Coupled elliptic systems depending on the gradient with nonlocal BCs in exterior domains

Abstract

We study the existence and multiplicity of positive radial solutions for a coupled elliptic system in exterior domains where the nonlinearities depend on the gradients and the boundary conditions are nonlocal. We use a new cone to establish the existence of solutions by means of fixed point index theory.

1 Introduction

In this paper, we study the existence and multiplicity of positive radial solutions for the coupled elliptic system

$$ \textstyle\begin{cases} -\Delta u= h_{1}( \vert x \vert ) f_{1}(u,v, \vert \nabla u \vert , \vert \nabla v \vert ),& \vert x \vert \in [r_{0},+\infty ), \\ -\Delta v= h_{2}( \vert x \vert ) f_{2}(u,v, \vert \nabla u \vert , \vert \nabla v \vert ),& \vert x \vert \in [r_{0},+\infty ), \\ \lim_{ \vert x \vert \to \infty }u(x)=\alpha _{1}[u],\qquad c_{1}u+\tilde{d_{1}}\frac{\partial u}{\partial r}= \beta _{1}[u] & \text{for } \vert x \vert =r_{0}, \\ \lim_{ \vert x \vert \to \infty }v(x)=\alpha _{2}[v],\qquad c_{2}v+\tilde{d_{2}}\frac{\partial v}{\partial r}= \beta _{2}[v] & \text{for } \vert x \vert =r_{0}, \end{cases} $$

(1.1)

where $\alpha _{i}[\cdot ]$, and $\beta _{i}[\cdot ]$ are bounded linear functionals, $h_{i}$ and $f_{i}$ are nonnegative functions, $c_{i}\geq 0$, $\tilde{d}_{i} \leq 0$, $r_{0}>0$, and $\frac{\partial }{\partial r}$ denotes (as in [23]) differentiation in the radial direction $r=|x|$. The functions $f_{i}$ are continuous, and every singularity is captured by the term $h_{i}\in L^{1}$, which may have pointwise singularities.

Many papers study the existence of radial solutions of elliptic equations in the exterior part of a ball. A variety of methods have been used, for instance, when the boundary conditions (BCs) are homogeneous; a priori estimates were utilized by Castro et al. [7], sub and super solutions were used by Djedali and Orpel [16] and Sankar et al. [39]; variational methods were used by Orpel [37], and topological methods where employed by Abebe and coauthors [1], do Ó et al. [17], Hai and Shivaji [29], Han and Wang [30], Lee [35], Orpel [38], and Stanczy [40].

In particular, recently, Hai and Shivaji [29] proved the existence and multiplicity of positive radial solutions for the superlinear elliptic system

$$ \textstyle\begin{cases} -\Delta u=\lambda h_{1}( \vert x \vert ) f_{1}(v), & \vert x \vert \in [r_{0},+\infty ), \\ -\Delta v=\lambda h_{2}( \vert x \vert ) f_{2}(u), & \vert x \vert \in [r_{0},+\infty ), \\ \lim_{ \vert x \vert \to \infty }u(x)=0,\qquad d_{1} \frac{\partial u}{\partial n}+\tilde{c}_{1}(u)u=0 & \text{for } \vert x \vert =r_{0}, \\ \lim_{ \vert x \vert \to \infty }v(x)=0,\qquad d_{2} \frac{\partial v}{\partial n}+\tilde{c}_{2}(v)v=0 & \text{for } \vert x \vert =r_{0}, \end{cases} $$

using a fixed point result of Krasnoselskii type applied to suitable completely continuous integral operators on $C[0,1]\times C[0,1]$. These results seem to be the first ones proving the multiplicity of positive solutions for this kind of systems.

On the other hand, in the context of nonhomogeneous BCs, elliptic problems were studied by Aftalion and Busca [2], Butler et al. [6], Cianciaruso and al. [10], Dhanya et al. [15], do Ó et al. [18–21], Goodrich [26, 27], Ko and et al. [31], and Lee et al. [34].

The existence of positive radial solutions of elliptic equations with nonlinearities depending on the gradient subject to Neumann, Dirichlet, or Robin boundary conditions has been investigated by a number of authors; see, for example, Averna et al. [5], Cianciaruso et al. [8, 9, 12], De Figueiredo et al. [13, 14], Faria et al. [22], and Montreanu et al. [36].

Our system (1.1) is quite general:

• (1) the nonlinearities $f_{i}$ depend on the functions u and v and their gradients; no monotonicity hypotheses are supposed.

• (2) the boundary conditions are nonlocal and represent feedback mechanisms. They have been deeply studied for ordinary differential equations, for example, in [11, 24, 25, 41, 42]).

To search solutions of the elliptic PDE

$$ -\Delta w=g\bigl( \vert x \vert \bigr)f\bigl(w, \vert \nabla w \vert \bigr) $$

with some boundary conditions, a topological approach is associating, by using standard transformations, an integral operator of the form

$$ Sw(t)= \int _{0}^{1} G(t,s)g\bigl(r(s)\bigr)\tilde{f} \bigl(w(s), \bigl\vert w'(s) \bigr\vert \bigr)\,ds. $$

It is straightforward, in the local problems, to find the Green’s function G by integration and by using the BCs. However, let us remark that, in the nonlocal problems, this is a long and technical calculation, often resulting in a sum of terms of different signs.

Here, as in [41], we treat the nonlocal problem as the perturbation of the simpler local problem. In such a way, we handle the positivity properties of the simpler Green’s function of the local problem.

Often, the associated integral operator is studied in the cone of nonnegative functions in the space $C^{1}[0,1]$ or in a weighed space of differential functions as in [3]. In our case and in particular when seeking for multiple solutions, it is suitable to work in a smaller cone: we will introduce a new cone in which we will use Harnack-type inequalities.

Moreover, since we are interested in positive solutions, the functionals $\alpha _{i}$ and $\beta _{i}$ must satisfy some positivity conditions; we will not suppose this in the whole space, but we choose to include the requirement in the definition of the cone.

We show that, under suitable conditions on the nonlinear terms, the fixed point index is 0 on certain open bounded subsets of the cone and 1 on the others; the choice of these subsets allows us to have more freedom on the conditions of the growth of the nonlinearities. These conditions relate the upper and lower bounds of the nonlinearities $f_{i}$ on stripes and some constants, depending on the kernel of the integral operator and on the nonlocal BCs that are easily estimable as we show in an example.

2 The associate integral operator

Consider in $\mathbb{R}^{n}$, $n\ge 3$, the equation

$$ -\triangle w= h\bigl( \vert x \vert \bigr)f\bigl(w, \vert \nabla w \vert \bigr), \quad \vert x \vert \in [r_{0},+\infty ). $$

(2.1)

Since we are interested in radial solutions $w=w(r)$, $r=|x|$, following [6], we rewrite (2.1) as

$$ -w''(r) - \frac{n-1}{r}w'(r)= h(r)f\bigl(w(r), \bigl\vert w'(r) \bigr\vert \bigr), \quad r\in [r_{0}, +\infty ). $$

(2.2)

By using the transformation

$$ r(t):=r_{0} t^{\frac{1}{2-n}},\quad t\in (0,1], $$

equation (2.2) becomes

$$ w''\bigl(r(t)\bigr)+g(t) f \biggl(w\bigl(r(t)\bigr), \frac{ \vert w'(r(t)) \vert }{ \vert r'(t) \vert } \biggr)=0, \quad t\in (0,1], $$

with

$$ g(t)= \frac{r_{0}^{2}}{(n-2)^{2}}t^{\frac{2n-2}{2-n}} h\bigl(r(t)\bigr). $$

Consider in $\mathbb{R}^{n}$ the system of boundary value problems

$$ \textstyle\begin{cases} -\Delta u= h_{1}( \vert x \vert ) f_{1}(u,v, \vert \nabla u \vert , \vert \nabla v \vert ),& \vert x \vert \in [r_{0},+\infty ), \\ -\Delta v= h_{2}( \vert x \vert ) f_{2}(u,v, \vert \nabla u \vert , \vert \nabla v \vert ),& \vert x \vert \in [r_{0},+\infty ), \\ \lim_{ \vert x \vert \to \infty }u(x)=\alpha _{1}[u],\qquad c_{1}u+\tilde{d_{1}}\frac{\partial u}{\partial r}= \beta _{1}[u] & \text{for } \vert x \vert =r_{0}, \\ \lim_{ \vert x \vert \to \infty }v(x)=\alpha _{2}[v],\qquad c_{2}v+\tilde{d_{2}}\frac{\partial v}{\partial r}= \beta _{2}[v] & \text{for } \vert x \vert =r_{0}. \end{cases} $$

