1 Introduction

In this paper we study non-existence, existence and multiplicity of positive solutions for systems having the form

$$\begin{aligned} \left\{ \begin{array}{ll} {\mathcal {M}}(\text{ u })+\lambda _1\mu _1(|x|)f_1(\text{ u },\text{ v })=0 &{} { \text{ in } {\mathcal {B}}(R)},\\ {\mathcal {M}}(\text{ v })+\lambda _2\mu _2(|x|)f_2(\text{ u },\text{ v })=0 &{} { \text{ in } {\mathcal {B}}(R)},\\ \text{ u }|_{\partial {\mathcal {B}}(R)}=0=\text{ v }|_{\partial {\mathcal {B}}(R),} \end{array} \right. \end{aligned}$$
(1.1)

where \({\mathcal {B}}(R)=\{x\in {\mathbb {R}}^N : |x|<R\}\) (\(R>0\), \(N\ge 2\)), \({\mathcal {M}}\) stands for the mean curvature operator in Minkowski space

$$\begin{aligned} {\mathcal {M}}(\text{ w })=\text{ div } \left( \frac{\nabla \text{ w }}{\sqrt{1-|\nabla \text{ w }|^2}}\right) \! , \end{aligned}$$

the parameters \(\lambda _1,\lambda _2\) are positive, the functions \(\mu _1,\; \mu _2:[0,R]\rightarrow [0,\infty )\) are continuous with \(\mu _1(r)>0<\mu _2(r)\) for all \(r\in (0,R],\) under the following hypothesis on the continuous functions \(f_1, f_2:[0,+\infty )^2\rightarrow [0,+\infty ):\)

(H):

(i) \(f_1(s,t), f_2(s,t)\) are quasi-monotone nondecreasing with respect to both s and t; (ii) there exist constants \(c>0\), \(p_1, q_2>1\) and \(q_1, p_2> 0\) such that

$$\begin{aligned} \begin{array}{ll} 0<f_1(s,t)\le c s^{p_1}t^{q_1},\\ 0<f_2(s,t)\le c s^{p_2}t^{q_2}, \end{array} \end{aligned}$$
(1.2)

for all \(s,t>0\).

Recall, a function \(g(s,t):[0,\infty )^2\rightarrow [0,\infty )\) is said to be quasi-monotone nondecreasing with respect to t (resp. s) if for fixed s (resp. t) one has

$$\begin{aligned} g(s, t_1 )\le g(s,t_2)\text { as } t_1\le t_2\quad (\text {resp. } g(s_1, t )\le g(s_2,t)\text { as } s_1\le s_2). \end{aligned}$$

In recent years, many papers were devoted to the study of Dirichlet problems for a single equation with operator \({\mathcal {M}}\) in a ball in \({\mathbb {R}}^N\) [1,2,3, 5, 7, 8, 13], while at our best knowledge, for systems with such an operator the study was recently initiated in [9]. So, in [7], for systems involving Lane-Emden type perturbations of the operator \({\mathcal {M}}\) and having a variational structure:

$$\begin{aligned} \left\{ \begin{array}{ll} {\mathcal {M}}(\text{ u })+\lambda \mu (|x|)(p+1)\text{ u }^p\text{ v }^{q+1}=0, &{} { \text{ in } {\mathcal {B}}(R)},\\ {\mathcal {M}}(\text{ v })+\lambda \mu (|x|)(q+1)\text{ u }^{p+1}\text{ v }^q=0, &{} { \text{ in } {\mathcal {B}}(R)},\\ \text{ u }|_{\partial {\mathcal {B}}(R)}=0=\text{ v }|_{\partial {\mathcal {B}}(R)}, \end{array} \right. \end{aligned}$$
(1.3)

where the positive exponents \(p,\ q\) satisfy \(\max \{p,q\}>1\) and the function \(\mu :[0,R]\rightarrow [0, \infty )\) is continuous and \(\mu (r)>0\) for all \(r\in (0,R]\), it was shown that there exists \(\varLambda >0\) such that (1.3) has zero, at least one or at least two positive solutions according to \(\lambda \in (0,\varLambda ),\) \(\lambda =\varLambda \) or \(\lambda > \varLambda \). This result extends the corresponding one obtained in [3] in the case of a single equation.

