1 Introduction

The purpose of this paper is to formulate conditions which guarantee the existence of continua of positive solutions of the following system involving perturbed Laplace operators

$$\begin{aligned} \left\{ \begin{array}{c} \mathrm{div}(a_{1}(||x||)\nabla u_{1}(x))+f_{1}(x,u_{1}(x),u_{2}(x))+g_{1}(x)x\cdot \nabla u_{1}(x)=0 \\ \mathrm{div}(a_{2}(||x||)\nabla u_{2}(x))+f_{2}(x,u_{1}(x),u_{2}(x))+g_{2}(x)x\cdot \nabla u_{2}(x)=0 \end{array} \right. \end{aligned}$$
(1)

for \(x\in G_{R}=\left\{ x\in \mathbb {R}^{n},||x||>R\right\} \) with \(n>2,\)\( R>0\) and \(||x||:=\sqrt{\sum \nolimits _{i=1}^{n}x_{i}^{2}}.\) We are interested in global solutions vanishing at infinity, i.e.

$$\begin{aligned} {\lim }{||x||\rightarrow \infty }u_{i}(x)=0\quad \text {for }i=1,2, \end{aligned}$$
(2)

which are often called evanescent solutions.

Many problems modeled by similar systems arise in various areas of applied mathematics, in biological, chemical or physical phenomena, for example in pseudoplastic fluids [9], reaction–diffusion processes, chemical heterogeneous catalysts [4] or heat conduction in electrically conducting materials [18]. Recently the existence and multiplicity of solutions for such elliptic systems considered also in unbounded domains has been widely discussed in the literature (see e.g. [5, 11,12,13,14, 23,24,25, 27, 34, 36] and the references therein).

The existence or nonexistence of radially symmetric solutions for the Emden-Fowler system involving p-Laplace operators and some real parameters was discussed e.g. in [8]. There the approach is based on suitable transformations which play a crucial role in the reduction of the main problem to a quadratic system. In [20] and [21] results concerning more general problems associated with the existence of weak solutions of elliptic inequalities can be found. In Covei’s paper [13] the following Lane-Emden-Fowler system is investigated

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u(x)=a_{1}(x)F_{1}(x,u(x),v(x)) \\ -\Delta v(x)=a_{2}(x)F_{2}(x,u(x),v(x)) \\ u=v=0\text { in }\partial \Omega \end{array} \right. \end{aligned}$$

for bounded domains \(\Omega \subset {\mathbb {R}}^{n}\) or \(\Omega ={\mathbb {R}} ^{n},\) in the case when nonlinearities \(F_{1},F_{2}\) are positive and satisfy some growth condition with respect to the second and third variables. The multiplicity of solutions for system of similar form and their additional properties were discussed, among others, in [11]. In that paper the variational approach allowed the authors to show the existence of at least nine solutions in the case when \(\Omega \) is a bounded regular domain in \({\mathbb {R}}^{n},\) the right-hand side is a Carathé odory function and satisfies, among others, some growth conditions (see Th.1.1). Precisely, these solutions \(U=(u_{1},u_{2}),\) satisfy the following sign conditions: both \(u_{1}\) and \(u_{2}\) are strictly positive or negative in the first four solutions; four others are such that one of the two coordinates is of the one sign while the other is of changing sign, and finally both coordinates change their sing in the ninth solution. There are many results concerning the existence of infinitely many solutions for the case when the system does not depend on \(\nabla u_{i}.\) For example, vanishing solutions for Schrödinger–Poisson system (also called Schrödinger–Maxwell equations), was discussed in [24], where an approximation methods were employed. Such systems are motivated by many problems arising in quantum mechanic, in plasma physics, semiconductor theory or nonlinear optics (see e.g. [6, 7, 15,16,17, 30] and the references therein). Based on variational methods Ambrosetti and Ruiz proved the existence of infinitely many radial solutions for the problem

$$\begin{aligned} \left\{ \begin{array}{l} -\triangle u+u+\lambda vu=|u|^{p-1}u \\ -\triangle v=K(x)u^{2}, \end{array} \right. \text { in }{\mathbb {R}}^{3} \end{aligned}$$

when \(p\in \left( 2,5\right) \) and \(\lambda >0\) is sufficiently large (see [2] and [3]). In the case when \(p\in (1,2]\) and \(\lambda >0\) is small enough, the existence of finite number of such solutions was proved there. In 2011 d’Avenia, Pomponio and Vaira constructed infinitely many nonradial solutions of the above system with the first equation with the function K(x) instead of constant parameter \(\lambda \) for \(p\in \left( 1,5\right) \) and K decaying at infinity. In [10, 26] and [33] the following general problem

$$\begin{aligned} \left\{ \begin{array}{l} -\triangle u+\lambda V(x)u+K(x)vu=f(x,u) \\ -\triangle v=K(x)u^{2} \end{array} \right. \text { in }{\mathbb {R}}^{3}, \end{aligned}$$

was considered for the positive potential V. Applying the variant of the fountain theorem the authors obtain infinitely many high-energy solutions for superlinear f. In the sublinear case and \(K\equiv 1,\) Sun constructed infinitely many small-energy solutions for such problem [32]. Similar multiplicity results for sublinear and odd f and positive nonconstant K was proved in [34]. The authors obtained the existence of finite number of nontrivial solution for superlinear case. The existence of a sequence of solutions decaying at infinity of the above system was obtained in [35] in the case when \(K\equiv 1,\)\(\lambda =1\), f is odd and satisfies particular estimates. We want to join in the discussion concerning the multiplicity of global solutions for (1) under quite mild conditions concerning nonlinearities. We focus on positive solutions with minimal growth and finite energy in a neighborhood of infinity. The aim of this paper is twofold. On the one hand we want to formulate sufficient conditions for the existence of infinitely many unbounded connected sets of global solutions for our problem. Here we have to emphasize that this results will be obtained without assumptions concerning the oddness or growth of nonlinearity \(f_{i}\) with respect to the second and/or third variables. These condition are often met in the literature. On the other hand we describe precisely the asymptotics of solutions and their gradients.

By a solution of our problem we understand a vector function \(\mathbf {u=} \left( u_{1},u_{2}\right) \in \left( C_{loc}^{2+\alpha }(G_{R})\right) ^{2}:=C_{loc}^{2+\alpha }(G_{R})\times C_{loc}^{2+\alpha }(G_{R})\) satisfying (1) and (2). Moreover, we say that \(\mathbf {u=} \left( u_{1},u_{2}\right) \) is a finite energy solution (or a solution with finite energy) in a neighborhood of infinity when there exits a nonnegative radial function \(\psi :=(\psi _{1},\psi _{2})\in C^{1}\left( \Omega _{R}\right) \) with \(\psi _{i}(x)\equiv 1,\)\(i=1,2,\) for ||x|| sufficiently large and such that \(\psi _{i}u_{i}\in D_{0}^{1,2}(\Omega _{R}),\) where \( D_{0}^{1,2}(\Omega _{R})\) denotes the completion of \(C_{0}^{\infty }\left( \Omega _{R}\right) \) in the norm \(||\varphi ||:=||\nabla \varphi ||_{L^{2}(\Omega _{R})}\) (see, among others, e.g. [27]).

