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# The Keller–Osserman-type conditions for the study of a semilinear elliptic system

## Abstract

We study the following system of equations:

$$\textstyle\begin{cases} \Delta u_{1}=p_{1} ( \vert x \vert ) f_{1} ( u_{1},u_{2} ) &\text{in }\mathbb{R}^{N}, \\ \Delta u_{2}=p_{2} ( \vert x \vert ) f_{2} ( u_{1},u_{2} )& \text{in }\mathbb{R}^{N}. \end{cases}$$
(0.1)

Here $$f_{1}$$, $$f_{2}$$ are continuous nonlinear functions that satisfy Keller–Osserman-type conditions, and $$p_{1}$$ and $$p_{2}$$ are continuous weight functions. We establish the existence of radial solutions for this system under various boundary conditions.

## Introduction

In this paper, we study the existence and asymptotic behavior of positive radial solutions of the following semilinear elliptic system:

$$\textstyle\begin{cases} \Delta u_{1}=p_{1} ( \vert x \vert ) f_{1} ( u_{1},u_{2} ) , \\ \Delta u_{2}=p_{2} ( \vert x \vert ) f_{2} ( u_{1},u_{2} ) , \end{cases}\displaystyle \mbox{for } x\in \mathbb{R}^{N}\ (N\geqslant 3).$$
(1.1)

Systems of type (1.1) arise from the study of Lotka–Volterra equations of predator–prey and competitive type under a zero Dirichlet-type condition and variable coefficients (possibly vanishing on subdomains of $$\mathbb{R}^{N}$$, for more on this, see [3, 4, 13, 16, 17]). Moreover, there are several connections between the diffusion–reaction system we consider and the modeling of some problems in physics; see .

We study system (1.1) under three different sets of boundary conditions:

• Finite Case: Both components $$(u_{1},u_{2})$$ are bounded, that is,

$$\lim_{ \vert x \vert \rightarrow \infty }u_{1} \bigl( \vert x \vert \bigr) < \infty \quad \text{and}\quad \lim_{ \vert x \vert \rightarrow \infty }u_{2} \bigl( \vert x \vert \bigr) < \infty .$$
(1.2)
• Infinite Case: Both components $$(u_{1},u_{2})$$ are large, that is,

$$\lim_{ \vert x \vert \rightarrow \infty }u_{1} \bigl( \vert x \vert \bigr) =\infty \quad \text{and} \quad \lim_{ \vert x \vert \rightarrow \infty }u_{2} \bigl( \vert x \vert \bigr) =\infty .$$
(1.3)
• Semifinite Case: One of the components is bounded, whereas the other is large, that is,

$$\lim_{ \vert x \vert \rightarrow \infty }u_{1} \bigl( \vert x \vert \bigr) < \infty \quad \text{and}\quad \lim_{ \vert x \vert \rightarrow \infty }u_{2} \bigl( \vert x \vert \bigr) =\infty ,$$
(1.4)

or

$$\lim_{ \vert x \vert \rightarrow \infty }u_{1} \bigl( \vert x \vert \bigr) =\infty \quad \text{and}\quad \lim_{ \vert x \vert \rightarrow \infty }u_{2} \bigl( \vert x \vert \bigr) < \infty .$$
(1.5)

Let us present some existing literature on this topic. The works of García-Melián, Rossi, and Sabina de Lis  and Lair and Mohammed  deal with the existence of solutions to the following system:

$$\textstyle\begin{cases} \Delta u_{1}=p_{1} ( x ) u_{1}^{\alpha }u_{2}^{\beta } & \text{in } B_{R}, \\ \Delta u_{2}=p_{2} ( x ) u_{1}^{\gamma }u_{2}^{\nu } & \text{in } B_{R}, \\ u_{1} ( x ) \rightarrow \infty & \text{as } \vert x \vert \rightarrow R, \\ u_{2} ( x ) \rightarrow \infty & \text{as } \vert x \vert \rightarrow R, \end{cases}$$
(1.6)

where $$B_{R}$$ is the ball of radius R in $$\mathbb{R}^{N}$$ (bounded or unbounded) centered at the origin, $$p_{1}$$ and $$p_{2}$$ are Hölder continuous positive functions, and α, β, γ, ν are nonnegative constants. If $$R=\infty$$, then $$B_{R}= \mathbb{R}^{N}$$, and the limit in (1.6) should be taken as $$\vert x \vert \rightarrow \infty$$. In the particular case of $$R=\infty$$ and $$p_{1} ( x ) =p_{2} ( x ) =1$$, Lair and Mohammed  proved that system (1.6) has a positive entire large radial solution if and only if

$$0\leqslant \max \{\alpha ,\nu \}\leqslant 1 \quad \text{and} \quad \beta \gamma \leqslant (1- \alpha ) (1-\nu ).$$
(1.7)

Let us point out that our system (1.1) is more general than system (1.6) considered in the aforementioned works. The goal of our paper is to obtain necessary and sufficient conditions for the existence of positive solutions to system (1.1) under conditions of the Keller–Osserman type [6, 14]. Another contribution of our work is estimates of the solutions, which generalizes similar results obtained in [7,8,9, 11, 15]. Let us finish this introduction by mentioning that some of the basic ideas underlying the present paper were already developed in our earlier works [1, 2].

## The mathematical results

### Definition 1

A solution $$(u_{1},u_{2})\in C^{2}( [ 0,\infty ) )\times C^{2}( [ 0,\infty ) )$$ of system (1.1) is called an entire bounded solution if condition (1.2) holds; it is called an entire large solution if condition (1.3) holds; it is called a semifinite entire large solution when (1.4) or (1.5) hold.

For clarity and ease of presentation, we introduce the following notations:

\begin{aligned}& r = \vert x \vert \quad \text{and}\quad a,b, \overline{c}_{1}, \overline{c}_{2},\varepsilon _{1},\varepsilon _{2} \in ( 0,\infty ) \quad \text{are suitably chosen,} \\& G_{1} ( r ) = \int _{0}^{r}t^{1-N} \int _{0}^{t}s ^{N-1}p_{1} ( s )\,ds\,dt, \\& G_{2} ( r ) = \int _{0}^{r}t^{1-N} \int _{0}^{t}s^{N-1}p_{2} ( s ) \,ds\,dt, \\& H ( r ) = \int _{a+b}^{r}\frac{1}{ ( f_{1}+f_{2} ) ( t,t ) }\,dt,\qquad H ( \infty ) = \lim_{r\rightarrow \infty }H ( r ) , \\& G ( r ) =a+b+G_{1} ( r ) +G_{2} ( r ) ,\qquad G_{i} ( \infty ) =\lim_{r\rightarrow \infty }G_{i} ( r ) , \quad i=1,2, \\& \omega _{1} ( r ) =f_{1}\biggl(1,\frac{b}{f_{2} ( a,a ) }+ \overline{c}_{2}f_{2}\biggl(1,1+\frac{1}{a}H^{-1} \bigl( G ( r ) \bigr) \biggr)G_{2} ( r ) \biggr), \\& \omega _{2} ( r ) =f_{2}\biggl(\frac{a}{f_{1} ( b,b ) }+ \overline{c}_{1}f_{1}\biggl(1+\frac{1}{b}H^{-1} \bigl(G ( r ) \bigr),1\biggr)G _{1} ( r ),1\biggr), \\& P_{1} ( r ) = \int _{0}^{r}y^{1-N} \int _{0}^{y}t^{N-1}p_{1} ( t ) f_{1} \bigl( a,b+f_{2} ( a,b ) G_{2} ( t ) \bigr) \,dt\,dy, \\& Q_{1} ( r ) = \int _{0}^{r}y^{1-N} \int _{0}^{y}t^{N-1}p_{2} ( t ) f_{2} \bigl( a+f_{1} ( a,b ) G_{1} ( t ) ,b \bigr)\,dt\,dy, \\& P_{2} ( r ) = \int _{0}^{r}\overline{c}_{1}z^{1+\varepsilon _{1}}p _{1} ( z ) \omega _{1} ( z )\,dz, \\& Q_{2} ( r ) = \int _{0}^{r}\overline{c}_{2}z^{1+\varepsilon _{2}}p _{2} ( z ) \omega _{2} ( z )\,dz, \\& P_{3} ( r ) = \int _{0}^{r}\sqrt{2\overline{c}_{1}\phi _{1} ( z ) \omega _{1} ( z ) }\,dz,\quad \phi _{1} ( z ) =\max \bigl\{ p_{1} ( t ) \vert 0\leqslant t \leqslant z \bigr\} , \\& Q_{3} ( r ) = \int _{0}^{r}\sqrt{2\overline{c}_{2}\phi _{2} ( z ) \omega _{2} ( z ) }\,dz,\quad \phi _{2} ( z ) =\max \bigl\{ p_{2} ( t ) \vert 0\leqslant t\leqslant z \bigr\} , \\& F_{1} ( r ) = \int _{a}^{r}\frac{1}{\sqrt{\int _{0}^{s}f _{1} ( t, f_{2} ( t,t ) )\,dt}}\,ds,\qquad F_{1} ( \infty ) =\lim_{r\rightarrow \infty }F_{1} ( r ) , \\& F_{2} ( r ) = \int _{b}^{r}\frac{1}{\sqrt{\int _{0}^{s}f _{2} ( f_{1} ( t,t ) ,t )\,dt}}\,ds,\qquad F_{2} ( \infty ) =\lim_{r\rightarrow \infty }F_{2} ( r ) , \\& P_{i} ( \infty ) =\lim_{r\rightarrow \infty }P_{i} ( r ) ,\qquad Q_{i} ( \infty ) = \lim_{r\rightarrow \infty }Q_{i} ( r )\quad \text{for }i=1,2,3 . \end{aligned}

