Skip to main content

Advertisement

We’d like to understand how you use our websites in order to improve them. Register your interest.

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.

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. [1821], 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. (1)

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

  2. (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.

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. (1)

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

  2. (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

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

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.

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. □

References

  1. 1.

    Abebe, A., Chhetri, M., Sankar, L., Shivaji, R.: Positive solutions for a class of superlinear semipositone systems on exterior domains. Bound. Value Probl. 2014 198, 9 pp. (2014)

  2. 2.

    Aftalion, A., Busca, J.: Symétrie radiale pour des problèmes elliptiques surdéterminés posés dans des domaines extérieurs. C. R. Acad. Sci., Sér. 1 Math. 324, 633–638 (1997)

  3. 3.

    Agarwal, R.P., O’Regan, D., Yan, B.: Multiple positive solutions of singular Dirichlet second order boundary-value problems with derivative dependence. J. Dyn. Control Syst. 15, 1–26 (2009)

  4. 4.

    Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709 (1976)

  5. 5.

    Averna, D., Motreanu, D., Tornatore, E.: Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence. Appl. Math. Lett. 61, 102–107 (2016)

  6. 6.

    Butler, D., Ko, E., Lee, E.K., Kyoung, E., Shivaji, R.: Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Commun. Pure Appl. Anal. 13, 2713–2731 (2014)

  7. 7.

    Castro, A., Sankar, L., Shivaji, R.: Uniqueness of nonnegative solutions for semipositone problems on exterior domains. J. Math. Anal. Appl. 394, 432–437 (2012)

  8. 8.

    Cianciaruso, F.: Existence of solutions of semilinear systems with gradient dependence via eigenvalue criteria. J. Math. Anal. Appl. 482(1), 1–21 (2020)

  9. 9.

    Cianciaruso, F., Infante, G., Pietramala, P.: Multiple positive radial solutions for Neumann elliptic systems with gradient dependence. Math. Methods Appl. Sci. 41(16), 6358–6367 (2018)

  10. 10.

    Cianciaruso, F., Infante, G., Pietramala, P.: Nonzero radial solutions for elliptic systems with coupled functional BCs in exterior domains. Proc. Edinb. Math. Soc. 62(3), 747–769 (2019)

  11. 11.

    Cianciaruso, F., Pietramala, P.: Multiple positive solutions of a \((p_{1},p_{2})\)-Laplacian system with nonlinear BCs. Bound. Value Probl. 2015, 163, 18 pp. (2015)

  12. 12.

    Cianciaruso, F., Pietramala, P.: Semilinear elliptic systems with dependence on the gradient. Mediterr. J. Math. 15(4), 152, 13 pp. (2018)

  13. 13.

    De Figueiredo, D.G., Sánchez, J., Ubilla, P.: Quasilinear equations with dependence on the gradient. Nonlinear Anal. 71, 4862–4868 (2009)

  14. 14.

    De Figueiredo, D.G., Ubilla, P.: Superlinear systems of second-order ODE’s. Nonlinear Anal. 68, 1765–1773 (2008)

  15. 15.

    Dhanya, R., Morris, Q., Shivaji, R.: Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball. J. Math. Anal. Appl. 434, 1533–1548 (2016)

  16. 16.

    Djebali, S., Orpel, A.: The continuous dependence on parameters of solutions for a class of elliptic problems on exterior domains. Nonlinear Anal. 73, 660–672 (2010)

  17. 17.

    do Ó, J.M., Lorca, S., Sánchez, J., Ubilla, P.: Positive radial solutions for some quasilinear elliptic systems in exterior domains. Commun. Pure Appl. Anal. 5, 571–581 (2006)

  18. 18.

    do Ó, J.M., Lorca, S., Sánchez, J., Ubilla, P.: Non-homogeneous elliptic equations in exterior domains. Proc. R. Soc. Edinb. A 136, 139–147 (2006)

  19. 19.

    do Ó, J.M., Lorca, S., Sánchez, J., Ubilla, P.: Positive solutions for a class of multiparameter ordinary elliptic systems. J. Math. Anal. Appl. 332, 1249–1266 (2007)

  20. 20.

    do Ó, J.M., Lorca, S., Sánchez, J., Ubilla, P.: Superlinear ordinary elliptic systems involving parameters. Mat. Contemp. 32, 107–127 (2007)

  21. 21.

    do Ó, J.M., Lorca, S., Sánchez, J., Ubilla, P.: Positive solutions for some nonlocal and nonvariational elliptic systems. Complex Var. Elliptic Equ. 61, 297–314 (2016)

  22. 22.

