Open Access

Existence of convex solutions for systems of Monge-Ampère equations

Boundary Value Problems20152015:128

https://doi.org/10.1186/s13661-015-0390-9

Received: 21 May 2015

Accepted: 14 July 2015

Published: 1 August 2015

The Erratum to this article has been published in Boundary Value Problems 2016 2016:193

Abstract

We study the existence, multiplicity, and nonexistence of convex solutions for systems of Monge-Ampère equations with multiparameters. The proof of the results is based on the method of upper and lower solutions and the fixed point index theory.

Keywords

convex solutions Monge-Ampère equations upper and lower solutions fixed point index theory

1 Introduction

In this paper, we consider the existence, multiplicity, and nonexistence of convex solutions for the following boundary value problem:
$$ \left \{ \textstyle\begin{array}{l} ((u'_{1}(r))^{N})'=\lambda_{1} Nr^{N-1}f^{1}(-u_{1},-u_{2},\ldots,-u_{n}), \\ ((u'_{2}(r))^{N})'=\lambda_{2} Nr^{N-1}f^{2}(-u_{1},-u_{2},\ldots,-u_{n}), \\ \ldots, \\ ((u'_{n}(r))^{N})'=\lambda_{n} Nr^{N-1}f^{n}(-u_{1},-u_{2},\ldots,-u_{n}), \\ u'_{i}(0)=u_{i}(1)=0, \quad i=1,2,\ldots,n, 0< r< 1, \end{array}\displaystyle \displaystyle \displaystyle \right . $$
(1.1)
where \(N\geq1\). Let \(\mathbb{R}_{+}=:[0,\infty)\). Throughout this paper, we assume that \(f^{i}\in C(\mathbb{R}_{+}^{n},\mathbb{R}_{+})\) (\(i=1,2,\ldots,n\)). Such a problem arises in the study of the existence of convex radial solutions for the following Dirichlet problem of the Monge-Ampère equations:
$$ \left \{ \textstyle\begin{array}{l} \det(D^{2}u_{1})=\lambda f^{1}(-u_{1},-u_{2},\ldots,-u_{n}), \\ \det(D^{2}u_{2})=\lambda f^{2}(-u_{1},-u_{2},\ldots,-u_{n}), \\ \ldots, \\ \det(D^{2}u_{n})=\lambda f^{n}(-u_{1},-u_{2},\ldots,-u_{n}), \\ u_{i}=0 \quad \mbox{on } \partial B, i=1,2,\ldots,n, \end{array}\displaystyle \right . $$
(1.2)
where \(D^{2}u_{i}=(\frac{\partial u_{i}}{\partial x_{i}\, \partial x_{j}})\) is the Hessian matrix of \(u_{i}\) and \(B=\{x\in\mathbb{R}^{N}:|x|<1\}\) is the unit ball in \(\mathbb{R}^{N}\).
For the scalar equation, Kutev [1] obtained the existence of a unique nontrivial convex radially symmetric solution of
$$ \left \{ \textstyle\begin{array}{l} \det(D^{2}u)=\lambda f(-u) \quad \mbox{in }B, \\ u=0 \quad \mbox{on }\partial B, \end{array}\displaystyle \right . $$
(1.3)
with \(f(u)=u^{p}\) based on the Schauder fixed point theorem for positive, compact operators. Hu and Wang [2] established several criteria for the existence, multiplicity, and nonexistence of strictly convex solutions for (1.3) with or without an eigenvalue parameter based on the fixed point index, due to Krasnoselskii. For systems, the problem (1.2) has been studied by Wang [3]. They considered the existence, multiplicity, and nonexistence of nontrivial radial convex solutions with superlinearity or sublinearity assumptions based on Krasnoselskii’s fixed point theorem in a cone. Therefore, it seems to be interesting to consider the convex radial solutions when the problem has multiparameters.
For the multiparameter problem, Dunninger and Wang [4, 5] considered the existence and multiplicity of positive radial solutions for the elliptic systems
$$ \left \{ \textstyle\begin{array}{l} \Delta u+\lambda k_{1}(|x|)f(u,v)=0, \\ \Delta v+\mu k_{2}(|x|)g(u,v)=0 \quad \mbox{in } \Omega, \\ u=v=0\quad \mbox{on }\partial\Omega, \end{array}\displaystyle \right . $$
(1.4)
where \(\Omega=\{x\in\mathbb{R}^{n}:R_{1}<|x|<R_{2}\}\), \(R_{1},R_{2}>0\), \(n\geq3\), \((\lambda,\mu)\in\mathbb{R}_{+}^{2}\setminus\{(0,0)\}\), \(k_{i}\in C([R_{1},R_{2}], \mathbb{R}_{+})\), not vanishing identically on any subinterval of \([R_{1},R_{2}]\) and \(f,g\in C(\mathbb{R}_{+}^{2},\mathbb{R}_{+}\setminus\{0\})\). In particular, Dunninger and Wang [4] considered problem (1.4) for the case \(f(0,0)>0\), \(g(0,0)>0\) and the following two conditions are satisfied:
(A1): 
f and g are nondecreasing on \(\mathbb{R}_{+}^{2}\), i.e.,
$$f(u_{1},v_{1})\leq f(u_{2},v_{2}) \quad \mbox{and}\quad g(u_{1},v_{1})\leq g(u_{2},v_{2}) $$
whenever \((u_{1},v_{1})\leq(u_{2},v_{2})\), where the inequality on \(\mathbb{R}_{+}^{2}\) can be understood componentwise;
(A2): 

\(f_{\infty}=:\lim_{(u,v)\rightarrow \infty} \frac{f(u,v)}{u+v}=\infty\), \(g_{\infty}=:\lim_{(u,v)\rightarrow\infty}\frac {g(u,v)}{u+v}=\infty\).

They proved for the case \(\lambda=\mu\) that there exists \(\lambda^{*}>0\) such that problem (1.4) has at least two, at least one, or no positive radial solutions according to \(0<\lambda<\lambda^{*}\), \(\lambda=\lambda^{*}\), or \(\lambda>\lambda^{*}\). Among other results, they considered the same problem for the case \(f(0,0)=g(0,0)=0\) in [5]. They proved under the assumptions \(f_{0}=g_{0}=0\) and \(f_{\infty}=g_{\infty}=\infty\) that problem (1.4) has at least one positive radial solution for all \(\lambda,\mu>0\), where
$$f_{0}=\lim_{(u,v)\rightarrow0}\frac{f(u,v)}{u+v},\qquad g_{0}=\lim_{(u,v)\rightarrow0}\frac{g(u,v)}{u+v}. $$
Lee [6] considered the multiplicity when the problem (1.4) has multiparameters for the first case and also when the problem has a perturbed boundary condition for the second case. Yang [7] proved the existence of positive solutions for Dirichlet boundary value problem of 2m-order nonlinear differential systems with n different parameters based on the method of upper and lower solutions and the fixed point index theory. Inspired by these references, we will study the existence, multiplicity, and nonexistence of convex solutions for systems of Monge-Ampère equations with multiparameters.