Next, we state our working assumptions.

### Standing Assumption

The weight functions $$p_{1}$$, $$p_{2}$$ and the nonlinearities $$f_{1}$$, $$f_{2}$$ satisfy:

1. (P1)

$$p_{1},p_{2}: [ 0,\infty ) \rightarrow [ 0, \infty )$$ are nontrivial radial continuous functions (radial, i.e., $$p_{1} ( x ) =p_{1} ( \vert x \vert )$$ and $$p_{2} ( x ) =p _{2} ( \vert x \vert )$$);

2. (C1)

$$f_{1}$$, $$f_{2}: [ 0,\infty ) \times [ 0, \infty ) \rightarrow [ 0,\infty )$$ are continuous and nondecreasing in both arguments, and $$f_{1} ( x,y ) >0$$, $$f_{2} ( x,y ) >0$$ for all $$x,y>0$$;

3. (C2)

for fixed parameters $$a,b\in ( 0,\infty )$$, there exist $$\overline{c}_{1},\overline{c}_{2}\in ( 0,\infty )$$ such that:

• for all $$t\geqslant \min \{a,b\}$$, $$w\geqslant 1$$, $$u\geqslant \min \{b,f_{2}(a,a)\}$$, $$v\geqslant \min \{1,\frac{b}{f_{2}(a,a)}\}$$, we have

$$f_{1} ( tw,uv ) \leqslant \overline{c}_{1}f_{1} ( t,u ) f_{1} ( w,v ) \quad \text{and} \quad f_{1} ( a,b ) \geqslant 1 ;$$
(2.1)
• for all $$t\geqslant \min \{a,b\}$$, $$w\geqslant 1$$, $$u\geqslant \min \{a,f_{1}(b,b)\}$$, $$v\geqslant \min \{1,\frac{a}{f_{1}(b,b)}\}$$, we have

$$f_{2} ( tw,uv ) \leqslant \overline{c}_{2}f_{2} ( t,u ) f_{2} ( w,v ) \quad \text{and} \quad f_{2} ( a,b ) \geqslant 1.$$
(2.2)

At this point we are ready to state our main results. The first result concerns the existence of entire solutions of (1.1) in the case $$F_{1} ( \infty ) =F_{2} ( \infty ) =\infty$$. Our findings here are summarized in the next theorem.

### Theorem 2

Assume that $$F_{1} ( \infty ) =F_{2} ( \infty ) =\infty$$. Then there exists an entire positive radial solution $$( u_{1},u_{2} ) \in C^{2} ( [ 0,\infty ) ) \times C^{2} ( [ 0,\infty ) )$$ of (1.1) such that $$u_{1} ( 0 ) =a$$, $$u_{2} ( 0 ) =b$$. Moreover, the following properties hold:

1. (1)

If $$r^{2N-2}p_{1} ( r )$$, $$r ^{2N-2}p_{2} ( r )$$ are nondecreasing for large r and there exist $$\varepsilon _{1}$$, $$\varepsilon _{2} \in ( 0,\infty )$$ such that $$p_{1}$$, $$p_{2}$$ satisfy

$$P_{2} ( \infty ) < \infty \quad \textit{and}\quad Q_{2} ( \infty ) < \infty ,$$
(2.3)

then the nonnegative radial solution $$( u_{1},u_{2} )$$ of (1.1) satisfies condition (1.2).

2. (2)

If $$p_{1}$$ and $$p_{2}$$ satisfy

$$P_{1} ( \infty ) = Q_{1} ( \infty ) = \infty ,$$
(2.4)

then the nonnegative radial solution $$( u_{1},u_{2} )$$ of (1.1) satisfies condition (1.3).

3. (3)

If $$r^{2N-2}p_{1} ( r )$$ is nondecreasing for large r and there exists $$\varepsilon _{1}\in ( 0,\infty )$$ such that $$p_{1}$$, $$p_{2}$$ satisfy

$$P_{2} ( \infty ) < \infty \quad \textit{and}\quad Q_{1} ( \infty ) =\infty ,$$
(2.5)

then the nonnegative radial solution $$( u_{1},u_{2} )$$ of (1.1) satisfies condition (1.4).

4. (4)

If $$r^{2N-2}p_{2} ( r )$$ is nondecreasing for large r and there exists $$\varepsilon _{2}\in ( 0,\infty )$$ such that $$p_{1}$$, $$p_{2}$$ satisfy

$$P_{1} ( \infty ) =\infty \quad \textit{and}\quad Q_{2} ( \infty ) < \infty ,$$
(2.6)

then the nonnegative radial solution $$( u_{1},u_{2} )$$ of (1.1) satisfies condition (1.5).

5. (5)

If (1.1) has a nonnegative entire large solution $$( u_{1},u_{2} )$$ such that $$u_{1} ( 0 ) =a$$, $$u_{2} ( 0 ) =b$$ and $$r^{2N-2}p_{1} ( r )$$, $$r^{2N-2}p_{2} ( r )$$ are nondecreasing for large r, then $$p_{1}$$ and $$p_{2}$$ satisfy

$$P_{2} ( \infty ) =\infty \quad \textit{and}\quad Q_{2} ( \infty ) =\infty ,$$
(2.7)

for all $$\varepsilon _{1},\varepsilon _{2}>0$$.

Our next result concerns the existence of entire solutions (1.1) in the case $$F_{1} ( \infty ) \leqslant \infty$$ and $$F_{2} ( \infty ) \leqslant \infty$$. Our findings are summarized in the next theorem.

### Theorem 3

The following statements hold:

1. (i)

If $$P_{3} ( \infty ) < F_{1} ( \infty ) <\infty$$ and $$Q_{3} ( \infty ) < F_{2} ( \infty ) <\infty$$, then system (1.1) has a positive bounded radial solution $$( u_{1},u_{2} ) \in C^{2} ( [ 0,\infty ) ) \times C^{2} ( [ 0,\infty ) )$$ with $$u_{1} ( 0 ) =a$$ and $$u_{2} ( 0 ) =b$$ such that

$$\textstyle\begin{cases} a+P_{1} ( r ) \leqslant u_{1} ( r ) \leqslant F_{1}^{-1} ( P_{3} ( r ) ) , \\ b+Q_{1} ( r ) \leqslant u_{2} ( r ) \leqslant F_{2}^{-1} ( Q_{3} ( r ) ). \end{cases}$$
2. (ii)

If $$F_{1} ( \infty ) =\infty$$, $$P_{1} ( \infty ) =\infty$$, and $$Q_{3} ( \infty ) < F_{2} ( \infty ) <\infty$$, then system (1.1) has a positive radial solution

$$( u_{1},u_{2} ) \in C^{2} \bigl( [ 0,\infty ) \bigr) \times C^{2} \bigl( [ 0,\infty) \bigr) ,$$
(2.8)

such that $$u_{1} ( 0 ) =a$$, $$u_{2} ( 0 ) =b$$, and (1.5) holds.

3. (iii)

If $$P_{3} ( \infty ) < F_{1} ( \infty ) <\infty$$ and $$F_{2} ( \infty ) =\infty$$, $$Q_{1} ( \infty ) =\infty$$, then system (1.1) has a positive radial solution

$$( u_{1},u_{2} ) \in C^{2} \bigl( [ 0,\infty ) \bigr) \times C^{2} \bigl( [ 0,\infty) \bigr) ,$$
(2.9)

such that $$u_{1} ( 0 ) =a$$, $$u_{2} ( 0 ) =b$$, and (1.4) holds.