Then, in the recent paper [8] are considered non-potential radial systems having the form

$$\begin{aligned} \left\{ \begin{array}{ll} {\mathcal {M}}(\text{ u })+\lambda _1\mu _1(|x|)\text{ u }^{p_1}\text{ v }^{q_1}=0, &{} { \text{ in } {\mathcal {B}}(R)},\\ {\mathcal {M}}(\text{ v })+\lambda _2\mu _2(|x|)\text{ u }^{p_2}\text{ v }^{q_2}=0, &{} { \text{ in } {\mathcal {B}}(R)},\\ u|_{\partial {\mathcal {B}}(R)}=0=v|_{\partial {\mathcal {B}}(R)}, \end{array} \right. \end{aligned}$$
(1.4)

where \(\lambda _1,\lambda _2\) are two positive parameters, \(p_1,p_2,q_1,q_2\) are positive exponents with \(\min \{p_1, q_2\}>1 \) and the weight functions \(\mu _1,\; \mu _2:[0,R]\rightarrow [0,\infty )\) are assumed to be continuous with \(\mu _1(r)>0<\mu _2(r)\) for all \(r\in (0,R]\). Using fixed point index estimations and lower and upper solutions method, it was proved the existence of a continuous curve \(\varGamma \) splitting the first quadrant into two disjoint open sets \({\mathcal {O}}_1\) and \({\mathcal {O}}_2\) such that the system (1.4) has zero, at least one or at least two positive, radial solutions according to \((\lambda _1, \lambda _2)\in {\mathcal {O}}_1,\) \((\lambda _1, \lambda _2)\in \varGamma \) or \((\lambda _1, \lambda _2)\in {\mathcal {O}}_2,\) respectively.

On the other hand, in paper [10] are studied multiparameter Dirichlet systems having the form

$$\begin{aligned} \left\{ \begin{array}{ll} {\mathcal {M}}(u)+\lambda _1f_1(u,v)=0, &{} \hbox {in }\varOmega ,\\ {\mathcal {M}}(v)+\lambda _2f_2(u,v)=0, &{} \hbox {in } \varOmega ,\\ u|_{\partial \varOmega }=0=v|_{\partial \varOmega }, \end{array} \right. \end{aligned}$$

where \(\varOmega \) is a general bounded smooth domain in \({\mathbb {R}}^N\) and the continuous functions \(f_1,f_2\) satisfy some sign, growth and quasi-monotonicity conditions. For such systems it has been obtained the existence of a hyperbola like curve which separates the first quadrant in two disjoint sets, an open one \({\mathcal {O}}\) and a closed one \({\mathcal {F}}\), such that the system has zero or at least one strictly positive solution, according to \((\lambda _1, \lambda _2)\in {\mathcal {O}}\) or \((\lambda _1, \lambda _2)\in {\mathcal {F}}\). Moreover, it has been showed that inside of \({\mathcal {F}}\) there exists an infinite rectangle in which the parameters being, the system has at least two strictly positive solutions. The approaches are based on a lower and upper solutions method and topological degree type arguments. This result extends, in some sense, to non-radial systems the existence/non-existence and multiplicity result obtained in [8] for the radial case.

In view of the above, the aim of this paper is two fold: firstly to complete the result obtained in [8] by showing that this still remains valid for more general systems of type (1.1) and secondly, to show that, at least in the radial case, a sharper result as the one in [8] can be obtained for such more general nonlinearities.

Since the solvability of (1.1) is guaranteed by [8, Corrolary 2.1], the main interest concerns the non-existence, existence and multiplicity of positive radial solutions. In this direction, the techniques employed in [8] for Lane-Emden systems will be adapted for system (1.1); we point out that the growth conditions on \(f_1, f_2\) from hypothesis (H) play a key role in the proof of the main result.

Therefore, we show that there exists a continuous curve \(\varGamma \) splitting the first quadrant into two disjoint unbounded, open sets \({\mathcal {O}}_1\) and \({\mathcal {O}}_2\) such that the system (1.1) has zero, at least one or at least two positive radial solutions according to \((\lambda _1, \lambda _2)\in {\mathcal {O}}_1,\) \((\lambda _1, \lambda _2)\in \varGamma \) or \((\lambda _1, \lambda _2)\in {\mathcal {O}}_2,\) respectively. The set \({\mathcal {O}}_1\) is adjacent to the coordinates axes \(0 \lambda _1\) and \(0 \lambda _2\) and the curve \(\varGamma \) approaches asymptotically to two lines parallel to the axes \(0 \lambda _1\) and \(0 \lambda _2\) (Theorem 3.1).

Also, notice that, at least when speaking about radial solutions, this result is sharper then the one obtained in [10] due to the fact that here, the curve \(\varGamma \) which delimitates the multiplicity of solutions is the optimal one. Moreover, the conditions in hypothesis (H) are satisfied by Lane-Emden nonlinearities, so the result in [8] is recovered.

The rest of the paper is organized as follows. In Section 2 we reduce problem (1.1) to a homogeneous mixed boundary value problem and also we recall some results concerning lower (and upper) solutions method and some fixed point index estimations proved in [8]. The main non-existence, existence and multiplicity result for the multiparameter system (1.1) is stated and proved in Section 3. An example of nonlinearities different from the ones in [8] is provided.

