1 Introduction

Boundary value problems involving the p-Laplacian arise from many branches of pure mathematics as in the theory of quasiregular and quasiconformal mapping (for example, see [1]) as well as from various problems in mathematical physics notably the flow of non-Newtonian fluids.Semipositone problems are well known to be mathematically challenging to study positive solutions, (see [2] and [3]). Nonetheless the nonsingular case is well studied when the nonlinearities satisfy certain sublinear growth at infinity. We consider an elliptic system of the form

$$\begin{aligned} \left\{ \begin{array}{lll} -M_{1}\Big (\int _{\Omega }|\nabla u|^{p}dx\Big )\Delta _{p} u=\lambda _{1} f_{1}(u)+ \mu _{1} \frac{g_{1}(v)}{v^{\alpha _{1}}} &{} in &{} \Omega ;\\ -M_{2}\Big (\int _{\Omega }|\nabla v|^{q}dx\Big )\Delta _q v=\lambda _{2} \frac{f_{2}(u)}{u^{\alpha _{2}}} + \mu _{2} g_{2}(v) &{} in &{} \Omega ; \\ u=v=0 &{} on &{} \partial \Omega , \end{array}\right. \end{aligned}$$
(1.1)

where \(M_{i}:\mathbb R^{+}_{0}\rightarrow \mathbb R^{+}\), \(i=1,2\) are two continuous and increasing functions , \( \Delta _m w:=\vert \nabla w \vert ^{m-2} \nabla w \) is the m-Laplacian operator for \( m > 1 \) and \( \Omega \subset \mathbb {R}^N; N\ge 1 \), is a bounded domain with smooth boundary (a bounded interval if \( N = 1 \)). For \( i = 1, 2 \), \( 0\le \alpha _i <1 \) are fixed constants and \( \lambda _i , \mu _i >0 \) are parameters.

The nonlinearities \( f_i, g_i : [0,\infty ) \rightarrow \mathbb {R}\) are continuous functions such that \( g_{1}(0) < 0 \) and \( f_{2}(0) < 0 \). Let \( \tilde{g}_{1}(s):= \frac{g_{1}(s)}{s^\alpha _{1}} \) and \( \tilde{f}_{2}(s):= \frac{f_{2}(s)}{s^\alpha _{2}} \) . Problem (1.1) is called nonlocal because of the term \(-M(\int _{\Omega }|\nabla u|^{r}dx)\) which implies that the first two equations in (1.1) are no longer pointwise equalities. This phenomenon causes some mathematical difficulties which makes the study of such a class of problem particularly interesting. Also, such a problem has physical motivation. Moreover, system (1.1) is related to the stationary version of the Kirchhoff equation

$$\begin{aligned} \rho \frac{\partial ^{2}u}{\partial t^{2}}-\Big (\frac{P_{0}}{h} + \frac{E}{2L}\int _{0}^{L}|\frac{\partial u}{\partial x}|^{2}dx\Big )\frac{\partial ^{2}u}{\partial x^{2}}=0 \end{aligned}$$
(1.2)

presented by Kirchhoff [4]. This equation extends the classical d’Alembert’s wave equation by considering the effects of the changes in the length of the strings during the vibrations. The parameters in (1.2) have the following meanings: L is the length of the string, h is the area of cross section, E is the Young’s modulus of the material, \(\rho \) is the mass density, and \(P_{0}\) is the initial tension.

When an elastic string with fixed ends is subjected to transverse vibrations, its length varies with the time: this introduces changes of the tension in the string. This induced Kirchhoff to propose a nonlinear correction of the classical D’Alembert’s equation. Later on, Woinowsky-Krieger (Nash-Modeer) incorporated this correction in the classical Euler-Bernoulli equation for the beam (plate) with hinged ends. See, for example, [5, 6] and the references therein.

Nonlocal problems also appear in other fields: for example, biological systems where u and v describe a process which depends on the average of itself (for instance, population density). See [7,8,9,10,11] and the references therein. In recent years, problems involving Kirchhoff type operators have been studied in many papers, we refer to [12,13,14,15,16,17,18], in which the authors have used different methods to prove the existence of solutions.