The paper is organized as follows. In Section 2, we introduce the upper and lower solutions method for systems and the fixed point index theory. In Section 3, we state and prove the existence, multiplicity, and nonexistence results.

2 Preliminaries

A nontrivial convex solution of (1.1) is negative on \([0,1)\). With the simple transformation \(v_{i}=-u_{i}\), (1.1) can be written as
$$ \left \{ \textstyle\begin{array}{l} ((-v'_{1}(t))^{N})'=\lambda_{1} Nt^{N-1}f^{1}(v_{1},v_{2},\ldots,v_{n}), \\ ((-v'_{2}(t))^{N})'=\lambda_{2} Nt^{N-1}f^{2}(v_{1},v_{2},\ldots,v_{n}), \\ \ldots, \\ ((-v'_{n}(t))^{N})'=\lambda_{n} Nt^{N-1}f^{n}(v_{1},v_{2},\ldots,v_{n}), \\ v'_{i}(0)=v_{i}(1)=0, \quad i=1,2,\ldots,n. \end{array}\displaystyle \right . $$
(2.1)
Therefore, throughout this paper we shall study the positive concave solution of (2.1).
Let \(\varphi(t)=t^{N}\), \(t\geq0\). For \(\mathbf{v}=(v_{1},v_{2},\ldots,v_{n})\), define the operators \(T_{\lambda_{i}}\) and \(T_{\lambda}\) as
$$\begin{aligned}& T_{\lambda_{i}}\mathbf{v}(t) =\int_{t}^{1} \varphi^{-1}\biggl(\int_{0}^{s} \lambda_{i}N\tau^{N-1} f^{i}\bigl(\mathbf{v}(\tau) \bigr)\, d\tau\biggr)\, ds,\quad i=1,2,\ldots,n, \\& T_{\lambda}\mathbf{v}(t)=\bigl(T_{\lambda_{1}}\mathbf{v}(t), T_{\lambda_{2}}\mathbf{v}(t),\ldots, T_{\lambda_{n}}\mathbf{v}(t)\bigr). \end{aligned}$$
Problem (2.1) is equivalent to
$$T_{\lambda}\mathbf{v}(t)=\mathbf{v}(t),\quad t\in[0,1]. $$
It implies that
$$v''_{i}(t)=-\frac{1}{N}\biggl(\int _{0}^{t}\lambda_{i}N\tau^{N-1} f^{i}\bigl(\mathbf{v}(\tau)\bigr)\, d\tau\biggr)^{\frac{1}{N}-1} \bigl( \lambda_{i}Nt^{N-1}f^{i}\bigl(\mathbf{v}(t)\bigr) \bigr)\leq0 $$
for \(t\in(0,1)\). Thus each component \(u_{i}\) must be convex.
Let X be the Banach space \(\underbrace{C[0,1]\times\cdots\times C[0,1]}_{n}\) with the norm \(\|\mathbf{v}\|=\sum_{i=1}^{n}\|v_{i}\|\) and \(\|v_{i}\|=\max_{t\in[0,1]}|v_{i}(t)|\), \(i=1,2,\ldots,n\). Let K be a cone in X defined as
$$K=\Biggl\{ \mathbf{v}=(v_{1},v_{2},\ldots,v_{n})\in X: v_{i}(t)\geq0, t\in[0,1]\mbox{ and }\min_{\frac{1}{4}\leq t\leq\frac{3}{4}}\sum _{i=1}^{n}v_{i}(t) \geq \frac{1}{4}\|\mathbf{v}\|\Biggr\} . $$
It follows similarly from [4, 5], we can get the following lemma.

Lemma 2.1

\(T_{\lambda}(K)\subset K\) and \(T_{\lambda}\) is completely continuous on X.

Consider the following boundary value problem:
$$ \left \{ \textstyle\begin{array}{l} ((-x'_{1}(t))^{N})'= F^{1}(t,x_{1},x_{2},\ldots,x_{n}), \\ ((-x'_{2}(t))^{N})'= F^{2}(t,x_{1},x_{2},\ldots,x_{n}), \\ \ldots, \\ ((-x'_{n}(t))^{N})'= F^{n}(t,x_{1},x_{2},\ldots,x_{n}), \\ x'_{i}(0)=x_{i}(1)=0, \quad i=1,2,\ldots,n, \end{array}\displaystyle \right . $$
(2.2)
where \(t\in(0,1)\), \(F^{i}:D\rightarrow\mathbb{R}\) is continuous with \(D\subset[0,1]\times\mathbb{R}^{n}\), \(i=1,2,\ldots,n\).

Definition 2.1

Let \(\alpha_{i}\in C^{2}([0,1],\mathbb{R})\), \(i=1,2,\ldots,n\), we say \((\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\) is a lower solution of (2.2) if \((t,\alpha_{1}(t),\alpha_{2}(t),\ldots,\alpha _{n}(t))\in D\) for all \(t\in(0,1)\) and
$$ \left \{ \textstyle\begin{array}{ll} ((-\alpha'_{i}(t))^{N})'\leq F^{i}(t,\alpha_{1},\alpha_{2},\ldots,\alpha_{n}), \\ \alpha'_{i}(0)=0,\qquad \alpha_{i}(1)\leq0,\quad i=1,2,\ldots,n. \end{array}\displaystyle \right . $$
(2.3)

Definition 2.2

Let \(\beta_{i}\in C^{2}([0,1],\mathbb{R})\), \(i=1,2,\ldots,n\), we say \((\beta_{1},\beta_{2},\ldots,\beta_{n})\) is an upper solution of (2.2) if \((t,\beta_{1}(t),\beta_{2}(t),\ldots,\beta_{n}(t))\in D\) for all \(t\in(0,1)\) and
$$ \left \{ \textstyle\begin{array}{l} ((-\beta'_{i}(t))^{N})'\geq F^{i}(t,\beta_{1},\beta_{2},\ldots,\beta_{n}), \\ \beta'_{i}(0)=0,\qquad \beta_{i}(1)\geq0, \quad i=1,2,\ldots,n. \end{array}\displaystyle \right . $$
(2.4)

Let \(D_{\alpha}^{\beta}=\{(t,x_{1},x_{2},\ldots,x_{n})\in[0,1]\times \mathbb{R}^{n}:\alpha_{i}(t)\leq x_{i}\leq\beta_{i}(t),i=1,2,\ldots,n\}\). We give a fundamental lemma of upper and lower solutions method.

