Solvability Criterion for a System Arising from Monge-Amp`ere Equations with Two Parameters

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Solvability Criterion for a System Arising from Monge-Amp`ere Equations with Two Parameters

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Submitted:

31 January 2024

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31 January 2024

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Abstract

Monge-Ampere equations have important research significance in many fields such as geometry, convex geometry and mathematical physics. In this paper, under some superlinear and sublinear conditions, the existence of nontrivial solutions for a system arising from Monge-Ampere equations with two parameters is investigated based on Guo-Krasnosel'skii fixed point theorem. In the end, two examples are given to illustrate our main results.

Keywords:

Subject: Computer Science and Mathematics - Mathematics

1. Introduction

In this paper, we consider the existence of nontrivial solutions for the following boundary value problem:

\{\begin{matrix} {({(u^{'} (s))}^{N})}^{'} = λ N r^{N - 1} f (- u (s), - v (s)), 0 < s < 1, \\ {({(v^{'} (s))}^{N})}^{'} = μ N r^{N - 1} g (- u (s), - v (s)), 0 < s < 1, \\ u^{'} (0) = u (1) = 0, v^{'} (0) = v (1) = 0, \end{matrix}

(1)

where

N \geq 1,

f, g : [0, 1] \times [0, + \infty) \times [0, + \infty) \to [0, + \infty)

are continuous,

λ

and

μ

are two positive parameters. The problem (1.1) arises in the study of the existence of nontrivial solutions to the following Dirichlet problem of the Monge-Amp

\overset{`}{e}

re equations:

\{\begin{matrix} \det (D^{2} u) = λ f (- u, - v) in B, \\ \det (D^{2} v) = μ g (- u, - v) in B, \\ u = v = 0 on \partial B, \end{matrix}

where

D^{2} u = (\frac{\partial^{2} u}{\partial x_{i} \partial x_{j}})

is the Hessian matrix of u,

D^{2} v = (\frac{\partial^{2} v}{\partial x_{i} \partial x_{j}})

is the Hessian matrix of v,

B = {x \in R^{N} | | x | < 1}

Monge-Amp

\overset{`}{e}

re equations play an important role in the study of geometric problems, fluid mechanics, and various other applied fields(see[1]). Many researchers have done some investigations related to Monge-Amp

\overset{`}{e}

re equations. Some scholars have studied the existence of nontrivial radial convex solutions for a single Monge-Amp

\overset{`}{e}

re equation or systems of Monge-Amp

\overset{`}{e}

re equations, utilizing the theory of topological degree, bifurcation techniques,the method of upper and lower solution, and so on. For further details, see [2-9,14-15,17-18,20-23] and the references therein.

For example, in [3], Ma and Gao investigated the following boundary value problem:

\{\begin{matrix} {({(u_{1}^{'} (t))}^{n})}^{'} = λ n t^{n - 1} f (- u (t)), 0 < t < 1, \\ u^{'} (0) = u (1) = 0 . \end{matrix} (1.2)

The boundary value problem (1.2) arose from the following Monge-Amp

\overset{`}{e}

re equation:

\{\begin{matrix} \det (D^{2} u) = λ f (- u) in B, \\ u = 0 on \partial B, \end{matrix} (1.3)

where

D^{2} u = (\frac{\partial^{2} u}{\partial x_{i} \partial x_{j}})

is the Hessian matrix of u,

B = {x \in R^{n} | | x | < 1}

. The global bifurcation technique was applied to determine the optimal intervals for parameter

λ

, ensuring the existence of single or multiple solutions of boundary value problem (1.2).

In [4], Wang established two solvability criteria for weakly coupled system:

\{\begin{matrix} {({(u_{1}^{'} (t))}^{N})}^{'} = N t^{N - 1} f (- u_{2} (t)), 0 < t < 1, \\ {({(u_{2}^{'} (t))}^{N})}^{'} = N t^{N - 1} g (- u_{1} (t)), 0 < t < 1, \\ u_{1}^{'} (0) = u_{2}^{'} (0) = 0, u_{1} (1) = u_{2} (1) = 0, \end{matrix} (1.4)

where

N \geq 1

. The system (1.4) arose from the following Monge-Amp

\overset{`}{e}

re equations:

\{\begin{matrix} \det (D^{2} u_{1}) = f (- u_{2}) in B, \\ \det (D^{2} u_{2}) = g (- u_{1}) in B, \\ u_{1} = u_{2} = 0 on \partial B, \end{matrix}

where

B = {x \in R^{N} | | x | < 1}

, and

D^{2} u_{i}

is the determinant of the Hessian matrix

(\frac{\partial^{2} u_{i}}{\partial x_{m} \partial x_{n}})

u_{i}

. The existence of convex radial solutions of the weakly coupled system (1.4) in superlinear and sublinear cases was obtained based on fixed point theorems in a cone.

In [5], Wang and An discussed the following system of the Monge-Amp

\overset{`}{e}

re equations:

\{\begin{matrix} \det (D^{2} u_{1}) = f_{1} (- u_{1}, \dots, - u_{n}) in B, \\ \dots \\ \det (D^{2} u_{n}) = f_{n} (- u_{1}, \dots, - u_{n}) in B, \\ u (x) = 0 on \partial B, \end{matrix} (1.5)

where

D^{2} u_{i} = (\frac{\partial^{2} u_{i}}{\partial x_{i} \partial x_{j}})

is the Hessian matrix of

u_{i}

B = {x \in R^{N} | | x | < 1}

. The system (1.5) can be easily changed into the following boundary value problem:

\{\begin{matrix} {({(u_{1}^{'} (r))}^{N})}^{'} = N r^{N - 1} f_{1} (- u_{1}, \dots, - u_{n}), 0 < r < 1, \\ \dots \\ {({(u_{n}^{'} (r))}^{N})}^{'} = N r^{N - 1} f_{n} (- u_{1}, \dots, - u_{n}), 0 < r < 1, \\ u_{i}^{'} (0) = u_{i} (1) = 0, i = 1, \dots, n, \end{matrix}

where

N \geq 1

. The existence of triple nontrivial radial convex solutions was obtained by using the Leggett-Williams fixed point theorem.