Here we also recall standard definitions of a super-solution and a sub-solution of (1)–(2) (see e.g. [25]). Precisely, by a super-solution of (1)–(2) in \(G_{R}\) we understand a vector function \(\overline{{\mathbf {u}}}=(\overline{u}_{1}, \overline{u}_{2})\in \left( C_{loc}^{2+\alpha }(G_{R})\right) ^{2}\) satisfying the following differential inequalities

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{div}(a_{1}(||x||)\nabla \overline{u}_{1}(x))+f_{1}(x,\overline{u} _{1}(x),\overline{u}_{2}(x))+g_{1}(x)x\cdot \nabla \overline{u}_{1}(x)\le 0, \,\,x\in G_{R}, \\ \mathrm{div}(a_{2}(||x||)\nabla \overline{u}_{2}(x))+f_{2}(x,\overline{u} _{1}(x),\overline{u}_{2}(x))+g_{2}(x)x\cdot \nabla \overline{u}_{2}(x)\le 0, \,\,x\in G_{R}, \\ {\lim }_{||x||\rightarrow \infty }\overline{u}_{i}(x)=0\text { for } i=1,2. \end{array} \right. \end{aligned}$$

Analogously, as for a sub-solution \(\underline{\mathbf {u}}=(\underline{u} _{1},\underline{u}_{2})\) of (1)–(2) in \(G_{R}\), the sign of the inequality should be reversed.

To show the existence of continua of positive solutions with finite energy we assume the following hypotheses

(H1):

\(g_{i}:G_{l_{0}}\rightarrow \mathbb {(}0,+\infty ),\) where \(1<l_{0}\le R,\) is locally Hölder continuous with exponent \(\alpha \in \left( 0,1\right) ,\)\(i=1,2;\)

(H2):

\(a_{i}:[1,+\infty )\rightarrow (c_{1},c_{2})\) belongs to \(C^{1+\alpha }([1,+\infty )),\) where \(0<c_{1}<c_{2},\)\( i=1,2;\)

(H3):

\(f_{1},f_{2}\) are continuously differentiable in \(G_{R}\times (\alpha _{1},\beta _{1})\times (\alpha _{2},\beta _{2}),\) where \(\alpha _{i}<0<\beta _{i},\)\(i=1,2,\)

(a):

\(f_{1}(x,0,0)\), \(f_{2}(x,0,0)\) are positive in \( G_{R},\)

(b):

\(f_{1}\) is nondecreasing in \( u _{2}\) and \(f_{2}\) is nondecreasing in \(u_{1}\);

(c):

for each \(i=1,2,\)there exists \(d_{i}\in (0,\beta _{i}),\)\(m_{i}<4(n-2)d_{i}c_{1}^{2}c_{2}^{-1}\)such that for all \( r\in [1,+\infty ),\)

$$\begin{aligned} \underset{\mathbf {(}u_{1},u_{2})\in [0,d_{1}]\times [0,d_{2}]}{ \sup }\underset{||x||=r}{\sup }f_{i}(x,u_{1},u_{2})\le m_{i}r^{2-2n} \end{aligned}$$

and there exist\(K_{1},K_{2}>1\)such that the following inequality holds

$$\begin{aligned} \left[ \frac{1}{2}\left( 1-K_{i}^{2-n}\right) ^{2}\frac{1}{K_{i}^{2}}\right] ^{-\frac{1}{n-2}}\le R. \end{aligned}$$
(3)

It is worth emphasizing that we need the monotonicity and differentiability of \(f_{i}\) only on some right-hand side neighborhood of the origin. Moreover, we can consider both sublinear and superlinear \( f_{i}.\) It is associated with the fact that we have to control only the value of nonlinearities \(f_{i}(x,\cdot ,\cdot )\) in the rectangle \( [0,d_{1}]\times [0,d_{2}]\). Thus we can omit growth conditions concerning variable \(\mathbf {u=}\left( u_{1},u_{2}\right) \). Our main tool is Theorem 1 (given below) which says that the existence of a sub-solution \(\underline{\mathbf {u}}=(\underline{u}_{1},\underline{u}_{2})\) and a super-solution \(\overline{\mathbf {u}}=(\overline{u}_{1},\overline{u} _{2})\) of our problem such that \(0\le \underline{u}_{i}\le \overline{u} _{i}\le d_{i}\) for \(i=1,2,\) implies the existence of a solution \(\mathbf {u= }\left( u_{1},u_{2}\right) \) of (1) which is squeezed between them. The proof of Theorem 1 is based on classical ideas associated with the Sattinger’s monotone iteration procedure (see [31]) and the approach described by Kawano [25] for the system which does not contain the gradient term. The similar theorem was also proved in the [29] (Th. 1.2) but only for the special case when \(a_{i}\equiv 1,\)\(i=1,2.\) Therefore we can not use it directly. For the reader’s convenience we present the proof of this result in the “Appendix”.

Now we formulate precisely the theorem which will be our main tool in the proof of the existence result. The proof of Theorem 1 is given at the end of the paper (in “Appendix”).

Theorem 1

Assume that (H1), (H2), (H3) hold and \(\overline{\mathbf {u}}=(\overline{u}_{1},\overline{u}_{2})\) and \(\underline{\mathbf {u}}=(\underline{u}_{1},\underline{u}_{2})\) are, respectively, a positive super-solution and a nonnegative sub-solution of (1)–(2) such that

$$\begin{aligned} \underline{u}_{i}\le \overline{u}_{i}\le d_{i}\,\, for\,\, i=1,2 \, {and}\, {all}\, x\in \,\overline{G}_{R}. \end{aligned}$$

Then there exists a vector function \(\mathbf {u}=\left( u_{1},u_{2}\right) \in \left( C_{loc}^{2+\alpha }(G_{R})\right) ^{2}\) satisfying (1)–(2) and such that for \(i=1,2\), \(\underline{u}_{i}\le u_{i}\le \overline{u}_{i}\) in \(\overline{G }_{R}\) and \(u_{i}(x)=\overline{u}_{i}(x)\) on \(\partial G_{R}.\)

This approach allows us to prove two main results: the first one is associated with the existence of sequences of uncountable sets of solutions and the other one gives additional information concerning the asymptotics of solutions and their gradients.

