 439–449. [Google Scholar] [CrossRef] [Green Version]
30. Lair, A.V. A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems. J. Math. Anal. Appl. 2010, 365, 103–108. [Google Scholar] [CrossRef]
31. Lair, A.V.; Wood, A.W. Existence of entire large positive solutions of semilinear elliptic systems. J. Differ. Equ. 2000, 164, 380–394. [Google Scholar] [CrossRef] [Green Version]
32. Li, H.; Zhang, P.; Zhang, Z. A remark on the existence of entire positve solutions for a class of semilinear elliptic system. J. Math. Anal. Appl. 2010, 365, 338–341. [Google Scholar] [CrossRef] [Green Version]
33. Ghanmi, A.; Maagli, H.; Radulescu, V.; Zeddini, N. Large and bounded solutions for a class of nonlinear Schrödinger stationary systems. Anal. Appl. 2009, 7, 391–404. [Google Scholar] [CrossRef]
34. Covei, D.-P. The Keller-Osserman-type conditions for the study of a semilinear elliptic system. Bound. Value Probl. 2019, 2019, 104. [Google Scholar] [CrossRef]
 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Share and Cite

MDPI and ACS Style

Wang, G.; Zhang, Z.; Yang, Z. The Existence of Radial Solutions to the Schrödinger System Containing a Nonlinear Operator. Axioms 2022, 11, 282. https://doi.org/10.3390/axioms11060282

AMA Style

Wang G, Zhang Z, Yang Z. The Existence of Radial Solutions to the Schrödinger System Containing a Nonlinear Operator. Axioms. 2022; 11(6):282. https://doi.org/10.3390/axioms11060282

Chicago/Turabian Style

Wang, Guotao, Zhuobin Zhang, and Zedong Yang. 2022. "The Existence of Radial Solutions to the Schrödinger System Containing a Nonlinear Operator" Axioms 11, no. 6: 282. https://doi.org/10.3390/axioms11060282

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.