The nonlinearities \( \tilde{g}_{1} \) and \( \tilde{f}_{2} \) are of infinite semipositone nature due to their singular behavior near the origin, namely \( \lim _{s\rightarrow 0^{+}}\tilde{g}_{1}(s)=\lim _{s\rightarrow 0^{+}}\tilde{f}_{2}(s)=-\infty \). Semipositone problems (even the nonsingular case \( \alpha _i=0 \)) are well known to be mathematically challenging to study positive solutions, (see [2] and [3]).The main tool used in this study is the method of sub and super solutions. Our results in this note improve the previous one [19] in which \(M_{1}(t)\equiv M_{2}(t)\equiv 1\).To our best knowledge, this is an interesting and new research topic for (p, q)-Kirchhoff type systems. One can refer to [20,21,22] for some recent existence results of infinite semipositone systems.

2 Existence of Solutions

In this section, we shall establish our existence result via the method of sub- super-solution [23]. For the system

$$\begin{aligned} -M_{1}\Big (\int _{\Omega }|\nabla u|^{p}dx\Big )\Delta _{p} u= & {} h_{1}(x,u,v), \quad x\in \Omega ,\\ -M_{2}\Big (\int _{\Omega }|\nabla v|^{q}dx\Big )\Delta _q v= & {} h_{2}(x,u,v), \quad x\in \Omega ,\\ u= & {} v = 0 , \quad x\in \partial \Omega , \end{aligned}$$

a pair of functions \((\psi _{1},\psi _{2})\in W^{1,p}\cap C(\overline{\Omega })\times W^{1,q}\cap C(\overline{\Omega })\) and \((z_{1},z_{2})\in W^{1,p}\cap C(\overline{\Omega })\times W^{1,q}\cap C(\overline{\Omega })\) are called a subsolution and supersolution if they satisfy \((\psi _{1},\psi _{2})=(0,0)=(z_{1},z_{2})\) on \(\partial \Omega \),

$$\begin{aligned}&M_{1}\Big (\int _{\Omega }|\nabla \psi _{1}|^{p}dx\Big ) \int _{\Omega }\ |\nabla \psi _{1}|^{p-2}\nabla \psi _{1}\cdot \nabla w\ dx\le \int _{\Omega }\ h_{1}(x,\psi _{1},\psi _{2})w\ dx, \\&\quad M_{2}\Big (\int _{\Omega }|\nabla \psi _{2}|^{q}dx\Big ) \int _{\Omega }\ |\nabla \psi _{2}|^{q-2}\nabla \psi _{2}\cdot \nabla w\ dx\le \int _{\Omega }\ h_{2}(x,\psi _{1},\psi _{2})w\ dx \end{aligned}$$

and

$$\begin{aligned}&M_{1}\Big (\int _{\Omega }|\nabla z_{1}|^{p}dx\Big ) \int _{\Omega }\ |\nabla z_{1}|^{p-2}\nabla z_{1}\cdot \nabla w\ dx\ge \int _{\Omega }\ h_{1}(x,z_{1},z_{2})w\ dx,\\&\quad M_{2}\Big (\int _{\Omega }|\nabla z_{2}|^{q}dx\Big ) \int _{\Omega }\ |\nabla z_{2}|^{q-2}\nabla z_{2}\cdot \nabla w\ dx\ge \int _{\Omega }\ h_{2}(x,z_{1},z_{2})w\ dx. \end{aligned}$$

for all \( w\in W=\{w\in C_{0}^{\infty }(\Omega )| w\ge 0, x\in \Omega \}\). Then we prove the following existence result.

Lemma 2.1

[14] Suppose there exist sub- and super-solutions \((\psi _{1},\psi _{2})\) and \((z_{1},z_{2})\) respectively of (1.1) such that \((\psi _{1},\psi _{2})\le (z_{1},z_{2})\). Then (1.1) has a solution (u, v) such that (u, v) \(\in [(\psi _{1},\psi _{2}),(z_{1},z_{2})]\).

We make the following assumptions:

  1. (H1)

    \( \tilde{g}_{1} \) and \( \tilde{f}_{2} \) are nondecreasing.