2 Preliminaries

As usual, when we are seeking for radial solutions of (1.1), by setting \(r=|x|\) and \(\text{ u }(x)=u(r)\), \(\text{ v }(x)=v(r)\), the Dirichlet problem (1.1) reduces to the homogeneous mixed boundary value problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} {[} r^{N-1}\varphi (u') ]^{'}+r^{N-1}\lambda _1\mu _1(r)f_1(u,v)=0, \\ {[} r^{N-1}\varphi (v') ]^{'}+r^{N-1}\lambda _2\mu _2(r)f_2(u,v)=0, \\ u'(0)=u(R)=0=v(R)=v'(0). \end{array}\right. } \end{aligned}$$
(2.1)

where

$$\begin{aligned} \varphi (y) = \frac{y}{\sqrt{1-y^2}} \quad (y\in {\mathbb {R}}, \; |y|<1). \end{aligned}$$

By a solution of (2.1) we mean a couple of nonnegative functions \((u,v)\in C^1[0,R]\times C^1[0,R]\) with \(||u'||_{\infty }<1,\) \(||v'||_{\infty }<1\) and \(r\mapsto r^{N-1}\varphi (u'(r)),\) \(r\mapsto r^{N-1}\varphi (v'(r))\) of class \(C^1\) on [0, R], which satisfies problem (2.1). Here and below, \(\Vert \cdot \Vert _{\infty }\) stands for the usual sup-norm on \(C:=C[0,R]\). We say that \(u\in C\) is positive if \(u>0\) on [0, R). By a positive solution of (2.1) we understand a solution (u, v) with both u and v positive.

Throughout this paper, the space \(C^1:=C^1[0,R]\) will be understood with the norm \(\Vert u\Vert _1=\Vert u\Vert _{\infty } + \Vert u'\Vert _{\infty },\) while the product space \(C^1\times C^1\) will be endowed with the norm \(||(u,v)||=\max \{||u||_{\infty }, ||v||_{\infty }\} + \max \{||u'||_{\infty } , ||v'||_{\infty }\} .\) We consider the closed subspace

$$\begin{aligned} C_M^1:=\{(u,v)\in C^1\times C^1:u'(0)=u(R)=0=v(R)=v'(0)\} \end{aligned}$$

and its closed, convex cone

$$\begin{aligned} K:=\{(u,v)\in C_M^1 : u\ge 0 \le v\text { on }[0,R]\}. \end{aligned}$$

Also we denote \(B(\rho ):=\{(u,v)\in K : \Vert (u,v)\Vert <\rho \}\).

Let us define the linear operators

$$\begin{aligned}&S:C\rightarrow C,\quad Su(r) = \frac{1}{r^{N-1}}\int _0^r t^{N-1}u(t)dt \quad (r\in (0,R]), \quad Su(0)=0; \\&P:C\rightarrow C^1, \quad Pu(r) = \int _r^R u(t)dt \quad (r\in [0,R]). \end{aligned}$$

It is easy to see that P is bounded and S is compact. Hence, the nonlinear operator \(P\circ \varphi ^{-1}\circ S:C\rightarrow C^1\) is compact. Denoting by \(N_{\lambda _i}\) the Nemytskii operator associated to \(\lambda _i\mu _if_i\) \((i=1,2),\) i.e.,

$$\begin{aligned} N_{\lambda _i}:C\times C\rightarrow C,\text { } N_{\lambda _i}(u,v) = \lambda _i\mu _i(\cdot )f_i(u_+(\cdot ), v_+(\cdot )) \quad (u,v \in C), \end{aligned}$$

\((s_+:=\max \{s,0\})\) we have that \(N_{\lambda _i}\) is continuous and takes bounded sets into bounded sets.

If A is a subset of K, we set

$$\begin{aligned} {\mathcal {K}}(A):=\{T \;|\; T:A\rightarrow K\text { is a compact operator}\}. \end{aligned}$$

Also, given a bounded open (in K) subset \(\mathcal{{O}}\) of K, we denote by \(i(T,\mathcal{{O}})\) the fixed point index of the operator \(T\in {\mathcal {K}}(\overline{\mathcal{{O}}})\) on \(\mathcal{{O}}\) with respect to K [6].

The following proposition follows from Propositions 2.1 and 2.2 in [8].

Proposition 2.1

  1. (i)

    A couple of functions \((u,v)\in K\) is a solution of (2.1) iff it is a fixed point of the compact nonlinear operator

    $$\begin{aligned} {\mathcal {D}}_{\lambda _1, \lambda _2}:K\rightarrow K,\quad {\mathcal {D}}_{\lambda _1,\lambda _2}=\left( P\circ \varphi ^{-1}\circ S \circ N_{\lambda _1}, P\circ \varphi ^{-1}\circ S \circ N_{\lambda _2} \right) . \end{aligned}$$

    In addition, for all \((u,v)\in K,\) it holds

    $$\begin{aligned} \Vert {\mathcal {D}}_{\lambda _1,\lambda _2}(u,v)\Vert < R+1. \end{aligned}$$
    (2.2)
  2. (ii)

    For all \(d\ge R+1\) it holds

    $$\begin{aligned} i({\mathcal {D}}_{\lambda _1,\lambda _2}, B(d)) = 1. \end{aligned}$$
    (2.3)

    In particular, problem (2.1) always has a solution.