In [6], the author studied the following system:

\{\begin{matrix} {({(u_{1}^{'} (r))}^{N})}^{'} = λ N r^{N - 1} f_{1} (- u_{1}, \dots, - u_{n}), 0 < r < 1, \\ \dots \\ {({(u_{n}^{'} (r))}^{N})}^{'} = λ N r^{N - 1} f_{n} (- u_{1}, \dots, - u_{n}), 0 < r < 1, \\ u_{i}^{'} (0) = u_{i} (1) = 0, i = 1, \dots, n, \end{matrix} (1.6)

where

N \geq 1

. The system (1.6) arose from the following system:

\{\begin{matrix} \det (D^{2} u_{1}) = λ f_{1} (- u_{1}, \dots, - u_{n}) in B, \\ \dots \\ \det (D^{2} u_{n}) = λ f_{n} (- u_{1}, \dots, - u_{n}) in B, \\ u_{i} = 0 on \partial B, i = 1, \dots, n, \end{matrix}

where

D^{2} u_{i} = (\frac{\partial^{2} u_{i}}{\partial x_{i} \partial x_{j}})

is the Hessian matrix of

u_{i}

B = {x \in R^{N} | | x | < 1}

Using fixed point theorems and considering sublinear and superlinear conditions, Wang explored the existence and two nontrivial radial solutions for the system (1.6) with a carefully selected parameter.

In [7], Gao and Wang considered the following boundary value problem:

\{\begin{matrix} {({(u_{1}^{'} (r))}^{N})}^{'} = λ_{1} N r^{N - 1} f_{1} (- u_{1}, - u_{2}, \dots, - u_{n}), \\ {({(u_{2}^{'} (r))}^{N})}^{'} = λ_{2} N r^{N - 1} f_{2} (- u_{1}, - u_{2}, \dots, - u_{n}), \\ \dots \\ {({(u_{n}^{'} (r))}^{N})}^{'} = λ_{n} N r^{N - 1} f_{n} (- u_{1}, - u_{2}, \dots, - u_{n}), \\ u_{i}^{'} (0) = u_{i} (1) = 0, i = 1, 2, \dots, n, 0 < r < 1, \end{matrix} (1.7)

where

N \geq 1

. The system (1.7) arose from the following system:

\{\begin{matrix} \det (D^{2} u_{1}) = λ_{1} f_{1} (- u_{1}, \dots, - u_{n}) in B, \\ \det (D^{2} u_{2}) = λ_{2} f_{2} (- u_{1}, \dots, - u_{n}) in B, \\ \dots \\ \det (D^{2} u_{n}) = λ_{n} f_{n} (- u_{1}, \dots, - u_{n}) in B, \\ u_{i} = 0 on \partial B, i = 1, \dots, n, \end{matrix}

where

D^{2} u_{i} = (\frac{\partial^{2} u_{i}}{\partial x_{i} \partial x_{j}})

is the Hessian matrix of

u_{i}

, and

B = {x \in R^{N} | | x | < 1}

is the unit ball in

R^{N}

. The existence, multiplicity, and nonexistence of convex solutions for systems of Monge-Amp

\overset{`}{e}

re equations with multiparameters were established via the upper and lower solutions method and the fixed point index theory.

In [18], Feng has continued to consider the existence and uniqueness of nontrivial radial convex solutions to the Monge-Amp

\overset{`}{e}

re equation (1.3). And the author also studied the following system:

\{\begin{matrix} \det (D^{2} u_{1}) = λ_{1} f_{1} (- u_{2}) in B, \\ \det (D^{2} u_{2}) = λ_{2} f_{2} (- u_{3}) in B, \\ \dots \\ \det (D^{2} u_{n}) = λ_{n} f_{n} (- u_{1}) in B, \\ u_{1} = u_{2} \dots = u_{n} = 0 on \partial B, \end{matrix} (1.8)

where

λ_{i} (i = 1, 2, \dots, n)

are positive parameters. By defining composite operators and using the eigenvalue theory in cones, the author obtained some new existence results of nontrivial radial convex solutions to the system (1.8), and also analyzed the asymptotic behavior of solutions to the system (1.8).

Meanwhile, in recent years, some authors have studied the existence of nontrivial solutions to other differential equations with parameters. For example, in [10], Hao et al. considered the existence of positive solutions for a system of nonlinear fractional differential equations nonlocal boundary value problems with parameters and p-Laplacian operator via the Guo-Krasnosel’skii fixed point theorem. In [11], by means of the method of upper and lower solutions and the fixed point index theory, Yang proved the existence of positive solutions for Dirichlet boundary value problem of 2m-order nonlinear differential systems with multiple different parameters. In [12], Jiang and Zhai investigated a class of nonlinear fourth-order systems with coupled integral boundary conditions and two parameters based on the Guo–Krasnosel’skii fixed point theorem and the Green’s functions.

Inspired by literatures [4-7,10-12,18], we consider the problem (1.1). In this paper, under some different combinations of superlinearity and sublinearity of the nonlinear terms, we use the Guo-Krasnosel’skii fixed point theorem to study the existence results of the system (1.1) and establish some existence results of nontrivial solutions based on various different values values of

λ

and

μ

. Here we extend the study in literature [4], and the main results are different from literatures [4,7,18].

2. Preliminaries

In this section, we give some preliminaries that will be used to prove existence results in Section 3. For further background knowledge of cone, we refer readers to the papers [4,16] for more details.

Lemma 1.

(see [16]) Let E be a Banach space, and

P \subset E

be a cone. Assume that

Ω_{1}

and

Ω_{2}

are bounded open sets in E,

θ \in Ω_{1}, {\bar{Ω}}_{1} \subset Ω_{2}

, the operator

A : P \cap ({\bar{Ω}}_{2} ∖ Ω_{1}) \to P

is completely continuous. If the following conditions are satisfied:

\begin{matrix} (i) & ∥ A x ∥ \leq ∥ x ∥, \forall x \in P \cap \partial Ω_{1}, ∥ A x ∥ \geq ∥ x ∥, \forall x \in P \cap \partial Ω_{2}, o r \\ (ii) & ∥ A x ∥ \geq ∥ x ∥, \forall x \in P \cap \partial Ω_{1}, ∥ A x ∥ \leq ∥ x ∥, \forall x \in P \cap \partial Ω_{2}, \end{matrix}

In order to solve the system (1.1), we give a simple transformation

x (s) = - u (s)

y (s) = - v (s)

in the system (1.1), then the system (1.1) can be changed to the following system:

\{\begin{matrix} {({(- x^{'} (s))}^{N})}^{'} = λ N s^{N - 1} f (x (s), y (s)), 0 < s < 1, \\ {({(- y^{'} (s))}^{N})}^{'} = μ N s^{N - 1} g (x (s), y (s)), 0 < s < 1, \\ x^{'} (0) = x (1) = 0, y^{'} (0) = y (1) = 0 . \end{matrix}

(2)

In the following, we treat the existence of positive solutions of the system (2.1).

Let

E = C [0, 1] \times C [0, 1]

with the norm

{∥ (x, y) ∥}_{E} = ∥ x ∥ + ∥ y ∥

, where

∥ x ∥ = max_{s \in [0, 1]} | x (s) |

and

∥ y ∥ = max_{s \in [0, 1]} | y (s) |

Define

P = {(x, y) \in E : x (s) \geq 0, y (s) \geq 0, \forall s \in [0, 1], min_{s \in [\frac{1}{4}, \frac{3}{4}]} (x (s) + y (s)) \geq \frac{1}{4} ∥ (x, y) ∥_{E}} .