  2. (H2)

    \( \lim _{s\rightarrow \infty }f_{1}(s)=\lim _{s\rightarrow \infty } g_{2}(s) =\lim _{s\rightarrow \infty }\tilde{g}_{1}(s)=\lim _{s\rightarrow \infty }\tilde{f}_{2}(s)=\infty , \)

  3. (H3)

    \( \lim _{s\rightarrow \infty } \frac{f_{1}(s)}{s^{p-1}}=\lim _{s\rightarrow \infty } \frac{q_{2}(s)}{s^{q-1}}=0 , \)

  4. (H4)

    \( \lim _{s\rightarrow \infty } \frac{\tilde{g}_{1}(M(\tilde{f}_{2}(s))^{\frac{1}{q-1}})}{s^{p-1}}=0 , \) for all \( M>0 , \)

  5. (H5)

    \(M_{i}: \mathbb R^{+}_{0}\rightarrow \mathbb R^{+}\) are continuous and increasing functions and \( m_{i} \le M_{i} \le m_{i,\infty } \),\(i=1,2\), for all \(t\in \mathbb R^{+}_{0}\), where \(\mathbb R^{+}_{0}:=[0,+\infty )\)

Our main result read as follows.

Theorem 2.2

Let (H1)–(H5) hold. Then (1.1) has a positive solution (u, v) provided \(\lambda _{1}+\mu _{1}\) and \(\lambda _{2}+\mu _{2}\) are large .

Proof

For \( \theta \in \mathbb {R}\), define

$$\begin{aligned} P(\theta ) :=p\theta ^2 +[p(q-1)-\alpha _{1} q]\theta -\alpha _{1} q(p-1). \end{aligned}$$

Then \( P (\theta ) \) has two distinct real roots

$$\begin{aligned} \theta _{1,P}:=\dfrac{\alpha _{1}q-p(q-1)-\sqrt{[\alpha _{1} q-p(q-1)]^2+4pq\alpha _{1}(p-1)}}{2p}<0\ , \end{aligned}$$

and

$$\begin{aligned} \theta _{2,P}:=\dfrac{\alpha _{1} q -p(q-1)+\sqrt{[\alpha _{1} q-p(q-1)]^2+4pq\alpha _{1}(p-1)}}{2p}>0\ . \end{aligned}$$

However, \( P (1) = p + p(q - 1)- \alpha _{1} q -\alpha _{1} q(p - 1) = pq(1 - \alpha _{1}) > 0 \) and hence \( 0< \theta _{2,P} < 1 \). Similarly,

$$\begin{aligned} Q(\theta ):=q\theta ^2+[q(p-1)-\alpha _{2}p]\theta -\alpha _{2}p(q-1) \end{aligned}$$

has two distinct real roots \( \theta _{1,Q}< 0< \theta _{2,Q} < 1 \). Let

$$\begin{aligned} \beta \in (\max \{\theta _{2,P}, \theta _{2,Q}\},1)\ . \end{aligned}$$

Then since \( P (\beta ) > 0 \) and \( Q(\beta ) > 0 \), we have

$$\begin{aligned} \left\{ \begin{array}{l} \dfrac{\beta _p}{p-1+\beta }> \dfrac{\alpha _{1}q}{q-1+\beta }, \\ \dfrac{\beta _q}{q-1+\beta }> \dfrac{\alpha _{2}p}{p-1+\beta }. \end{array}\right. \end{aligned}$$
(2.1)

Now let \( \nu _m \) be the principal eigenvalue of

$$\begin{aligned} \left\{ \begin{array}{lll} -\Delta _m \phi = \nu \vert \phi \vert ^{m-2}\phi &{} in &{} \Omega \ ; \\ \phi =0 &{} in &{} \Omega \ . \end{array}\right. \end{aligned}$$
(2.2)