4. (iv)

If $$r^{2N-2}p_{1} ( r )$$ is nondecreasing for large r, $$F_{1} ( \infty ) =\infty$$, and there exists $$\varepsilon _{1}\in ( 0,\infty )$$ such that $$P_{2} ( \infty ) <\infty$$ and $$Q_{3} ( \infty ) < F_{2} ( \infty ) <\infty$$, then system (1.1) has a positive radial solution

$$( u_{1},u_{2} ) \in C^{2} \bigl( [ 0,\infty ) \bigr) \times C^{2} \bigl( [ 0,\infty) \bigr) ,$$
(2.10)

such that $$u_{1} ( 0 ) =a$$, $$u_{2} ( 0 ) =b$$, and (1.2) holds.

5. (v)

If $$r^{2N-2}p_{2} ( r )$$ is nondecreasing for large r, $$P_{3} ( \infty ) < F_{1} ( \infty ) <\infty$$, $$F_{2} ( \infty ) = \infty$$, and there exists $$\varepsilon _{2}\in ( 0, \infty )$$ such that $$Q_{2} ( \infty ) <\infty$$, then system (1.1) has a positive radial solution

$$( u_{1},u_{2} ) \in C^{2} \bigl( [ 0,\infty ) \bigr) \times C^{2} \bigl( [ 0,\infty) \bigr) ,$$
(2.11)

such that $$u_{1} ( 0 ) =a$$, $$u_{2} ( 0 ) =b$$, and (1.2) holds.

### Remark 4

If $$f_{1} ( u_{1},u_{2} ) =u_{1}^{\alpha }u_{2}^{\beta }$$ and $$f_{2} ( u_{1},u_{2} ) =u_{1}^{\gamma }u_{2}^{ \nu }$$, where α, β, γ, ν are nonnegative constants such that $$\alpha + ( \gamma +\nu ) \beta \leqslant 1$$ and $$( \alpha +\beta ) \gamma +\upsilon \leqslant 1$$, then $$F_{1} ( \infty ) =F_{2} ( \infty ) =\infty$$. If $$G_{1} ( \infty ) =G_{2} ( \infty ) = \infty$$, then $$P_{1} ( \infty ) =Q_{1} ( \infty ) =\infty$$, but the converse is not true. If $$G_{1} ( \infty ) <\infty$$ and $$G_{2} ( \infty ) <\infty$$, then $$P_{1} ( \infty ) <\infty$$ and $$Q_{1} ( \infty ) <\infty$$, but the converse is not true.

## Proofs of theorems

The main idea in proving our results is reducing system (1.1) to a system of second-order ODEs and giving a complete classification of its solutions. Among many possible methods to establish the existence of radial solutions to system (1.1), we will follow here the one based on a successive approximation as in . In the radial setting, system (1.1) becomes a system of differential equations of the form

$$\textstyle\begin{cases} ( r^{N-1}u_{1}^{\prime } ( r ) ) ^{\prime }=r ^{N-1}p_{1} ( r ) f_{1} ( u_{1} ( r ) ,u _{2} ( r ) ) , & r\in {}[ 0,\infty ), \\ ( r^{N-1}u_{2}^{\prime } ( r ) ) ^{\prime }=r ^{N-1}p_{2} ( r ) f_{2} ( u_{1} ( r ) ,u _{2} ( r ) ) , & r\in {}[ 0,\infty ), \end{cases}$$
(3.1)

which can be solved subject to the initial boundary conditions $$u_{1} ( 0 ) =a$$, $$u_{2} ( 0 ) =b$$, and $$u_{1}^{\prime } ( 0 ) =u_{2}^{\prime } ( 0 ) =0$$. The differential equations and initial conditions in (3.1) are equivalent to the integral equations

$$\textstyle\begin{cases} u_{1} ( r ) =a+\int _{0}^{r}t^{1-N}\int _{0}^{t}s^{N-1}p_{1} ( s ) f_{1} ( u_{1} ( s ) ,u_{2} ( s ) )\,ds\,dt, \\ u_{2} ( r ) =b+\int _{0}^{r}t^{1-N}\int _{0}^{t}s^{N-1}p_{2} ( s ) f_{2} ( u_{1} ( s ) ,u_{2} ( s ) )\,ds\,dt. \end{cases}$$
(3.2)

To construct a solution to this system, we define the sequences $$\{ u_{1}^{k} ( r ) \} _{k\geqslant 0}$$ and $$\{ u_{2}^{k} ( r ) \} _{k\geqslant 0}$$ on $$[ 0,\infty )$$ by

$$\textstyle\begin{cases} u_{1}^{0}=a \quad \text{and}\quad u_{2}^{0}=b, \quad r\geqslant 0, \\ u_{1}^{k} ( r ) =a+\int _{0}^{r}t^{1-N}\int _{0}^{t}s^{N-1}p _{1} ( s ) f_{1} ( u_{1}^{k-1} ( s ) ,u _{2}^{k-1} ( s ) )\,ds\,dt, \\ u_{2}^{k} ( r ) =b+\int _{0}^{r}t^{1-N}\int _{0}^{t}s^{N-1}p _{2} ( s ) f_{2} ( u_{1}^{k-1} ( s ) ,u _{2}^{k-1} ( s ) )\,ds\,dt. \end{cases}$$
(3.3)

Obviously, for all $$r\geqslant 0$$ and $$k\in {\mathbb{N}}$$, we have $$u_{1}^{k} ( r ) \geqslant a$$ and $$u_{2}^{k} ( r ) \geqslant b$$. Our assumptions yield $$u_{1}^{0}(r)\leqslant u_{1}^{1}(r)$$ and $$u_{2}^{0}(r)\leqslant u_{2}^{1}(r)$$ for all $$r\geqslant 0$$. From these inequalities we can easily get $$u_{1}^{1}(r)\leqslant u_{1}^{2}(r)$$ and $$u_{2}^{1}(r)\leqslant u_{2}^{2}(r)$$ for all $$r\geqslant 0$$. Continuing this reasoning, we obtain that the sequences $$\{ u_{1}^{k} ( r ) \} _{k\geqslant 0}$$ and $$\{ u_{2}^{k} ( r ) \} _{k\geqslant 0}$$ are nondecreasing on $$[ 0,\infty )$$. Thus there exist

$$u_{1} ( r ) =\lim_{k\rightarrow \infty }u_{1}^{k} ( r ) \leqslant \infty \quad \text{and}\quad u_{2} ( r ) =\lim _{k\rightarrow \infty }u_{2}^{k} ( r ) \leqslant \infty , \quad r \in [ 0,\infty ) .$$
(3.4)

We will next establish “upper bounds” for this sequences. To do this, we note that $$\{ u_{1}^{k} ( r ) \} _{k\geqslant 0}$$ and $$\{ u_{2}^{k} ( r ) \} _{k\geqslant 0}$$ satisfy

$$\textstyle\begin{cases} {}[ r^{N-1} ( u_{1}^{k} ( r ) ) ^{\prime }]^{ \prime }=r^{N-1}p_{1} ( r ) f_{1} ( u_{1}^{k-1} ( r ) ,u_{2}^{k-1} ( r ) ) , \\ {}[ r^{N-1} ( u_{2}^{k} ( r ) ) ^{\prime }]^{ \prime }=r^{N-1}p_{2} ( r ) f_{2} ( u_{1}^{k-1} ( r ) ,u_{2}^{k-1} ( r ) ) . \end{cases}$$
(3.5)

Using the monotonicity of $$\{ u_{1}^{k} ( r ) \} _{k\geqslant 0}$$ and $$\{ u_{2}^{k} ( r ) \} _{k\geqslant 0}$$, we obtain the inequalities

$$\textstyle\begin{cases} {}[ r^{N-1} ( u_{1}^{k} ( r ) ) ^{\prime }]^{ \prime }=r^{N-1}p_{1} ( r ) f_{1} ( u_{1}^{k-1} ( r ) ,u_{2}^{k-1} ( r ) ) \\ \hphantom{{}[ r^{N-1} ( u_{1}^{k} ( r ) ) ^{\prime }]^{ \prime }}\leqslant r^{N-1}p_{1} ( r ) f_{1} ( u_{1}^{k} ( r ) ,u_{2}^{k} ( r ) ) , \\ {}[ r^{N-1} ( u_{2}^{k} ( r ) ) ^{\prime }]^{ \prime }=r^{N-1}p_{2} ( r ) f_{2} ( u_{1}^{k-1} ( r ) ,u_{2}^{k-1} ( r ) ) \\ \hphantom{{}[ r^{N-1} ( u_{2}^{k} ( r ) ) ^{\prime }]^{ \prime }}\leqslant r^{N-1}p_{2} ( r ) f_{2} ( u_{1}^{k} ( r ) ,u_{2}^{k} ( r ) ) . \end{cases}$$
(3.6)