Then P is a cone of E.

By literature [4], we define the operators

A_{1}

A_{2}

and A as follows:

A_{1} (x, y) (s) = \int_{s}^{1} {(\int_{0}^{u} λ N τ^{N - 1} f (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u, s \in [0, 1],

A_{2} (x, y) (s) = \int_{s}^{1} {(\int_{0}^{u} μ N τ^{N - 1} g (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u, s \in [0, 1] .

and

A (x, y) = (A_{1} (x, y), A_{2} (x, y)), (x, y) \in E

. It is obvious that the fixed points of the operator A are solutions of the system (2.1).

Similar to the proof of Lemma 2.3 in literature [4], we have the following lemma.

Lemma 2.

A : P \to P

is completely continuous.

3. Main Results

Denote

f_{0} = \underset{x + y \to 0^{+}}{lim sup} \frac{f (x, y)}{{(x + y)}^{N}}, g_{0} = \underset{x + y \to 0^{+}}{lim sup} \frac{g (x, y)}{{(x + y)}^{N}},

f_{\infty} = \underset{x + y \to \infty}{lim inf} \frac{f (x, y)}{{(x + y)}^{N}}, g_{\infty} = \underset{x + y \to \infty}{lim inf} \frac{g (x, y)}{{(x + y)}^{N}},

{\hat{f}}_{0} = \underset{x + y \to 0^{+}}{lim inf} \frac{f (x, y)}{{(x + y)}^{N}}, {\hat{g}}_{0} = \underset{x + y \to 0^{+}}{lim inf} \frac{g (x, y)}{{(x + y)}^{N}},

{\hat{f}}_{\infty} = \underset{x + y \to \infty}{lim sup} \frac{f (x, y)}{{(x + y)}^{N}}, {\hat{g}}_{\infty} = \underset{x + y \to \infty}{lim sup} \frac{g (x, y)}{{(x + y)}^{N}} .

F = \int_{0}^{1} {(\int_{0}^{1} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u, G = \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u .

For

f_{0}, g_{0}, f_{\infty}, g_{\infty} \in (0, \infty)

, we define the symbols below:

M_{1} = \frac{2^{N}}{G^{N} f_{\infty}}, M_{2} = \frac{1}{2^{N} F^{N} f_{0}},

M_{3} = \frac{2^{N}}{G^{N} g_{\infty}}, M_{4} = \frac{1}{2^{N} F^{N} g_{0}} .

Theorem 1.

(1) Assume that

f_{0}, g_{0}, f_{\infty}, g_{\infty} \in (0, \infty), M_{1} < M_{2}, M_{3} < M_{4}

, then for

λ \in (M_{1}, M_{2})

and

μ \in (M_{3}, M_{4})

, the system (2.1) has at least one positive solution.

(2) Assume that

f_{0} = 0, g_{0}, f_{\infty}, g_{\infty} \in (0, \infty), M_{3} < M_{4}

, then for

λ \in (M_{1}, \infty)

and

μ \in (M_{3}, M_{4})

, the system (2.1) has at least one positive solution.

(3) Assume that

f_{0}, f_{\infty}, g_{\infty} \in (0, \infty), g_{0} = 0, M_{1} < M_{2}

, then for

λ \in (M_{1}, M_{2})

and

μ \in (M_{3}, \infty)

, the system (2.1) has at least one positive solution.

(4) Assume that

f_{0} = g_{0} = 0, f_{\infty}, g_{\infty} \in (0, \infty)

, then for

λ \in (M_{1}, \infty)

and

μ \in (M_{3}, \infty)

, the system (2.1) has at least one positive solution.

(5) Assume that

f_{0}, g_{0} \in (0, \infty), f_{\infty} = \infty

f_{0}, g_{0} \in (0, \infty), g_{\infty} = \infty

, then for

λ \in (0, M_{2})

and

μ \in (0, M_{4})

, the system (2.1) has at least one positive solution.

(6) Assume that

f_{0} = 0, g_{0} \in (0, \infty), g_{\infty} = \infty

f_{0} = 0, g_{0} \in (0, \infty), f_{\infty} = \infty

, then for

λ \in (0, \infty)

and

μ \in (0, M_{4})

, the system (2.1) has at least one positive solution.

(7) Assume that

f_{0} \in (0, \infty), g_{0} = 0, g_{\infty} = \infty

f_{0} \in (0, \infty), g_{0} = 0, f_{\infty} = \infty

, then for

λ \in (0, M_{2})

and

μ \in (0, \infty)

, the system (2.1) has at least one positive solution.

(8) Assume that

f_{0} = g_{0} = 0, g_{\infty} = \infty

f_{0} = g_{0} = 0, f_{\infty} = \infty

, then for

λ \in (0, \infty)

and

μ \in (0, \infty)

, the system (2.1) has at least one positive solution.

Proof.

Due to the similarity in the proofs of the above cases, we will demonstrate the case (1) and the case (6).

(1) For each

λ \in (M_{1}, M_{2})

and

μ \in (M_{3}, M_{4})

, there exists

ε > 0

such that

\frac{2^{N}}{G^{N} (f_{\infty} - ε)} \leq λ \leq \frac{1}{2^{N} F^{N} (f_{0} + ε)},

\frac{2^{N}}{G^{N} (g_{\infty} - ε)} \leq μ \leq \frac{1}{2^{N} F^{N} (g_{0} + ε)} .

By the definitions of

f_{0}

and

g_{0}

, we know that there exists

r_{1} > 0

such that

f (x, y) < (f_{0} + ε) {(x + y)}^{N}, 0 \leq x + y \leq r_{1},

g (x, y) < (g_{0} + ε) {(x + y)}^{N}, 0 \leq x + y \leq r_{1} .

We define the set

Ω_{1} = {{(x, y) \in E : ∥ (x, y) ∥}_{E} < r_{1}}

, for any

(x, y) \in P \cap \partial Ω_{1}

, we get

0 \leq x (s) + y (s) \leq ∥ x ∥ + ∥ y ∥ = {∥ (x, y) ∥}_{E} = r_{1}, \forall s \in [0, 1],

then

\begin{array}{l} A_{1} (x, y) (s) & = \int_{s}^{1} {(\int_{0}^{u} λ N τ^{N - 1} f (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \leq \int_{0}^{1} {(\int_{0}^{1} λ N τ^{N - 1} f (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \leq \int_{0}^{1} {(\int_{0}^{1} λ N τ^{N - 1} (f_{0} + ε) {(x (τ) + y (τ))}^{N} d τ)}^{\frac{1}{N}} d u \\ \leq {(f_{0} + ε)}^{\frac{1}{N}} \int_{0}^{1} (\int_{0}^{1} λ N τ^{N - 1} (∥ x ∥ + {∥ y ∥)}^{N} {d τ)}^{\frac{1}{N}} d u \\ = {(f_{0} + ε)}^{\frac{1}{N}} λ^{\frac{1}{N}} \int_{0}^{1} {(\int_{0}^{1} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot {∥ (x, y) ∥}_{E} \\ \leq \frac{{∥ (x, y) ∥}_{E}}{2} . \end{array}

Therefore,

∥ A_{1} (x, y) ∥ \leq \frac{1}{2} {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{1} .