Theorem 2

Suppose that (H1), (H2) and (H3) hold. Then there exist infinitely many connected sets of positive solutions \(\{(u_{1}^{q,l},u_{2}^{q,l}),q\in {\mathbb {R}} \}_{l\in {\mathbb {N}}}\) of (1)–(2), where \( {\mathbb {N}}:=\{1,2,3,...\}\). For all \(q\in {\mathbb {R}},\) the sequence \(\{(u_{1}^{q,l},u_{2}^{q,l})\}_{l\in {\mathbb {N}}}\) is nondecreasing in \(G_{R}\) and increasing on \(\partial G_{R}\) (with respect to each coordinate). Moreover

  1. 1.

    there exists at least one sequence of continua \(\{S_{l}^{0}\}_{l\in {\mathbb {N}}}\) consisting of all solutions \((u_{1}^{q,l},u_{2}^{q,l})\) such that both coordinates \(u_{1}^{q,l}\) and \(u_{2}^{q,l}\) have different value on \(\partial G_{R}, \) i.e.

    $$\begin{aligned} S_{l}^{0}:=\{(u_{1}^{q,l},u_{2}^{q,l}),q\in {\mathbb {R}}\, {\textit{ such that }}\, u_{1}^{q,l}\ne u_{2}^{q,l}{ {on} }\,\partial G_{R}\}\, {\textit{ for all}}\,\, l\in {\mathbb {N}}, \end{aligned}$$
  2. 2.

    there exists at least one sequence of continua \(\{S_{l}^{1}\}_{l\in {\mathbb {N}}}\) consisting of all solutions \((u_{1}^{q,l},u_{2}^{q,l})\) such that both coordinates \(u_{1}^{q,l}\) and \(u_{2}^{q,l}\) have the same value on \(\partial G_{R},\) i.e.

    $$\begin{aligned} S_{l}^{1}:=\{(u_{1}^{q,l},u_{2}^{q,l});q\in {\mathbb {R}}{\textit{ such that }} u_{1}^{q,l}=u_{2}^{q,l}{ {on} }\,\,\partial G_{R}\} {\textit{ for all }} l\in {\mathbb {N}}, \end{aligned}$$
  3. 3.

    each of such sequences \(\{(u_{1}^{q,l},u_{2}^{q,l})\}_{l\in {\mathbb {N}}}\) generates another solution \(({\widetilde{u}}_{1}^{q}, {\widetilde{u}}_{2}^{q})\) of (1)–(2) and, in consequence, \(\{({\widetilde{u}}_{1}^{q},{\widetilde{u}}_{2}^{q}),q\in {\mathbb {R}}\}\) is also a connected set of positive solutions of (1)–(2).

Theorem 3

Under hypotheses (H1), (H2) and (H3) ,solutions \((u_{1}^{q,l},u_{2}^{q,l})\), where \(q\in {\mathbb {R}}\) and \(l\in {\mathbb {N}}\), described in Theorem 2, satisfy the following conditions:

  1. 1.

    \((u_{1}^{q,l},u_{2}^{q,l})\) has finite energy in neighborhood of infinity,

  2. 2.

    for \(i\in \{1,2\}\),

    $$\begin{aligned} u_{i}^{q,l}(x)=O\left( \frac{1}{||x||^{n-2}}\right) \,\mathrm{as }\,\Vert x\Vert \rightarrow +\infty \end{aligned}$$
    (4)

    (which means that \((u_{1}^{q,l},u_{2}^{q,l})\) has the minimal growth) and for all \(\phi \in C^{1}(1,+\infty )\) such that \({\lim }_{{r\rightarrow +\infty }}\phi (r)=0\) and \({\lim }_{r\rightarrow +\infty }\phi ^{\prime }(r)r^{n-1}=+\infty \),

    $$\begin{aligned} u_{i}^{q,l}(x)=o\left( \phi (||x||)\right) \,\mathrm{as }\,\Vert x\Vert \rightarrow +\infty . \end{aligned}$$
    (5)

2 Existence of continua of super-solutions

Taking into account the fact that \(f(x,0,0)\ge 0\) in \(\Omega _{R}\) we obtain the trivial sub-solution \(\underline{\mathbf {u}}\equiv (0,0)\) of (1)–(2). Therefore it suffices to construct an increasing sequences of positive super-solutions for (1)–(2). To this effect we consider a sequence of auxiliary linear problems. It appears that their radial positive solutions give us a sequence of super-solutions for our main problem. Let \(q\in {\mathbb {R}}\) and \(l\in {\mathbb {N}}\) be fixed. We consider the following two independent linear problems

$$\begin{aligned} \left\{ \begin{array}{l} -\mathrm{div}(a_{i}(||x||)\nabla \overline{u}_{i}(x))={\widetilde{f}} _{i}^{q,l}(||x||),\text {for }x\in \Omega _{1}, \\ \overline{u}_{i}(x)=0\text { for }||x||=1,\underset{||x||\rightarrow \infty }{ \lim }\overline{u}_{i}(x)=0,\,\,i=1,2, \end{array} \right. \end{aligned}$$
(6)

where \(i=1,2,\)

$$\begin{aligned} {\widetilde{f}}_{i}^{q,l}(r):=\frac{1}{r^{2n}}\left( r^{2}m_{i}+p_{i}^{q,l}\right) \end{aligned}$$

and

$$\begin{aligned} p_{i}^{q,l}=\left( 4(n-2)d_{i}c_{1}^{2}c_{2}^{-1}-\frac{m_{i}}{n-2}\right) \left( 1-\frac{q}{q+1}\frac{1}{l}\right) \end{aligned}$$

for all \(l\in {\mathbb {N}},\)\(i=1,2.\) Now we describe some properties of \( {\widetilde{f}}_{i}^{q,l}\) and \(p_{i}^{q,l}\) .

Remark 1

For each q\(\in {\mathbb {R}}\) and \(i=1,2 \mathbf {,}\)

  1. 1.

    \(\{p_{i}^{q,l}\}_{l\in N}\) is an increasing sequence of positive numbers,

  2. 2.

    for all \(l\in {\mathbb {N}}\), the following inequality holds

    $$\begin{aligned} \int _{1}^{\infty }r^{n-1}{\widetilde{f}} _{i}^{q,l}(r)dr<4(n-2)d_{i}c_{1}^{2}c_{2}^{-1},\,\,i=1,2, \end{aligned}$$
    (7)
  3. 3.

    there exists \(K_{i}>1\) such that for all \(l\in {\mathbb {N}}\), the following inequality holds

    $$\begin{aligned} R^{2-n}\le \frac{1}{2}\left( 1-K_{i}^{2-n}\right) ^{2}\left[ \frac{\underset{r\in [1,K_{i}]}{\inf }r^{2n-2}{\widetilde{f}}_{i}^{q,l}(r)}{\underset{ r\in [1,+\infty )}{\sup }r^{2n-2}{\widetilde{f}}_{i}^{q,l}(r)}\right] . \end{aligned}$$
    (8)

Proof

Taking into account the definition of \(p_{i}^{q,l}\) we get that \(\{p_{i}^{q,l}\}_{l\in {\mathbb {N}}}\subset \left( 0,+\infty \right) \) and it is an increasing sequence. Moreover, (H3) part (c) implies for all \(\ i\in \{1,2\},\) the following chain of inequalities

$$\begin{aligned} \int _{1}^{\infty }r^{n-1}{\widetilde{f}}_{i}^{q,k}(r)dr<\frac{m_{i}}{n-2} +4(n-2)d_{i}c_{1}^{2}c_{2}^{-1}-\frac{m_{i}}{n-2} =4(n-2)d_{i}c_{1}^{2}c_{2}^{-1}. \end{aligned}$$