(2.3)

Set $u(t)=u(r(t))$ and $v(t)=v(r(t))$. Thus with system (2.3) we associate the system of ODEs

$$ \textstyle\begin{cases} u''(t) + g_{1}(t) f_{1} (u(t),v(t), \frac{ \vert u'(t) \vert }{ \vert r'(t) \vert }, \frac{ \vert v'(t) \vert }{ \vert r'(t) \vert } ) = 0,\quad t\in (0,1), \\ v''(t) + g_{2}(t) f_{2} (u(t),v(t), \frac{ \vert u'(t) \vert }{ \vert r'(t) \vert }, \frac{ \vert v'(t) \vert }{ \vert r'(t) \vert } ) = 0, \quad t\in (0,1), \\ u(0)=\alpha _{1}[u],\qquad c_{1}u(1)+d_{1} u'(1)=\beta _{1}[u], \\ v(0)=\alpha _{2}[v],\qquad c_{2}v(1)+d_{2} v'(1)=\beta _{2}[v], \end{cases} $$

(2.4)

where $d_{i}=\frac{r_{0}}{2-n}\tilde{d}_{i}$ and $g_{i}(t)=\frac{r_{0}^{2}}{(n-2)^{2}}t^{ \frac{2n-2}{2-n}} h_{i}(r(t))$.

We study the existence of positive solutions of system (2.4) by means of the associated system of perturbed Hammerstein integral equations

$$ \textstyle\begin{cases} u(t)=\gamma _{1}(t)\alpha _{1}[u]+\delta _{1}(t)\beta _{1}[u]+ \int _{0}^{1}k_{1}(t,s)g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds, \\ v(t)=\gamma _{2}(t)\alpha _{2}[v]+\delta _{2}(t) \beta _{2}[v]+ \int _{0}^{1}k_{2}(t,s)g_{2}(s)f_{2} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds,\end{cases} $$

where $\gamma _{i}$ is the solution of the BVP $w''(t)=0$, $w(0)=1$, $c_{i}w(1)+d_{i} w'(1)=0$, that is,

$$ \gamma _{i}(t)= 1-\frac{c_{i}t}{d_{i}+c_{i} }; $$

$\delta _{i}$ is the solution of the BVP $w''(t)=0$, $w(0)=0$, $c_{i}w(1)+d_{i} w'(1)=1$, that is,

$$ \delta _{i}(t)= \frac{t}{d_{i}+c_{i} }; $$

and $k_{i}$ is the Green’s function associated with the homogeneous problem in which $\alpha _{i}[w]=\beta _{i}[w]=0$, that is,

$$ k_{i}(t,s):= \textstyle\begin{cases} s (1-\frac{c_{i} t}{d_{i}+c_{i}} ), & s \le t, \cr t (1-\frac{c_{i} s}{d_{i}+c_{i}} ),&s>t. \end{cases} $$

In the following proposition, we resume the properties of the functions $\gamma _{i}$, $\delta _{i}$ and $k_{i}$, which will be further useful.

Proposition 2.1

We have, for$i=1,2$:

• The functions$\gamma _{i}$, $\delta _{i}$are in$C^{1} [0,1]$; moreover, for$t \in [a_{i},b_{i}] \subset (0,1)$, with$a_{i}+b_{i}<1$,

  $$\begin{aligned}& \Vert \gamma _{i} \Vert _{\infty }=1\quad \textit{and}\quad \gamma _{i}(t) \geq 1-t\geq 1-b_{i}= (1-b_{i}) \Vert \gamma _{i} \Vert _{\infty }>a_{i} \Vert \gamma _{i} \Vert _{\infty }; \\& \Vert \delta _{i} \Vert _{\infty }= \frac{1}{d_{i}+c_{i}}\quad \textit{and}\quad \delta _{i}(t)=t \Vert \delta _{i} \Vert _{\infty }\geq a_{i} \Vert \delta _{i} \Vert _{\infty }. \end{aligned}$$

• The kernels$k_{i}$are nonnegative and continuous in$[0,1]\times [0,1]$. Moreover, for$t\in [a_{i},b_{i}]$, we have

  $$\begin{aligned} &k_{i}(t,s) \leq \phi _{i}(s)\quad \textit{for } (t,s)\in [0,1] \times [0,1], \quad \textit{and} \\ &k_{i}(t,s) \geq a_{i} \phi _{i}(s)\quad \textit{for }(t,s)\in [a_{i},b_{i}] \times [0,1], \end{aligned}$$

  with$\phi _{i}(s):=s (1-\frac{c_{i} }{d_{i}+c_{i}}s )$.

Let $\omega (t)=t(1-t)$. Our setting will be the Banach space (see [3])

$$ C_{\omega }^{1}[0,1]=\Bigl\{ w \in C[0,1]\cap C^{1}(0,1) : \sup_{t \in (0,1)} \omega (t) \bigl\vert w'(t) \bigr\vert < +\infty \Bigr\} $$

endowed with the norm

$$ \Vert w \Vert : =\max \bigl\{ \Vert w \Vert _{\infty }, \bigl\Vert w' \bigr\Vert _{\omega } \bigr\} , $$

where $\| w\| _{\infty }:=\max_{t\in [ 0, 1] }|w(t)|$ and $\|w'\|_{\omega }:=\sup_{t\in (0,1)}\omega (t)|w'(t)|$.

For $i=1,2$ and fixed $[a_{i},b_{i}] \subset (0,1)$ such that $a_{i}+b_{i}<1$, we consider the cones

$$ \begin{aligned} K_{i}&:= \Bigl\{ w\in C_{\omega }^{1}[0, 1]:w\geq 0, \min_{t\in [ a_{i},b_{i}]}w(t)\geq a_{i} \Vert w \Vert _{\infty },\\ &\quad \bigl\Vert w' \bigr\Vert _{\omega } \leq 4w(1/2), \alpha _{i}[w]\geq 0, \beta _{i}[w]\geq 0 \Bigr\} \end{aligned} $$

in $C_{\omega }^{1}[0,1]$ and the cone

$$ K:=K_{1}\times K_{2} $$

in $C_{\omega }^{1}[0,1]\times C_{\omega }^{1}[0,1]$.

Note that the functions in $K_{i}$ are strictly positive on the subinterval $[a_{i},b_{i}]$ and that for $w \in K_{i}$, we have the inequalities $\|w'\|_{\omega }\leq \|w\|\le 4 \|w\|_{\infty }$.

Set

$$ F_{i}(u,v) (t):= \int _{0}^{1}k_{i}(t,s)g_{i}(s)f_{i} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds $$

and

$$\begin{aligned}& \begin{aligned} T(u,v) (t):= \begin{pmatrix} \gamma _{1}(t)\alpha _{1}[u]+\delta _{1}(t)\beta _{1}[u]+F_{1}(u,v)(t) \\ \gamma _{2}(t)\alpha _{2}[v]+\delta _{2}(t) \beta _{2}[v]+F_{2}(u,v)(t)\end{pmatrix} := \begin{pmatrix} T_{1}(u,v)(t) \\ T_{2}(u,v)(t)\end{pmatrix}. \end{aligned} \end{aligned}$$

(2.5)

We will further assume that, for $i=1,2$,

• $f_{i}: [0,+\infty )^{4} \to [0,+\infty )$ is continuous, and there exist $\psi _{i}:[0,+\infty )^{4} \to [0,+\infty )$ such that $f_{i} (u,v,\frac{|u'|}{|r'|},\frac{|v'|}{|r'|} )\leq \psi _{i} (u,v, \frac{\omega |u'|}{|r'|}, \frac{\omega |v'|}{|r'|} )$ for all $(u,v) \in K$;

• $h_{i}:[r_{0},+\infty ) \to [0,+\infty )$ is continuous, and $h_{i}(r)\leq \frac{1}{r^{n+\mu _{i}}}$ as $r\to +\infty$ for some $\mu _{i}>0$.