(3)

By applying the same method, we deduce

\begin{array}{l} A_{2} (x, y) (s) & = \int_{s}^{1} {(\int_{0}^{u} μ N τ^{N - 1} g (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \leq \int_{0}^{1} {(\int_{0}^{1} μ N τ^{N - 1} g (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \leq \int_{0}^{1} {(\int_{0}^{1} μ N τ^{N - 1} (g_{0} + ε) {(x (τ) + y (τ))}^{N} d τ)}^{\frac{1}{N}} d u \\ \leq {(g_{0} + ε)}^{\frac{1}{N}} \int_{0}^{1} (\int_{0}^{1} μ N τ^{N - 1} (∥ x ∥ + {∥ y ∥)}^{N} {d τ)}^{\frac{1}{N}} d u \\ = {(g_{0} + ε)}^{\frac{1}{N}} μ^{\frac{1}{N}} \int_{0}^{1} {(\int_{0}^{1} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot {∥ (x, y) ∥}_{E} \\ \leq \frac{{∥ (x, y) ∥}_{E}}{2} . \end{array}

Therefore,

∥ A_{2} (x, y) ∥ \leq \frac{1}{2} {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{1} .

(4)

By (3.1) and (3.2) we have

{∥ A (x, y) ∥}_{E} = ∥ A_{1} (x, y) ∥ + ∥ A_{2} {(x, y) ∥ \leq ∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{1} .

(5)

On the other hand, considering the definitions of

f_{\infty}

and

g_{\infty}

, there exists

{\bar{r}}_{2} > 0

such that

f (x, y) \geq (f_{\infty} - ε) {(x + y)}^{N}, x + y \geq {\bar{r}}_{2},

g (x, y) \geq (g_{\infty} - ε) {(x + y)}^{N}, x + y \geq {\bar{r}}_{2} .

We take

r_{2} = max {2 r_{1}, 4 {\bar{r}}_{2}}

and denote

Ω_{2} = {{(x, y) \in E : ∥ (x, y) ∥}_{E} < r_{2}}

. For any

(x, y) \in P \cap \partial Ω_{2}

, we get

min_{s \in [\frac{1}{4}, \frac{3}{4}]} (x (s) + y (s)) \geq \frac{1}{4} {∥ (x, y) ∥}_{E} = \frac{1}{4} r_{2} \geq {\bar{r}}_{2},

then

\begin{array}{l} A_{1} (x, y) (\frac{1}{4}) & = \int_{\frac{1}{4}}^{1} {(\int_{0}^{u} λ N τ^{N - 1} f (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \geq \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} λ N τ^{N - 1} f (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \geq \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} λ N τ^{N - 1} (f_{\infty} - ε) {(x (τ) + y (τ))}^{N} d τ)}^{\frac{1}{N}} d u \\ \geq {(f_{\infty} - ε)}^{\frac{1}{N}} \int_{\frac{1}{4}}^{\frac{3}{4}} (\int_{\frac{1}{4}}^{u} λ N τ^{N - 1} (\frac{1}{4} ∥ (x, y) ∥_{E})^{N} {d τ)}^{\frac{1}{N}} d u \\ = \frac{1}{4} {(f_{\infty} - ε)}^{\frac{1}{N}} λ^{\frac{1}{N}} \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot {∥ (x, y) ∥}_{E} \\ \geq \frac{{∥ (x, y) ∥}_{E}}{2} . \end{array}

Therefore,

∥ A_{1} (x, y) ∥ \geq \frac{1}{2} {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{2} .

(6)

In a similar manner, for any

(x, y) \in P \cap \partial Ω_{2}

, we obtain

\begin{array}{l} A_{2} (x, y) (\frac{1}{4}) & = \int_{\frac{1}{4}}^{1} {(\int_{0}^{u} μ N τ^{N - 1} g (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \geq \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} μ N τ^{N - 1} g (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \geq \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} μ N τ^{N - 1} (g_{\infty} - ε) {(x (τ) + y (τ))}^{N} d τ)}^{\frac{1}{N}} d u \\ \geq {(g_{\infty} - ε)}^{\frac{1}{N}} \int_{\frac{1}{4}}^{\frac{3}{4}} (\int_{\frac{1}{4}}^{u} μ N τ^{N - 1} (\frac{1}{4} ∥ (x, y) ∥_{E})^{N} {d τ)}^{\frac{1}{N}} d u \\ = \frac{1}{4} {(g_{\infty} - ε)}^{\frac{1}{N}} μ^{\frac{1}{N}} \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot {∥ (x, y) ∥}_{E} \\ \geq \frac{{∥ (x, y) ∥}_{E}}{2} . \end{array}

Therefore,

∥ A_{2} (x, y) ∥ \geq \frac{1}{2} {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{2} .

(7)

By (3.4) and (3.5) we have

{∥ A (x, y) ∥}_{E} = ∥ A_{1} (x, y) ∥ + ∥ A_{2} {(x, y) ∥ \geq ∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{2} .

(8)

From (3.3), (3.6) and Lemma 2.1, we get that A has at least one fixed point

(x, y) \in P \cap ({\bar{Ω}}_{2} ∖ Ω_{1})

such that

r_{1} \leq {∥ (x, y) ∥}_{E} \leq r_{2}

, so the system (2.1) has at least one positive solution. The proof of the case (1) is completed.

(6) Assume

f_{0} = 0, g_{0} \in (0, \infty), g_{\infty} = \infty

, then for each

λ \in (0, \infty)

and

μ \in (0, M_{4})

, there exists

ε > 0

such that

0 < λ < \frac{1}{2^{N} F^{N} ε}, \frac{4^{N} ε}{G^{N}} < μ < \frac{1}{2^{N} F^{N} (g_{0} + ε)} .

Considering the definitions of

f_{0}

and

g_{0}

, we know that there exists

r_{3} > 0

such that

f (x, y) < ε {(x + y)}^{N}, 0 \leq x + y \leq r_{3},

g (x, y) < (g_{0} + ε) {(x + y)}^{N}, 0 \leq x + y \leq r_{3} .