Taking into account (3.6), we easily to see that

\begin{aligned}& {}\bigl[ r^{N-1} \bigl( u_{1}^{k} ( r ) +u_{2}^{k} ( r ) \bigr) ^{\prime } \bigr]^{\prime } \\& \quad \leqslant r^{N-1} \bigl( p_{1} ( r ) f_{1} \bigl( u_{1}^{k} ( r ) ,u_{2}^{k} ( r ) \bigr) +p_{2} ( r ) f_{2} \bigl( u_{1}^{k} ( r ) ,u_{2}^{k} ( r ) \bigr) \bigr) \\& \quad \leqslant r^{N-1} \bigl( p_{1} ( r ) +p_{2} ( r ) \bigr) \bigl( ( f_{1}+f_{2} ) \bigl( u_{1}^{k} ( r ) +u_{2}^{k} ( r ) ,u_{1}^{k} ( r ) +u _{2}^{k} ( r ) \bigr) \bigr). \end{aligned}
(3.7)

Integrating this inequality yields

$$\frac{ ( u_{1}^{k} ( r ) +u_{2}^{k} ( r ) ) ^{\prime }}{ ( f_{1}+f_{2} ) ( u_{1}^{k} ( r ) +u_{2}^{k} ( r ) ,u_{1}^{k} ( r ) +u_{2}^{k} ( r ) ) }\leqslant G_{1}^{\prime } ( r ) +G_{2}^{\prime } ( r ) .$$
(3.8)

Integrating (3.8) between 0 and r, we get

$$\int _{a+b}^{u_{1}^{k} ( r ) +u_{2}^{k} ( r ) }\frac{dt}{ ( f_{1}+f_{2} ) ( t,t ) }\leqslant G ( r ) ,$$
(3.9)

or, in the H notation,

$$H \bigl( u_{1}^{k} ( r ) +u_{2}^{k} ( r ) \bigr) \leqslant G ( r ) .$$
(3.10)

It follows from this estimate and the fact that H is a bijection (with the inverse denoted $$H^{-1}$$) that

$$u_{1}^{k} ( r ) +u_{2}^{k} ( r ) \leqslant H^{-1} \bigl( G ( r ) \bigr) .$$
(3.11)

This occurs on bounded intervals, since

\begin{aligned} u_{1}^{k} ( r ) +u_{2}^{k} ( r ) \geqslant & a+b+f _{1} ( a,b ) G_{1} ( r ) +f_{2} ( a,b ) G_{2} ( r ) \\ \geqslant & a+b+G_{1} ( r ) +G_{2} ( r ) =G ( r ) \end{aligned}
(3.12)

by (3.3) and (C1). Recalling that $$\{ u_{1}^{k} ( r ) \} _{k\geqslant 0}$$ and $$\{ u_{2}^{k} ( r ) \} _{k\geqslant 0}$$ are nondecreasing sequences on $$[ 0, \infty )$$, the above estimate yields

\begin{aligned} u_{1}^{k} ( r ) \leqslant & a+ \int _{0}^{r}t^{1-N} \int _{0}^{t}s^{N-1}p_{1} ( s ) f_{1} \bigl( u_{1}^{k} ( s ) ,u_{2}^{k} ( s ) \bigr)\,ds\,dt \\ \leqslant & a+f_{1} \bigl( u_{1}^{k} ( r ) ,u_{2}^{k} ( r ) \bigr) G_{1} ( r ) \\ \leqslant & a+f_{1} \bigl( u_{2}^{k} ( r ) +H^{-1} \bigl( G ( r ) \bigr) ,u_{2}^{k} ( r ) \bigr) G_{1} ( r ) \\ =&a+f_{1}\biggl(u_{2}^{k} ( r ) \biggl(1+ \frac{1}{u_{2}^{k} ( r ) }H^{-1} \bigl( G ( r ) \bigr) \biggr),u_{2}^{k} ( r ) \biggr)G_{1} ( r ) \\ \leqslant & a+f_{1}\biggl(u_{2}^{k} ( r ) \biggl(1+ \frac{1}{b}H^{-1} \bigl( G ( r ) \bigr) \biggr),u_{2}^{k} ( r ) \biggr)G_{1} ( r ) \\ \leqslant & a+\overline{c}_{1}f_{1} \bigl( u_{2}^{k} ( r ) ,u_{2} ^{k} ( r ) \bigr) f_{1}\biggl(1+ \frac{1}{b}H^{-1} \bigl( G ( r ) \bigr) ,1\biggr)G_{1} ( r ) \\ =&f_{1} \bigl( u_{2}^{k} ( r ) ,u_{2}^{k} ( r ) \bigr) \biggl(\frac{a}{f_{1} ( u_{2}^{k} ( r ) ,u_{2}^{k} ( r ) ) }+ \overline{c}_{1}f_{1}\biggl(1+\frac{1}{b}H^{-1} \bigl( G ( r ) \bigr) ,1\biggr)G_{1} ( r ) \biggr) \\ \leqslant & f_{1} \bigl( u_{2}^{k} ( r ) ,u_{2}^{k} ( r ) \bigr) \biggl(\frac{a}{f_{1} ( b,b ) }+ \overline{c}_{1}f_{1}\biggl(1+ \frac{1}{b}H^{-1} \bigl( G ( r ) \bigr) ,1\biggr)G_{1} ( r ) \biggr) \end{aligned}
(3.13)

and

\begin{aligned} u_{2}^{k} ( r ) \leqslant & b+ \int _{0}^{r}t^{1-N} \int _{0}^{t}s^{N-1}p_{2} ( s ) f_{2} \bigl( u_{1}^{k} ( s ) ,u_{2}^{k} ( s ) \bigr)\,ds\,dt \\ \leqslant & b+f_{2} \bigl( u_{1}^{k} ( r ) ,u_{2}^{k} ( r ) \bigr) G_{2} ( r ) \\ \leqslant & b+f_{2} \bigl( u_{1}^{k} ( r ) ,u_{1}^{k} ( r ) +H^{-1} \bigl( G ( r ) \bigr) \bigr) G_{2} ( r ) \\ =&b+f_{2}\biggl(u_{1}^{k} ( r ) ,u_{1}^{k} ( r ) \biggl(1+\frac{1}{u_{1}^{k} ( r ) }H^{-1} \bigl( G ( r ) \bigr) \biggr)\biggr)G_{2} ( r ) \\ \leqslant & b+f_{2}\biggl(u_{1}^{k} ( r ) ,u_{1}^{k} ( r ) \biggl(1+\frac{1}{a}H^{-1} \bigl( G ( r ) \bigr) \biggr)\biggr)G_{2} ( r ) \\ \leqslant & b+\overline{c}_{2}f_{2}\bigl(u_{1}^{k} ( r ) ,u_{1}^{k} ( r ) \bigr)G_{2} ( r ) f_{2}\biggl(1,1+\frac{1}{a}H^{-1}\bigl(G ( r ) \bigr) \biggr) \\ \leqslant & f_{2} \bigl( u_{1}^{k} ( r ) ,u_{1}^{k} ( r ) \bigr) \biggl(\frac{b}{f_{2} ( a,a ) }+ \overline{c}_{2}f_{2}\biggl(1,1+\frac{1}{a} H^{-1} \bigl( G ( r ) \bigr) \biggr)G_{2} ( r ) \biggr). \end{aligned}
(3.14)