Coming to the last part let us note that

$$\begin{aligned} \underset{r\in [1,+\infty )}{\sup }\left( r^{2n-2}{\widetilde{f}} _{i}^{q,l}(r)\right) =\underset{r\in [1,+\infty )}{\sup }\left( m_{i}+ \frac{p_{i}^{q,l}}{r^{2}}\right) =m_{i}+p_{i}^{q,l} \end{aligned}$$

and

$$\begin{aligned} \underset{r\in [1,K_{i}]}{\inf }\left( r^{2n-2}{\widetilde{f}} _{i}^{q,k}(r)\right) =\underset{r\in [1,K_{i}]}{\inf }\left( m_{i}+ \frac{p_{i}^{q,l}}{r^{2}}\right) =m_{i}+\frac{p_{i}^{q,l}}{K_{i}^{2}}. \end{aligned}$$

Thus, by (H3) part (c), \(\left[ \frac{1}{2}\left( 1-K_{i}^{2-n}\right) ^{2}\frac{1}{K_{i}^{2}}\right] \ge R^{2-n}\) and further

$$\begin{aligned} R^{2-n}\underset{r\in [1,+\infty )}{\sup }\left( r^{2n-2}{\widetilde{f}} _{i}^{q,l}(r)\right)\le & {} \frac{1}{2}\left( 1-K_{i}^{2-n}\right) ^{2}\frac{ 1}{K_{i}^{2}}\left( m_{i}+p_{i}^{q,l}\right) \\\le & {} \frac{1}{2}\left( 1-K_{i}^{2-n}\right) ^{2}\underset{r\in [1,K_{i}]}{\inf }\left( r^{2n-2}{\widetilde{f}}_{i}^{q,k}(r)\right) \end{aligned}$$

which gives (8). \(\square \)

Applying the transformation given by diffeomorphism \(\psi :[0,1)\rightarrow [1,+\infty ),\) where \(\psi (t)=\left( 1-t\right) ^{\frac{1}{2-n}},\) we use the well-known fact that the solution \((\overline{u}_{1}^{q,l}, \overline{u}_{2}^{q,l})\) of (6) can be described as \(\overline{ u}_{i}^{q,l}(x)=z_{i}^{q,l}(\psi ^{-1}(||x||))\) for \(i=1,2,\) namely \( \overline{u}_{i}^{q,k}(x)=z_{i}^{q,k}(1-||x||^{2-n})\), where \(z_{i}^{q,l},\)\( i=1,2,\) are solutions of the following two independent Dirichlet problems with singularity at the end-point 1

$$\begin{aligned} \left\{ \begin{array}{ll} -({\widetilde{a}}_{i}(t)z_{i}^{\prime }(t))^{\prime }=h_{i}^{q,l}(t)\,\, \mathbf {,} &{} \text {in }\mathbf {(}0,1) \\ z_{i}(0)=z_{i}(1)=0, &{} \end{array} \right. \end{aligned}$$
(9)

with

$$\begin{aligned} h_{i}^{q,l}(t)= & {} \frac{1}{\left( n-2\right) ^{2}}\left( 1-t\right) ^{\frac{ 2n-2}{2-n}}{\widetilde{f}}_{i}^{q,l}(\left( 1-t\right) ^{\frac{1}{2-n}}), \\ {\widetilde{a}}_{i}(t)= & {} a_{i}(\left( 1-t\right) ^{\frac{1}{2-n}}). \end{aligned}$$

Remark 2

Taking into account assumptions made on \(f_{i}\) we state that for each \(i=1,2\), \(h_{i}^{q,l}\) satisfies the following conditions

  1. 1.

    for all \(l\in {\mathbb {N}}\) and \(q\in {\mathbb {R}}\), \(h_{i}^{q,l}\) is continuous, \(h_{i}^{q,l}(\cdot )>0\), \(h_{i}^{q,l}<h_{i}^{q,l+1}\) and

    $$\begin{aligned} \int _{0}^{1}h_{i}^{q,l}(r)dr\le 4d_{i}c_{1}^{2}c_{2}^{-1}, \end{aligned}$$
    (10)
  2. 2.

    there exists \(0<\varepsilon _{i}<1\) such that for all \(l\in {\mathbb {N}}\) and \(q\in {\mathbb {R}}\)

    $$\begin{aligned} (1-\overline{t})\left( \underset{t\in [0,1)}{\sup } h_{i}^{q,l}(t)\right) \le \frac{1}{2}\varepsilon _{i}^{2}\left( \underset{ t\in [0,\varepsilon _{i}]}{\inf }h_{i}^{q,l}(t)\right) , \end{aligned}$$
    (11)

    where \(\overline{t}:=1-R^{2-n}\in \left( 0,1\right) .\)

Lemma 4

For each \(q\in {\mathbb {R}}\) and \(l\in {\mathbb {N}} \) there exists a super-solution \((\overline{u}_{1}^{q,l},\overline{ u}_{2}^{q,l})\) of our problem such that

$$\begin{aligned} \overline{u}_{i}^{q,l}<\overline{u}_{i}^{q,l+1} {\textit{ in }} G_{R}, \,\,i=1,2 \end{aligned}$$
(12)

and for all \(l\in {\mathbb {N}}\) and \(q_{1},q_{2}\in {\mathbb {R}}\)

$$\begin{aligned} {\textit{if } }q_{1}> & {} q_{2}{ \textit{ then } }\,\overline{u} _{i}^{q_{1},l}<\overline{u}_{i}^{q_{2},l},{ \textit{in } }G_{R},\,\, i=1,2, \nonumber \\ {\textit{if } }q_{1}< & {} q_{2}{ \textit{ then } }\,\overline{u} _{i}^{q_{1},l}>\overline{u}_{i}^{q_{2},l}\,\,,{ \textit{in } }G_{R}, \,\,i=1,2. \end{aligned}$$
(13)

Precisely, for each \(q\in R\) there exist two types of sequences of super-solutions \(\left\{ (\overline{u}_{1}^{q,l},\overline{u} _{2}^{q,l})\right\} _{l\in {\mathbb {N}}}\) : in the first one for all \(l\in {\mathbb {N}}\), \(\overline{u}_{1}^{q,l}\equiv \overline{u} _{2}^{q,l}\) in \(G_{R}\), in the other one for all \(l\in {\mathbb {N}}\), \(\overline{u}_{1}^{q,l}\ne \overline{u}_{2}^{q,l}\).