• $0\leq \alpha _{i}[\gamma _{i}]<1$ and $0\leq \beta _{i}[ \delta _{i}]<1$;

• $\alpha _{i}[k_{i}]:= \alpha _{i}[k_{i}(\cdot ,s)]$ and $\beta _{i}[k_{i}]:= \beta _{i}[k_{i}(\cdot ,s)]$ are nonnegative;

• $D_{i}=(1-\alpha _{i}[\gamma _{i}])(1-\beta _{i}[\delta _{i}])- \alpha _{i}[\delta _{i}]\beta _{i}[\gamma _{i}]>0$.

Note that the hypotheses on the nonlinearities $f_{i}$ are used in [3] and are satisfied, for example, if the functions $f_{i}$ are continuous and, with respect to the last two variables, decreasing or bounded.

We now prove that T leaves the cone K invariant and is completely continuous.

Theorem 2.2

The operatorTmapsKintoKand is completely continuous.

Proof

To prove that T leaves the cone K invariant, it suffices to prove that $T_{i}K\subset K_{i}$.

For $(u_{1},u_{2})\in K$, we have

$$\begin{aligned} \bigl\Vert T_{i}(u_{1},u_{2}) \bigr\Vert _{\infty }& \leq \Vert \gamma _{i} \Vert _{\infty }\alpha _{i}[u_{i}]+ \Vert \delta _{i} \Vert _{\infty }\beta _{i}[u_{i}] \\ &\quad{}+ \int _{0}^{1}\phi _{i}(s)g_{i}(s)f_{i} \biggl(u_{1}(s),u_{2}(s), \frac{ \vert u_{1}'(s) \vert }{ \vert r'(s) \vert }, \frac{ \vert u'_{2}(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ & \leq \Vert \gamma _{i} \Vert _{\infty }\alpha _{i}[u_{i}]+ \Vert \delta _{i} \Vert _{\infty }\beta _{i}[u_{i}] \\ & \quad {}+ \int _{0}^{1}\phi _{i}(s)g_{i}(s) \psi _{i} \biggl(u_{1}(s),u_{2}(s), \frac{\omega (s) \vert u_{1}'(s) \vert }{ \vert r'(s) \vert }, \frac{\omega (s) \vert u'_{2}(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ &< \infty . \end{aligned}$$

(2.6)

On the other hand, we have

$$\begin{aligned} \min_{t\in [a_{i},b_{i}]}T_{i}(u_{1},u_{2}) (t) &\geq a_{i} \biggl( \Vert \gamma _{i} \Vert _{\infty }\alpha _{i}[u_{i}]+ \Vert \delta _{i} \Vert _{\infty }\beta _{i}[u_{i}] \\ &\quad{}+ \int _{0}^{1}\phi _{i}(s)g_{i}(s)f_{i} \biggl(u_{1}(s),u_{2}(s), \frac{ \vert u_{1}'(s) \vert }{ \vert r'(s) \vert }, \frac{ \vert u_{2}'(s) \vert }{ \vert r'(s) \vert } \biggr) \,ds \biggr) \\ &\geq a_{i} \bigl\Vert T_{i}(u_{1},u_{2}) \bigr\Vert _{\infty }. \end{aligned}$$

Now we prove that, for every $(u_{1},u_{2})\in K$,

$$ \bigl\Vert \bigl(F_{i}(u_{1},u_{2}) \bigr)' \bigr\Vert _{\omega }\le 4 F_{i}(u_{1},u_{2}) (1/2). $$

We have

$$\begin{aligned} &\omega (t) \bigl\vert \bigl(F_{i}(u_{1},u_{2}) \bigr)'(t) \bigr\vert \\ &\quad = \biggl\vert -t(1-t) \int _{0}^{t} \frac{c_{i} s}{d_{i}+c_{i}}g_{i}(s)f_{i} \biggl(u_{1}(s),u_{2}(s), \frac{ \vert u_{1}'(s) \vert }{ \vert r'(s) \vert }, \frac{ \vert u_{2}'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ &\qquad{}+t(1-t) \int _{t}^{1} \biggl(1- \frac{c_{i} s}{d_{i}+c_{i}} \biggr)g_{i}(s)f_{i} \biggl(u_{1}(s),u_{2}(s), \frac{ \vert u_{1}'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert u_{2}'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \biggr\vert \\ &\quad \leq \int _{0}^{t} t(1-t) \frac{c_{i} s}{d_{1}+c_{i}}g_{i}(s)f_{i} \biggl(u_{1}(s),u_{2}(s),\frac{ \vert u_{1}'(s) \vert }{ \vert r'(s) \vert }, \frac{ \vert u_{2}'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ &\qquad{}+ \int _{t}^{1} t(1-t) \biggl(1- \frac{c_{i} s}{d_{i}+c_{i}} \biggr)g_{i}(s)f_{i} \biggl(u_{1}(s),u_{2}(s), \frac{ \vert u_{1}'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert u_{2}'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds. \end{aligned}$$

Since $\frac{c_{i} t}{d_{i}+c_{i}}\leq t$, we have that $(1-t)\leq (1-\frac{c_{i} t}{d_{i}+c_{i}} )$, and consequently

$$\begin{aligned} \omega (t) \bigl\vert \bigl(F_{i}(u_{1},u_{2}) \bigr)'(t) \bigr\vert &\leq \int _{0}^{t} s \biggl(1- \frac{c_{i} t}{d_{i}+c_{i}} \biggr)g_{i}(s)f_{i} \biggl(u_{1}(s),u_{2}(s), \frac{ \vert u_{1}'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert u_{2}'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ &\quad{}+ \int _{t}^{1} t \biggl(1-\frac{c_{i} s}{d_{i}+c_{i}} \biggr)g_{i}(s)f_{i} \biggl(u_{1}(s),u_{2}(s), \frac{ \vert u_{1}'(s) \vert }{ \vert r'(s) \vert }, \frac{ \vert u_{2}'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ &= \int _{0}^{1}k_{i}(t,s)g_{i}(s)f_{i} \biggl(u_{1}(s),u_{2}(s), \frac{ \vert u_{1}'(s) \vert }{ \vert r'(s) \vert }, \frac{ \vert u_{2}'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ &=F_{i}(u_{1},u_{2}) (t)\le \bigl\Vert F_{i}(u_{1},u_{2}) \bigr\Vert _{\infty }. \end{aligned}$$

Let $\tau _{i} \in [0,1]$ be such that

$$ F_{i}(u_{1},u_{2}) (\tau _{i})= \bigl\Vert F_{i}(u_{1},u_{2}) \bigr\Vert _{\infty }. $$

For any $t\in [0,1]$, we can easily compute that

$$ \frac{k_{i}(t,s)}{k_{i}(\tau _{i},s)}= \textstyle\begin{cases} t/\tau _{i}, &\tau _{i},t\leq s, \\ (1-\frac{c_{i} t}{d_{i}+c_{i}} ) (1- \frac{c_{i} \tau _{i}}{d_{i}+c_{i}} )^{-1}, &\tau _{i},t> s, \\ t (1-\frac{c_{i} s}{d_{i}+c_{i}} )s^{-1} (1-\frac{c_{i} \tau _{i}}{d_{i}+c_{i}} )^{-1}, &t\leq s \leq \tau _{i}, \\ s (1-\frac{c_{i} t}{d_{i}+c_{i}} )\tau _{i}^{-1} (1-\frac{c_{i} s}{d_{i}+c_{i}} )^{-1}, & \tau _{i}\leq s \leq t, \end{cases} $$

and that $\frac{k_{i}(t,s)}{k_{i}(\tau _{i},s)}\geq t(1-t)$ for $t,s\in [0,1]$. Then, for all $t\in [0,1]$, we have

$$\begin{aligned} F_{i}(u_{1},u_{2}) (t) =& \int _{0}^{1} \frac{k_{i}(t,s)}{k_{i}(\tau _{i},s)}k_{i}( \tau _{i},s)g_{i}(s)f_{i} \biggl(u_{1}(s),u_{2}(s), \frac{ \vert u_{1}'(s) \vert }{ \vert r'(s) \vert }, \frac{ \vert u_{2}'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ \geq & t(1-t) \int _{0}^{1}k_{i}(\tau _{i},s)g_{i}(s)f_{i} \biggl(u_{1}(s),u_{2}(s), \frac{ \vert u_{1}'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert u_{2}'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ =&t(1-t) \bigl\Vert F_{i}(u_{1},u_{2}) \bigr\Vert _{\infty }. \end{aligned}$$