Substituting (3.13) and (3.14) into (3.6), we obtain

\begin{aligned} \bigl[ r^{N-1} \bigl( u_{1}^{k} ( r ) \bigr) ^{\prime } \bigr] ^{\prime } \leqslant & r^{N-1}p_{1} ( r ) f _{1} \bigl( u_{1}^{k} ( r ) ,u_{2}^{k} ( r ) \bigr) \\ \leqslant & r^{N-1}p_{1} ( r ) f_{1}(u_{1}^{k} ( r ) ,f _{2} \bigl( u_{1}^{k} ( r ) ,u_{1}^{k} ( r ) \bigr) \biggl(\frac{b}{f_{2} ( a,a ) } \\ &{}+\overline{c}_{2}f_{2}\biggl(1,\frac{a+H^{-1} ( G ( r ) ) }{a} \biggr)G_{2} ( r ) \biggr) \\ \leqslant & r^{N-1}p_{1} ( r ) \overline{c}_{1}f_{1} \bigl( u _{1}^{k} ( r ) ,f_{2} \bigl( u_{1}^{k} ( r ) ,u _{1}^{k} ( r ) \bigr) \bigr) \omega _{1} ( r ) \end{aligned}
(3.15)

and

\begin{aligned} \bigl[ r^{N-1} \bigl( u_{2}^{k} ( r ) \bigr) ^{\prime } \bigr] ^{\prime } \leqslant & r^{N-1}p_{2} ( r ) f _{2} \bigl( u_{1}^{k} ( r ) ,u_{2}^{k} ( r ) \bigr) \\ \leqslant & r^{N-1}p_{2} ( r ) f_{2} \biggl(f_{1} \bigl( u_{2}^{k} ( r ) ,u_{2}^{k} ( r ) \bigr) \biggl(\frac{a}{f_{1} ( b,b ) } \\ &{}+\overline{c}_{1}f_{1}\biggl(\frac{b+H^{-1} ( G ( r ) ) }{b},1 \biggr)G_{1} ( r ) \biggr),u_{2}^{k} ( r ) \biggr) \\ \leqslant & r^{N-1}p_{2} ( r ) \overline{c}_{2}f_{2} \bigl( f _{1} \bigl( u_{2}^{k} ( r ) ,u_{2}^{k} ( r ) \bigr) ,u_{2}^{k} ( r ) \bigr) \omega _{2} ( r ) . \end{aligned}
(3.16)

In summary, we get

$$\textstyle\begin{cases} r^{N-1} ( u_{1}^{k} ) ^{\prime \prime }\leqslant ( N-1 ) r^{N-2} ( u_{1}^{k} ) ^{\prime }+r^{N-1} ( u_{1}^{k} ) ^{\prime \prime }= [ r^{N-1} ( u_{1}^{k} ) ^{\prime } ] ^{\prime } \\ \hphantom{r^{N-1} ( u_{1}^{k} ) ^{\prime \prime }}\leqslant r^{N-1}p_{1} ( r ) \overline{c}_{1}f_{1} ( u_{1} ^{k} ( r ) ,f_{2} ( u_{1}^{k} ( r ) ,u_{1} ^{k} ( r ) ) ) \omega _{1} ( r ) , \\ r^{N-1} ( u_{2}^{k} ) ^{\prime \prime }\leqslant [ r^{N-1} ( u_{2}^{k} ) ^{\prime } ] ^{\prime } \\ \hphantom{r^{N-1} ( u_{2}^{k} ) ^{\prime \prime }}\leqslant r^{N-1}p_{2} ( r ) \overline{c}_{2}f_{2} ( f_{1} ( u_{2}^{k} ( r ) ,u_{2}^{k} ( r ) ) ,u_{2}^{k} ( r ) ) \omega _{2} ( r ) . \end{cases}$$
(3.17)

Multiplying the first inequality in (3.17) by $$( u_{1}^{k} ( r ) ) ^{\prime }$$ and the second by $$( u _{2}^{k} ( r ) ) ^{\prime }$$, we arrive at

$$\textstyle\begin{cases} \{ [ ( u_{1}^{k} ( r ) ) ^{\prime } ] ^{2} \} ^{\prime }\leqslant 2p_{1} ( r ) \overline{c}_{1}f_{1} ( u_{1}^{k} ( r ) ,f_{2} ( u_{1}^{k} ( r ) ,u_{1}^{k} ( r ) ) ) ( u_{1}^{k} ( r ) ) ^{\prime }\omega _{1} ( r ) , \\ \{ [ ( u_{2}^{k} ( r ) ) ^{\prime } ] ^{2} \} ^{\prime }\leqslant 2p_{2} ( r ) \overline{c}_{2}f_{2} ( f_{1} ( u_{2}^{k} ( r ) ,u _{2}^{k} ( r ) ) ,u_{2}^{k} ( r ) ) ( u_{2}^{k} ( r ) ) ^{\prime }\omega _{2} ( r ) . \end{cases}$$
(3.18)

Integrating in (3.18) from 0 to r, we get

$$\textstyle\begin{cases} [ ( u_{1}^{k} ( r ) ) ^{\prime } ] ^{2}\leqslant \int _{0}^{r}2p_{1} ( z ) \overline{c}_{1}f_{1} ( u _{1}^{k} ( z ) ,f_{2} ( u_{1}^{k} ( z ) ,u _{1}^{k} ( z ) ) ) ( u_{1}^{k} ( z ) ) ^{\prime }\omega _{1} ( z )\,dz, \\ [ ( u_{2}^{k} ( r ) ) ^{\prime } ] ^{2}\leqslant \int _{0}^{r}2p_{2} ( z ) \overline{c}_{2}f_{2} ( f _{1} ( u_{2}^{k} ( z ) ,u_{2}^{k} ( z ) ) ,u_{2}^{k} ( z ) ) ( u_{2}^{k} ( z ) ) ^{\prime }\omega _{2} ( z )\,dz. \end{cases}$$
(3.19)

Now set

\begin{aligned} &\phi _{1} ( r ) =\max \bigl\{ p_{1} ( z ) \vert 0\leqslant z\leqslant r \bigr\} , \\ &\phi _{2} ( r ) =\max \bigl\{ p_{2} ( z ) \vert 0\leqslant z\leqslant r \bigr\} . \end{aligned}
(3.20)

By the definition of $$\phi _{1} ( r )$$ and $$\phi _{2} ( r )$$ we get from inequalities (3.19) that

$$\textstyle\begin{cases} [ ( u_{1}^{k} ( r ) ) ^{\prime } ] ^{2}\leqslant 2\overline{c}_{1}\phi _{1} ( r ) \omega _{1} ( r ) \int _{0}^{r}f_{1} ( u_{1}^{k} ( z ) ,f_{2} ( u_{1}^{k} ( z ) ,u_{1}^{k} ( z ) ) ) ( u_{1}^{k} ( z ) ) ^{\prime }\,dz, \\ [ ( u_{2}^{k} ( r ) ) ^{\prime } ] ^{2}\leqslant 2\overline{c}_{2}\phi _{2} ( r ) \omega _{2} ( r ) \int _{0}^{r}f_{2} ( f_{1} ( u_{2}^{k} ( z ) ,u_{2}^{k} ( z ) ) ,u_{2}^{k} ( z ) ) ( u_{2}^{k} ( z ) ) ^{ \prime }\,dz. \end{cases}$$
(3.21)

As a consequence of (3.21), we get

$$\textstyle\begin{cases} ( u_{1}^{k} ( r ) ) ^{\prime }\leqslant \sqrt{2 \phi _{1} ( r ) \omega _{1} ( r ) } ( \int _{a}^{u_{1}^{k} ( r ) }\overline{c}_{1}f_{1} ( z,f _{2} ( z,z ) )\,dz ) ^{1/2}, \\ ( u_{2}^{k} ( r ) ) ^{\prime }\leqslant \sqrt{2 \phi _{2} ( r ) \omega _{2} ( r ) } ( \int _{b}^{u_{2}^{k} ( r ) }\overline{c}_{2}f_{2} ( f_{1} ( z,z ) ,z )\,dz ) ^{1/2}. \end{cases}$$
(3.22)

From this we easily deduce

$$\textstyle\begin{cases} \frac{ ( u_{1}^{k} ( r ) ) ^{\prime }}{ ( \int _{a}^{u_{1}^{k} ( r ) }f_{1} ( z,f_{2} ( z,z ) )\,dz ) ^{1/2}}\leqslant \overline{c}_{1}^{1/2}\sqrt{2 \phi _{1} ( r ) \omega _{1} ( r ) }, \\ \frac{ ( u_{2}^{k} ( r ) ) ^{\prime }}{ ( \int _{b}^{u_{2}^{k} ( r ) }f_{2} ( f_{1} ( z,z ) ,z )\,dz ) ^{1/2}}\leqslant \overline{c}_{2}^{1/2}\sqrt{2 \phi _{2} ( r ) \omega _{2} ( r ) }. \end{cases}$$
(3.23)