We define the set

Ω_{3} = {{(x, y) \in E : ∥ (x, y) ∥}_{E} < r_{3}}

, for any

(x, y) \in P \cap \partial Ω_{3}

, we deduce

\begin{array}{l} A_{1} (x, y) (s) & = \int_{s}^{1} {(\int_{0}^{u} λ N τ^{N - 1} f (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \leq \int_{0}^{1} {(\int_{0}^{1} λ N τ^{N - 1} f (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \leq \int_{0}^{1} {(\int_{0}^{1} λ N τ^{N - 1} ε {(x (τ) + y (τ))}^{N} d τ)}^{\frac{1}{N}} d u \\ \leq ε^{\frac{1}{N}} \int_{0}^{1} (\int_{0}^{1} λ N τ^{N - 1} (∥ x ∥ + {∥ y ∥)}^{N} {d τ)}^{\frac{1}{N}} d u \\ = ε^{\frac{1}{N}} λ^{\frac{1}{N}} \int_{0}^{1} {(\int_{0}^{1} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot {∥ (x, y) ∥}_{E} \\ < \frac{{∥ (x, y) ∥}_{E}}{2} . \end{array}

(9)

Therefore,

∥ A_{1} (x, y) ∥ \leq \frac{1}{2} {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{3} .

Similar to the proof of (3.7), we have

∥ A_{2} (x, y) ∥ \leq \frac{1}{2} {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{3},

then

{∥ A (x, y) ∥}_{E} \leq {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{3} .

(10)

On the other hand, since

g_{\infty} = \infty

, we know that there exists

{\bar{r}}_{4} > 0

such that

g (x, y) \geq \frac{1}{ε} {(x + y)}^{N}, x, y \geq 0, x + y \geq {\bar{r}}_{4} .

We take

r_{4} = max {2 r_{3}, 4 {\bar{r}}_{4}}

and denote

Ω_{4} = {{(x, y) \in E : ∥ (x, y) ∥}_{E} < r_{4}}

, for any

(x, y) \in P \cap \partial Ω_{4}

, we get

{min}_{s \in [\frac{1}{4}, \frac{3}{4}]} (x (s) + y (s)) \geq \frac{1}{4} {∥ (x, y) ∥}_{E} = \frac{1}{4} r_{4} \geq {\bar{r}}_{4},

then

\begin{array}{l} A_{2} (x, y) (\frac{1}{4}) & = \int_{\frac{1}{4}}^{1} {(\int_{0}^{u} μ N τ^{N - 1} g (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \geq \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} μ N τ^{N - 1} g (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \geq \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} μ N τ^{N - 1} \frac{1}{ε} {(x (τ) + y (τ))}^{N} d τ)}^{\frac{1}{N}} d u \\ \geq {(\frac{1}{ε})}^{\frac{1}{N}} \int_{\frac{1}{4}}^{\frac{3}{4}} (\int_{\frac{1}{4}}^{u} μ N τ^{N - 1} (\frac{1}{4} ∥ (x, y) ∥_{E})^{N} {d τ)}^{\frac{1}{N}} d u \\ = \frac{1}{4} {(\frac{1}{ε})}^{\frac{1}{N}} μ^{\frac{1}{N}} \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot ∥ (x, y) ∥_{E} \\ > {∥ (x, y) ∥}_{E} . \end{array}

Therefore,

{∥ A (x, y) ∥}_{E} \geq ∥ A_{2} {(x, y) ∥ \geq ∥ (x, y) ∥}_{E}, (x, y) \in P \cap \partial Ω_{4} .

(11)

From (3.8), (3.9) and Lemma 2.1, we get that A has at least one fixed point

(x, y) \in P \cap ({\bar{Ω}}_{4} ∖ Ω_{3})

such that

r_{3} \leq {∥ (x, y) ∥}_{E} \leq r_{4}

, that is,

(x, y)

is a positive solution for the system (2.1), so the proof is completed. □

For

{\hat{f}}_{0}, {\hat{g}}_{0}, {\hat{f}}_{\infty}, {\hat{g}}_{\infty} \in (0, \infty)

, we define the symbols below:

Q_{1} = \frac{2^{N}}{G^{N} {\hat{f}}_{0}}, Q_{2} = \frac{1}{2^{N} F^{N} {\hat{f}}_{\infty}},

Q_{3} = \frac{2^{N}}{G^{N} {\hat{g}}_{0}}, Q_{4} = \frac{1}{2^{N} F^{N} {\hat{g}}_{\infty}} .

Theorem 2.

(1) Assume that

{\hat{f}}_{0}, {\hat{g}}_{0}, {\hat{f}}_{\infty}, {\hat{g}}_{\infty} \in (0, \infty), Q_{1} < Q_{2}, Q_{3} < Q_{4}

, then for

λ \in (Q_{1}, Q_{2})

and

μ \in (Q_{3}, Q_{4})

, the system (2.1) has at least one positive solution.

(2) Assume that

{\hat{f}}_{0}, {\hat{g}}_{0}, {\hat{f}}_{\infty} \in (0, \infty), {\hat{g}}_{\infty} = 0,

and

Q_{1} < Q_{2}

, then for each

λ \in (Q_{1}, Q_{2})

and

μ \in (Q_{3}, \infty)

, the system (2.1) has at least one positive solution.

(3) Assume that

{\hat{f}}_{0}, {\hat{g}}_{0}, {\hat{g}}_{\infty} \in (0, \infty), {\hat{f}}_{\infty} = 0,

and

Q_{3} < Q_{4}

, then for each

λ \in (Q_{1}, \infty)

and

μ \in (Q_{3}, Q_{4})

, the system (2.1) has at least one positive solution.

(4) Assume that

{\hat{f}}_{0}, {\hat{g}}_{0} \in (0, \infty), {\hat{f}}_{\infty} = {\hat{g}}_{\infty} = 0,

then for each

λ \in (Q_{1}, \infty)

and

μ \in (Q_{3}, \infty)

, the system (2.1) has at least one positive solution.

(5) Assume that

{\hat{f}}_{\infty}, {\hat{g}}_{\infty} \in (0, \infty), {\hat{f}}_{0} = \infty

{\hat{f}}_{\infty}, {\hat{g}}_{\infty} \in (0, \infty), {\hat{g}}_{0} = \infty

, then for each

λ \in (0, Q_{2})

and

μ \in (0, Q_{4})

, the system (2.1) has at least one positive solution.

(6) Assume that

{\hat{f}}_{0} = \infty, {\hat{f}}_{\infty} \in (0, \infty), {\hat{g}}_{\infty} = 0

{\hat{f}}_{\infty} \in (0, \infty), {\hat{g}}_{\infty} = 0, {\hat{g}}_{0} = \infty

, then for each

λ \in (0, Q_{2})

and

μ \in (0, \infty)

, the system (2.1) has at least one positive solution.