For $t=\frac{1}{2}$, we obtain

$$ \bigl\Vert F_{i}(u_{1},u_{2}) \bigr\Vert _{\infty }\leq 4F_{i}(u_{1},u_{2}) \biggl( \frac{1}{2} \biggr). $$

Therefore we conclude that

$$ \bigl\Vert \bigl(F_{i}(u_{1},u_{2}) \bigr)' \bigr\Vert _{\omega }\le \bigl\Vert F_{i}(u_{1},u_{2}) \bigr\Vert _{ \infty } \leq 4 F_{i}(u_{1},u_{2}) \biggl(\frac{1}{2} \biggr). $$

Since $\gamma _{i}, \delta _{i} \in K_{i}$, we have

$$\begin{aligned} \bigl\vert \omega (t) \bigl(T_{i}(u_{1},u_{2}) \bigr)'(t) \bigr\vert \leq & \omega (t) \bigl\vert \gamma '_{i}(t) \bigr\vert \alpha _{i}[u_{i}]+ \omega (t) \bigl\vert \delta '_{i}(t) \bigr\vert \beta _{i}[u_{i}] \\ &{}+\omega (t) \bigl\vert \bigl(F_{i}(u_{i},u_{2}) \bigr)'(t) \bigr\vert \\ \leq & \bigl\Vert \gamma '_{i} \bigr\Vert _{\omega }\alpha _{i}[u_{i}]+ \bigl\Vert \delta '_{i} \bigr\Vert _{ \omega }\beta _{i}[u_{i}]+ \bigl\Vert F'_{i}(u_{1},u_{2}) \bigr\Vert _{\omega } \\ \leq &4\gamma _{i} \biggl(\frac{1}{2} \biggr)\alpha _{i}[u_{i}]+4 \delta _{i} \biggl( \frac{1}{2} \biggr)\beta _{i}[u_{i}]+4F_{i}(u_{i},u_{2}) \biggl(\frac{1}{2} \biggr) \\ =&4T_{i}(u_{1},u_{2}) \biggl(\frac{1}{2} \biggr). \end{aligned}$$

Taking the supremum on $[0,1]$, we obtain

$$ \bigl\Vert \bigl(T_{i}(u_{1},u_{2}) \bigr)' \bigr\Vert _{\omega }\le 4 T_{i}(u_{1},u_{2}) \biggl( \frac{1}{2} \biggr). $$

Since $\alpha _{i}$ and $\beta _{i}$ are linear functionals, it follows that $\alpha _{i}[T_{i}(u_{1},u_{2})]$ and $\beta _{i}[T_{i}(u_{1},u_{2})]$ are nonnegative.

Summarizing, we have $TK\subset K$.

To prove the complete continuity of T, let us note that the continuity of f, $k_{i}$, $\alpha _{i}$, and $\beta _{i}$ give the continuity of each $T_{i}$ and thus the continuity of T.

Let U be a bounded subset of K; from (2.6) it follows that $T(U)$ is bounded in K. Now we prove that $T(U)$ is relatively compact in K. It is a standard argument based on the uniform continuity of the kernels $k_{i}$ on $[0,1]\times [0,1]$ and on the Ascoli–Arzelà theorem that $T_{i}(U)$ is relatively compact in $C[0,1]$.

Now let $(u_{n},v_{n})_{n\in \mathbb{N}}$ be a sequence in U. Then $T_{i}(u_{n},v_{n})\subset K_{i}$.

There exists $(u_{n_{k}},v_{n_{k}})_{k\in \mathbb{N}}$ such that $(T_{1}(u_{n_{k}},v_{n_{k}}))_{k\in \mathbb{N}}$ converges in $C[0,1]$.

Since $T_{2}(U)$ is relatively compact, there exists $(u_{n_{k_{p}}},v_{n_{k_{p}}})_{p\in \mathbb{N}}:=(u_{n_{p}},v_{n_{p}})_{p \in \mathbb{N}}\subset (u_{n_{k}}, v_{n_{k}})_{k\in \mathbb{N}}$ such that $(T_{i}(u_{n_{p}},v_{n_{p}}))_{p\in \mathbb{N}}\to w_{i}\in C[0,1]$ for $i=1,2$. Since

$$ \bigl\Vert \bigl(T_{i}(u_{n_{p}},v_{n_{p}}) \bigr)'-\bigl(T_{i}(u_{n_{m}},v_{n_{m}}) \bigr)' \bigr\Vert _{ \omega }\leq 4 \bigl\Vert T_{i}(u_{n_{p}},v_{n_{p}})-T_{i}(u_{n_{m}},v_{n_{m}}) \bigr\Vert _{\infty }, $$

that is, $((T_{i}(u_{n_{p}},v_{n_{p}}))')_{p\in \mathbb{N}}$ is a Cauchy sequence in $\|\cdot \|_{\omega }$ for $i=1,2$. Then $(T_{i}(u_{n_{p}},v_{n_{p}}))_{p\in \mathbb{N}}$ is a Cauchy sequence in $C^{1}_{\omega }[0,1]$, and so it converges to $w_{i}\in C^{1}_{\omega }[0,1]$. The closedness of K implies that $(w_{1},w_{2})\in K$, and therefore $T(U)$ is relatively compact in K. □

To use the fixed point index, we utilize the following sets in K for $\rho _{1},\rho _{2}>0$:

$$\begin{aligned}& K_{\rho _{1},\rho _{2}}:=\bigl\{ (u,v)\in K: \Vert u \Vert _{\infty }< \rho _{1} \text{ and } \Vert v \Vert _{\infty }< \rho _{2} \bigr\} , \\& V_{\rho _{1},\rho _{2}}:=\Bigl\{ (u,v)\in K: {\min_{t\in [a_{1},b_{1}]}}u(t)< \rho _{1} \text{ and } {\min_{t\in [a_{2},b_{2}]}}v(t)< \rho _{2}\Bigr\} . \end{aligned}$$

Since $\|w'\|_{\omega }\leq 4\|w\|_{\infty }$ in K, we have $\|w\|\leq 4\|w\|_{\infty }$, and therefore $K_{\rho _{1},\rho _{2}}$ and $V_{\rho _{1},\rho _{2}}$ are open and bounded with respect to K. It is straightforward to verify that these sets satisfy the following properties:

$(P_{1})$:

$K_{\rho _{1},\rho _{2}}\subset V_{\rho _{1},\rho _{2}}\subset K_{ \rho _{1}/a_{1},\rho _{2}/a_{2}}$.

$(P_{2})$:

$(w_{1},w_{2})\in \partial K_{\rho _{1},\rho _{2}}$ if and only if $(w_{1},w_{2})\in K$ and for some $i\in \{1,2\}$$\|w_{i}\|_{\infty }=\rho _{i}$ and $a_{i} \rho _{i} \le w_{i}(t)\le \rho _{i}$ for $t\in [a_{i},b_{i}]$.

$(P_{3})$:

$(w_{1},w_{2}) \in \partial V_{\rho _{1},\rho _{2}}$ if and only if $(w_{1},w_{2})\in K$ and for some $i\in \{1,2\}$$\min_{t\in [a_{i},b_{i}]} w_{i}(t)= \rho _{i}$ and $\rho _{i} \le w_{i}(t) \le \rho _{i}/a_{i}$ for $t\in [a_{i},b_{i}]$.

The following theorem follows from classical results about fixed point index (more details can be seen, for example, in [4, 28]).

Theorem 2.3

LetKbe a cone in an ordered Banach spaceX. LetΩbe an open bounded subset with$0 \in \varOmega \cap K$and$\overline{\varOmega \cap K}\neq K$. Let$\varOmega ^{1}$be open inXwith$\overline{\varOmega ^{1}}\subset \varOmega \cap K$. Let$F:\overline{\varOmega \cap K}\rightarrow K$be a compact map. Suppose that

• (1) $Fx\neq \mu x$for all$x\in \partial ( \varOmega \cap K)$and$\mu \geq 1$.

• (2) There exists$h\in K\setminus \{0\}$such that$x\neq Fx+\lambda h$for all$x\in \partial (\varOmega ^{1} \cap K)$and$\lambda >0$.