Lemma 2.2

Let \((\alpha_{1}(t),\alpha_{2}(t),\ldots,\alpha_{n}(t))\) and \((\beta_{1}(t),\beta_{2}(t),\ldots,\beta_{n}(t))\) be lower and upper solutions of (2.2), respectively, such that
(h1): 

\((\alpha_{1}(t),\alpha_{2}(t),\ldots,\alpha_{n}(t)) \leq(\beta_{1}(t),\beta_{2}(t),\ldots,\beta_{n}(t))\), \(\forall t\in(0,1)\);

(h2): 

\(D_{\alpha}^{\beta}\subset D\);

(h3): 
\(F^{i}(t,x_{1},x_{2},\ldots,x_{n})\) is nondecreasing on \(\mathbb{R}^{n}\) for fixed \(t\in[0,1]\), that is,
$$F^{i}(t,x_{1},x_{2},\ldots,x_{n})\leq F^{i}(t,y_{1},y_{2},\ldots,y_{n}),\quad i=1,2,\ldots,n, $$
whenever \((x_{1},x_{2},\ldots,x_{n})\leq(y_{1},y_{2},\ldots,y_{n})\).
Then problem (2.2) has at least one solution \((x_{1}(t),x_{2}(t),\ldots,x_{n}(t))\) such that for all \(t\in(0,1)\),
$$\bigl(\alpha_{1}(t),\alpha_{2}(t),\ldots, \alpha_{n}(t)\bigr) \leq\bigl(x_{1}(t),x_{2}(t), \ldots,x_{n}(t)\bigr) \leq\bigl(\beta_{1}(t), \beta_{2}(t),\ldots,\beta_{n}(t)\bigr). $$

Proof

It is easy to verify that problem (2.2) is equivalent to the following system of integral equations:
$$x_{i}(t)=\int_{t}^{1} \varphi^{-1}\biggl(\int_{0}^{s} F^{i}\bigl(\tau,x_{1}(\tau),x_{2}(\tau), \ldots,x_{n}(\tau)\bigr)\, d\tau\biggr)\, ds,\quad i=1,2,\ldots,n, $$
where \(t\in[0,1]\). Define the function series \(\{x_{i}^{(k)}(t)\}_{k=0}^{\infty}\) by
$$ \left \{ \textstyle\begin{array}{l} x_{i}^{(0)}(t)=\alpha_{i}(t), \\ x_{i}^{(k+1)}(t)=\int_{t}^{1}\varphi^{-1} (\int_{0}^{s}F^{i}(\tau,x_{1}^{(k)}(\tau),x_{2}^{(k)}(\tau), \ldots,x_{n}^{(k)}(\tau))\, d\tau)\, ds, \\ \quad k=0,1,\ldots, i=1,2,\ldots,n. \end{array}\displaystyle \right . $$
(2.5)
The inequalities in (2.3) are equivalent to
$$\begin{aligned}& \alpha_{i}(t)\leq \int_{t}^{1} \varphi^{-1}\biggl(\int_{0}^{s}F^{i} \bigl(\tau,\alpha_{1}(\tau),\alpha _{2}(\tau), \ldots, \alpha_{n}(\tau)\bigr)\, d\tau\biggr)\, ds, \\& \quad t\in[0,1], i=1,2, \ldots,n. \end{aligned}$$
(2.6)
It follows from the above inequalities that
$$x_{i}^{(1)}(t)\geq x_{i}^{(0)}(t),\quad t \in[0,1], i=1,2,\ldots,n. $$
By induction, assume
$$x_{i}^{(k)}(t)\geq x_{i}^{(k-1)}(t),\quad t \in[0,1], i=1,2,\ldots,n. $$
Then for \(k\geq1\) and by (h3) and (2.5), we obtain
$$\begin{aligned} x_{i}^{(k+1)}(t)-x_{i}^{(k)}(t) =& \int _{t}^{1}\biggl(\varphi^{-1}\biggl(\int _{0}^{s}F^{i}\bigl(\tau,x_{1}^{(k)}( \tau ),x_{2}^{(k)}(\tau), \ldots,x_{n}^{(k)}( \tau)\bigr)\, d\tau\biggr) \\ &{} -\varphi^{-1}\biggl(\int_{0}^{s}F^{i} \bigl(\tau,x_{1}^{(k-1)} (\tau),x_{2}^{(k-1)}( \tau), \ldots,x_{n}^{(k-1)}(\tau)\bigr)\, d\tau\biggr)\biggr)\, ds \\ \geq&0,\quad t\in[0,1], i=1,2,\ldots,n, \end{aligned}$$
which implies that
$$\alpha_{i}(t)=x_{i}^{(0)}(t)\leq x_{i}^{(1)}(t)\leq\cdots\leq x_{i}^{(k)}(t) \leq\cdots. $$
Since \((\beta_{1}(t),\beta_{2}(t),\ldots,\beta_{n}(t))\) is an upper solution of (2.2), we have
$$\begin{aligned} \beta_{i}(t) \geq& \int_{t}^{1} \varphi^{-1}\biggl(\int_{0}^{s}F^{i} \bigl(\tau,\beta_{1}(\tau),\beta_{2}(\tau), \ldots, \beta_{n}(\tau)\bigr)\,d\tau\biggr)\,ds \\ \geq&\int_{t}^{1}\varphi^{-1}\biggl( \int_{0}^{s}F^{i}\bigl(\tau, \alpha_{1}(\tau ),\alpha_{2}(\tau), \ldots, \alpha_{n}(\tau)\bigr)\,d\tau\biggr)\,ds \\ \geq&\alpha_{i}(t)=x_{i}^{(0)}(t), \quad t \in[0,1], i=1,2,\ldots,n. \end{aligned}$$
Assume \(\beta_{i}(t)\geq x_{i}^{(k)}(t)\), \(t\in[0,1]\), \(i=1,2,\ldots,n\), \(k\in\mathbb{N}\), then by the definition of \(x_{i}^{(k)}(t)\) and the above inequalities, we obtain
$$\begin{aligned} \beta_{i}(t)-x_{i}^{(k+1)}(t) \geq& \int _{t}^{1}\biggl(\varphi^{-1}\biggl(\int _{0}^{s}F^{i}\bigl(\tau, \beta_{1}(\tau),\beta _{2}(\tau), \ldots,\beta_{n}( \tau)\bigr)\,d\tau\biggr) \\ &{} -\varphi^{-1}\biggl(\int_{0}^{s}F^{i} \bigl(\tau,x_{1}^{(k)}(\tau),x_{2}^{(k)}( \tau), \ldots,x_{n}^{(k)}(\tau)\bigr)\,d\tau\biggr)\biggr)\,ds \\ \geq&0, \quad t\in[0,1], i=1,2,\ldots,n, \end{aligned}$$
which implies that \(\{x_{i}^{(k)}(t)\}_{k=0}^{\infty}\) is bounded above by \(\beta_{i}(t)\), hence the limit \(x_{i}^{*}(t)= \lim_{k\rightarrow\infty}x_{i}^{(k)}(t)\) exists and satisfies
$$\alpha_{i}(t)\leq x_{i}^{*}(t)\leq\beta_{i}(t),\quad t\in[0,1], i=1,2,\ldots,n. $$
Moreover, by taking the limits in both sides of (2.5), we obtain
$$x_{i}^{*}(t)=\int_{t}^{1} \varphi^{-1}\biggl(\int_{0}^{s}F^{i} \bigl(\tau,x_{1}^{*}(\tau ),x_{2}^{*}(\tau), \ldots,x_{n}^{*}( \tau)\bigr)\,d\tau\biggr)\,ds, \quad t\in[0,1], i=1,2,\ldots,n, $$
which implies that \((x_{1}^{*}(t),x_{2}^{*}(t),\ldots,x_{n}^{*}(t))\) is a solution of (2.2). □