Integrating (3.23), we arrive at

$$\textstyle\begin{cases} \int _{a}^{u_{1}^{k} ( r ) }\frac{1}{\sqrt{\int _{0}^{s}f _{1} ( t,f_{2} ( t,t ) )\,dt}}\,ds\leqslant \int _{0} ^{r}\sqrt{2\overline{c}_{1}\phi _{1} ( z ) \omega _{1} ( z ) }\,dz, \\ \int _{b}^{u_{2}^{k} ( r ) }\frac{1}{\sqrt{\int _{0}^{s}f _{2} ( f_{1} ( t,t ) ,t )\,dt}}\,ds\leqslant \int _{0} ^{r}\sqrt{2\overline{c}_{2}\phi _{2} ( z ) \omega _{2} ( z ) }\,dz. \end{cases}$$
(3.24)

Now (3.24) can be written as

$$\textstyle\begin{cases} F_{1} ( u_{1}^{k} ( r ) ) \leqslant P_{3} ( r ) , \\ F_{2} ( u_{2}^{k} ( r ) ) \leqslant Q_{3} ( r ) . \end{cases}$$
(3.25)

Finally, using the fact that $$F_{1}^{-1}$$, $$F_{2}^{-1}$$ are strictly increasing on $$[ 0,F_{1} ( \infty ) )$$ and $$[ 0,F_{2} ( \infty ) )$$, we get

$$\textstyle\begin{cases} u_{1}^{k} ( r ) \leqslant F_{1}^{-1} ( P_{3} ( r ) ) , \\ u_{2}^{k} ( r ) \leqslant F_{2}^{-1} ( Q_{3} ( r ) ) . \end{cases}$$
(3.26)

These inequalities are independent of k. We claim that the sequences $$\{ u_{1}^{k} ( r ) \} _{k\geqslant 0}$$ and $$\{ u_{2}^{k} ( r ) \} _{k\geqslant 0}$$ are bounded on $$[ 0,c_{0} ]$$ for arbitrary $$c_{0}>0$$. Indeed, since

$$\bigl( u_{1}^{k} ( r ) \bigr) ^{\prime }\geqslant 0 \quad \text{and}\quad \bigl( u_{2}^{k} ( r ) \bigr) ^{\prime } \geqslant 0, \quad r\geqslant 0,$$
(3.27)

it follows that

$$u_{1}^{k} ( r ) \leqslant u_{1}^{k} ( c_{0} ) \leqslant C_{1} \quad \text{and}\quad u_{2}^{k} ( r ) \leqslant u_{2}^{k} ( c_{0} ) \leqslant C_{2}\quad \text{on } [ 0,c_{0} ] .$$
(3.28)

Here $$C_{1}=F_{{1}}^{-1} ( P_{3} ( c_{0} ) )$$ and $$C_{2}=F_{2}^{-1} ( Q_{3} ( c_{0} ) )$$ are positive constants. Thus the sequences $$\{ u_{1}^{k} ( r ) \} _{k\geqslant 0}$$ and $$\{ u_{2}^{k} ( r ) \} _{k\geqslant 0}$$ are bounded and equicontinuous on $$[ 0,c _{0} ]$$ for arbitrary $$c_{0}>0$$. By the Arzelà–Ascoli theorem there exist subsequences of $$\{ u_{1}^{k} ( r ) \} _{k\geqslant 0}$$ and $$\{ u_{2}^{k} ( r ) \} _{k\geqslant 0}$$ converging uniformly to $$u_{1} ( r )$$ and $$u_{2} ( r )$$ on $$[ 0,c_{0} ]$$. Since $$\{ u_{1}^{k} ( r ) \} _{k\geqslant 0}$$ and $$\{ u_{2}^{k} ( r ) \} _{k\geqslant 0}$$ are nondecreasing on $$[ 0,\infty )$$, we see that $$\{ u _{1}^{k} ( r ) \} _{k\geqslant 0}$$ and $$\{ u_{2} ^{k} ( r ) \} _{k\geqslant 0}$$ converge uniformly to $$u_{1} ( r )$$ and $$u_{2} ( r )$$ on $$[ 0, \infty )$$. By the arbitrariness of $$c_{0}$$ we deduce that $$( u_{1},u_{2} )$$ is the desired solution of (3.2). Since $$( u_{1}^{k},u_{2}^{k} )$$ is spherically symmetric, then $$( u_{1},u_{2} )$$, obtained as a limit, is also spherically symmetric. Then

$$u_{1} ( r ) =\lim_{k\rightarrow \infty }u_{1}^{k} ( r ) \quad \text{and}\quad u_{2} ( r ) =\lim_{k\rightarrow \infty }u_{2}^{k} ( r ) ,\quad r\geqslant 0,$$
(3.29)

are well defined. A straightforward calculation shows that radial solutions of (1.1) are solutions of the ordinary differential equation system (3.1). Then it follows that the radial solutions of (1.1) with $$u_{1} ( 0 ) =a$$ and $$u_{2} ( 0 ) =b$$ satisfy

\begin{aligned}& u_{1} ( r ) =a+ \int _{0}^{r}\frac{1}{t^{N-1}} \int _{0}^{t}s ^{N-1}p_{1} ( s ) f_{1} \bigl( u_{1} ( s ) ,u _{2} ( s ) \bigr) \,ds\,dt,\quad r\geqslant 0, \end{aligned}
(3.30)
\begin{aligned}& u_{2} ( r ) =b+ \int _{0}^{r}\frac{1}{t^{N-1}} \int _{0}^{t}s ^{N-1}p_{2} ( s ) f_{2} \bigl( u_{1} ( s ) ,u _{2} ( s ) \bigr)\,ds \,dt,\quad r\geqslant 0. \end{aligned}
(3.31)

Setting

\begin{aligned}& L_{1} ( r ) =r^{1-N} \int _{0}^{r}s^{N-1}p_{1} ( s ) f_{1} \bigl( u_{1} ( s ) ,u_{2} ( s ) \bigr)\,ds, \\& L_{2} ( r ) =r^{-N} ( 1-N ) \int _{0}^{r}s^{N-1}p _{1} ( s ) f_{1} \bigl( u_{1} ( s ) ,u_{2} ( s ) \bigr)\,ds, \\& L_{3} ( r ) =p_{1} ( r ) f_{1} \bigl( u_{1} ( s ) ,u_{2} ( s ) \bigr) \end{aligned}

and repeating the proof in , we can see that

\begin{aligned}& \lim_{r\rightarrow 0}u_{1}^{\prime } ( r ) = \lim _{r\rightarrow 0}L_{1} ( r ) =u_{1}^{\prime } ( 0 ) =0, \\& \lim_{r\rightarrow 0}u_{1}^{\prime \prime } ( r ) = \lim _{r\rightarrow 0} \bigl( L_{2} ( r ) +L_{3} ( r ) \bigr) =u_{1}^{\prime \prime } ( 0 ) =\frac{p _{1} ( 0 ) f_{1} ( u_{1} ( 0 ) ,u_{2} ( 0 ) ) }{N}, \end{aligned}

from which it follows that $$u_{1}^{\prime } ( r )$$ and $$u_{1}^{\prime \prime } ( r )$$ are continuous at $$r=0$$. In the same fashion, $$u_{2}^{\prime } ( r )$$ and $$u_{2}^{ \prime \prime } ( r )$$ are continuous at $$r=0$$. Clearly, $$( u_{1},u_{2} ) \in C^{2} [ 0,\infty ) \times C^{2} [ 0,\infty )$$.