ThenFhas at least one fixed point$x \in (\varOmega \cap K)\setminus \overline{(\varOmega ^{1}\cap K)}$.

Denoting by$i_{K}(F,U)$the fixed point index ofFin some$U\subset X$, we have

$$ i_{K}(F,\varOmega \cap K)=1 \quad \textit{and}\quad i_{K} \bigl(F,\varOmega ^{1} \cap K\bigr)=0 . $$

The same result holds if

$$ i_{K}(F,\varOmega \cap K)=0\quad \textit{and}\quad i_{K} \bigl(F,\varOmega ^{1} \cap K\bigr)=1. $$

3 A system of elliptic PDE

We define the following sets:

$$\begin{aligned} &\varOmega ^{\rho _{1},\rho _{2}}= [0, \rho _{1} ]\times [0, \rho _{2} ]\times [0, +\infty )\times [ 0, +\infty ), \\ &A_{1}^{s_{1},s_{2}}= \biggl[s_{1},\frac{s_{1}}{a_{1}} \biggr]\times \biggl[0,\frac{s_{2}}{a_{2}} \biggr]\times [0, +\infty ) \times [ 0, +\infty ), \\ &A_{2}^{s_{1},s_{2}}= \biggl[0,\frac{s_{1}}{a_{1}} \biggr]\times \biggl[s_{2},\frac{s_{2}}{a_{2}} \biggr]\times [0, +\infty )\times [ 0, + \infty ), \end{aligned}$$

and the numbers

$$\begin{aligned}& \begin{aligned} C_{i}&:= \biggl[\frac{1}{D_{i}} \biggl( \bigl[ \bigl(1-\beta _{i}[ \delta _{i}]\bigr)+ \Vert \delta _{i} \Vert _{\infty }\beta _{i}[\gamma _{i}] \bigr] \int _{0}^{1}\alpha _{i}[k_{i}]g_{i}(s) \,ds \\ &\quad {} + \bigl[\alpha _{i}[\delta _{i}]+ \Vert \delta _{i} \Vert _{ \infty } \bigl(1-\alpha _{i}[ \gamma _{i}]\bigr) \bigr] \int _{0}^{1}\beta _{i}[k_{i}]g_{i}(s) \,ds \biggr) +\sup_{t\in [0,1]} \int _{0}^{1}k_{i}(t,s)g_{i}(s) \,ds \biggr]^{-1}, \end{aligned} \\& \begin{aligned} M_{i}&= \biggl[\frac{1}{D_{i}} \biggl( \bigl[a_{i}\bigl(1-\beta _{i}[\delta _{i}] \bigr)+a_{i} \Vert \delta _{i} \Vert _{\infty } \beta _{i}[\gamma _{i}] \bigr] \int _{a_{i}}^{b_{i}} \alpha _{i}[k_{i}]g_{i}(s) \,ds \\ &\quad {}+ \bigl[a_{i}\alpha _{i}[\delta _{i}]+a_{i} \Vert \delta _{i} \Vert _{\infty } \bigl(1-\alpha _{i}[\gamma _{i}]\bigr) \bigr] \int _{a_{i}}^{b_{i}} \beta _{1}[k_{i}]g_{i}(s) \,ds \biggr)\\ &\quad {}+\inf_{t \in [a_{i},b_{i}]} \int _{a_{i}}^{b_{i}}k_{i}(t,s)g_{i}(s) \,ds) \biggr]^{-1}. \end{aligned} \end{aligned}$$

Theorem 3.1

Suppose that there exist$\rho _{1}, \rho _{2}, s_{1},s_{2}\in (0,+\infty )$, with$\rho _{i}< s _{i}$, $i=1,2$, such that

$$ \sup_{\varOmega ^{\rho _{1},\rho _{2}}} f_{i}(w_{1},w_{2},z_{1},z_{2})< C_{i} \rho _{i} $$

(3.1)

and

$$ \inf_{A_{i}^{s_{1},s_{2}}} f_{i}(w_{1},w_{2},z_{1},z_{2})>M_{i}s_{i}. $$

(3.2)

Then system (2.3) has at least one positive radial solution.

Proof

Note that the choice of the numbers $\rho _{i}$ and $s_{i}$ ensures the compatibility of conditions (3.1) and (3.2).

We want to show that $i_{K}(T, K_{\rho _{1},\rho _{2}})=1$ and $i_{K}(T, V_{s_{1},s_{2}})=0$, so that from Theorem 2.3 it follows that the completely continuous operator T has a fixed point in $V_{s_{1},s_{2}}\setminus \overline{K}_{\rho _{1},\rho _{2}}$. Then system (2.3) admits a positive radial solution.

First, we claim that $\lambda (u,v)\neq T(u,v)$ for all $(u,v)\in \partial K_{\rho _{1},\rho _{2}}$ and $\lambda \geq 1$, which implies that the index of T is 1 on $K_{\rho _{1},\rho _{2}}$. Suppose this is not true. Let $\lambda \geq 1$, and let $(u,v)\in \partial K_{\rho _{1},\rho _{2}}$ be such that

$$ \lambda (u,v)=T(u,v). $$

In view of $(P_{2})$, without loss in generality, let us suppose that $\|u\|_{\infty }=\rho _{1}$. Then

$$\begin{aligned} \lambda u(t) =&\gamma _{1}(t)\alpha _{1}[u]+\delta _{1}(t)\beta _{1}[u] \\ &{}+ \int _{0}^{1}k_{1}(t,s)g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds. \end{aligned}$$

(3.3)

Applying $\alpha _{1}$ to both terms, we have

$$ \lambda \alpha _{1}[u]=\alpha _{1}[\gamma _{1}] \alpha _{1}[u]+\alpha _{1}[ \delta _{1}]\beta _{1}[u]+ \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s),\frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds, $$

wich implies

$$ \bigl(\lambda -\alpha _{1}[\gamma _{1}]\bigr)\alpha _{1}[u]-\alpha _{1}[\delta _{1}] \beta _{1}[u]= \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds. $$

In a similar way, applying $\beta _{1}$, we obtain

$$ \bigl(\lambda -\beta _{1}[\delta _{1}]\bigr)\beta _{1}[u]-\beta _{1}[\gamma _{1}] \alpha _{1}[u]= \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds. $$

Denoting

$$ N_{1}^{\lambda }:= \begin{pmatrix} \lambda -\alpha _{1}[\gamma _{1}] & -\alpha _{1}[\delta _{1}] \\ -\beta _{1}[\gamma _{1}] & \lambda -\beta _{1}[\delta _{1}] \end{pmatrix},\qquad N^{1}_{1}:=N_{1},\quad \text{and}\quad D_{1}:= \det N_{1}, $$

we can write the previous conditions as

$$ N_{1}^{\lambda } \begin{pmatrix} \alpha _{1}[u] \\ \beta _{1}[u] \end{pmatrix}= \begin{pmatrix} \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds \\ \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds \end{pmatrix}. $$

Therefore we get that

$$\begin{aligned} \begin{pmatrix} \alpha _{1}[u] \\ \beta _{1}[u] \end{pmatrix}=\bigl(N_{1}^{\lambda } \bigr)^{-1} \begin{pmatrix} \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds \\ \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds \end{pmatrix} \\ \leq (N_{1})^{-1} \begin{pmatrix} \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds \\ \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds \end{pmatrix}, \end{aligned}$$