Then the corresponding eigenfunction, \( \phi _m \in C^1(\bar{\Omega }) \) is of one sign in \( \Omega \) and \( \frac{\partial \phi _m}{\partial \eta }<0 \) on \( \partial \Omega \). Without loss of generality, we normalize \( \phi _m \) so that \( \phi _m >0 \) in \( \Omega \) and \( \Vert \phi _m \Vert _\infty =1 \). Furthermore, since \( \vert \nabla \phi _m \vert \ne 0 \) near \( \partial \Omega \), and \( \phi _m >0 \) in \( \Omega \), there exist \( \delta , a >0 \) and \( 0< \sigma < 1 \) such that for \( m = p, q \)

$$\begin{aligned} \left\{ \begin{array}{lll} \nu _m \phi _{m}^{m}-\frac{(m-1)(1-\beta )}{m-1+\beta }\vert \nabla \phi _m \vert ^m \le -a &{} on &{} \bar{\Omega }_\delta \ ; \\ \phi _m \ge \sigma &{} on &{} \Omega {\setminus } \bar{\Omega }_\delta \ , \end{array}\right. \end{aligned}$$
(2.3)

where \( \Omega _\delta :=\{x\in \Omega : \mathrm{dist}(x,\partial \Omega ) <\delta \} \). Moreover, there exist domain constants \( C_i:=C_i(\Omega )>0 \), for \( i = 1, 2 \), such that \( C_{1}\phi _p \le \phi _q \) and \( C_{2}\phi _q \le \phi _p \) in \( \Omega \). Let \( (\psi _{1},\psi _{2}) \) be the pair given by

$$\begin{aligned} \psi _{1}:=\left[ \dfrac{(\lambda _{1}+\mu _{1})}{m_{1,\infty }a}K_{0} \right] ^{\frac{1}{p-1}}\ \dfrac{p-1+\beta }{p}\phi _{p}^{\frac{p}{p-1+\beta }} \end{aligned}$$

and

$$\begin{aligned} \psi _{2}:=\left[ \dfrac{(\lambda _{2}+\mu _{2})}{m_{2,\infty }a}\bar{K}_{0} \right] ^{\frac{1}{q-1}}\ \dfrac{q-1+\beta }{q}\phi _{q}^{\frac{q}{q-1+\beta }}\ , \end{aligned}$$

where \( K_{0} \) and \( \bar{K}_{0} \) are positive constants defined by \( -K_{0}:=\min _{s\ge 0}\{f_{1}(s),g_{1}(s)\} \) and \( -\bar{K}_{0} :=\min _{s\ge 0}\{f_{2}(s),g_{2}(s)\} \). Observe that for \( z_m:= A\frac{m-1+\beta }{m}\phi _{m}^{\frac{m}{m-1+\beta }} \) we have \( \nabla z_m =A\phi _{m}^{\frac{1-\beta }{m-1+\beta }}\nabla \phi _m \). Therefore, using the weak formulation of (2.2), for all \( \xi \in W \) , we get

$$\begin{aligned}&\int _{\Omega }\vert \nabla z_m \vert ^{m-2} \nabla z_m \cdot \nabla \xi dx\\ &=A^{m-1}\int _{\Omega }\phi _{m}^{\frac{(1-\beta )(m-1)}{m-1+\beta }}\vert \nabla \phi _m \vert ^{m-2}\nabla \phi _m \cdot \nabla \xi dx\\ &=A^{m-1}\int _{\Omega }\vert \nabla \phi _{m} \vert ^{m-2}\nabla \phi _m \cdot \Big [\nabla ( \xi \phi _{m}^{\frac{(1-\beta )(m-1)}{m-1+\beta }})\\&\quad -\xi \dfrac{(1-\beta )(m-1)}{m-1+\beta }\phi _{m}^{\frac{-\beta m}{m-1+\beta }}\Big ] dx\\ &=A^{m-1}\int _{\Omega }\Big [\vert \nabla \phi _m \vert ^{m-2}\nabla \phi _m \cdot \nabla ( \xi \phi _{m}^{\frac{(1-\beta )(m-1)}{m-1+\beta }})\\&\quad - \dfrac{(1-\beta )(m-1)}{m-1+\beta }\vert \nabla \phi _m \vert ^{m-2}\phi _{m}^{\frac{-\beta m}{m-1+\beta }}\xi \nabla \phi _m \Big ] dx\\ &=A^{m-1}\int _{\Omega }\Big [\nu _m \phi _{m}^{m-1}(\phi _{m}^{\frac{(1-\beta )(m-1)}{m-1+\beta }}\xi )\\&\quad -\dfrac{(1-\beta )(m-1)}{m-1+\beta }\vert \nabla \phi _m \vert ^{m-2}\phi _{m}^{\frac{-\beta m}{m-1+\beta }}\xi \nabla \phi _m \Big ] dx\\ &=A^{m-1}\int _{\Omega }\phi _{m}^{\frac{-\beta m}{m-1+\beta }}\Big [\nu _m \phi _{m}^{m}-\frac{(1-\beta )(m-1)}{m-1+\beta }\vert \nabla \phi _m \vert ^m\Big ]\xi )dx\ . \end{aligned}$$