The following well-known results of the fixed point index are crucial in our arguments.

Lemma 2.3

[8]

Let X be a Banach space, K a cone in X and Ω bounded open in X. Let \(0\in\Omega\) and \(T:K\cap\bar{\Omega}\rightarrow K\) be condensing. Suppose that \(Tx\neq\lambda x\) for all \(x\in K\cap\partial\Omega\) and all \(\lambda\geq1\). Then
$$i(T,K\cap\Omega,K)=1. $$

Lemma 2.4

[8]

Let X be a Banach space and K a cone in X. For \(r>0\), define \(K_{r}=\{x\in K: \|x\|< r\}\). Assume that \(T:\bar{K}_{r}\rightarrow K\) is a compact map such that \(Tx\neq x\) for \(x\in\partial K_{r}\). If \(\|x\|\leq\|Tx\|\) for all \(x\in\partial K_{r}\), then
$$i(T,K_{r},K)=0. $$

3 Main results

Theorem 3.1

Assume for all \(i=1,2,\ldots,n\),
(H1): 

\((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in \mathbb{R}_{+}^{n}\setminus\{(0,0,\ldots,0)\}\);

(H2): 
\(f^{i}\in C(\mathbb{R}^{n}_{+},\mathbb{R}_{+})\) is nondecreasing on \(\mathbb{R}^{n}_{+}\), that is,
$$f^{i}(u_{1},u_{2},\ldots,u_{n})\leq f^{i}(v_{1},v_{2},\ldots,v_{n}), \quad \textit{if } (u_{1},u_{2},\ldots,u_{n})\leq (v_{1},v_{2},\ldots,v_{n}) $$
and there exists at least one \(j\in\{1,2,\ldots,n\}\), such that \(f^{j}(0,0,\ldots,0)>0\);
(H3): 
there exist constants \(m_{i}>0\) such that
$$f^{i}(\mathbf{v})\geq m_{i}\varphi\Biggl(\sum _{i=1}^{n}v_{i}\Biggr); $$
(H4): 
$$\lim_{\|\mathbf{v}\|\rightarrow\infty} \frac{f^{i}(\mathbf{v})}{\varphi(\sum_{i=1}^{n}v_{i})}=\infty. $$
Then there exists a bounded and continuous surface Γ separating \(\mathbb{R}_{+}^{n}\setminus\{(0,0,\ldots,0)\}\) into two disjoint subsets \(\Omega_{1}\) and \(\Omega_{2}\) such that problem (1.1) has at least two convex solutions for \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\Omega_{1}\), at least one convex solution for \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\Gamma\) and no solution for \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\Omega_{2}\). Moreover, let \(\Gamma_{+}\cup\Gamma_{0}\) be the parametric representation of Γ, where
$$\Gamma_{+}\mbox{: } \lambda_{n}=\lambda_{n}(\lambda_{1}, \lambda_{2},\ldots, \lambda_{n-1})>0,\qquad \Gamma_{0}\mbox{: } \lambda_{n}=\lambda_{n}( \lambda_{1},\lambda_{2},\ldots,\lambda_{n-1})=0. $$
Then on \(\Gamma_{+}\), the function \(\lambda_{n}=\lambda_{n}(\lambda_{1},\lambda_{2},\ldots, \lambda_{n-1})\) is continuous and nonincreasing on \(\mathbb{R}_{+}^{n-1}\setminus\{(0,0,\ldots,0)\}\), that is, if \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n-1}) \leq(\lambda'_{1},\lambda'_{2},\ldots,\lambda'_{n-1})\), then
$$\lambda_{n}(\lambda_{1},\lambda_{2},\ldots, \lambda_{n-1}) \geq\lambda_{n}\bigl(\lambda'_{1}, \lambda'_{2},\ldots,\lambda'_{n-1} \bigr) $$
and on \(\Gamma_{0}\), the function \(\lambda_{n-1}=\lambda_{n-1}(\lambda_{1},\lambda_{2},\ldots,\lambda_{n-2})\) is continuous and nonincreasing on \(\mathbb{R}_{+}^{n-2}\setminus\{(0,0,\ldots,0)\}\).

We need some lemmas to prove Theorem 3.1. The following lemma is a prior estimate for solutions of problem (2.1).

Lemma 3.1

Assume (H1)-(H4) hold. Let Σ be a compact subset of \(\mathbb{R}_{+}^{n}\setminus \{(0,0,\ldots,0)\}\). Then there exists a constant \(C_{\Sigma}>0\) such that for all \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\Sigma\) and all possible positive solutions \(\mathbf{v}=(v_{1},v_{2},\ldots,v_{n})\) of (2.1) at \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\), one has
$$\|\mathbf{v}\|\leq C_{\Sigma}. $$