(7) Assume that

{\hat{f}}_{0} = \infty, {\hat{g}}_{\infty} \in (0, \infty), {\hat{f}}_{\infty} = 0

{\hat{g}}_{\infty} \in (0, \infty), {\hat{g}}_{0} = \infty, {\hat{f}}_{\infty} = 0

, then for each

λ \in (0, \infty)

and

μ \in (0, Q_{4})

, the system (2.1) has at least one positive solution.

(8) Assume that

{\hat{f}}_{\infty} = {\hat{g}}_{\infty} = 0, {\hat{f}}_{0} = \infty

{\hat{f}}_{\infty} = {\hat{g}}_{\infty} = 0, {\hat{g}}_{0} = \infty

, then for each

λ \in (0, \infty)

and

μ \in (0, \infty)

, the system (2.1) has at least one positive solution.

Proof.

Due to the similarity in the proofs of the above cases, we will demonstrate the case (1) and the case (6).

(1) For each

λ \in (Q_{1}, Q_{2})

and

μ \in (Q_{3}, Q_{4})

, there exists

ε > 0

such that

\frac{2^{N}}{G^{N} ({\hat{f}}_{0} - ε)} \leq λ \leq \frac{1}{2^{N} F^{N} ({\hat{f}}_{\infty} + ε)},

\frac{2^{N}}{G^{N} ({\hat{g}}_{0} - ε)} \leq μ \leq \frac{1}{2^{N} F^{N} ({\hat{g}}_{\infty} + ε)} .

By the definitions of

{\hat{f}}_{0}

and

{\hat{g}}_{0}

, we know that there exists

r_{1} > 0

such that

f (x, y) \geq ({\hat{f}}_{0} - ε) {(x + y)}^{N}, x, y \geq 0, x + y \leq r_{1},

g (x, y) \geq ({\hat{g}}_{0} - ε) {(x + y)}^{N}, x, y \geq 0, x + y \leq r_{1} .

We define the set

Ω_{1} = {{(x, y) \in E : ∥ (x, y) ∥}_{E} < r_{1}}

, for any

(x, y) \in P \cap \partial Ω_{1}

, we get

\begin{array}{l} A_{1} (x, y) (\frac{1}{4}) & = \int_{\frac{1}{4}}^{1} {(\int_{0}^{u} λ N τ^{N - 1} f (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \geq \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} λ N τ^{N - 1} f (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \geq \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} λ N τ^{N - 1} ({\hat{f}}_{0} - ε) {(x (τ) + y (τ))}^{N} d τ)}^{\frac{1}{N}} d u \\ \geq {({\hat{f}}_{0} - ε)}^{\frac{1}{N}} \int_{\frac{1}{4}}^{\frac{3}{4}} (\int_{\frac{1}{4}}^{u} λ N τ^{N - 1} (\frac{1}{4} ∥ (x, y) ∥_{E})^{N} {d τ)}^{\frac{1}{N}} d u \\ = \frac{1}{4} {({\hat{f}}_{0} - ε)}^{\frac{1}{N}} λ^{\frac{1}{N}} \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot {∥ (x, y) ∥}_{E} \\ \geq \frac{{∥ (x, y) ∥}_{E}}{2} . \end{array}

Therefore,

∥ A_{1} (x, y) ∥ \geq \frac{1}{2} {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{1} .

(12)

In a similar manner, for any

(x, y) \in P \cap \partial Ω_{1}

, we deduce

\begin{array}{l} A_{2} (x, y) (\frac{1}{4}) & = \int_{\frac{1}{4}}^{1} {(\int_{0}^{u} μ N τ^{N - 1} g (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \geq \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} μ N τ^{N - 1} g (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \geq \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} μ N τ^{N - 1} ({\hat{g}}_{0} - ε) {(x (τ) + y (τ))}^{N} d τ)}^{\frac{1}{N}} d u \\ \geq {({\hat{g}}_{0} - ε)}^{\frac{1}{N}} \int_{\frac{1}{4}}^{\frac{3}{4}} (\int_{\frac{1}{4}}^{u} μ N τ^{N - 1} (\frac{1}{4} ∥ (x, y) ∥_{E})^{N} {d τ)}^{\frac{1}{N}} d u \\ = \frac{1}{4} {({\hat{g}}_{0} - ε)}^{\frac{1}{N}} μ^{\frac{1}{N}} \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot {∥ (x, y) ∥}_{E} \\ \geq \frac{{∥ (x, y) ∥}_{E}}{2} . \end{array}

Therefore,

∥ A_{2} (x, y) ∥ \geq \frac{1}{2} {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{1} .

(13)

From (3.10) and (3.11) we deduce

{∥ A (x, y) ∥}_{E} = ∥ A_{1} (x, y) ∥ + ∥ A_{2} {(x, y) ∥ \geq ∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{1} .

(14)

Let

f^{*} (u) = max_{_{0 \leq x + y \leq u}} f (x, y)

g^{*} (u) = max_{_{0 \leq x + y \leq u}} g (x, y)

, then we have

f (x, y) \leq f^{*} (u), x, y \geq 0, x + y \leq u,

g (x, y) \leq g^{*} (u), x, y \geq 0, x + y \leq u .

Similar to the proof of [10], we have

\underset{u \to + \infty}{lim sup} \frac{f^{*} (u)}{u^{N}} \leq {\hat{f}}_{\infty}, \underset{u \to + \infty}{lim sup} \frac{g^{*} (u)}{u^{N}} \leq {\hat{g}}_{\infty} .

According to the above inequality, we know that there exists

{\bar{r}}_{2} > 0

such that

\frac{f^{*} (u)}{u^{N}} \leq \underset{u \to + \infty}{lim sup} \frac{f^{*} (u)}{u^{N}} + ε \leq {\hat{f}}_{\infty} + ε, u \geq {\bar{r}}_{2},

\frac{g^{*} (u)}{u^{N}} \leq \underset{u \to + \infty}{lim sup} \frac{g^{*} (u)}{u^{N}} + ε \leq {\hat{g}}_{\infty} + ε, u \geq {\bar{r}}_{2},

then

f^{*} (u) \leq ({\hat{f}}_{\infty} + ε) u^{N}, g^{*} (u) \leq ({\hat{g}}_{\infty} + ε) u^{N}, u \geq {\bar{r}}_{2} .