Then, for all \( \xi \in W \) , \( (\psi _{1},\psi _{2}) \) satisfies

$$\begin{aligned}&M_{1}\Big (\int _{\Omega }|\nabla \psi _{1}|^{p}dx\Big ) \int _{\Omega }\ |\nabla \psi _{1}|^{p-2}\nabla \psi _{1}\cdot \nabla w\ dx\\ &=\dfrac{\lambda _{1}+\mu _{1}}{m_{1,\infty }a}K_{0} M_{1}\Big (\int _{\Omega }|\nabla \psi _{1}|^{p}dx\Big )\int _{\Omega }\phi _{p}^{\frac{-\beta p}{p-1+\beta }}\Big [\nu _p\phi _{p}^{p}-\dfrac{(1-\beta )(p-1)}{p-1+\beta }\vert \nabla \phi _p \vert ^{p}\Big ] \xi dx \end{aligned}$$

and

$$\begin{aligned}&M_{2}\Big (\int _{\Omega }|\nabla \psi _{2}|^{q}dx\Big ) \int _{\Omega _\delta }\ |\nabla \psi _{2}|^{q-2}\nabla \psi _{2}\cdot \nabla w\ dx\\ &=\dfrac{\lambda _{2}+\mu _{2}}{m_{2,\infty }a}\bar{K}_{0} M_{2}\Big (\int _{\Omega _\delta }|\nabla \psi _{2}|^{q}dx\Big )\int _{\Omega }\phi _{q}^{\frac{-\beta q}{q-1+\beta }}\Big [\nu _q\phi _{q}^{q}-\dfrac{(1-\beta )(q-1)}{q-1+\beta }\vert \nabla \phi _q \vert ^{q}\Big ] \xi dx\ . \end{aligned}$$

Now on \( \bar{\Omega }_\delta \), since \( \Vert \phi _p \Vert _\infty =1=\Vert \phi _q \Vert _\infty \), using (2.1), (2.3) and the inequality \( \phi _q\ge C_{1} \phi _p \), for large \( \lambda _{2}+\mu _{2} \), we have