Proof

Suppose by contradiction that there exists a sequence \(\{(v_{1}^{(m)},v_{2}^{(m)},\ldots,v_{n}^{(m)})\}_{m=1}^{\infty}\) of positive solutions of (2.1) at \((\lambda_{1}^{(m)},\lambda_{2}^{(m)},\ldots,\lambda_{n}^{(m)})\) such that \((\lambda_{1}^{(m)},\lambda_{2}^{(m)},\ldots,\lambda_{n}^{(m)})\in\Sigma\) for all m and
$$\bigl\Vert \bigl(v_{1}^{(m)},v_{2}^{(m)}, \ldots,v_{n}^{(m)}\bigr)\bigr\Vert \rightarrow\infty. $$
Then \(\mathbf{v}^{(m)}=(v_{1}^{(m)},v_{2}^{(m)},\ldots,v_{n}^{(m)})\in K\) and thus
$$ \min_{\frac{1}{4}\leq t\leq\frac{3}{4}}\Biggl(\sum _{i=1}^{n}v_{i}^{(m)}(t)\Biggr) \geq\frac{1}{4}\bigl\Vert \mathbf{v}^{(m)}\bigr\Vert . $$
(3.1)
Since Σ is compact, the sequence \(\{(\lambda_{1}^{(m)},\lambda_{2}^{(m)},\ldots,\lambda_{n}^{(m)})\} _{m=1}^{\infty}\) has a convergent subsequence which we denote without loss of generality still by \(\{(\lambda_{1}^{(m)},\lambda_{2}^{(m)}, \ldots,\lambda_{n}^{(m)})\} _{m=1}^{\infty}\) such that
$$\lim_{m\rightarrow\infty}\lambda_{i}^{(m)}= \lambda_{i}^{*},\quad i=1,2,\ldots,n $$
and at least one \(\lambda_{j}^{*}>0\), hence for m sufficiently large, we have \(\lambda_{j}^{(m)}\geq\lambda_{j}^{*}/2>0\). Then from (H4), we may choose \(R_{j}>0\) such that
$$ f^{j}(\mathbf{v})\geq L_{1}\varphi\Biggl(\sum _{i=1}^{n}v_{i}\Biggr)\quad \mbox{for all }\sum_{i=1}^{n}v_{i} \geq R_{j}, $$
(3.2)
where \(L_{1}\) satisfies
$$\frac{1}{4}\varphi^{-1}\biggl(\frac{\lambda_{j}^{*}}{2}L_{1} \biggr) \int_{\frac{1}{4}}^{\frac{3}{4}} \varphi^{-1}\biggl( \int_{\frac{1}{4}}^{s}N\tau^{N-1}\,d\tau \biggr)\,ds>1. $$
Combining (3.1) with (3.2), we get
$$\begin{aligned} \bigl\Vert \mathbf{v}^{(m)}\bigr\Vert =& \bigl\Vert T_{\lambda^{(m)}}\mathbf{v}^{(m)}\bigr\Vert \\ \geq&\max_{t\in[0,1]}\bigl\vert T_{\lambda_{j}^{(m)}} \mathbf{v}^{(m)}(t)\bigr\vert \\ =&\int_{0}^{1}\varphi^{-1}\biggl(\int _{0}^{s}\lambda_{j}^{(m)}N \tau^{N-1} f^{j}\bigl(\mathbf{v}^{(m)}(\tau)\bigr)\,d \tau\biggr)\,ds \\ \geq&\int_{\frac{1}{4}}^{\frac{3}{4}} \varphi^{-1}\biggl( \int_{\frac{1}{4}}^{s}\frac{\lambda_{j}^{*}}{2}N \tau^{N-1} f^{j}\bigl(\mathbf{v}^{(m)}(\tau)\bigr)\,d \tau\biggr)\,ds \\ \geq&\int_{\frac{1}{4}}^{\frac{3}{4}} \varphi^{-1}\Biggl( \int_{\frac{1}{4}}^{s}\frac{\lambda_{j}^{*}}{2}N \tau^{N-1} L_{1}\varphi\Biggl(\sum _{i=1}^{n}v_{i}^{(m)}(\tau)\Biggr) \,d\tau\Biggr)\,ds \\ \geq&\int_{\frac{1}{4}}^{\frac{3}{4}} \varphi^{-1}\biggl( \int_{\frac{1}{4}}^{s}\frac{\lambda_{j}^{*}}{2}N \tau^{N-1} L_{1}\varphi\biggl(\frac{1}{4}\bigl\Vert \mathbf{v}^{(m)}\bigr\Vert \biggr)\,d\tau\biggr)\,ds \\ =&\frac{1}{4}\varphi^{-1}\biggl(\frac{\lambda_{j}^{*}L_{1}}{2}\biggr)\int _{\frac {1}{4}}^{\frac{3}{4}} \varphi^{-1}\biggl(\int _{\frac{1}{4}}^{s}N\tau^{N-1}\,d\tau\biggr) \,ds\cdot\bigl\Vert \mathbf {v}^{(m)}\bigr\Vert \\ >&\bigl\Vert \mathbf{v}^{(m)}\bigr\Vert \end{aligned}$$
for m sufficiently large. This is a contradiction. □

Lemma 3.2

Assume (H1)-(H2) hold. If (2.1) has a positive solution at \((\bar{\lambda}_{1},\bar{\lambda}_{2},\ldots,\bar{\lambda}_{n})\). Then (2.1) also has a positive solution at \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\) for all \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n}) \leq(\bar{\lambda}_{1},\bar{\lambda}_{2},\ldots,\bar{\lambda}_{n})\).

Proof

Let \((\bar{v}_{1},\bar{v}_{2},\ldots,\bar{v}_{n})\) be a positive solution of (2.1) at \((\bar{\lambda}_{1},\bar{\lambda}_{2},\ldots,\bar{\lambda}_{n})\) and let \((\lambda_{1},\lambda_{2}, \ldots,\lambda_{n})\in \mathbb{R}_{+}^{n}\setminus\{(0,0,\ldots,0)\}\) with \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n}) \leq(\bar{\lambda}_{1},\bar{\lambda}_{2},\ldots,\bar{\lambda}_{n})\). Then \((\bar{v}_{1},\bar{v}_{2},\ldots,\bar{v}_{n})\) is an upper solution and \((0,0,\ldots,0)\) is a lower solution of (2.1) at \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\), respectively. It is easy to see that \((\bar{v}_{1},\bar{v}_{2},\ldots,\bar{v}_{n})\neq(0,0,\ldots,0)\) and \((\bar{v}_{1},\bar{v}_{2},\ldots,\bar{v}_{n})\geq(0,0,\ldots,0)\). By (H2), we obtain \((0,0,\ldots,0)\) is not a solution of (2.1) at \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\), Lemma 2.2 implies that (2.1) has a positive solution at \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\). □

Lemma 3.3

Assume (H1)-(H3) hold. Then there exists \((\lambda_{1}^{*},\lambda_{2}^{*},\ldots,\lambda_{n}^{*})>(0,0,\ldots,0)\) such that (2.1) has a positive solution for all \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n}) \leq(\lambda_{1}^{*},\lambda_{2}^{*},\ldots,\lambda_{n}^{*})\).