Moreover for each \(q\in R\) and \(l\in {\mathbb {N}}\) the following assertions hold

$$\begin{aligned} \overline{u}_{i}^{q,l}(x)=O\left( \frac{1}{||x||^{n-2}}\right) \,\mathrm{as }\,\Vert x\Vert \rightarrow +\infty ,\,\,i=1,2 \end{aligned}$$
(14)

and

$$\begin{aligned} \overline{u}_{i}^{q,l}(x)=o\left( \phi (||x||)\right) \,\mathrm{as }\,\Vert x\Vert \rightarrow +\infty ,\,\,i=1,2 \end{aligned}$$
(15)

for all \(\phi \in C^{1}(1,+\infty )\) such that \({\lim }_{ r\rightarrow +\infty }\phi (r)=0\) and \({\lim }_{ r\rightarrow +\infty }\phi ^{\prime }(r)r^{n-1}=+\infty .\)

Proof

We start with the solutions of the sequence of Dirichlet problems (9). To this end we fix \(q\in {\mathbb {R}}\) and \( l\in {\mathbb {N}}\). It is obvious that taking

$$\begin{aligned} z_{i}^{q,l}(t)=\int _{0}^{1}\mathbf {G}_{i}(s,t)h_{i}^{q,l}(s)ds, \end{aligned}$$
(16)

where

$$\begin{aligned} \mathbf {G}_{i}(s,t):=\frac{1}{c_{i}}\left\{ \begin{array}{cc} \mathop {\displaystyle \int }\limits _{0}^{s}\frac{1}{\widetilde{a_{i}}(r)}dr\mathop {\displaystyle \int }\limits _{t}^{1} \frac{1}{{\widetilde{a}}_{i}(r)}dr &{} \text { for }0\le s\le t \\ \mathop {\displaystyle \int }\limits _{0}^{t}\frac{1}{{\widetilde{a}}_{i}(r)}dr\mathop {\displaystyle \int }\limits _{s}^{1} \frac{1}{{\widetilde{a}}_{i}(r)}dr &{} \text {for }t<s\le 1, \end{array} \right. \end{aligned}$$

with \(c_{i}:=\mathop {\displaystyle \int }\limits _{0}^{1}\frac{1}{{\widetilde{a}}_{i}(s)}dr,\) we obtain the solution of (9). By the definition of \(\mathbf {G}_{i} \mathbf {,}\) we get for all \(t\in [0,1],\)

$$\begin{aligned} 0\le z_{i}^{q,l}(t)=\int _{0}^{1}\mathbf {G}_{i}(s,t)h_{i}^{q,l}(s)ds\le \frac{1}{c_{i}}\frac{1}{c_{1}^{2}}t\left( 1-t\right) \int _{0}^{1}h_{i}^{q,l}(s)ds\le d_{i}. \end{aligned}$$

Let us note that for all \(l\in {\mathbb {N}},\)\(i=1,2,\)\( h_{i}^{q,l}(t)<h_{i}^{q,l+1}(t)\) for all \(t\in (0,1),\) which implies

$$\begin{aligned} -\left[ a_{i}(t)\left( \left( z_{i}^{q,l+1}\right) ^{\prime }(t)-\left( z_{i}^{q,l}\right) ^{\prime }(t)\right) \right] ^{\prime }=h_{i}^{q,l+1}(t)-h_{i}^{q,l}(t)>0. \end{aligned}$$

The above assertion and the boundary condition \( z_{i}^{q,l}(0)=z_{i}^{q,l+1}(0)=z_{i}^{q,l}(1)=z_{i}^{q,l+1}(1)=0\) lead to the inequality

$$\begin{aligned} z_{i}^{q,l}(t)<z_{i}^{q,l+1}(t)\text { in }(0,1). \end{aligned}$$
(17)

Applying the same reasoning we obtain for all \(l\in {\mathbb {N}}\) and \(q_{1},q_{2}\in {\mathbb {R}},\)

$$\begin{aligned} \text {if }q_{1}> & {} q_{2}\text { then }z_{i}^{q_{1},l}<z_{i}^{q_{2},l},\,\, i=1,2,\text { in }G_{R} \nonumber \\ \text {if }q_{1}< & {} q_{2}\text { then }z_{i}^{q_{1},l}>z_{i}^{q_{2},l}\,\,, \,\,i=1,2,\text { in }G_{R}. \end{aligned}$$
(18)

Taking into account the facts that the unique solution \(z_{i}^{q,l}\) of (9) is nontrivial and concave, we can state that \(z_{i}^{q,l}\) is positive in (0, 1). Moreover we get

$$\begin{aligned} (z_{i}^{q,l})^{\prime }(t)\le 0\text { for all }t\in [\overline{t},1). \end{aligned}$$
(19)

Indeed, owing to (11) we obtain the following chain of assertions

$$\begin{aligned} \left( z_{i}^{q,l}\right) ^{\prime }(\overline{t})\le & {} -\left( \underset{ s\in (0,\varepsilon _{i})}{\inf }h_{i}^{q,l}(s)\right) \int _{0}^{\varepsilon _{i}}sds+\int _{\overline{t}}^{1}\left( \underset{s\in (0,1)}{\sup } h_{i}^{q,l}(s)\right) ds \\= & {} -\frac{1}{2}\varepsilon _{i}^{2}\left( \underset{s\in (0,\varepsilon _{i}) }{\inf }h_{i}^{q,l}(s)\right) +(1-\overline{t})\left( \underset{s\in (0,1)}{ \sup }h_{i}^{q,l}(s)\right) \le 0. \end{aligned}$$

Moreover, we can describe the behavior of \(z_{i}^{q,l}\) as \(t\rightarrow \)\( 1^{-}.\) Since \(\left( z_{i}^{q,l}\right) ^{\prime }(t)=-\int _{0}^{1}sh_{i}^{q,l}(s)ds+\int _{t}^{1}h_{i}^{q,l}(s)ds\) we get

$$\begin{aligned} (0,+\infty )\ni M_{i}^{q,l}:=\int _{0}^{1}sh_{i}^{q,l}(s)ds=-{\lim }_{ t\rightarrow 1^{-}}(z_{i}^{q,l})^{\prime }(t)={\lim }_{t\rightarrow 1^{-}}\frac{z_{i}^{q,l}(t)}{(1-t)} \end{aligned}$$
(20)

and further

$$\begin{aligned} z_{i}^{q,l}(t)=O(1-t)\,\,for\,\,t\rightarrow 1^{-}. \end{aligned}$$
(21)