so that formula (3.3) becomes, for $t \in [0,1]$,

$$\begin{aligned} u(t) \leq &\frac{1}{D_{1}} \biggl[\gamma _{1}(t) \bigl(1-\beta _{1}[\delta _{1}]\bigr) \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ & {}+\gamma _{1}(t)\alpha _{1}[\delta _{1}] \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s),\frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ & {}+\delta _{1}(t)\beta _{1}[\gamma _{1}] \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s),\frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ & {}+\delta _{1}(t) \bigl(1-\alpha _{1}[ \gamma _{1}]\bigr) \int _{0}^{1} \beta _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \biggr] \\ &{}+ \int _{0}^{1}k_{1}(t,s)g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ =&\frac{1}{D_{1}} \biggl[ \bigl[\gamma _{1}(t) \bigl(1- \beta _{1}[\delta _{1}]\bigr)+ \delta _{1}(t)\beta _{1}[\gamma _{1}] \bigr]\\ &{}\times \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s),\frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ &{}+ \bigl[\gamma _{1}(t)\alpha _{1}[ \delta _{1}]+\delta _{1}(t) \bigl(1- \alpha _{1}[ \gamma _{1}]\bigr) \bigr]\\ &{}\times \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s),\frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \biggr] \\ &{}+ \int _{0}^{1}k_{1}(t,s)g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ \leq &\sup_{\varOmega ^{\rho _{1},\rho _{2}}} f_{1}(w_{1},w_{2},z_{1},z_{2}) \biggl[\frac{1}{D_{1}} \biggl[ \bigl(\gamma _{1}(t) \bigl(1-\beta _{1}[ \delta _{1}]\bigr)+\delta _{1}(t)\beta _{1}[\gamma _{1}] \bigr) \int _{0}^{1} \alpha _{1}[k_{1}]g_{1}(s) \,ds \\ &{}+ \bigl(\gamma _{1}(t)\alpha _{1}[ \delta _{1}]+\delta _{1}(t) \bigl(1- \alpha _{1}[ \gamma _{1}]\bigr) \bigr) \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s) \,ds \biggr] \\ &{}+ \int _{0}^{1}k_{1}(t,s)g_{1}(s) \,ds \biggr]. \end{aligned}$$

(3.4)

Taking the supremum on $[0,1]$ in the last inequality, it follows that

$$\begin{aligned} \rho _{1} =& \Vert u \Vert _{\infty }\\ \leq& \sup _{\varOmega ^{\rho _{1},\rho _{2}}} f_{1}(w_{1},w_{2},z_{1},z_{2}) \biggl[\frac{1}{D_{1}} \biggl[ \bigl[\bigl(1-\beta _{1}[\delta _{1}]\bigr)+ \Vert \delta _{1} \Vert _{\infty }\beta _{1}[\gamma _{1}] \bigr] \int _{0}^{1} \alpha _{1}[k_{1}]g_{1}(s) \,ds \\ &{}+ \bigl[\alpha _{1}[\delta _{1}]+ \Vert \delta _{1} \Vert _{ \infty } \bigl(1-\alpha _{1}[\gamma _{1}]\bigr) \bigr] \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s) \,ds \biggr] \\ &{}+\sup_{t \in [0,1]} \int _{0}^{1}k_{1}(t,s)g_{1}(s) \,ds \biggr]\\ =&\frac{1}{C_{1}}\sup_{\varOmega ^{\rho _{1},\rho _{2}}}f_{1}(w_{1},w_{2},z_{1},z_{2})< \rho _{1}, \end{aligned}$$

which is a contradiction.

Now we show that that the index of T is 0 on $V_{s_{1},s_{2}}$.

Consider $l(t)=1$ for $t\in [ 0,1]$ and note that $(l,l)\in K$. Now we claim that

$$ (u,v)\neq T(u,v)+\lambda (l,l)\quad \text{for }(u,v)\in \partial V_{s_{1},s_{2}} \text{ and } \lambda \geq 0. $$

Assume, by contradiction, that there exist $(u,v)\in \partial V_{s_{1},s_{2}}$ and $\lambda \geq 0$ such that $(u,v)=T(u,v)+\lambda (l,l)$.

Without loss of generality, we can assume that $\min_{t\in [a_{1},b_{1}]} u(t)= s_{1}$ and $s_{1}\leq u(t)\leq {s_{1}/a_{1}}$ for $t\in [a_{1},b_{1}]$. Then, for $t\in [ a_{1},b_{1}]$, we obtain

$$ u(t) =\gamma _{1}(t)\alpha _{1}[u]+\delta _{1}(t)\beta _{1}[u]+ \int _{0}^{1}k_{1}(t,s)g_{1}(s)f_{1} \biggl(u(s),v(s),\frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds+\lambda . $$

(3.5)

Applying $\alpha _{1}$ and $\beta _{1}$ to both sides of (3.5) gives

$$\begin{aligned}& \begin{aligned} \alpha _{1}[u]&=\alpha _{1}[\gamma _{1}]\alpha _{1}[u]+\alpha _{1}[ \delta _{1}]\beta _{1}[u]\\ &\quad {}+ \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s),\frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds+\lambda \alpha _{1}[1], \end{aligned} \\& \begin{aligned} \beta _{1}[u]&=\beta _{1}[\gamma _{1}]\alpha _{1}[u]+\beta _{1}[ \delta _{1}]\beta _{1}[u]\\ &\quad {}+ \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s),\frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds+\lambda \beta _{1}[1]. \end{aligned} \end{aligned}$$

Thus we have

$$\begin{aligned} &\bigl(1-\alpha _{1}[\gamma _{1}]\bigr)\alpha _{1}[u]-\alpha _{1}[\delta _{1}] \beta _{1}[u]\\ &\quad = \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds+\lambda \alpha _{1}[1], \\ &{-}\beta _{1}[\gamma _{1}]\alpha _{1}[u]+\bigl(1- \beta _{1}[\delta _{1}]\bigr) \beta _{1}[u]\\ &\quad = \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds+\lambda \beta _{1}[1]. \end{aligned}$$

Therefore

$$ N_{1} \begin{pmatrix} \alpha _{1}[u] \\ \beta _{1}[u] \end{pmatrix}= \begin{pmatrix} \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds+\lambda \alpha _{1}[1] \\ \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds+\lambda \beta _{1}[1] \end{pmatrix}. $$

Applying the matrix $(N_{1})^{-1}$ to both sides of the last equality, we obtain

$$\begin{aligned} \begin{pmatrix} \alpha _{1}[u] \\ \beta _{1}[u] \end{pmatrix}&=(N_{1})^{-1} \begin{pmatrix} \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds+\lambda \alpha _{1}[1] \\ \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds+\lambda \beta _{1}[1] \end{pmatrix} \\ &\geq (N_{1})^{-1} \begin{pmatrix} \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds \\ \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)f_{1} (u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } )\,ds \end{pmatrix}. \end{aligned}$$

Thus, as in the previous step, we have

$$\begin{aligned} u(t)&\geq \frac{1}{D_{1}} \biggl[ \bigl[\gamma _{1}(t) \bigl(1- \beta _{1}[ \delta _{1}]\bigr)+\delta _{1}(t)\beta _{1}[\gamma _{1}] \bigr]\\ &\quad {}\times \int _{0}^{1} \alpha _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ &\quad{}+ \bigl[\gamma _{1}(t)\alpha _{1}[\delta _{1}]+\delta _{1}(t) \bigl(1- \alpha _{1}[\gamma _{1}]\bigr) \bigr]\\ &\quad {}\times \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s),\frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \biggr] \\ &\quad{}+ \int _{0}^{1}k_{1}(t,s)g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds+\lambda . \end{aligned}$$

Then, for $t \in [a_{1},b_{1}]$, we obtain

$$\begin{aligned} u(t)&\geq \frac{1}{D_{1}} \biggl[ \bigl[\gamma _{1}(t) \bigl(1- \beta _{1}[ \delta _{1}]\bigr)+\delta _{1}(t)\beta _{1}[\gamma _{1}] \bigr]\\ &\quad {}\times \int _{a_{1}}^{b_{1}} \alpha _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ &\quad{}+ \bigl[\gamma _{1}(t)\alpha _{1}[\delta _{1}]+\delta _{1}(t) \bigl(1- \alpha _{1}[\gamma _{1}]\bigr) \bigr]\\ &\quad {}\times \int _{a_{1}}^{b_{1}}\beta _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s),\frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \biggr] \\ &\quad{}+ \int _{a_{1}}^{b_{1}}k_{1}(t,s)g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds+\lambda \\ &\geq \inf_{A_{1}^{s_{1},s_{2}}}f_{1}(w_{1},w_{2},z_{1},z_{2}) \biggl[\frac{1}{D_{1}} \biggl[ \bigl[\gamma _{1}(t) \bigl(1-\beta _{1}[ \delta _{1}]\bigr)+\delta _{1}(t)\beta _{1}[\gamma _{1}] \bigr] \int _{a_{1}}^{b_{1}} \alpha _{1}[k_{1}]g_{1}(s) \,ds \\ &\quad{}+ \bigl[\gamma _{1}(t)\alpha _{1}[\delta _{1}]+\delta _{1}(t) \bigl(1- \alpha _{1}[\gamma _{1}]\bigr) \bigr] \int _{a_{1}}^{b_{1}}\beta _{1}[k_{1}]g_{1}(s) \,ds \biggr] \\ &\quad {}+ \int _{a_{1}}^{b_{1}}k_{1}(t,s)g_{1}(s) \,ds \biggr]+\lambda . \end{aligned}$$