Proof

Let \(\beta_{i}(t)=\int_{t}^{1}\varphi^{-1}(\int_{0}^{s}N\tau^{N-1}\,d\tau)\,ds =\frac{1}{2}(1-t^{2})\), \(t\in[0,1]\), \(i=1,2,\ldots,n\), be the unique solution of
$$\left \{ \textstyle\begin{array}{l} ((-v'_{i}(t))^{N})'=Nt^{N-1}, \quad i=1,2,\ldots,n, \\ v'_{i}(0)=v_{i}(1)=0. \end{array}\displaystyle \right . $$
Let \(M_{i}=\max_{t\in[0,1]}f^{i}(\beta_{1}(t),\beta_{2}(t),\ldots ,\beta_{n}(t))\), then by (H3), \(M_{i}>0\), \(i=1,2,\ldots,n\), and at \((\lambda_{1}^{*},\lambda_{2}^{*},\ldots,\lambda_{n}^{*}) =(\frac{1}{M_{1}},\frac{1}{M_{2}},\ldots,\frac{1}{M_{n}})\), we get
$$\begin{aligned}& \bigl(\bigl(-\beta'_{i}(t)\bigr)^{N} \bigr)'-\lambda_{i}^{*}Nt^{N-1} f^{i} \bigl(\beta_{1}(t),\beta_{2}(t),\ldots,\beta_{n}(t) \bigr) \\& \quad = Nt^{N-1}-\lambda_{i}^{*}Nt^{N-1} f^{i}\bigl(\beta_{1}(t),\beta_{2}(t),\ldots, \beta_{n}(t)\bigr) \\& \quad = -Nt^{N-1}\bigl[\lambda_{i}^{*}f^{i}\bigl( \beta_{1}(t),\beta_{2}(t),\ldots,\beta _{n}(t) \bigr)-1\bigr] \\& \quad \geq 0,\quad t\in(0,1), i=1,2,\ldots,n. \end{aligned}$$
This shows that \((\beta_{1}(t),\beta_{2}(t),\ldots,\beta_{n}(t))\) is an upper solution of (2.1) at \((\lambda_{1}^{*},\lambda_{2}^{*},\ldots,\lambda_{n}^{*})\). On the other hand, \((0,0,\ldots,0)\) is obviously a lower solution and \((0,0,\ldots,0)\leq(\beta_{1}(t),\beta_{2}(t),\ldots, \beta_{n}(t))\). Thus by Lemma 2.2, (2.1) has a positive solution at \((\lambda_{1}^{*},\lambda_{2}^{*},\ldots,\lambda_{n}^{*})\), and by Lemma 3.2 we complete the proof. □
Define
$$\begin{aligned} S =&\bigl\{ (\lambda_{1},\lambda_{2},\ldots, \lambda_{n}) \in\mathbb{R}_{+}^{n}\setminus\bigl\{ (0,0,\ldots,0) \bigr\} : (2.1)\ \mbox{has a positive solution at} \\ &(\lambda_{1}, \lambda_{2},\ldots,\lambda_{n})\bigr\} . \end{aligned}$$
Then by Lemma 3.3, \(S\neq\emptyset\), and it is easy to see that \((S,\leq)\) is a partially ordered set.

Lemma 3.4

Assume (H1)-(H3) hold. Then \((S,\leq)\) is bounded above.

Proof

Let \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in S\) and \(\mathbf{v}=(v_{1},v_{2},\ldots,v_{n})\) be a positive solution of (2.1) at \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\). Then by (H3), we get
$$\begin{aligned} \|v_{i}\| =&\int_{0}^{1} \varphi^{-1}\biggl(\int_{0}^{s} \lambda_{i}N\tau^{N-1} f^{i}\bigl(\mathbf{v}(\tau) \bigr)\,d\tau\biggr)\,ds \\ \geq&\int_{\frac{1}{4}}^{\frac{3}{4}} \varphi^{-1}\biggl( \int_{\frac{1}{4}}^{s}\lambda_{i}N \tau^{N-1} f^{i}\bigl(\mathbf{v}(\tau)\bigr)\,d\tau\biggr)\,ds \\ \geq&\int_{\frac{1}{4}}^{\frac{3}{4}} \varphi^{-1}\Biggl( \int_{\frac{1}{4}}^{s}\lambda_{i}N \tau^{N-1} m_{i}\varphi\Biggl(\sum _{i=1}^{n}v_{i}(\tau)\Biggr)\,d\tau\Biggr) \,ds \\ \geq&\int_{\frac{1}{4}}^{\frac{3}{4}} \varphi^{-1}\biggl( \int_{\frac{1}{4}}^{s}\lambda_{i}N \tau^{N-1} m_{i}\varphi\biggl(\frac{1}{4}\|\mathbf{v}\| \biggr)\,d\tau\biggr)\,ds \\ =&\frac{1}{4}\varphi^{-1}(\lambda_{i}m_{i}) \int_{\frac{1}{4}}^{\frac{3}{4}} \varphi^{-1}\biggl(\int _{\frac{1}{4}}^{s}N\tau^{N-1}\,d\tau\biggr)\,ds \cdot\|\mathbf{v}\|. \end{aligned}$$
Thus
$$\lambda_{i}\leq\frac{1}{m_{i}}\varphi \biggl(\frac{4\|v_{i}\|}{\int_{\frac{1}{4}}^{\frac{3}{4}} \varphi^{-1}(\int_{\frac{1}{4}}^{s}N\tau^{N-1}\,d\tau)\,ds \|\mathbf{v}\|} \biggr)\leq\frac{d}{m_{i}},\quad i=1,2,\ldots,n, $$
where
$$d=\varphi \biggl(\frac{4}{\int_{\frac{1}{4}}^{\frac{3}{4}} \varphi^{-1}(\int_{\frac{1}{4}}^{s}N\tau^{N-1}\,d\tau)\,ds } \biggr). $$
Therefore S is bounded above by \((\bar{\lambda}_{1},\bar{\lambda}_{2},\ldots,\bar{\lambda}_{n}) =(\frac{d}{m_{1}},\frac{d}{m_{2}},\ldots,\frac{d}{m_{n}})\). □

Similar to Lemmas 2.6-2.8 in [7], we can prove the following lemmas.

Lemma 3.5

Assume (H1)-(H3) hold. Then every chain in S has a unique supremum in S.

Lemma 3.6

Assume (H1)-(H3) hold. Then there exists \(\tilde{\lambda}_{i}\in[\lambda_{i}^{*},\bar{\lambda}_{i}]\) such that (2.1) has a positive solution at \((0,\ldots,0,\lambda_{i},0,\ldots,0)\) for all \(0<\lambda_{i}\leq\tilde{\lambda}_{i}\) and no solution at \((0,\ldots,0,\lambda_{i},0, \ldots,0)\) for all \(\lambda_{i}>\tilde{\lambda}_{i}\).