### Proof of Theorem 2 completed

Choose $$R>0$$ such that $$r^{2N-2}p_{1} ( r )$$ and $$r^{2N-2}p_{2} ( r )$$ are nondecreasing for $$r\geqslant R$$. Using the same arguments as in (3.15) and (3.16), we can see that

$$\textstyle\begin{cases} [ r^{N-1} ( u_{1} ( r ) ) ^{\prime } ] ^{\prime }\leqslant r^{N-1}p_{1} ( r ) \overline{c}_{1}f_{1} ( u_{1} ( r ) ,f_{2} ( u _{1} ( r ) ,u_{1} ( r ) ) ) \omega _{1} ( r ) , \\ [ r^{N-1} ( u_{2} ( r ) ) ^{\prime } ] ^{\prime }\leqslant r^{N-1}p_{2} ( r ) \overline{c}_{2}f_{2} ( f_{1} ( u_{2} ( r ) ,u _{2} ( r ) ) ,u_{2} ( r ) ) \omega _{2} ( r ) . \end{cases}$$
(3.32)

Multiplying the first equation in (3.32) by $$r^{N-1} ( u _{1} ) ^{\prime }$$ and the second by $$r^{N-1} ( u_{2} ) ^{\prime }$$ and integrating from R to r yield, for $$r\geqslant R$$,

$$\textstyle\begin{cases} [ r^{N-1} ( u_{1} ( r ) ) ^{\prime } ] ^{2}\leqslant [ R^{N-1} ( u_{1} ( R ) ) ^{\prime } ] ^{2} \\ \hphantom{ [ r^{N-1} ( u_{1} ( r ) ) ^{\prime } ] ^{2}\leqslant }{}+2\int _{R}^{r}z^{2N-2}p_{1} ( z ) \overline{c}_{1}\omega _{1} ( z ) \frac{d}{dz}\int _{a}^{u_{1} ( z ) }f _{1} ( s,f_{2} ( s,s ) )\,ds\,dz, \\ [ r^{N-1} ( u_{2} ( r ) ) ^{\prime } ] ^{2}\leqslant [ R^{N-1} ( u_{2} ( R ) ) ^{\prime } ] ^{2} \\ \hphantom{ [ r^{N-1} ( u_{2} ( r ) ) ^{\prime } ] ^{2}\leqslant }{}+2\int _{R}^{r}z^{2N-2}p_{2} ( z ) \overline{c}_{2}\omega _{2} ( z ) \frac{d}{dz}\int _{b}^{u_{2} ( z ) }f _{2} ( f_{1} ( s,s ) ,s )\,ds\,dz, \end{cases}$$
(3.33)

from the monotonicity of $$z^{2N-2}p_{1} ( z )$$ and $$z^{2N-2}p_{2} ( z )$$ for $$r\geqslant z\geqslant R$$ we get that

$$\textstyle\begin{cases} [ r^{N-1} ( u_{1} ( r ) ) ^{\prime } ] ^{2}\leqslant C_{1}+2\overline{c}_{1}r^{2N-2}p_{1} ( r ) \omega _{1} ( r ) \overline{F}_{1} ( u_{1} ( r ) ) , \\ [ r^{N-1} ( u_{2} ( r ) ) ^{\prime } ] ^{2}\leqslant C_{2}+2\overline{c}_{2}r^{2N-2}p_{2} ( r ) \omega _{2} ( r ) \overline{F}_{2} ( u_{2} ( r ) ) , \end{cases}$$
(3.34)

where

\begin{aligned}& \begin{aligned}&C_{1}= \bigl[ R^{N-1} \bigl( u_{1} ( R ) \bigr) ^{\prime } \bigr] ^{2},\qquad C_{2}= \bigl[ R^{N-1} \bigl( u_{2} ( R ) \bigr) ^{\prime } \bigr] ^{2}, \\ &\overline{F}_{1} \bigl( u_{1} ( r ) \bigr) = \int _{0}^{u _{1} ( r ) }f_{1} \bigl( s,f_{2} ( s,s ) \bigr)\,ds, \\ &\overline{F}_{2} \bigl( u_{2} ( r ) \bigr) = \int _{0}^{u _{2} ( r ) }f_{2} \bigl( f_{1} ( s,s ) ,s \bigr)\,ds. \end{aligned} \end{aligned}
(3.35)

This implies that

$$\textstyle\begin{cases} \frac{ ( u_{1} ( r ) ) ^{\prime }}{\sqrt{ \overline{F}_{1} ( u_{1} ( r ) ) }}\leqslant \frac{\sqrt{C _{1}}r^{1-N}}{\sqrt{\overline{F}_{1} ( u_{1} ( r ) ) }}+\sqrt{2\overline{c}_{1}p_{1} ( r ) \omega _{1} ( r ) }, \\ \frac{ ( u_{2} ( r ) ) ^{\prime }}{\sqrt{ \overline{F}_{2} ( u_{2} ( r ) ) }}\leqslant \frac{\sqrt{C _{2}}r^{1-N}}{\sqrt{\overline{F}_{2} ( u_{2} ( r ) ) }}+\sqrt{2\overline{c}_{2}p_{2} ( r ) \omega _{2} ( r ) }. \end{cases}$$
(3.36)

In particular, integrating (3.36) from R to r, we arrive at the following inequality:

\begin{aligned}& \int _{u_{1} ( R ) }^{u_{1} ( r ) } \biggl[ \int _{0}^{t}f_{1} \bigl( z,f_{2} ( z,z ) \bigr)\,dz \biggr] ^{-1/2}\,dt \\& \quad =F_{1} \bigl( u_{1} ( r ) \bigr) -F_{1} \bigl( u_{1} ( R ) \bigr) \\& \quad \leqslant \int _{R}^{r}\frac{\sqrt{C_{1}}t^{1-N}}{(\int _{0}^{u_{1} ( t ) }f_{1} ( s,f_{2} ( s,s ) )\,ds)^{-1/2}}\,dt+ \int _{R}^{r}\sqrt{2\overline{c}_{1}p_{1} ( z ) \omega _{1} ( z ) }\,dz \\& \quad = \int _{R}^{r}\frac{\sqrt{C_{1}}t^{1-N}}{(\int _{0}^{u_{1} ( t ) }f_{1} ( s,f_{2} ( s,s ) )\,dz)^{-1/2}}\,dt+ \int _{R}^{r}z^{\frac{-1-\varepsilon _{1}}{2}}z^{\frac{1+\varepsilon _{1}}{2}}\bigl(2 \overline{c}_{1}p_{1} ( z ) \omega _{1} ( z ) \bigr)^{\frac{1}{2}}\,dz \\& \quad \leqslant \frac{\sqrt{C_{1}}\int _{R}^{r}t^{1-N}\,dt}{(\int _{0}^{u_{1} ( R ) }f_{1} ( z,f_{2} ( z,z ) )\,dz)^{1/2}} \\& \qquad {}+\biggl( \int _{R}^{r}z^{-1-\varepsilon _{1}}\,dz \biggr)^{\frac{1}{2}}\biggl( \int _{R}^{r}z^{1+ \varepsilon _{1}}2\overline{c}_{1}p_{1} ( z ) \omega _{1} ( z )\,dz\biggr)^{\frac{1}{2}} \\& \quad \leqslant \frac{\sqrt{C_{1}}\int _{R}^{r}t^{1-N}\,dt}{(\int _{0}^{u_{1} ( R ) }f_{1} ( z,f_{2} ( z,z ) )\,dz)^{1/2}}+\biggl(\frac{2P _{2} ( r ) }{\varepsilon _{1}R^{\varepsilon _{1}}}\biggr)^{ \frac{1}{2}}. \end{aligned}
(3.37)

We next turn to estimating the second solution. A similar calculation yields

$$F_{2} \bigl( u_{2} ( r ) \bigr) -F_{2} \bigl( u_{2} ( R ) \bigr) \leqslant \frac{\sqrt{C_{2}}\int _{R}^{r}t^{1-N}\,dt}{( \int _{0}^{u_{2} ( R ) }f_{2} ( f_{1} ( z,z ) ,z )\,dz)^{1/2}}+\biggl(\frac{2Q_{2} ( r ) }{\varepsilon _{2}R^{\varepsilon _{2}}} \biggr)^{\frac{1}{2}}.$$
(3.38)

Inequalities (3.37) and (3.38) are needed in proving the “boundedness” of the functions $$u_{1}$$ and $$u_{2}$$. Indeed, they can be written as

$$\textstyle\begin{cases} u_{1} ( r ) \leqslant F_{1}^{-1}(F_{1} ( u_{1} ( R ) ) +\frac{\sqrt{C_{1}}\int _{R}^{r}t^{1-N}\,dt}{(\int _{0}^{u_{1} ( R ) }f_{1} ( z,f_{2} ( z,z ) )\,dz)^{1/2}}+(\frac{2P_{2} ( r ) }{\varepsilon _{1}R ^{\varepsilon _{1}}})^{\frac{1}{2}}), \\ u_{2} ( r ) \leqslant F_{2}^{-1}(F_{2} ( u_{2} ( R ) ) +\frac{\sqrt{C_{2}}\int _{R}^{r}t^{1-N}\,dt}{(\int _{0}^{u_{2} ( R ) }f_{2} ( f_{1} ( z,z ) ,z )\,dz)^{1/2}}+(\frac{2Q_{2} ( r ) }{\varepsilon _{2}R^{\varepsilon _{2}}})^{\frac{1}{2}}). \end{cases}$$
(3.39)

Having discussed the “bounded” case, we now turn to the claims of the theorem.