Now we consider the vector function \(\left( \overline{u}_{1}^{q,l},\overline{ u}_{2}^{q,l}\right) ,\) where \(\overline{u} _{i}^{q,l}(x)=z_{i}^{q,l}(1-||x||^{2-n})\) in \(G_{R_{i}}\) which is a positive radial solution of (6). Taking into account (20) we obtain

$$\begin{aligned} \lim _{||x||\rightarrow \infty }\frac{\overline{u}_{i}^{q,l}(x)}{||x||^{2-n}} =M_{i}^{q,l}\in \left( 0,+\infty \right) \end{aligned}$$

which implies (14). To prove (15) we take an arbitrary \( \phi \in C^{1}(1,+\infty )\) such that \({\lim }_{r\rightarrow +\infty }\phi (r)=0\) and \({\lim }_{r\rightarrow +\infty }\phi ^{\prime }(r)r^{n-1}=+\infty .\) Then function \(\psi (t)=\phi ((1-t)^{\frac{1}{2-n}})\) satisfies conditions: \({\lim }_{t\rightarrow 1^{-}}\psi (t)=0\) and \( {\lim }_{t\rightarrow 1^{-}}\psi ^{\prime }(t)=+\infty .\) It is clear that \({\lim }_{t\rightarrow 1^{-}}\frac{(z_{i}^{q,l})^{\prime }(t)}{\psi ^{\prime }(t)}=0.\) Thus, applying again the de L’Hospital’s rule, we obtain \({\lim }_{t\rightarrow 1^{-}}\frac{z_{i}^{q,l}(t)}{\psi (t) }=0\) and further

$$\begin{aligned} \underset{||x||\rightarrow +\infty }{\lim }_\frac{\overline{u}_{i}^{q,l}(x)}{ \phi (||x||)}= & {} \underset{||x||\rightarrow +\infty }{\lim }_\frac{ z_{i}^{q,l}(1-||x||^{2-n})}{\phi (||x||)}=\underset{t\rightarrow 1^{-}}{\lim }\frac{z_{i}^{q,l}(t)}{\phi ((1-t)^{\frac{1}{2-n}})} \\= & {} \underset{t\rightarrow 1^{-}}{\lim }\frac{z_{i}^{q,l}(t)}{\psi (t)}=0 \end{aligned}$$

for \(i=1,2,\) which gives (15). \(\square \)

3 Proof of the main results

Now we can prove our main result concerning the existence of sequences of connected sets of positive evanescent solutions of (1). The proof of this theorem is based on the sub-solution and super-solutions method described in Theorem 1 and the iteration process applied in [28].

Proof (of Theorem 2) Suppose that (H1), (H2), (H3) hold. Let us fix \(q\in {\mathbb {R}}.\) Lemma 4 leads to the existence of a nondecreasing (with respect to each coordinate) sequence \(\{(\overline{u}_{1}^{q,l},\overline{u} _{2}^{q,l})\}_{l\in {\mathbb {N}}}\) of super-solutions of (1)–(2). Let us consider trivial vector function \(\left( 0,0\right) \) being a sub-solution of (1)–(2) and its super-solution \(( \overline{u}_{1}^{q,1},\overline{u}_{2}^{q,1}).\) Theorem 1 gives the existence of solution \((u_{1}^{q,1},u_{2}^{q,1})\) for our problem such that for \(i=1,2,\)

$$\begin{aligned} 0\le u_{i}^{q,1}\le \overline{u}_{i}^{q,1}\le d_{i}\ \text { in }G_{R} \text { and }u_{i}^{q,1}=\overline{u}_{i}^{q,1}\text { on }\partial G_{R}. \end{aligned}$$

Now we treat the above solution \((u_{1}^{q,1},u_{2}^{q,1})\) as a sub-solution of our problem. Since we have super-solution \((\overline{u} _{1}^{q,2},\overline{u}_{2}^{q,2})\) satisfying the following chain of inequalities for \(i=1,2,\)

$$\begin{aligned} u_{i}^{q,1}\le \overline{u}_{i}^{q,1}<\overline{u}_{i}^{q,2}\text { in } G_{R}, \end{aligned}$$

we can apply again Theorem 1 and derive the existence of a solution \( (u_{1}^{q,2},u_{2}^{q,2})\) of (1)–(2) such that for \( i=1,2,\)

$$\begin{aligned} u_{i}^{q,1}\le & {} u_{i}^{q,2}\le \overline{u}_{i}^{q,2}\le d_{i}\ \text { in }G_{R}, \\ u_{i}^{q,2}= & {} \overline{u}_{i}^{q,2}\text { on }\partial G_{R}. \end{aligned}$$

The last equality and the fact that \(\overline{u}_{i}^{q,1}<\overline{u} _{i}^{q,2}\) on \(\partial G_{R}\) imply that for \(i=1,2,\)\(u_{i}^{q,1}\ne u_{i}^{q,2}.\) Iterating this process and having constructed solution \( (u_{1}^{q,l},u_{2}^{q,l})\) of (1)–(2) we obtain the existence of another solution \((u_{1}^{q,l+1},u_{2}^{q,l+1})\) such that

$$\begin{aligned} u_{i}^{q,l}\le & {} u_{i}^{q,l+1}\le \overline{u}_{i}^{q,l+1}\le d_{i}\ \text { in }G_{R}, \\ u_{i}^{q,l+1}= & {} \overline{u}_{i}^{q,l+1}\text { on }\partial G_{R} \end{aligned}$$

for \(i=1,2.\) Since \(\overline{u}_{i}^{q,l}<\overline{u}_{i}^{q,l+1}\) on \( \partial G_{R}\) we obtain for \(i=1,2,\)\(u_{i}^{q,l}\ne u_{i}^{q,l+1}.\) Finally we obtain a sequence of bounded solutions \(\left\{ (u_{1}^{q,l},u_{2}^{q,l})\right\} _{l\in {\mathbb {N}}}\) which is nondecreasing in \(G_{R}\) and increasing on \(\partial G_{R}.\) Since for all \(q\in {\mathbb {R}} \), \(l\in {\mathbb {N}},\) and \(\ i=1,2,\)\({\lim }_{||x||\rightarrow \infty }u_{i}^{q,l}(x)=0\) we state that we get evanescent solutions.

Let us consider the case when for all \(l\in {\mathbb {N}},\) we take \({\widetilde{f}}_{1}^{q,l}\ne {\widetilde{f}}_{2}^{q,l},\) which implies \(\overline{u} _{1}^{q,l}\ne \overline{u}_{2}^{q,l}.\) Then \(u_{1}^{q,l}\ne u_{2}^{q,l}\) on \(\partial G_{R}\) and we can obtain the first sequence of continua \( \{S_{l}^{0}\}_{l\in {\mathbb {N}}},\) where

$$\begin{aligned} S_{l}^{0}:=\{(u_{1}^{q,l},u_{2}^{q,l}),q\in {\mathbb {R}}\text { such that } u_{1}^{q,l}\ne u_{2}^{q,l}\text { on }\partial G_{R}\} \end{aligned}$$

for all \(l\in {\mathbb {N}}.\) In the case when we consider \({\widetilde{f}} _{1}^{q,l}\equiv {\widetilde{f}}_{2}^{q,l}\) for all \(l\in {\mathbb {N}}, \) we get \(\overline{u}_{1}^{q,l}\equiv \overline{u}_{2}^{q,l}\) in \(G_{R}\) and further we obtain \(u_{1}^{q,l}=u_{2}^{q,l}\) on \(\partial G_{R}\) which allows us to construct the other sequence of continua \(\{S_{l}^{1}\}_{l\in {\mathbb {N}}}\) , where

$$\begin{aligned} S_{l}^{1}:=\{(u_{1}^{q,l},u_{2}^{q,l});q\in {\mathbb {R}}\text { such that } u_{1}^{q,l}=u_{2}^{q,l}\text { on }\partial G_{R}\}\text { for all }l\in {\mathbb {N}}. \end{aligned}$$