The following two lemmas are immediate consequences of [8, Lemma 3.2] and [7, Lemma 2.4 ].

Lemma 2.1

Assume (H). If there is some \(M>0\) such that either

$$\begin{aligned} \lim \limits _{s\rightarrow 0_+}\frac{f_1(s,t)}{s}=0\text { uniformly with }t\in [0,M], \end{aligned}$$
(2.4)

or

$$\begin{aligned} \lim \limits _{t\rightarrow 0_+}\frac{f_2(s,t)}{t}=0\text { uniformly with }s\in [0,M], \end{aligned}$$
(2.5)

then there exists \(\rho _0=\rho _0(\lambda _1,\lambda _2)>0\) such that

$$\begin{aligned} i({\mathcal {D}}_{\lambda _1,\lambda _2}, B(\rho )) =1\text { for all }0<\rho \le \rho _0. \end{aligned}$$

Lemma 2.2

Under assumption (H), if (u, v) is a nontrivial solution of problem (2.1), then (u, v) is a positive solution with both u and v strictly decreasing.

Definition 2.1

A lower solution of (2.1) is a couple of nonnegative functions \((\alpha _u,\alpha _v)\in C^1\times C^1,\) such that \(\Vert \alpha _{u}'\Vert _{\infty }<1,\) \(\Vert \alpha _{v}'\Vert _{\infty }<1, \) the mappings \(r\mapsto r^{N-1}\varphi (\alpha _u'(r)),\) \(r\mapsto r^{N-1}\varphi (\alpha _v'(r))\) are of class \(C^1\) on [0, R] and satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} {[} r^{N-1}\varphi (\alpha _{u}')]'+r^{N-1}\lambda _1\mu _1(r)f_1(\alpha _u,\alpha _v)\ge 0, \\ {[} r^{N-1}\varphi (\alpha _{v}') ]'+r^{N-1}\lambda _2\mu _2(r)f_2(\alpha _u,\alpha _v)\ge 0, \\ \alpha _u(R) = 0,\quad \alpha _v(R) = 0. \end{array}\right. } \end{aligned}$$
(2.6)

An upper solution \((\beta _u,\beta _v)\in C^1\times C^1\) is defined by reversing the first two inequalities in (2.6) and asking \(\beta _u(R)\ge 0,\;\beta _v(R)\ge 0\) instead of \(\alpha _u(R) = 0,\;\alpha _v(R) = 0\).

The following lemma is an immediate consequence of [8, Lemma 3.1].

Lemma 2.3

Assume that (2.1) has a lower solution \((\alpha _u, \alpha _v)\) and \(f_1(s,t)\) (resp. \(f_2(s,t))\) is quasi-monotone nondecreasing with respect to t (resp. s) and let

$$\begin{aligned} {\mathcal {A}}_{\alpha }={\mathcal {A}}_{(\alpha _u, \alpha _v)}:=\{(u,v)\in K : \alpha _u\le u, \; \alpha _v\le v\}. \end{aligned}$$

Then, the following hold true:

  1. (i)

    problem (2.1) has always a solution in \({\mathcal {A}}_{\alpha };\)

  2. (ii)

    if (2.1) has an unique solution \((u_0, v_0)\) in \({\mathcal {A}}_{\alpha }\) and there exists \(\rho _0 > 0\) such that \({\overline{B}}((u_0, v_0), \rho _0):=\{(u,v)\in K \; : \; \Vert (u-u_0,v-v_0)\Vert \le \rho _0 \}\subset {\mathcal {A}}_{\alpha },\) then

    $$\begin{aligned} i({\mathcal {D}}_{\lambda _1,\lambda _2}, B((u_0, v_0),\rho ))=1,\quad \text {for all } 0<\rho \le \rho _0. \end{aligned}$$

3 Non-existence, existence and multiplicity

In this section, under hypothesis (H), we study the existence and multiplicity of positive solutions for system (1.1). We employ here the technique used in [8] for the study of a system involving Lane-Emden nonlinearities to the more general system (1.1). For this, we consider the corresponding radial problem (2.1).

Setting

$$\begin{aligned} {\mathcal {S}} :=\{(\lambda _1, \lambda _2) : \; \lambda _1, \lambda _2 >0 \text { and } (2.1) \text { has at least one positive solution} \}, \end{aligned}$$

we know that \({\mathcal {S}}\) is nonempty and unbounded in both directions of the axes \(0\lambda _1\) and \(0\lambda _2\) (see [8, Theorem 2.3]).