Claim 1: When $$P_{2} ( \infty ) <\infty$$ and $$Q_{2} ( \infty ) <\infty$$, from (3.39) we find that

$$\textstyle\begin{cases} \lim_{r\rightarrow \infty }u_{1} ( r ) < \infty \\ \lim_{r\rightarrow \infty }u_{2} ( r ) < \infty \end{cases}\displaystyle \text{for all } r\geqslant 0,$$
(3.40)

and so $$( u_{1},u_{2} )$$ is bounded. We next consider:

Claim 2: Let $$( u_{1},u_{2} )$$ be a solution of (3.2). The case $$P_{1} ( \infty ) =Q_{1} ( \infty ) =\infty$$ is proved as follows:

\begin{aligned} u_{1} ( r ) =&a+ \int _{0}^{r}t^{1-N} \int _{0}^{t}s^{N-1}p_{1} ( s ) f_{1} \bigl( u_{1} ( s ) ,u_{2} ( s ) \bigr)\,ds \,dt \\ =&a+ \int _{0}^{r}y^{1-N} \int _{0}^{y}t^{N-1}p_{1} ( t ) f_{1}\biggl(a,b \\ &{}+ \int _{0}^{t}z^{1-N} \int _{0}^{z}s^{N-1}p_{2} ( s ) f_{2} \bigl( u_{1} ( s ) ,u_{2} ( s ) \bigr)\,ds \biggr)\,dz)\,dt\,dy \\ \geqslant & a+ \int _{0}^{r}y^{1-N} \int _{0}^{y}t^{N-1}p_{1} ( t ) f_{1} \bigl( a,b+f_{2} ( a,b ) G_{2} ( t ) \bigr) \,dt\,dy \\ \geqslant & \int _{0}^{r}y^{1-N} \int _{0}^{y}t^{N-1}p_{1} ( t ) f_{1} \bigl( a,b+f_{2} ( a,b ) G_{2} ( t ) \bigr) \,dt\,dy=P_{1} ( r ) . \end{aligned}
(3.41)

As in the preceding lines, we can prove

$$u_{2} ( r ) \geqslant Q_{1} ( r ) .$$
(3.42)

Passing to the limit in (3.41) and in the last inequality, we get

$$\lim_{r\rightarrow \infty }u_{1} ( r ) = \lim_{r\rightarrow \infty }u_{2} ( r ) =\infty ,$$
(3.43)

which yields the claim.

Claim 3: In a similar way as in Claim 1 and Claim 2, we have the estimates

\begin{aligned}& \lim_{r\rightarrow \infty }u_{1} ( r ) \\& \quad \leqslant F_{1}^{-1}\biggl(F_{1} \bigl( u_{1} ( R ) \bigr) +\frac{\sqrt{C _{1}}\int _{R}^{\infty }t^{1-N}\,dt}{(\int _{0}^{u_{1} ( R ) }f _{1} ( z,f_{2} ( z,z ) )\,dz)^{1/2}}+\sqrt{\frac{2P _{2} ( \infty ) }{\varepsilon _{1}R^{\varepsilon _{1}}}} \biggr) \\& \quad \leqslant F_{1}^{-1}\biggl(F_{1} \bigl( u_{1} ( R ) \bigr) +\frac{R ^{N-2}\sqrt{C_{1}}}{ ( N-2 ) (\int _{0}^{u_{1} ( R ) }f_{1} ( z,f_{2} ( z,z ) )\,dz)^{1/2}}+\sqrt{\frac{2P _{2} ( \infty ) }{\varepsilon _{2}R^{\varepsilon _{2}}}} \biggr)< \infty . \end{aligned}
(3.44)

Arguing as in (3.42) we have

$$\lim_{r\rightarrow \infty }u_{2} ( r ) =\infty .$$
(3.45)

Finally, since we know that

$$P_{2} ( \infty ) < \infty \quad \text{and}\quad Q_{1} ( \infty ) =\infty ,$$
(3.46)

the claim yields

$$\lim_{r\rightarrow \infty }u_{1} ( r ) < \infty \quad \text{and}\quad \lim_{r\rightarrow \infty }u_{2} ( r ) =\infty .$$
(3.47)

Claim 4: By a straightforward modification of the proof presented in the Claim 3 the results hold since any statement about $$P_{2} ( \infty )$$ can be translated into a statement about $$Q_{2} ( \infty )$$.

Claim 5: If $$( u_{1},u_{2} )$$ is a nonnegative non-trivial entire large solution of (1.1), then $$( u_{1},u _{2} )$$ satisfy

\begin{aligned}& u_{1} ( r ) \leqslant F_{1}^{-1}\biggl(F_{1} \bigl( u_{1} ( R ) \bigr) +\frac{\sqrt{C_{1}}\int _{R}^{r}t^{1-N}\,dt}{(\int _{0}^{u_{1} ( R ) }f_{1} ( z,f_{2} ( z,z ) )\,dz)^{1/2}}+\sqrt{ \frac{2P_{2} ( r ) }{ \varepsilon _{1}R^{\varepsilon _{1}}}}\biggr), \end{aligned}
(3.48)
\begin{aligned}& u_{2} ( r ) \leqslant F_{2}^{-1}\biggl(F_{2} \bigl( u_{2} ( R ) \bigr) +\frac{\sqrt{C_{2}}\int _{R}^{r}t^{1-N}\,dt}{(\int _{0}^{u_{2} ( R ) }f_{2} ( f_{1} ( z,z ) ,z )\,dz)^{1/2}}+\sqrt{\frac{2Q_{2} ( r ) }{ \varepsilon _{2}R^{\varepsilon _{2}}}} \biggr), \end{aligned}
(3.49)

where

$$C_{1}= \bigl[ R^{N-1} \bigl( u_{1} ( R ) \bigr) ^{\prime } \bigr] ^{2} \quad \text{and}\quad C_{2}= \bigl[ R^{N-1} \bigl( u_{2} ( R ) \bigr) ^{\prime } \bigr] ^{2}.$$
(3.50)

Next, assuming to the contrary that

$$P_{2} ( \infty ) < \infty \quad \text{and}\quad Q_{2} ( \infty ) < \infty ,$$
(3.51)

then (2.7) follows by taking $$r\rightarrow \infty$$ in (3.48) and (3.49). This concludes the last claim and concludes the proof of the theorem. □

### Proof of Theorem 3 completed

It follows from (3.25) and the conditions of the theorem that

\begin{aligned}& F_{1} \bigl( u_{1}^{k} ( r ) \bigr) \leqslant P_{3} ( \infty ) < F_{1} ( \infty ) < \infty , \end{aligned}
(3.52)
\begin{aligned}& F_{2} \bigl( u_{2}^{k} ( r ) \bigr) \leqslant Q_{3} ( \infty ) < F_{2} ( \infty ) < \infty . \end{aligned}
(3.53)

On the other hand, since $$F_{1}^{-1}$$, respectively $$F_{2}^{-1}$$, is strictly increasing on $$[ 0,F_{1} ( \infty ) )$$ respectively $$[ 0,F_{2} ( \infty ) )$$, we find that

$$u_{1}^{k} ( r ) \leqslant F_{1}^{-1} \bigl( P_{3} ( \infty ) \bigr) < \infty \quad \text{and}\quad u_{2}^{k} ( r ) \leqslant F_{2}^{-1} \bigl( Q_{3} ( \infty ) \bigr) < \infty ,$$
(3.54)

and then the nondecreasing sequences $$\{ u_{1}^{k} ( r ) \} _{k\geqslant 0}$$ and $$\{ u_{2}^{k} ( r ) \} _{k\geqslant 0}$$ are bounded above for all $$r\geqslant 0$$ and k. Combining these two facts, we conclude that $$( u_{1}^{k} ( r ) ,u_{2} ^{k} ( r ) ) \rightarrow ( u_{1} ( r ) ,u_{2} ( r ) )$$ as $$k\rightarrow \infty$$ and the limit functions $$u_{1}$$ and $$u_{2}$$ are positive entire bounded radial solutions of system (1.1). The remainder of the proof is similar to that of Theorem 2. □

## Conclusion

In this paper we investigate the existence of solutions on $$\mathbb{R}^{N}$$ for a system of partial differential equations under new Keller–Osserman-type conditions.

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## Funding

This work was supported by a mobility grant of the Romanian Ministery of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P1-1.1-MC-2019-0168, within PNCDI III.

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