Lemma 3.7

Assume (H1)-(H3) hold. Then there exists a continuous surface Γ separating \(\mathbb{R}_{+}^{n}\setminus\{(0,0,\ldots,0)\}\) into two disjoint subsets \(\Sigma_{1}\) and \(\Sigma_{2}\) such that \(\Sigma_{1}\) is bounded and \(\Sigma_{2}\) is unbounded, (2.1) has at least one solution for \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\Sigma_{1}\cup\Gamma\) and no solution for \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\Sigma_{2}\). The function \(\lambda_{n}=\lambda_{n}(\lambda_{1},\lambda_{2},\ldots,\lambda_{n-1})\) is nonincreasing, that is, if
$$(\lambda_{1},\lambda_{2},\ldots,\lambda_{n-1})\leq \bigl(\lambda'_{1},\lambda'_{2}, \ldots,\lambda'_{n-1}\bigr)\leq (\tilde{ \lambda}_{1},\tilde{\lambda}_{2},\ldots,\tilde{ \lambda}_{n-1}), $$
then
$$\lambda_{n}(\lambda_{1},\lambda_{2},\ldots, \lambda_{n-1})\geq \lambda_{n}\bigl(\lambda'_{1}, \lambda'_{2},\ldots,\lambda'_{n-1} \bigr). $$
Moreover, if \(\lambda_{n}=0\), then the function \(\lambda_{n-1}(\lambda_{1},\lambda_{2},\ldots,\lambda_{n-2})\) is nonincreasing.

Lemma 3.8

Assume (H1)-(H3) hold and let \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\Omega_{1}\). Then there exists \(\varepsilon_{0}>0\) such that \((v^{*}_{1}+\varepsilon,v^{*}_{2}+\varepsilon,\ldots,v^{*}_{n}+\varepsilon)\) is an upper solution of (2.1) at \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\) for all \(\varepsilon\in(0,\varepsilon_{0}]\), where \((v_{1}^{*},v_{2}^{*},\ldots,v_{n}^{*})\) is the positive solution of (2.1) corresponding to some \((\lambda_{1}^{*},\lambda_{2}^{*},\ldots,\lambda_{n}^{*})\in\Gamma\) satisfying
$$(\lambda_{1},\lambda_{2},\ldots,\lambda_{n}) < \bigl(\lambda_{1}^{*},\lambda_{2}^{*},\ldots,\lambda_{n}^{*} \bigr). $$

Proof

From (H3), there exists a constant \(M>0\) such that
$$0< M\leq\min_{t\in[0,1]}f^{i}\bigl(v^{*}_{1}(t),v^{*}_{2}(t), \ldots,v^{*}_{n}(t)\bigr),\quad i=1,2,\ldots,n. $$
Then by the uniform continuity of \(f^{i}\) on a compact set, there exists \(\varepsilon_{0}>0\) such that
$$\bigl\vert f^{i}\bigl(v^{*}_{1}+\varepsilon,v^{*}_{2}+ \varepsilon,\dots,v^{*}_{n}+\varepsilon\bigr) -f^{i} \bigl(v^{*}_{1},v^{*}_{2},\ldots,v^{*}_{n}\bigr)\bigr\vert < \frac{M(\lambda_{i}^{*}-\lambda _{i})}{\lambda_{i}} $$
for all \(t\in[0,1]\), \(i=1,2,\ldots,n\) and \(0<\varepsilon\leq\varepsilon_{0}\). Let \(\tilde{v}^{*}_{i}(t)=v^{*}_{i}(t)+\varepsilon\), \(i=1,2,\ldots,n\), then \(\tilde{v}_{i}^{*\prime}(0)=0\), \(\tilde{v}_{i}^{*}(1)>0\) and
$$\begin{aligned}& \bigl(\bigl(-\tilde{v}^{*\prime}_{i}(t)\bigr)^{N} \bigr)'-\lambda_{i} Nt^{N-1} f^{i} \bigl(\tilde{v}^{*}_{1}(t),\tilde{v}^{*}_{2}(t),\ldots, \tilde{v}^{*}_{n}(t)\bigr) \\& \quad = \lambda_{i}^{*}Nt^{N-1}f^{i} \bigl(v^{*}_{1}(t),v^{*}_{2}(t),\ldots,v^{*}_{n}(t)\bigr) -\lambda_{i} Nt^{N-1} f^{i}\bigl( \tilde{v}^{*}_{1}(t),\tilde{v}^{*}_{2}(t),\ldots, \tilde{v}^{*}_{n}(t)\bigr) \\& \quad = \lambda_{i} Nt^{N-1}\bigl[f^{i} \bigl(v^{*}_{1}(t),v^{*}_{2}(t),\ldots,v^{*}_{n}(t)\bigr) -f^{i}\bigl(\tilde{v}^{*}_{1}(t),\tilde{v}^{*}_{2}(t), \ldots,\tilde{v}^{*}_{n}(t)\bigr)\bigr] \\& \qquad {}+\bigl(\lambda_{i}^{*}-\lambda_{i} \bigr)Nt^{N-1}f^{i}\bigl(v^{*}_{1}(t),v^{*}_{2}(t), \ldots ,v^{*}_{n}(t)\bigr) \\& \quad > -Nt^{N-1}M\bigl(\lambda_{i}^{*}-\lambda_{i} \bigr)+ \bigl(\lambda_{i}^{*}-\lambda_{i}\bigr)Nt^{N-1}f^{i} \bigl(v^{*}_{1}(t),v^{*}_{2}(t),\ldots ,v^{*}_{n}(t)\bigr) \\& \quad = \bigl(\lambda_{i}^{*}-\lambda_{i}\bigr)Nt^{N-1} \bigl[f^{i}\bigl(v^{*}_{1}(t),v^{*}_{2}(t),\ldots ,v^{*}_{n}(t)\bigr)-M\bigr] \\& \quad \geq 0, \quad i=1,2,\ldots,n \end{aligned}$$
for all \(t\in[0,1]\), \(i=1,2,\ldots,n\). Hence \((\tilde{v}^{*}_{1}(t),\tilde{v}^{*}_{2}(t),\ldots,\tilde{v}^{*}_{n}(t))\) is an upper solution of (2.1) at \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\) for all \(\varepsilon\in(0,\varepsilon_{0}]\). □