Lemma 3.1

Assume (H). Then, the followings are true:

  1. (i)

    There exist \(\lambda ^*_1, \lambda ^*_2>0,\) such that \({\mathcal {S}} \subset [\lambda ^*_1, +\infty ) \times [\lambda ^*_2, +\infty )\) and for all \((\lambda _1, \lambda _2)\in (0,+\infty )^2\backslash ([\lambda ^*_1, +\infty )\times [\lambda ^*_2,+\infty )),\) problem (2.1) has only the trivial solution.

  2. (ii)

    If \(({\overline{\lambda }}_1, {\overline{\lambda }}_2)\in {\mathcal {S}},\) then \([{\overline{\lambda }}_1, +\infty ) \times [{\overline{\lambda }}_2, +\infty )\subset {\mathcal {S}}.\)

  3. (iii)

    If \(({\overline{\lambda }}_1, {\overline{\lambda }}_2)\in {\mathcal {S}},\) then for all \((\lambda _1, \lambda _2) \in ({\overline{\lambda }}_1, +\infty ) \times ({\overline{\lambda }}_2, +\infty ),\) problem (2.1) has at least two positive solutions.

Proof

This follows the outline of the proof of Lemma 4.1 in [8].

(i) Let \(\lambda _1, \lambda _2>0\) and (u, v) be a positive solution of (2.1). It follows from Lemma 2.2 that u and v are both strictly decreasing. Integrating the first equation in (2.1) on [0, r], one obtains

$$\begin{aligned} -r^{N-1}\varphi (u'(r))=\lambda _1 \int _0^r t^{N-1}\mu _1(t)f_1(u(t), v(t))dt, \; \text { for all }r\in [0,R]. \end{aligned}$$

Since u, v are strictly decreasing on [0, R] and using (1.2), we deduce

$$\begin{aligned} -r^{N-1}u'(r)&\le -r^{N-1}\varphi (u'(r))\\&\le \lambda _1 \int _0^r t^{N-1}\mu _1(t)c u^{p_1}(t)v^{q_1}(t)dt\\&\le \lambda _1\mu _1^Mc u^{p_1}(0)v^{q_1}(0)r^N/N, \end{aligned}$$

where \(\mu _i^M:=\max \nolimits _{[0,R]}\mu _i\;\; (i=1,2).\) Integrating on [0, R] we get

$$\begin{aligned} u(0)\le \lambda _1 \mu _1^M c u^{p_1}(0)v^{q_1}(0)R^2/(2N). \end{aligned}$$
(3.1)

Analogously, one has

$$\begin{aligned} v(0)\le \lambda _2 \mu _2^M c u^{p_2}(0)v^{q_2}(0)R^2/(2N). \end{aligned}$$
(3.2)

From \(0<u(0),v(0)<R\) and \(p_1, q_2>1\) we obtain

$$\begin{aligned} \lambda _i> 2N/(\mu _i^M c R^{p_i+q_i+1})>0 \quad (i=1,2). \end{aligned}$$
(3.3)

Consider now the nonempty sets

$$\begin{aligned} {\mathcal {S}}_1 :=&\{\lambda _1>0 : \; \exists \; \lambda _2>0 \text { such that }(\lambda _1, \lambda _2)\in {\mathcal {S}}\}, \\ {\mathcal {S}}_2 :=&\{\lambda _2>0 : \; \exists \; \lambda _1>0 \text { such that }(\lambda _1, \lambda _2)\in {\mathcal {S}}\} \end{aligned}$$

and let

$$\begin{aligned} (0<)\; \lambda ^{*}_i := \inf {\mathcal {S}}_i\; (<+\infty ) \quad (i=1,2). \end{aligned}$$

It follows that \({\mathcal {S}}\subset [\lambda ^*_1,+\infty )\times [\lambda ^*_2, +\infty )\) and for all \(\lambda _1, \lambda _2\in (0,+\infty )^2\backslash ([\lambda ^*_1,+\infty )\) \(\times [\lambda ^*_2, +\infty ))\), problem (2.1) has only the trivial solution (see Lemma 2.2).