$$\begin{aligned}&M_{1}\Big (\int _{\Omega }|\nabla \psi _{1}|^{p}dx\Big ) \int _{\Omega }\ |\nabla \psi _{1}|^{p-2}\nabla \psi _{1}\cdot \nabla w\ dx\\ &=\dfrac{\lambda _{1}+\mu _{1}}{m_{1,\infty }a}K_{0} M_{1}\Big (\int _{\Omega }|\nabla \psi _{1}|^{p}dx\Big )\int _{\Omega _\delta }\phi _{p}^{\frac{-\beta p}{p-1+\beta }} \Big [\lambda _p \phi _{p}^{p}-\dfrac{(1-\beta )(p-1)}{p-1+\beta }\vert \nabla \phi _p \vert ^{p}\Big ] \xi dx\\ \le&-(\lambda _{1}+\mu _{1}) K_{0} \int _{\Omega _\delta }\phi _{p}^{\frac{-\beta p}{p-1+\beta }}\xi dx\\ \le&\lambda _{1} \int _{\Omega _\delta }f_{1}(\psi _{1})\xi dx -\dfrac{K_{0} \mu _{1}}{\left[ \left( \frac{\lambda _{2}+\mu _{2}}{m_{1,\infty }a}\bar{K}_{0}\right) ^{\frac{1}{q-1}}\dfrac{q-1+\beta }{q}C_{1}^{\frac{q}{q-1+\beta }}\right] ^{\alpha _{1}}}\int _{\Omega _\delta }\dfrac{\xi }{\phi _{p}^{\frac{\beta _p}{p-1+\beta }}}dx\\ \le&\lambda _{1} \int _{\Omega _\delta }f_{1}(\psi _{1})\xi dx -\dfrac{K_{0} \mu _{1}}{\left[ \left( \frac{\lambda _{2}+\mu _{2}}{m_{1,\infty }a}\bar{K}_{0}\right) ^{\frac{1}{q-1}}\dfrac{q-1+\beta }{q}C_{1}^{\frac{q}{q-1+\beta }}\right] ^{\alpha _{1}}}\int _{\Omega _\delta }\dfrac{\xi }{\phi _{p}^{\frac{\alpha _{1q}}{q-1+\beta }}}dx\\ &=\lambda _{1} \int _{\Omega _\delta }f_{1}(\psi _{1})\xi dx -\dfrac{K_{0} \mu _{1}}{\left[ \left( \frac{\lambda _{2}+\mu _{2}}{m_{1,\infty }a}\bar{K}_{0}\right) ^{\frac{1}{q-1}}\dfrac{q-1+\beta }{q}\right] ^{\alpha _{1}}}\int _{\Omega _\delta }\dfrac{\xi }{(C_{1}\phi _{p})^{\frac{\alpha _{1q}}{q-1+\beta }}}dx\\ \le&\lambda _{1} \int _{\Omega _\delta }f_{1}(\psi _{1})\xi dx -\dfrac{K_{0} \mu _{1}}{\left[ \left( \frac{\lambda _{2}+\mu _{2}}{m_{1,\infty }a}\bar{K}_{0}\right) ^{\frac{1}{q-1}}\dfrac{q-1+\beta }{q}\right] ^{\alpha _{1}}}\int _{\Omega _\delta }\dfrac{\xi }{\phi _{q}^{\frac{\alpha _{1q}}{q-1+\beta }}}dx\\ \le&\lambda _{1} \int _{\Omega _\delta }f_{1}(\psi _{1})\xi dx +\mu _{1} \int _{\Omega _\delta }\dfrac{g_{1}(\psi _{2})}{\left[ \left( \frac{\lambda _{2}+\mu _{2}}{m_{1,\infty }a}\bar{K}_{0}\right) ^{\frac{1}{q-1}}\dfrac{q-1+\beta }{q}\phi _{q}^{\frac{q}{q-1+\beta }}\right] ^{\alpha _{1}}}\xi dx\\ &=\int _{\Omega _\delta }\left[ \lambda _{1} f_{1}(\psi _{1})+\mu _{1} \frac{g_{1}(\psi _{2})}{\psi _{2}^{\alpha _{1}}}\right] \xi dx =\int _{\Omega _\delta }[\lambda _{1} f_{1}(\psi _{1})+\mu _{1} \tilde{g}_{1} (\psi _{2})]\xi dx\ . \end{aligned}$$

Similarly for \( \lambda _{1}+\mu _{1} \) large, it can be shown that \( \psi _{2} \) satisfies

$$\begin{aligned} M_{2}\Big (\int _{\Omega }|\nabla \psi _{2}|^{q}dx\Big ) \int _{\Omega }\ |\nabla \psi _{2}|^{q-2}\nabla \psi _{2}\cdot \nabla w\ dx \le \int _{\Omega _\delta }\left[ \lambda _{2}\tilde{f}_{2}(\psi _{1})+\mu _{2} g_{2}(\psi _{2})\right] \xi dx \end{aligned}$$

for all \( \xi \in W \).

Next, in \( \Omega {\setminus } \bar{\Omega }_\delta \), since \( \phi _p, \phi _q \ge \sigma >0 \), by (H2) , the following estimate holds for \( \lambda _i+\mu _i \) sufficiently large for \( i = 1, 2 \)

$$\begin{aligned} f_{1}(\psi _{1}),\tilde{f}_{2}(\psi _{1}),\tilde{g}_{1} (\psi _{2}),g_{2}(\psi _{2}) \ge \max \{\frac{K_{0} \nu _p}{m_{1,\infty }a}, \frac{\bar{K}_{0} \nu _q}{m_{2,\infty }a}\} \end{aligned}$$