Proof of Theorem 3.1

Because we have proved the above lemmas, we only need to prove the existence of the second positive solution of (2.1) for \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\Omega_{1}\). Let \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\Omega_{1}\). Denote
$$\tilde{v}^{*}_{i}(t)=v^{*}_{i}(t)+\varepsilon, \quad i=1,2, \ldots,n, t\in{[0,1]}, $$
where ε is given in Lemma 3.8. Define the set
$$D=\bigl\{ \mathbf{v}=(v_{1},v_{2},\ldots,v_{n})\in X:-\varepsilon< v_{i}(t)< \tilde{v}^{*}_{i}(t), t\in[0,1], i=1,2, \ldots,n\bigr\} . $$
Then D is bounded open in X and \(0\in D\). The map \(T_{\lambda}:K\cap\bar{D}\rightarrow K\) is condensing, since it is completely continuous. Let \(\mathbf{v}\in K\cap \partial D\), then there exists \(t_{0}\in[0,1]\) such that \(v_{i}(t_{0})=\tilde{v}^{*}_{i}(t_{0})\) for some \(i\in\{1,2,\ldots,n\}\). Let \(\tilde{\mathbf{v}}^{*}=(\tilde{v}^{*}_{1},\tilde{v}^{*}_{2}, \ldots,\tilde{v}^{*}_{n})\), then by (H2) and Lemma 3.8, we have
$$\begin{aligned} \begin{aligned} T_{\lambda_{i}}\mathbf{v}(t_{0})&= \int_{t_{0}}^{1} \varphi^{-1}\biggl(\int_{0}^{s} \lambda_{i} N\tau^{N-1} f^{i}\bigl(\mathbf{v}(\tau) \bigr)\,d\tau\biggr)\,ds \\ &\leq\int_{t_{0}}^{1}\varphi^{-1}\biggl( \int_{0}^{s}\lambda_{i}^{*} N \tau^{N-1} f^{i}\bigl(\tilde{\mathbf{v}}^{*}(\tau)\bigr)\,d\tau \biggr)\,ds \\ &< \tilde{v}^{*}_{i}(t_{0})=v_{i}(t_{0}) \leq\theta v_{i}(t_{0}) \end{aligned} \end{aligned}$$
for all \(\theta\geq1\). Thus \(\mathbf{v}\neq\theta\mathbf{v}\) for all \(\mathbf{v}\in K\cap\partial D\) and all \(\theta\geq1\). Lemma 2.3 implies that
$$i(T_{\lambda},K\cap D,K)=1. $$
For some \(\lambda_{i}>0\), it follows from (H4) that there exists \(R_{i}>0\) such that
$$ f^{i}(\mathbf{v})\geq L_{2}\varphi\Biggl(\sum _{i=1}^{n}v_{i}\Biggr),\quad \forall\sum_{i=1}^{n}v_{i}\geq R_{i}, $$
(3.3)
where \(L_{2}\) satisfies
$$\frac{1}{4}\varphi^{-1}(\lambda_{i} L_{2}) \int_{\frac{1}{4}}^{\frac{3}{4}}\varphi^{-1} \biggl(\int _{\frac{1}{4}}^{s}N\tau^{N-1}\,d\tau\biggr)\,ds>1. $$
Let \(R^{*}=\max\{C_{\Sigma},4R_{i},\|\tilde{\mathbf{v}}^{*}\|\}\), where \(C_{\Sigma}\) is given in Lemma 3.1 with Σ a compact set in \(\mathbb{R}_{+}^{n}\setminus\{(0,0,\ldots,0)\}\) containing \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\). Let
$$K_{R^{*}}=\bigl\{ \mathbf{v}\in K:\|\mathbf{v}\|< R^{*}\bigr\} , $$
then by Lemma 3.1,
$$T_{\lambda}\mathbf{v}\neq\mathbf{v},\quad \forall\mathbf{v}\in\partial K_{R^{*}}. $$
Furthermore, if \(\mathbf{v}\in\partial K_{R^{*}}\), then
$$\min_{t\in[\frac{1}{4},\frac{3}{4}]}\Biggl(\sum_{i=1}^{n}v_{i}(t) \Biggr) \geq\frac{1}{4}\|\mathbf{v}\|\geq R_{i}. $$
Thus by (3.3),
$$f^{i}\bigl(\mathbf{v}(t)\bigr)\geq L_{2}\varphi\Biggl(\sum _{i=1}^{n}v_{i}(t)\Biggr),\quad \forall t\in\biggl[\frac{1}{4},\frac{3}{4}\biggr]. $$
Therefore
$$\begin{aligned} \bigl\Vert T_{\lambda}\mathbf{v}(t)\bigr\Vert \geq& \max _{t\in[0,1]}\bigl\vert T_{\lambda_{i}}\mathbf{v}(t)\bigr\vert \\ \geq&\int_{\frac{1}{4}}^{\frac{3}{4}}\varphi^{-1} \biggl( \int_{\frac{1}{4}}^{s}\lambda_{i} N \tau^{N-1}f^{i}\bigl(\mathbf{v}(\tau)\bigr)\,d\tau\biggr)\,ds \\ \geq&\int_{\frac{1}{4}}^{\frac{3}{4}}\varphi^{-1} \Biggl( \int_{\frac{1}{4}}^{s}\lambda_{i} N \tau^{N-1}L_{2}\varphi\Biggl(\sum_{i=1}^{n}v_{i}( \tau)\Biggr)\,d\tau\Biggr)\,ds \\ \geq&\int_{\frac{1}{4}}^{\frac{3}{4}}\varphi^{-1} \biggl( \int_{\frac{1}{4}}^{s}\lambda_{i} N \tau^{N-1}L_{2}\varphi\biggl(\frac{1}{4}\|\mathbf{v}\| \biggr)\,d\tau\biggr)\,ds \\ =&\frac{1}{4}\varphi^{-1}(\lambda_{i} L_{2}) \int_{\frac{1}{4}}^{\frac{3}{4}} \varphi^{-1}\biggl(\int_{\frac{1}{4}}^{s} N \tau^{N-1}\,d\tau\biggr)\,ds\cdot\|\mathbf{v}\| \\ >&\|\mathbf{v}\|. \end{aligned}$$
It follows from Lemma 2.4 that
$$i(T_{\lambda},K_{R^{*}},K)=0. $$
Consequently by the additivity of the fixed point index,
$$\begin{aligned} 0 =&i(T_{\lambda},K_{R^{*}},K) \\ =&i(T_{\lambda},K\cap D,K)+i(T_{\lambda},K_{R^{*}}\setminus \overline{K\cap D},K) \\ =&1+i(T_{\lambda},K_{R^{*}}\setminus\overline{K\cap D},K), \end{aligned}$$
which implies
$$i(T_{\lambda},K_{R^{*}}\setminus\overline{K\cap D},K)=-1. $$
Thus \(T_{\lambda}\) has at least one fixed point in \(K\cap D\) and another in \(K_{R^{*}}\setminus\overline{K\cap D}\). This implies that (2.1) has at least two positive solutions at \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\Omega_{1}\). Thus (1.1) has at least two negative solutions at \((\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\Omega_{1}\). □

Notes

Declarations

Acknowledgements

The authors are highly grateful for the referees’ careful reading and comments on this paper, which led to the improvement of the original manuscript.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, College of Science, Hohai University

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