    Faria, L.F.O., Miyagaki, O.H., Pereira, F.R.: Quasilinear elliptic system in exterior domains with dependence on the gradient. Math. Nachr. 287, 61–373 (2014)

  23. 23.

    Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979)

  24. 24.

    Goodrich, C.S.: The effect of a nonstandard cone on existence theorem applicability in nonlocal boundary value problems. J. Fixed Point Theory Appl. 19(4), 2629–2646 (2017)

  25. 25.

    Goodrich, C.S.: On semipositone non-local boundary-value problems with nonlinear or affine boundary conditions. Proc. Edinb. Math. Soc. 60(3), 635–649 (2017)

  26. 26.

    Goodrich, C.S.: New Harnack inequalities and existence theorems for radially symmetric solutions of elliptic PDEs with sign changing or vanishing Green’s function. J. Differ. Equ. 264(1), 236–262 (2018)

  27. 27.

    Goodrich, C.S.: Radially symmetric solutions of elliptic PDEs with uniformly negative weight. Ann. Mat. Pura Appl. 197(5), 1585–1611 (2018)

  28. 28.

    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988)

  29. 29.

    Hai, D.D., Shivaji, R.: Existence and multiplicity of positive radial solutions for singular superlinear elliptic systems in the exterior of a ball. J. Differ. Equ. 266(4), 2232–2243 (2019)

  30. 30.

    Han, G., Wang, J.: Multiple positive radial solutions of elliptic equations in an exterior domain. Monatshefte Math. 148, 217–228 (2006)

  31. 31.

    Ko, E., Ramaswamy, M., Shivaji, R.: Uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball. J. Math. Anal. Appl. 423, 399–409 (2015)

  32. 32.

    Lan, K.Q., Lin, W.: Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli. Nonlinear Anal. 74, 7184–7197 (2011)

  33. 33.

    Lan, K.Q., Webb, J.R.L.: Positive solutions of semilinear differential equations with singularities. J. Differ. Equ. 148, 407–421 (1998)

  34. 34.

    Lee, E.K., Shivaji, R., Son, B.: Positive radial solutions to classes of singular problems on the exterior domain of a ball. J. Math. Anal. Appl. 434, 1597–1611 (2016)

  35. 35.

    Lee, Y.H.: A multiplicity result of positive radial solutions for a multiparameter elliptic system on an exterior domain. Nonlinear Anal. 45, 597–611 (2001)

  36. 36.

    Motreanu, D., Sciammetta, A., Tornatore, E.: A sub-supersolution approach for Neumann boundary value problems with gradient dependence. Nonlinear Anal., Real World Appl. 54, 103096 (2020) 12 pp.

  37. 37.

    Orpel, A.: Nonlinear BVPS with functional parameters. J. Differ. Equ. 246, 1500–1522 (2009)

  38. 38.

    Orpel, A.: Increasing sequences of positive evanescent solutions of nonlinear elliptic equations. J. Differ. Equ. 259, 1743–1756 (2015)

  39. 39.

    Sankar, L., Sasi, S., Shivaji, R.: Semipositone problems with falling zeros on exterior domains. J. Math. Anal. Appl. 401, 146–153 (2013)

  40. 40.

    Stanczy, R.: Positive solutions for superlinear elliptic equations. J. Math. Anal. Appl. 283, 159–166 (2003)

  41. 41.

    Webb, J.R.L., Infante, G.: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. 74(3), 673–693 (2006)

  42. 42.

    Webb, J.R.L., Infante, G.: Non-local boundary value problems of arbitrary order. J. Lond. Math. Soc. 79(1), 238–258 (2009)

Download references

Acknowledgements

The authors thank the referees for carefully reading the manuscript and providing helpful suggestions and comments.

Availability of data and materials

Not applicable.

Funding

Not applicable.

Author information

Affiliations

Authors

Contributions

The authors declare that the study was realized in collaboration with the same responsibility. All the authors read and approved the final manuscript.

Corresponding author

Correspondence to Luigi Muglia.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cianciaruso, F., Muglia, L. & Pietramala, P. Coupled elliptic systems depending on the gradient with nonlocal BCs in exterior domains. Bound Value Probl 2020, 87 (2020). https://doi.org/10.1186/s13661-020-01384-7

Download citation

MSC

  • 35J66
  • 45G15

Keywords

  • Elliptic system
  • Dependence on the gradient
  • Nonlocal boundary conditions
  • Fixed point index
  • Cone
  • Positive solution