We take

r_{2} = max {2 r_{1}, {\bar{r}}_{2}}

and denote

Ω_{2} = {{(x, y) \in E : ∥ (x, y) ∥}_{E} < r_{2}}

, for any

(x, y) \in P \cap \partial Ω_{2}

, we get

f (x (s) + y (s)) \leq f^{*} {(∥ (x, y) ∥}_{E}), g (x (s) + y (s)) \leq g^{*} {(∥ (x, y) ∥}_{E}),

then

\begin{array}{l} A_{1} (x, y) (s) & \leq \int_{0}^{1} (\int_{0}^{1} λ N τ^{N - 1} f^{*} {(∥ (x, y) ∥}_{E} {) d τ)}^{\frac{1}{N}} d u \\ \leq \int_{0}^{1} (\int_{0}^{1} λ N τ^{N - 1} ({\hat{f}}_{\infty} + ε) (∥ (x, y) ∥_{E})^{N} {d τ)}^{\frac{1}{N}} d u \\ = {({\hat{f}}_{\infty} + ε)}^{\frac{1}{N}} λ^{\frac{1}{N}} \int_{0}^{1} {(\int_{0}^{1} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot {∥ (x, y) ∥}_{E} \\ \leq \frac{{∥ (x, y) ∥}_{E}}{2} . \end{array}

Therefore,

∥ A_{1} (x, y) ∥ \leq \frac{1}{2} {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{2} .

(15)

In a similar manner, for any

(x, y) \in P \cap \partial Ω_{2}

, we have

\begin{array}{l} A_{2} (x, y) (s) & \leq \int_{0}^{1} (\int_{0}^{1} μ N τ^{N - 1} g^{*} {(∥ (x, y) ∥}_{E} {) d τ)}^{\frac{1}{N}} d u \\ \leq \int_{0}^{1} (\int_{0}^{1} μ N τ^{N - 1} ({\hat{g}}_{\infty} + ε) (∥ (x, y) ∥_{E})^{N} {d τ)}^{\frac{1}{N}} d u \\ = {({\hat{g}}_{\infty} + ε)}^{\frac{1}{N}} μ^{\frac{1}{N}} \int_{0}^{1} {(\int_{0}^{1} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot {∥ (x, y) ∥}_{E} \\ \leq \frac{{∥ (x, y) ∥}_{E}}{2} . \end{array}

Therefore,

∥ A_{2} (x, y) ∥ \leq \frac{1}{2} {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{2} .

(16)

From (3.13) and (3.14) we deduce

{∥ A (x, y) ∥}_{E} = ∥ A_{1} (x, y) ∥ + ∥ A_{2} {(x, y) ∥ \leq ∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{2} .

(17)

Hence, by using (3.12), (3.15) and Lemma 2.1, we conclude that A has at least one fixed point

(x, y) \in P \cap ({\bar{Ω}}_{2} ∖ Ω_{1})

such that

r_{1} \leq ∥ (x, y) ∥ \leq r_{2} .

(6) Assume

{\hat{f}}_{0} = \infty, {\hat{f}}_{\infty} \in (0, \infty), {\hat{g}}_{\infty} = 0

, then for any

λ \in (0, Q_{2})

and

μ \in (0, \infty)

, there exists

ε > 0

such that

\frac{4^{N} ε}{G^{N}} < λ < \frac{1}{2^{N} F^{N} ({\hat{f}}_{\infty} + ε)}, 0 < μ < \frac{1}{2^{N} F^{N} ε} .

Since

{\hat{f}}_{0} = \infty

, we know that there exists

r_{3} > 0

such that

f (x, y) \geq \frac{1}{ε} {(x + y)}^{N}, x, y \geq 0, 0 \leq x + y \leq r_{3} .

We choose the set

Ω_{3} = {{(x, y) \in E : ∥ (x, y) ∥}_{E} < r_{3}}

, for any

(x, y) \in P \cap \partial Ω_{3}

, we deduce

\begin{array}{l} A_{1} (x, y) (\frac{1}{4}) & \geq \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} λ N τ^{N - 1} f (x (τ), y (τ)) d τ)}^{\frac{1}{N}} d u \\ \geq \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} λ N τ^{N - 1} \frac{1}{ε} {(x (τ) + y (τ))}^{N} d τ)}^{\frac{1}{N}} d u \\ \geq {(\frac{1}{ε})}^{\frac{1}{N}} \int_{\frac{1}{4}}^{\frac{3}{4}} (\int_{\frac{1}{4}}^{u} λ N τ^{N - 1} (\frac{1}{4} ∥ (x, y) ∥_{E})^{N} {d τ)}^{\frac{1}{N}} d u \\ = \frac{1}{4} {(\frac{1}{ε})}^{\frac{1}{N}} λ^{\frac{1}{N}} \int_{\frac{1}{4}}^{\frac{3}{4}} {(\int_{\frac{1}{4}}^{u} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot {∥ (x, y) ∥}_{E} \\ \geq {∥ (x, y) ∥}_{E} . \end{array}

Therefore,

{∥ A (x, y) ∥}_{E} \geq ∥ A_{1} {(x, y) ∥ \geq ∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{3} .

(18)

Let

f^{*} (u) = max_{_{0 \leq x + y \leq u}} f (x, y)

g^{*} (u) = max_{_{0 \leq x + y \leq u}} g (x, y) .

Similar to the proof of [10], we have

\underset{u \to + \infty}{lim sup} \frac{f^{*} (u)}{u^{N}} \leq {\hat{f}}_{\infty}, \underset{u \to + \infty}{lim sup} \frac{g^{*} (u)}{u^{N}} = 0 .

Hence, for above

ε > 0

, there exists

{\bar{r}}_{4} > 0

such that

\frac{f^{*} (u)}{u^{N}} \leq \underset{u \to + \infty}{lim sup} \frac{f^{*} (u)}{u^{N}} + ε \leq {\hat{f}}_{\infty} + ε, u \geq {\bar{r}}_{4},

\frac{g^{*} (u)}{u^{N}} \leq \underset{u \to + \infty}{lim sup} \frac{g^{*} (u)}{u^{N}} + ε = ε, u \geq {\bar{r}}_{4},

then

f^{*} (u) \leq ({\hat{f}}_{\infty} + ε) u^{N}, g^{*} (u) \leq ε u^{N}, u \geq {\bar{r}}_{4} .

We take

r_{4} = max {2 r_{3}, {\bar{r}}_{4}}

and denote

Ω_{4} = {{(x, y) \in E : ∥ (x, y) ∥}_{E} < r_{4}}

, for any

(x, y) \in P \cap \partial Ω_{4}

, we get

f (x (s) + y (s)) \leq f^{*} {(∥ (x, y) ∥}_{E}), g (x (s) + y (s)) \leq g^{*} {(∥ (x, y) ∥}_{E}),

then

\begin{array}{l} A_{1} (x, y) (s) & \leq \int_{0}^{1} (\int_{0}^{1} λ N τ^{N - 1} f^{*} {(∥ (x, y) ∥}_{E} {) d τ)}^{\frac{1}{N}} d u \\ \leq \int_{0}^{1} (\int_{0}^{1} λ N τ^{N - 1} ({\hat{f}}_{\infty} + ε) (∥ (x, y) ∥_{E})^{N} {d τ)}^{\frac{1}{N}} d u \\ = {({\hat{f}}_{\infty} + ε)}^{\frac{1}{N}} λ^{\frac{1}{N}} \int_{0}^{1} {(\int_{0}^{1} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot {∥ (x, y) ∥}_{E} \\ \leq \frac{{∥ (x, y) ∥}_{E}}{2} . \end{array}

Therefore,

∥ A_{1} (x, y) ∥ \leq \frac{1}{2} {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{4} .