Taking the minimum over $[a_{1},b_{1}]$ gives

$$\begin{aligned} s_{1}&\geq \inf_{A_{1}^{s_{1},s_{2}}}f_{1}(w_{1},w_{2},z_{1},z_{2}) \biggl[\frac{1}{D_{1}} \biggl[ \bigl[a_{1}\bigl(1-\beta _{1}[\delta _{1}]\bigr)+a_{1} \Vert \delta _{1} \Vert _{\infty }\beta _{1}[\gamma _{1}] \bigr] \int _{a_{1}}^{b_{1}} \alpha _{1}[k_{1}]g_{1}(s) \,ds \\ &\quad{}+ \bigl[a_{1}\alpha _{1}[\delta _{1}]+a_{1} \Vert \delta _{1} \Vert _{ \infty } \bigl(1-\alpha _{1}[\gamma _{1}]\bigr) \bigr] \int _{a_{1}}^{b_{1}} \beta _{1}[k_{1}]g_{1}(s) \,ds \biggr] \\ &\quad {}+\inf_{t \in [a_{1},b_{1}]} \int _{a_{1}}^{b_{1}}k_{1}(t,s)g_{1}(s) \,ds \biggr]+\lambda \\ &>M_{1}s_{1}\frac{1}{M_{1}}+ \lambda , \end{aligned}$$

a contradiction. □

By means of Theorem 3.1 and the fixed point index properties in Theorem 2.3 we can state results on the existence of multiple positive solutions for system (2.3). Here we enunciate a result on the existence of two positive solutions (see [32, 33] for the conditions that ensure three or more positive results).

Theorem 3.2

Suppose that there exist$\rho _{i},s _{i},\theta _{i}\in (0,\infty )$with$\rho _{i}< s_{i}$and$\frac{s_{i}}{a_{i}}<\theta _{i}$, $i=1,2$, such that

$$\begin{aligned} &\sup_{\varOmega ^{\rho _{1},\rho _{2}}} f_{i}(w_{1},w_{2},z_{1},z_{2})< C_{i} \rho _{i}, \\ &\inf_{A_{i}^{s_{1},s_{2}}} f_{i}(w_{1},w_{2},z_{1},z_{2})>M_{i}s_{i}, \\ &\sup_{\varOmega ^{\theta _{1},\theta _{2}}} f_{i}(w_{1},w_{2},z_{1},z_{2})< C_{i}, \theta _{i}. \end{aligned}$$

Then system (2.3) has at least two positive radial solutions.

Example 3.3

Note that Theorems 3.1 and 3.2 can be applied when the nonlinearities $f_{i}$ are of the type

$$ f_{i}\bigl(u,v, \vert \nabla u \vert , \vert \nabla v \vert \bigr)=\bigl(\delta _{i} u^{\alpha _{i}}+ \gamma _{i} v^{\beta _{i}}\bigr)\zeta _{i}\bigl(u,v, \vert \nabla u \vert , \vert \nabla v \vert \bigr) $$

with continuous functions $\zeta _{i}$ bounded by a strictly positive constant, $\alpha _{i},\beta _{i}>1$, and suitable $\delta _{i},\gamma _{i} \geq 0$.

Setting

$$ q(w)=w^{3}\chi _{[0,40]}(w)+ \bigl\vert 65\text{,}600-w^{2} \bigr\vert \chi _{(40,360)}(w)+ \frac{100}{101} \vert 65\text{,}000-w \vert \chi _{[360,+\infty )}(w), $$

we can consider in $\mathbb{R}^{3}$ the system of BVPs

$$\begin{aligned}& \textstyle\begin{cases} -\Delta u = \frac{1}{30 \vert x \vert ^{4}} (2-\sin ( \vert \nabla u \vert ^{2}+ \vert \nabla v \vert ^{2}) q(u) &\text{in } \varOmega , \\ -\Delta v= \frac{1}{\pi \vert x \vert ^{4}} \arctan (1+ \vert \nabla u \vert ^{2}+ \vert \nabla v \vert ^{2} ) q(v)&\text{in } \varOmega , \\ \lim_{ \vert x \vert \to \infty }u(x)=u (\frac{1}{4} ), \qquad 2 u-4\frac{\partial u}{\partial r}=u ( \frac{1}{2} )& \text{for } \vert x \vert =1, \\ \lim_{ \vert x \vert \to \infty }v(x)=v (\frac{1}{4} ), \qquad 3v- 2\frac{\partial v}{\partial r}=v (\frac{1}{2} )& \text{for } \vert x \vert =1. \end{cases}\displaystyle \end{aligned}$$

(3.6)

Let $[a_{1},b_{1}]=[a_{2},b_{2}]= [\frac{1}{4},\frac{1}{2} ]$. By direct computation we obtain

$$\begin{aligned} &D_{i}=\frac{c_{i}-1}{4(c_{i}+d_{i})}; \\ &\sup_{t\in [0,1]} \int _{0}^{1}k_{i}(t,s)g_{i}(s) \,ds= \frac{(c_{i}+2d_{i})^{2}}{8(c_{i}+d_{i})^{2}};\qquad \inf_{t\in [ \frac{1}{4},\frac{1}{2} ]} \int _{\frac{1}{4}}^{\frac{1}{2}}k_{i}(t,s)g_{i}(s) \,ds= \frac{5c_{i}+8d_{i}}{128 (c_{i} + d_{i})}; \\ & \int _{0}^{1}\alpha _{i}[k_{i}]g_{i}(s) \,ds= \frac{(3c_{i}+7d_{i})}{32(c_{i}+d_{i})}; \qquad \int _{0}^{1}\beta _{i}[k_{i}]g_{i}(s) \,ds= \frac{(c_{i}+3d_{i})}{8(c_{i}+d_{i})}; \\ & \int _{a_{i}}^{b_{i}}\alpha _{i}[k_{i}]g_{i}(s) \,ds= \frac{5c_{i}+8d_{i}}{128 (c_{i} + d_{i})};\qquad \int _{a_{i}}^{b_{i}} \beta _{i}[k_{i}]g_{i}(s) \,ds=\frac{3}{32 } \biggl(1 - \frac{c_{i}}{2 (c_{i} + d_{i})} \biggr). \end{aligned}$$

Since in our example the mixed perturbed conditions state that $c_{1}=2$, $d_{1}=4$, $c_{2}=3$, and $d_{2}=2$, we easily compute $C_{i}$ and $M_{i}$:

$$\begin{aligned} C_{1}=\frac{9}{47},\qquad C_{2}=\frac{25}{48}, \qquad M_{1}=\frac{96}{41}, \qquad M_{2}= \frac{16}{3}. \end{aligned}$$

Choosing $\rho _{1}=\rho _{2}=1$, $s_{1}=9$, $s_{2}=5$, $\theta _{1}=40\text{,}000$, and $\theta _{2}=65\text{,}000$, we have

$$\begin{aligned} &\sup_{\varOmega ^{\rho _{1},\rho _{2}}} f_{1} \leq \frac{1}{10} \sup _{u \in [0,\rho _{1}]} q(u)=\frac{1}{10}q(\rho _{1})= \frac{1}{10}< \frac{9}{47}=C_{1}\rho _{1}, \\ & \inf_{A_{1}^{s_{1},s_{2}}} f_{1} \geq \frac{1}{30}\inf _{u \in [s_{1},4s_{1}]} q(u)=\frac{1}{30}q(s_{1})= \frac{243}{10}>\frac{192}{41}=M_{1}s_{1}, \\ &\sup_{\varOmega ^{\theta _{1},\theta _{2}}} f_{1} \leq \frac{1}{10}\sup _{u \in [0,\theta _{1}]} q(u)=\frac{1}{10}q(40)=6400< \frac{36\text{,}000}{47}=C_{1} \theta _{1}, \\ &\sup_{\varOmega ^{\rho _{1},\rho _{2}}} f_{2} \leq \frac{1}{2}\sup _{v \in [0,\rho _{2}]} q(v)=\frac{1}{2}q(\rho _{2})= \frac{1}{2}< \frac{25}{48}=C_{2}\rho _{2}, \\ &\inf_{A_{2}^{s_{1},s_{2}}} f_{2} \geq \frac{1}{4}\inf _{v \in [s_{2},4s_{2}]} q(v)=\frac{1}{4}q(s_{2})= \frac{125}{4}>\frac{80}{3}=M_{2}s_{2}, \\ &\sup_{\varOmega ^{\theta _{1},\theta _{2}}} f_{2} \leq \frac{1}{2} \sup _{v \in [0,\theta _{2}]} q(v)=\frac{1}{2}q(40)=3200< \frac{203\text{,}125}{6}=C_{2} \theta _{2}, \end{aligned}$$