(ii) Let \((\lambda ^0_1, \lambda ^0_2) \in [{\overline{\lambda }}_1, +\infty )\times [{\overline{\lambda }}_2, +\infty ) \) be arbitrarily chosen and \(({\overline{u}}, {\overline{v}})\) be a positive solution for (2.1) with \(\lambda _1={\overline{\lambda }}_1\) and \(\lambda _2={\overline{\lambda }}_2\). Then, \(({\overline{u}}, {\overline{v}})\) is a lower solution of (2.1) with \(\lambda _1={\lambda ^0_1}\) and \(\lambda _2={\lambda ^0_2}\). From Proposition 2.3 (i) and the fact that \(({\overline{u}}, {\overline{v}})\) is positive, we obtain \((\lambda ^0_1,\lambda ^0_2)\in {\mathcal {S}}.\)

(iii) From (ii) we get that \(({\overline{\lambda }}_1,+\infty )\times ({\overline{\lambda }}_2,+\infty )\subset {\mathcal {S}}\) and let \((\lambda ^0_1, \lambda ^0_2) \in ({\overline{\lambda }}_1, +\infty )\times ({\overline{\lambda }}_2, +\infty ) \). It remains to show that problem (2.1) with \(\lambda _1=\lambda ^0_1\) and \(\lambda _2=\lambda ^0_2\) has a second positive solution. For this, let \(({\overline{u}},{\overline{v}})\) be the lower solution constructed as above. We fix \((u_0,v_0)\) a positive solution of (2.1) with \(\lambda _1 = \lambda ^0_1\) and \(\lambda _2 = \lambda ^0_2\) such that \((u_0,v_0)\in {\mathcal {A}}:= {\mathcal {A}}_{({\overline{u}},{\overline{v}})}\).

Now, we claim that there exists \(\varepsilon >0\) such that \({\overline{B}}((u_0,v_0),\varepsilon )\subset {\mathcal {A}}.\) By using the quasi-monotonicity of the functions \(f_1\) and \(f_2\), for all \(r\in [0,R/2]\), we have

$$\begin{aligned} {\overline{u}}(r)&=\int _r^{R}\varphi ^{-1}\left( \frac{1}{t^{N-1}}\int _0^ts^{N-1}[{\overline{\lambda }}_1\mu _1(s)f_1({\overline{u}}(s), {\overline{v}}(s))]ds\right) dt\\&<\int _r^{R}\varphi ^{-1}\left( \frac{1}{t^{N-1}}\int _0^ts^{N-1}[\lambda _1^0\mu _1(s)f_1(u_0(s), v_0(s))]ds\right) dt\\&= u_0(r). \end{aligned}$$

Analogously we obtain that \({\overline{v}}(r)<v_0(r)\text { on }[0,R/2].\) So, we can find \(\varepsilon _1>0\) such that if \((u,v)\in K\) then

$$\begin{aligned} \Vert u-u_0\Vert _{\infty }\le \varepsilon _1 \Rightarrow {\overline{u}}\le u \; \text{ and } \; \Vert v-v_0\Vert _{\infty }\le \varepsilon _1 \Rightarrow {\overline{v}}\le v\text { on }[0,R/2]. \end{aligned}$$
(3.4)

On the other hand, for \(r\in [R/2, R]\) one obtains \(u_0'(r)<{\overline{u}}'(r)\) and \(v_0'(r)<{\overline{v}}'(r).\) Thus, there is some \(\varepsilon _2\in (0,\varepsilon _1)\) such that if \((u,v)\in K\), then

$$\begin{aligned} \Vert u'-u_0'\Vert _{\infty }\le \varepsilon _2 \Rightarrow {\overline{u}}'> u' \; \text{ and } \; \Vert v'-v_0'\Vert _{\infty }\le \varepsilon _2 \Rightarrow {\overline{v}}'> v' \text{ on } [R/2,R]. \end{aligned}$$

From

$$\begin{aligned} u(r)=-\int _r^Ru'(s)ds>-\int _r^R{\bar{u}}'(s)ds= {\bar{u}}(r) \end{aligned}$$

we have that \(u>{\overline{u}}\) (and, similarly \(v>{\overline{v}}\)) on [R/2, R). This means that

$$\begin{aligned} \Vert u'-u_0'\Vert _{\infty }\le \varepsilon _2 \Rightarrow {\overline{u}}\le u \; \text{ and } \; \Vert v'-v_0'\Vert _{\infty }\le \varepsilon _2 \Rightarrow {\overline{v}}\le v\text { on }[R/2,R].\quad \end{aligned}$$
(3.5)

The claim follows from (3.4) and (3.5), by taking \(\varepsilon \in (0,\varepsilon _2).\)

Next, if (2.1) has a second solution contained in \({\mathcal {A}},\) then it is nontrivial and the proof is complete. If not, by Lemma 2.3 we infer that

$$\begin{aligned} i({\mathcal {D}}_{\lambda ^0_1, \lambda ^0_2}, B((u_0,v_0),\rho ))=1\text { for all }0<\rho \le \varepsilon , \end{aligned}$$

where \({\mathcal {D}}_{\lambda ^0_1, \lambda ^0_2}\) stands for the fixed point operator associated to problem (2.1) with \(\lambda _1=\lambda ^0_1\) and \(\lambda _2=\lambda ^0_2\). Also, from Proposition 2.1 (ii) we have

$$\begin{aligned} i({\mathcal {D}}_{\lambda ^0_1, \lambda ^0_2}, B(\rho ))=1\text { for all }\rho \ge R+1, \end{aligned}$$

and from Lemma 2.1 we get

$$\begin{aligned} i({\mathcal {D}}_{\lambda ^0_1, \lambda ^0_2}, B(\rho ))=1\text { for all }\rho >0\text { sufficiently small.} \end{aligned}$$