(19)

In a similar manner, for any

(x, y) \in P \cap \partial Ω_{4}

, we have

\begin{array}{l} A_{2} (x, y) (s) & \leq \int_{0}^{1} (\int_{0}^{1} μ N τ^{N - 1} g^{*} {(∥ (x, y) ∥}_{E} {) d τ)}^{\frac{1}{N}} d u \\ \leq \int_{0}^{1} (\int_{0}^{1} μ N τ^{N - 1} {ε (∥ (x, y) ∥}_{E})^{N} {d τ)}^{\frac{1}{N}} d u \\ = ε^{\frac{1}{N}} μ^{\frac{1}{N}} \int_{0}^{1} {(\int_{0}^{1} N τ^{N - 1} d τ)}^{\frac{1}{N}} d u \cdot {∥ (x, y) ∥}_{E} \\ \leq \frac{{∥ (x, y) ∥}_{E}}{2} . \end{array}

Therefore,

∥ A_{2} (x, y) ∥ \leq \frac{1}{2} {∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{4} .

(20)

From (3.17) and (3.18) we deduce

{∥ A (x, y) ∥}_{E} = ∥ A_{1} (x, y) ∥ + ∥ A_{2} {(x, y) ∥ \leq ∥ (x, y) ∥}_{E}, \forall (x, y) \in P \cap \partial Ω_{4} .

(21)

Hence, by using (3.16), (3.19) and Lemma 2.1, we conclude that A has at least one fixed point

(x, y) \in P \cap ({\bar{Ω}}_{4} ∖ Ω_{3})

such that

r_{3} \leq {∥ (x, y) ∥}_{E} \leq r_{4},

that is,

(x, y)

is a positive solution for the system (2.1). □

4. Applications

Example 1.

We consider the following boundary value problem:

\{\begin{matrix} {({(- x^{'} (s))}^{3})}^{'} = 3 λ s^{2} f (x (s), y (s)), 0 < s < 1, \\ {({(- y^{'} (s))}^{3})}^{'} = 3 μ s^{2} g (x (s), y (s)), 0 < s < 1, \\ x^{'} (0) = x (1) = 0, y^{'} (0) = y (1) = 0, \end{matrix}

(32)

Take

f (x, y) = {(x + y)}^{N + 2}

g (x, y) = {(x + y)}^{N} + {(x + y)}^{N} e^{x + y}

, where

N = 3

. By a simple calculation we get

M_{4} \approx 0.0625

, and

f_{0} = \underset{x + y \to 0^{+}}{lim sup} \frac{f (x, y)}{{(x + y)}^{N}} = \underset{x + y \to 0^{+}}{lim sup} {(x + y)}^{2} = 0,

g_{0} = \underset{x + y \to 0^{+}}{lim sup} \frac{g (x, y)}{{(x + y)}^{N}} = \underset{x + y \to 0^{+}}{lim sup} (1 + e^{x + y}) = 2,

f_{\infty} = \underset{x + y \to \infty}{lim inf} \frac{f (x, y)}{{(x + y)}^{N}} = \underset{x + y \to \infty}{lim inf} {(x + y)}^{2} = \infty .

Then, for each

λ \in (0, \infty)

and

μ \in (0, 0.0625)

, by Theorem 3.1(6) we obtain that the system (4.1) has at least one positive solution.

Example 2.

We consider the following boundary value problem:

\{\begin{matrix} {({(- x^{'} (s))}^{3})}^{'} = 3 λ s^{2} f (x (s), y (s)), 0 < s < 1, \\ {({(- y^{'} (s))}^{3})}^{'} = 3 μ s^{2} g (x (s), y (s)), 0 < s < 1, \\ x^{'} (0) = x (1) = 0, y^{'} (0) = y (1) = 0, \end{matrix}

(23)

Take

f (x, y) = \frac{{(x + y)}^{N}}{tan {(x + y)}^{N}}

g (x, y) = \frac{1}{x + y}

, where

N = 3

. By a simple calculation we get

Q_{2} \approx 0.1962

, and

{\hat{f}}_{0} = \underset{x + y \to 0^{+}}{lim inf} \frac{f (x, y)}{{(x + y)}^{N}} = \underset{x + y \to 0^{+}}{lim inf} \frac{1}{arctan {(x + y)}^{N}} = \infty,

{\hat{g}}_{\infty} = \underset{x + y \to \infty}{lim sup} \frac{g (x, y)}{{(x + y)}^{N}} = \underset{x + y \to \infty}{lim sup} \frac{1}{{(x + y)}^{N + 1}} = 0,

{\hat{f}}_{\infty} = \underset{x + y \to \infty}{lim sup} \frac{f (x, y)}{{(x + y)}^{N}} = \underset{x + y \to \infty}{lim sup} \frac{1}{{arctan (x + y)}^{N}} = \frac{2}{π} .

Then, for each

λ \in (0, 0.1962)

and

μ \in (0, \infty)

, by Theorem 3.2(6) we obtain that the system (4.2) has at least one positive solution.

5. Conclusion

The system of Monge-Amp

\overset{`}{e}

re equations is significant in various fields of study, including geometry, mathematical physics, materials science, and others. In this paper, by considering some combinations of superlinearity and sublinearlity of the functions f and g, we use the Guo-Krasnosel’skii fixed point theorem to study the existence of nontrivial solutions for a system of Monge-Amp

\overset{`}{e}

re equations with two parameters and establish diverse existence outcomes for nontrivial solutions based on various values of

λ

and

μ

, which enrich the theories for the system of Monge-Amp

\overset{`}{e}

re equations. The research in this paper is different from reference [4], and can be said to be its generalization.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Funding

The project is supported by the National Natural Science Foundation of China (11801322) and Shandong Natural Science Foundation(ZR2021MA064).

Data Availability Statement

The data in the article is not applicable.

Acknowledgments

The authors would like to thank reviewers for their valuable comments, which help to enrich the content of this paper.

Conflicts of Interest

The authors declare there is no conflicts of interest.

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Solvability Criterion for a System Arising from Monge-Amp`ere Equations with Two Parameters

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Applications

5. Conclusion

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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