where the suprema and infima are computed on

$$\begin{aligned} &\varOmega ^{\rho _{1},\rho _{2}}=\varOmega ^{1,1}= [0,1]\times [0,1] \times [0,+\infty )\times [0,+\infty ); \\ &A_{1}^{s_{1},s_{2}}=A_{1}^{9,5}= [9,36]\times [0,20]\times [0,+ \infty )\times [0,+\infty ); \\ &A_{2}^{s_{1},s_{2}}=A_{2}^{9,5}= [0,36]\times [5,20]\times [0,+ \infty )\times [0,+\infty ); \\ &\varOmega ^{\theta_{1},\theta _{2}}=\varOmega ^{40\text{,}000,60\text{,}000}= [0,40\text{,}000] \times [0,60\text{,}000] \times [0,+\infty )\times [0,+\infty ). \end{aligned}$$

Then the hypotheses of Theorem 3.2 are satisfied, and hence system (3.6) has at least two positive solutions.

4 Nonexistence results

We now show a nonexistence result for the system of elliptic equations (2.3) when the the functions $f_{i}$ have an enough “small” or “large” growth.

Theorem 4.1

Assume that one of following conditions holds:

$$\begin{aligned}& f_{i}(w_{1},w_{2},z_{1},z_{2})< C_{i} w_{i} ,\quad w_{i}>0 \textit{ for }i=1,2, \end{aligned}$$

(4.1)

$$\begin{aligned}& f_{i}(w_{1},w_{2}, z_{1},z_{2})>M_{i} w_{i} ,\quad w_{i}>0 \textit{ for }i=1,2. \end{aligned}$$

(4.2)

Then the only possible positive solution of system (2.3) is the zero one.

Proof

Suppose that (4.1) holds and assume that there exists a solution $(\bar{u},\bar{v})$ of (2.3), $(\bar{u},\bar{v})\neq (0,0)$; then $(u,v):=(\bar{u}\circ r,\bar{v}\circ r)$ is a fixed point of T. Let, for example, $\|(u,v)\|=\|u\| \leq 4\|u\|_{\infty }\neq 0$.

Then, for $t\in [0,1]$, taking into account (3.4), we have

$$\begin{aligned} u(t) &< C_{1} \biggl(\frac{1}{D_{1}} \biggl[ \bigl[\bigl(1-\beta _{1}[\delta _{1}]\bigr)+ \Vert \delta _{1} \Vert _{\infty }\beta _{1}[\gamma _{1}] \bigr] \int _{0}^{1} \alpha _{1}[k_{1}]g_{1}(s)u(s) \,ds \\ &\quad{}+ \bigl[\alpha _{1}[\delta _{1}]+ \Vert \delta _{1} \Vert _{\infty } \bigl(1- \alpha _{1}[ \gamma _{1}]\bigr) \bigr] \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s)u(s) \,ds \biggr] + \int _{0}^{1}k_{1}(t,s)g_{1}(s)u(s) \,ds \biggr) \\ &\leq C_{1} \Vert u \Vert _{\infty } \biggl( \frac{1}{D_{1}} \biggl[ \bigl[\bigl(1- \beta _{1}[\delta _{1}]\bigr)+ \Vert \delta _{1} \Vert _{\infty }\beta _{1}[\gamma _{1}] \bigr] \int _{0}^{1}\alpha _{1}[k_{1}]g_{1}(s) \,ds \\ &\quad{}+ \bigl[\alpha _{1}[\delta _{1}]+ \Vert \delta _{1} \Vert _{\infty } \bigl(1- \alpha _{1}[ \gamma _{1}]\bigr) \bigr] \int _{0}^{1}\beta _{1}[k_{1}]g_{1}(s) \,ds \biggr] + \int _{0}^{1}k_{1}(t,s)g_{1}(s) \,ds \biggr). \end{aligned}$$

For $u>0$, taking the supremum for $t\in [0,1]$, we have $\|u\|_{\infty }<\|u\|_{\infty }$, a contradiction.

Suppose that (4.2) holds and assume that there exists $(u,v)\in K$ such that $(u,v)=T(u,v)$ and $(u,v)\neq (0,0)$. Let, for example, $\|u\|_{\infty }\neq 0$; then $\sigma :=\min_{t\in [a_{1},b_{1}]}u(t)>0$ since $u \in K_{1}$. Thus, as in the proof of Theorem 3.1, we have, for $t \in [a_{1},b_{1}]$,

$$\begin{aligned} u(t)&\geq \frac{1}{D_{1}} \biggl[ \bigl[a_{1}\bigl(1-\beta _{1}[\delta _{1}]\bigr)+a_{1} \Vert \delta _{1} \Vert _{\infty }\beta _{1}[\gamma _{1}] \bigr]\\ &\quad {}\times \int _{a_{1}}^{b_{1}} \alpha _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ &\quad{}+ \bigl[a_{1}\alpha _{1}[\delta _{1}]+a_{1} \Vert \delta _{1} \Vert _{ \infty } \bigl(1-\alpha _{1}[\gamma _{1}]\bigr) \bigr]\\ &\quad {}\times \int _{a_{1}}^{b_{1}} \beta _{1}[k_{1}]g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \biggr] \\ &\quad{}+ \int _{a_{1}}^{b_{1}}k_{1}(t,s)g_{1}(s)f_{1} \biggl(u(s),v(s), \frac{ \vert u'(s) \vert }{ \vert r'(s) \vert },\frac{ \vert v'(s) \vert }{ \vert r'(s) \vert } \biggr)\,ds \\ &>M_{1} \biggl(\frac{1}{D_{1}} \biggl[ \bigl[a_{1} \bigl(1-\beta _{1}[\delta _{1}]\bigr)+a_{1} \Vert \delta _{1} \Vert _{\infty }\beta _{1}[\gamma _{1}] \bigr] \int _{a_{1}}^{b_{1}} \alpha _{1}[k_{1}]g_{1}(s)u(s) \,ds \\ &\quad{}+ \bigl[a_{1}\alpha _{1}[\delta _{1}]+a_{1} \Vert \delta _{1} \Vert _{ \infty } \bigl(1-\alpha _{1}[\gamma _{1}]\bigr) \bigr] \int _{a_{1}}^{b_{1}} \beta _{1}[k_{1}]g_{1}(s)u(s) \,ds \biggr] \\ &\quad {}+ \int _{a_{1}}^{b_{1}}k_{1}(t,s)g_{1}(s)u(s) \,ds \biggr) \\ &\geq M_{1} \sigma \biggl(\frac{1}{D_{1}} \biggl[ \bigl[a_{1}\bigl(1-\beta _{1}[ \delta _{1}] \bigr)+a_{1} \Vert \delta _{1} \Vert _{\infty }\beta _{1}[\gamma _{1}] \bigr] \int _{a_{1}}^{b_{1}}\alpha _{1}[k_{1}]g_{1}(s) \,ds \\ &\quad{}+ \bigl[a_{1}\alpha _{1}[\delta _{1}]+a_{1} \Vert \delta _{1} \Vert _{ \infty } \bigl(1-\alpha _{1}[\gamma _{1}]\bigr) \bigr] \int _{a_{1}}^{b_{1}} \beta _{1}[k_{1}]g_{1}(s) \,ds \biggr] + \int _{a_{1}}^{b_{1}}k_{1}(t,s)g_{1}(s) \,ds \biggr). \end{aligned}$$

For $u>0$, taking the infimum for $t\in [a_{1},b_{1}]$, we obtain $\sigma >\sigma$, a contradiction. □