Let \(\rho _1,\rho _2>0\) be sufficiently small and \(\rho _3\ge R+1\) be such that \({\overline{B}}((u_0,v_0),\rho _1)\cap {\overline{B}}(\rho _2)=\emptyset \) and \({\overline{B}}((u_0,v_0),\rho _1)\cup {\overline{B}}(\rho _2)\subset B(\rho _3).\) From the additivity-excision property of the fixed point index it follows that

$$\begin{aligned} i({\mathcal {D}}_{\lambda ^0_1, \lambda ^0_2}, B(\rho _3)\backslash [{\overline{B}}((u_0,v_0),\rho _1)\cup {\overline{B}}(\rho _2)])=-1. \end{aligned}$$

Therefore, \({\mathcal {D}}_{\lambda ^0_1, \lambda ^0_2}\) has a fixed point \((u,v)\in B(\rho _3)\backslash [{\overline{B}}((u_0,v_0),\rho _1)\cup {\overline{B}}(\rho _2)].\) But this means that (2.1) has a second positive solution. \(\square \)

Now, for \(\theta \in \left( 0,{\pi }/{2}\right) \), we denote

$$\begin{aligned} {\mathcal {L}}(\theta ):=\{\lambda >0:\ (\lambda \cos \theta ,\lambda \sin \theta )\in {\mathcal {S}}\}, \end{aligned}$$

which is a nonempty set, and we rewrite problem (2.1) in the form

$$\begin{aligned} {\left\{ \begin{array}{ll} {[} r^{N-1}\varphi (u') ]^{'}+r^{N-1}\lambda \cos \theta \; \mu _1(r) f_1(u,v)=0, \\ {[} r^{N-1}\varphi (v') ]^{'}+r^{N-1}\lambda \sin \theta \; \mu _2(r) f_2(u,v)=0, \\ u'(0)=u(R)=0=v(R)=v'(0), \end{array}\right. } \end{aligned}$$
(3.6)

where \(\lambda >0\) is a real parameter.

Proposition 3.1

There exists a continuous function \(\varLambda :\left( 0,{\pi }/{2}\right) \rightarrow (0,\infty )\) such that

$$\begin{aligned} \lim \limits _{\theta \rightarrow 0}\varLambda (\theta ) \sin \theta -\lambda ^*_2=0=\lim \limits _{\theta \rightarrow {\pi }/{2}}\varLambda (\theta ) \cos \theta -\lambda ^*_1 \end{aligned}$$
(3.7)

and the followings hold true:

  1. (i)

    \(\varLambda (\theta )\in {\mathcal {L}}(\theta )\), for every \(\theta \in \left( 0,{\pi }/{2}\right) \);

  2. (ii)

    system (2.1) has at least two positive solutions, for all \((\lambda _1,\lambda _2)\in (\varLambda (\theta )\cos \theta , \) \(+\infty )\times ( \varLambda (\theta )\sin \theta , + \infty )\).

Proof

This follows the outline of the proof of Proposition 4.1 in [8]. For each \(\theta \in \left( 0,{\pi }/{2}\right) \), let

$$\begin{aligned} \varLambda (\theta ):=\inf {\mathcal {L}}(\theta ). \end{aligned}$$
(3.8)

Note that \(\varLambda (\theta )\) is \(<\infty \) because \({\mathcal {L}}(\theta ) \ne \emptyset \) and \(>0\) by Lemma 3.1 (i). We first prove statements (i) and (ii).

(i) Let \(\{\lambda ^k\}\subset {\mathcal {L}}(\theta )\) be a decreasing sequence converging to \(\varLambda (\theta )\) and \((u_k,v_k)\in K\) with \(u_k>0<v_k\) on [0, R) be such that

$$\begin{aligned} u_k&=P\circ \varphi ^{-1}\circ S\circ [\lambda ^k \cos \theta \; \mu _1 f_1(u_k, v_k)],\\ v_k&= P\circ \varphi ^{-1}\circ S\circ [\lambda ^k \sin \theta \; \mu _2 f_2(u_k, v_k)]. \end{aligned}$$