Coming to the proof of the last part of Theorem 2 we fix \( q\in {\mathbb {R}}\) and consider the sequence \(\{(u_{1}^{q,l},u_{2}^{q,l})\}_{l \in {\mathbb {N}}}\) of solutions. Based on the classical estimate for solutions of elliptic PDE (see e.g. [27]-Lemma 3.2) in the following annulus \( \Omega _{j,R}:=\{x\in {\mathbb {R}}^{n},\)\(R+\frac{1}{2j}<||x||<R+j\},\)\(j\in {\mathbb {N}},\) we state the existence of \(D>0\) independent of l such that

$$\begin{aligned} ||u_{i}^{q,l}||_{C^{2,\alpha }(\overline{\Omega }_{j,R})}\le D\text { for all }j\ge 1. \end{aligned}$$

The compactness of the injection \(C^{2,\alpha }(\overline{\Omega } _{1,R})\rightarrow C^{2}(\overline{\Omega }_{1,R}),\) gives the existence of a subsequence \(\{(u_{1}^{q,l},u_{2}^{q,l})\}_{l\in {\mathbb {N}}}\) which tends to \(({\widetilde{u}}_{1}^{1,q},{\widetilde{u}}_{2}^{1,q})\) in \(\overline{\Omega } _{1,R}\) in the \(\left( C^{2}(\overline{\Omega }_{1,R})\right) ^{2}\) norm. Therefore \(({\widetilde{u}}_{1}^{1,q},{\widetilde{u}}_{2}^{1,q})\) is also a solution of (1) in \(\overline{\Omega }_{1,R}.\) It is clear that the subsequence \(\{(u_{1}^{q,l_{k}},u_{2}^{q,l_{k}})\}_{k\in {\mathbb {N}}},\) also satisfies the above estimate, and consequently there exists a subsequence of \(\{(u_{1}^{q,l_{k}},u_{2}^{q,l_{k}})\}_{k\in {\mathbb {N}}}\) which converges in the \(\left( C^{2}(\overline{\Omega }_{2,R})\right) ^{2}\) norm to \(({\widetilde{u}}_{1}^{2,q},{\widetilde{u}}_{2}^{2,q})\), which is a solution of (1) in \(\overline{\Omega }_{2,R}\) and for \(i=1,2,\)\( {\widetilde{u}}_{i}^{2,q}={\widetilde{u}}_{i}^{1,q}\) in \(\overline{\Omega } _{1,R}.\) Iterating this schema, we can construct inductively a sequence \( \left\{ ({\widetilde{u}}_{1}^{j,q},{\widetilde{u}}_{2}^{j,q})\right\} _{j\in {\mathbb {N}}}\) of solutions of (1) in \(\overline{\Omega }_{j,R}\) such that \({\widetilde{u}}_{i}^{j+1,q}={\widetilde{u}}_{i}^{j,q}\) in \(\overline{ \Omega }_{j,R}.\) This property allows us to consider a vector function (\( {\widetilde{u}}_{1}^{q},{\widetilde{u}}_{2}^{q})\) given as follows

$$\begin{aligned} {\widetilde{u}}_{i}^{q}:={\widetilde{u}}_{i}^{j,q}\text { in }\Omega _{j,R}\text { for all }j\ge 1,\,\,i=1,2 \end{aligned}$$

and state that it satisfies (1). Let us consider an arbitrary bounded set \(\overline{M}\subset \Omega _{R}.\) It is clear that there exists \(j\in {\mathbb {N}}\) such that \(\overline{M}\subset \Omega _{j,R}.\) Applying the above results for \(\Omega _{j,R}\) we derive that (\({\widetilde{u}}_{1}^{q}, {\widetilde{u}}_{2}^{q})\in \)\(\left( C^{2}(\overline{M})\right) ^{2}.\) Now, the regularity arguments associated with the Schauder’s estimates imply that (\({\widetilde{u}}_{1}^{q},{\widetilde{u}}_{2}^{q})\in \left( C^{2,\alpha }( \overline{M})\right) ^{2},\) consequently, (\({\widetilde{u}}_{1}^{q}, {\widetilde{u}}_{2}^{q})\in \left( C_{loc}^{2,\alpha }(\Omega _{R})\right) ^{2} \) and satisfies (1)–(2). Finally, the properties of (\( u_{1}^{q,l},u_{2}^{q,l})\) allowed us to prove that (\({\widetilde{u}}_{1}^{q}, {\widetilde{u}}_{2}^{q})\) is a positive evanescent solution of (1). \(\square \)

Our task is now to prove the other result concerning the asymptotics of solutions and their gradients. Here we employee the ideas described in [19, 27, 28].

Proof (of Theorem 3) Suppose that (H1), (H2), (H3) hold. Let us fix \(q\in {\mathbb {R}}\), \(l\in {\mathbb {N}},\) and consider the solution \((u_{1}^{q,l},u_{2}^{q,l})\) given in Theorem 2. We prove that each \((u_{1}^{q,l},u_{2}^{q,l})\) has finite energy applying the standard approach (see e.g. [27]). To this end we consider \(x\in \Omega _{R}\) such that \(||x||\ge 2R,\) and a ball B(x; r / 2) of center x and radius r / 2,  where \(r=||x||.\) Then, applying (14 ) and the estimates for solutions of elliptic problems ( [22], Th. 6.2), we obtain the existence of \(L_{i}^{q,l}>R,\)\(C_{i}^{q,l}>0\) such that for all \(x\in {\mathbb {R}}^{n},\)\(||x||>L_{i}^{q,l},\)

$$\begin{aligned} \frac{r}{2}|\nabla u_{i}^{q,l}(x)|\le & {} C_{i}^{q,l}\left( ||u_{i}^{q,l}||_{C(B(x;r/2))}+3\left( \frac{r}{2}\right) ^{2}||f_{i}^{q,l}||_{C(B(x;r/2))}\right) \\\le & {} C_{i}^{q,l}\left( ||u_{i}^{q,l}||_{C(B(x;r/2))}+\frac{3m}{4}\right) r^{2-n}, \end{aligned}$$

where \(m:=\max \{m_{1},m_{2}\}\). Finally, taking into account (14 ), we obtain the existence of \(M_{i}^{q,l}>0\) and \(L_{i}^{q,l}>1\) such that for all \(x\in {\mathbb {R}}^{n},\)\(||x||>L_{i}^{q,l},\)

$$\begin{aligned} |\nabla u_{i}^{q,l}(x)|\le C_{i}^{q,l}\left( \frac{3M_{i}^{q,l}}{2}+\frac{3m }{4}\right) ||x||^{1-n}. \end{aligned}$$

According to the definition given in the first section, this assertion implies that \((u_{1}^{q,l},u_{2}^{q,l})\) has finite energy in a neighborhood of infinity.