From (2.2) and Arzela-Ascoli theorem we obtain that there exists \((u,v)\in K\) such that, passing eventually to a subsequence, \(\{(u_k,v_k)\}\) converges to (u, v) in \(C\times C\) – with the usual product topology. Hence, \(u\ge 0 \le v\) and

$$\begin{aligned} u&=P\circ \varphi ^{-1}\circ S\circ [\varLambda (\theta ) \cos \theta \; \mu _1 f_1(u, v)],\\ v&= P\circ \varphi ^{-1}\circ S\circ [\varLambda (\theta ) \sin \theta \; \mu _2 f_2(u, v)]. \end{aligned}$$

From (3.1) and (3.2) we have that

$$\begin{aligned} u_k(0)\le \lambda ^k \cos \theta \; \mu _1^M c u_k^{p_1}(0)v_k^{q_1}(0)R^2/(2N) \end{aligned}$$

and

$$\begin{aligned} v_k(0)\le \lambda ^k \sin \theta \; \mu _2^M c u_k^{p_2}(0)v_k^{q_2}(0)R^2/(2N), \end{aligned}$$

which, taking into account that \(0<u_k(0), v_k(0)<R,\) imply

$$\begin{aligned} u_k^{p_1-1}(0)>\frac{2N}{\lambda ^k \mu _1^M c R^{q_1+2}\cos \theta } \end{aligned}$$

and

$$\begin{aligned} v_k^{q_2-1}(0)>\frac{2N}{\lambda ^k\mu _2^M c R^{p_2+2}\sin \theta }. \end{aligned}$$

These ensure that there is a constant \(c_1>0\) such that \(u_k(0), v_k(0)>c_1\) for all k. This leads to \(u(0), v(0)\ge c_1,\) hence by Lemma 2.2 we get \(u>0<v\) on [0, R). Consequently, \(\varLambda (\theta )\in {\mathcal {L}}(\theta ).\)

(ii) This follows from statement (iii) in Lemma 3.1.

The continuity of \(\varLambda \) and the equalities in (3.7) can be proved in the same manner as it is done in the proof of Proposition 4.1 in [8]. \(\square \)

Theorem 3.1

Assume (H). Then, there exist \(\lambda ^*_1, \lambda ^*_2>0\) and a continuous function \(\varLambda :(0,\pi /2)\rightarrow (0,+\infty ),\) generating the curve

$$\begin{aligned} (\varGamma ) \left\{ \begin{array}{ll} \lambda _1(\theta ) = \varLambda (\theta )\cos \theta \\ \lambda _2(\theta ) = \varLambda (\theta )\sin \theta \\ \end{array} \right. ,\qquad \theta \in (0,\pi /2) \end{aligned}$$

such that

  1. (i)

    \(\varGamma \subset [\lambda ^*_1, +\infty )\times [\lambda ^*_2, +\infty );\)

  2. (ii)

    the following asymptotic behaviors hold

    $$\begin{aligned}&\lim \limits _{\theta \rightarrow \pi /2}\lambda _2(\theta )=+\infty =\lim \limits _{\theta \rightarrow 0}\lambda _1(\theta ), \end{aligned}$$
    (3.9)
    $$\begin{aligned}&\lim \limits _{\theta \rightarrow 0}\lambda _2(\theta )-\lambda ^*_2=0=\lim \limits _{\theta \rightarrow \pi /2}\lambda _1(\theta )-\lambda ^*_1; \end{aligned}$$
    (3.10)
  3. (iii)

    \(\varGamma \) separates the first quadrant \((0,+\infty )\times (0,+\infty ) \) in two disjoint sets \({\mathcal {O}}_1\) and \({\mathcal {O}}_2\) such that problem (1.1) has zero, at least one or at least two radial positive solutions, according to \((\lambda _1, \lambda _2)\in {\mathcal {O}}_1,\) \((\lambda _1, \lambda _2)\in \varGamma \) or \((\lambda _1, \lambda _2)\in {\mathcal {O}}_2.\)

Proof

This follows from Lemma 3.1 and Proposition 3.1. \(\square \)

Example 3.1

Let \(p_1, q_2>1\) and \(q_1,p_2>0\). The conclusion of Theorem 3.1 is obtained for the following choices of \(f_1\) and \(f_2\) in problem (1.1):

  1. (i)

    \(f_1(\text{ u,v })=\text{ u }^{p_1}\text{ v }^{q_1}\) and \(f_2(\text{ u,v })=\text{ u }^{p_2}\text{ v }^{q_2}\) – Lane-Emden type nonlinearities;

  2. (ii)

    \(f_1(\text{ u,v })=\text{ u }^{p_1}\ln (1+\text{ v }^{q_1})\) and \(f_2(\text{ u,v })=\text{ v }^{q_2}\ln (1+\text{ u }^{p_2})\);

  3. (iii)

    \(f_1(\text{ u,v })=\text{ u }^{p_1}\text{ v }^{q_1}\text{ arctg }(\text{ v })\) and \(f_2(\text{ u,v })=\text{ u }^{p_2}\text{ v }^{q_2}\text{ arctg }(\text{ u })\).