Assertions (4) and (5) are a simple consequence of (14) and (15). \(\square \)

4 Final remarks and examples

Employing the Kawano’s approach (see [25]) we can obtain the same conclusion as in Theorem 2 also in the case when condition ( b) in (H3) is replaced by the following condition

(b’):

\(f_{1}\) is nonincreasing in \(u_{2}\) and \( f_{2}\) is nonincreasing in \(u_{1}\) in \(G_{R}\times [0,d_{1}]\times [0,d_{2}]\).

Then in the proof of Theorem 1 we have to choose different starting point of the monotone procedure and consider a super-subsolution \((\overline{ u}_{1},\underline{u}_{2})\) of our problem instead of super-solution. In this case we can also prove the existence of sequences of connected sets of solutions.

Now two natural questions appear. The first one concerns the existence of functions for which (H3) holds. It turns out that it is easy to find many examples of \(f_{1},f_{2}\) satisfying (H3) among functions of the form \(f_{i}(x,u_{1},u_{2})=\overline{f}_{i}(u_{1},u_{2})\left( ||x||^{q}+k(x)\right) ^{-1},\) where \(q>2n-2,\)k is positive and sufficiently smooth and \(\overline{f}_{i}\) is a polynomial, exponential or rational function or their combinations, e.g. \(\ \overline{f} _{i}(u_{1},u_{2})=c(u_{1}^{5}+u_{1}^{4}+(u_{1}+u_{2})^{2}+1)\) or \(\overline{f }_{i}(u_{1},u_{2})=c(e^{u_{1}+u_{2}}+\frac{u_{1}^{3}+u_{2}^{3}}{ (4-u_{1})(5-u_{2})}).\) With help of our approach we can also investigate problems of the Emden-Fowler type when \(\overline{f} _{i}(u_{1},u_{2})=c(u_{1}^{\alpha }+u_{2}^{\beta }+M)\) with \(\alpha ,\beta ,M>0.\) At the end of this paper an example of the problem with superlinear \( f_{i}(x,\cdot ,\cdot )\) will be discussed more precisely.

The other question is associated with R. Because of condition (3), the radius R has to be sufficiently large. For \(n=3\) we can consider \( \Omega _{R}\) being the complement of large balls because in this case R has to be greater than or equal to 32. But for \(n=4,\) we can consider \( \Omega _{R}\) with all \(R\ge 3.\, 674\,2.\) It is easy to note that for high dimension n,  radius R can be close to 1. This fact is associated with the behavior of each function \(t_{n}(k)=\left[ \frac{ 2k^{2n-2}}{\left( k^{n-2}-1\right) ^{2}}\right] ^{\frac{1}{n-2}}\) in \( (1,+\infty ),\) where \(n\in {{\mathbb {N}}}\). We can note that for each \(n\in {\mathbb {N}}\) the minimal value of \(t_{n}\left( \cdot \right) \) in \( (1,+\infty )\) is attained at \(k_{n}=\root n-2 \of {\left( n-1\right) }\) and then

$$\begin{aligned} \lim _{n\rightarrow \infty }t_{n}(k_{n})=\lim _{n\rightarrow \infty }\left[ \frac{2\left( n-1\right) \left( n-1\right) ^{\frac{2}{n-2}}}{\left( n-2\right) ^{2}}\right] ^{\frac{1}{n-2}}= 1. \end{aligned}$$

Condition (H3) part (c) can be rewritten as follows \( R>t_{n}(k_{n}).\) Below is the table with values \(t_{n}(k_{n})\) for higher n

n

5

10

50

100

200

400

\(t_{n}(k_{n})\)

\(2.\,077\,0\)

\(1.\,202\,9\)

\(1.\,018\,9\)

\(1.\,008\,3\)

\(1.\,003\,8\)

\(1.\,001\,8.\)

Example 1

Let us consider the following system

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{div}(a_{1}(||x||)u_{1}(x))+\frac{1}{2} (2^{u_{1}(x)+u_{2}(x)}+u_{1}^{2}(x)+u_{2}^{2}(x))\left( ||x||^{6}+1\right) ^{-1} \\ \quad +g_{1}(x)x\cdot \nabla u_{1}(x)=0, \\ \mathrm{div}(a_{2}(||x||)u_{2}(x))+\frac{ u_{1}^{5}(x)+u_{2}^{3}(x)+1}{(3-u_{1}(x))(7-u_{2}(x))}\left( ||x||^{8}+1\right) ^{-1}+g_{2}(x)x\cdot \nabla u_{2}(x)=0 \\ \underset{||x||\rightarrow \infty }{\lim }u_{i}(x)=0\text { for }i=1,2, \end{array} \right. \end{aligned}$$
(22)

for \(x\in G_{R}\), where \(G_{R}=\left\{ x\in {\mathbb {R}}^{3},||x||>R\right\} \) and for \(i=1,2,\)\(g_{i}(x)=\frac{1}{||x||-\frac{1}{2}\sin x_{i}},\)\( a_{i}(||x||)=1+\frac{||x||^{4}}{x_{i}^{2}+||x||^{4}}\). We show that Theorems 2 and 3 can be applied for the system given above. To this end we note that for \(g_{i}\) and \(a_{i},\)\(i=1,2\), (H1) and (H2) hold with \(c_{1}=1,\)\(c_{2}=2\). Since, in our case

$$\begin{aligned} f_{1}(x,u_{1},u_{2})=\frac{1}{2}(2^{u_{1}+u_{2}}+u_{1}^{2}+u_{2}^{2})\left( ||x||^{6}+1\right) ^{-1} \end{aligned}$$

and

$$\begin{aligned} f_{2}(x,u_{1},u_{2})=\frac{1+u_{1}^{5}+u_{2}^{3}}{(3-u_{1})(7-u_{2})}\left( ||x||^{8}+1\right) ^{-1} \end{aligned}$$

we see that for \(d_{1}=d_{2}=1,\) simple calculations allows us to obtain for \(r\ge 1,\)

$$\begin{aligned} \underset{(u_{1},u_{2})\in [0,d_{1}]\times \left[ 0,d_{2}\right] }{ \sup }\underset{||x||=r}{\sup }f_{1}(x,u_{1},u_{2})\le 2r^{-6} \end{aligned}$$

and

$$\begin{aligned} \underset{(u_{1},u_{2})\in [0,d_{1}]\times \left[ 0,d_{2}\right] }{ \sup }\underset{||x||=r}{\sup }f_{2}(x,u_{1},u_{2})\le r^{-8}. \end{aligned}$$

Therefore, it suffices to take \(m_{1}=2\) and \(m_{2}:=1.\) It is clear that \( m_{i}\le 2=4(n-2)d_{i}c_{1}^{2}c_{2}^{-1}.\) Then we can state that ( H3) also holds. Finally, Theorem 2 leads to the existence of sequences of connected sets of positive solutions (22) with asymptotics described in Theorem 3.