Therefore for \( \xi \in W \) , \(\psi _{1}\) satisfies

$$\begin{aligned}&M_{1}\Big (\int _{\Omega {\setminus } \bar{\Omega }_\delta }|\nabla \psi _{1}|^{p}dx\Big ) \int _{\Omega {\setminus } \bar{\Omega }_\delta }\ |\nabla \psi _{1}|^{p-2}\nabla \psi _{1}\cdot \nabla \xi ,dx \\ \le&K_{0} \nu _p \dfrac{\lambda _{1}+\mu _{1}}{m_{1,\infty }a}M_{1}\Big (\int _{\Omega {\setminus } \bar{\Omega }_\delta }|\nabla \psi _{1}|^{p}dx\Big ) \int _{\Omega {\setminus }\bar{\Omega }_\delta }\phi _{p}^{\frac{-\beta _p}{p-1+\beta }}\phi _{p}^{p} \xi dx\\ &=K_{0} \nu _p \dfrac{\lambda _{1}+\mu _{1}}{m_{1,\infty }a}M_{1}\Big (\int _{\Omega {\setminus } \bar{\Omega }_\delta }|\nabla \psi _{1}|^{p}dx\Big )\int _{\Omega {\setminus }\bar{\Omega }_\delta }\phi _{p}^{\frac{p(p-1)}{p-1+\beta }}\xi dx\\ \le&K_{0} \nu _p \dfrac{\lambda _{1}+\mu _{1}}{m_{1,\infty }a}M_{1}\Big (\int _{\Omega {\setminus } \bar{\Omega }_\delta }|\nabla \psi _{1}|^{p}dx\Big )\int _{\Omega {\setminus }\bar{\Omega }_\delta }\xi dx\\ \le&\int _{\Omega {\setminus }\bar{\Omega }_\delta }[\lambda _{1} f_{1}(\psi _{1})+\mu _{1} \tilde{g}_{1}(\psi _{2})]\xi dx\ . \end{aligned}$$

Similarly for \( \xi \in W \) , \(\psi _{2}\) satisfies

$$\begin{aligned} M_{2}\Big (\int _{\Omega {\setminus } \bar{\Omega }_\delta }|\nabla \psi _{2}|^{q}dx\Big ) \int _{\Omega {\setminus } \bar{\Omega }_\delta }\ |\nabla \psi _{2}|^{q-2}\nabla \psi _{2}\cdot \nabla \xi ,dx \le \int _{\Omega {\setminus }\bar{\Omega }_\delta }[\lambda _{2} \tilde{f}_{2}(\psi _{1})+\mu _{2} g_{2}(\psi _{2})]\xi dx\ . \end{aligned}$$

Therefore, \( (\psi _{1},\psi _{2}) \) is a subsolution of (1.1) for \( \lambda _i+\mu _i \) large for \( i = 1, 2 \). Now we will construct a supersolution of (1.1). For \( m = p, q \), let \( e_m\in C^1(\bar{\Omega }) \) be the unique solution of

$$\begin{aligned} \left\{ \begin{array}{lll} -\Delta _m e =1 &{} in &{} \Omega \ ;\\ e=0 &{} in &{} \Omega \ . \end{array}\right. \end{aligned}$$
(2.4)

It is well known that \( e_m > 0 \) in \( \Omega \) and that \( \frac{\partial e_m}{\partial \eta }<0 \) on \( \partial \Omega \) where \( \eta \) is the outward normal on the boundary \( \partial \Omega \). Then we set

$$\begin{aligned} (z_{1},z_{2}):= \left( Ce_p, [\frac{(\lambda _{2}+\mu _{2})}{m_{2}}\tilde{f}_{2}(C\Vert e_p \Vert _{\infty })] ^{\frac{1}{q-1}}e_q\right) \ . \end{aligned}$$

For all \( \xi \in W \), \( z_{1} \) satisfies

$$\begin{aligned}&M_{1}\Big (\int _{\Omega }|\nabla z_{1}|^{p}dx\Big )\int _{\Omega }\ |\nabla z_{1}|^{p-2}\nabla z_{1}\cdot \nabla w\ dx\nonumber \\&\ge m_{1}C^{p-1}\int _{\Omega }\vert \nabla e_p \vert ^{p-2}\nabla e_p\cdot \nabla \xi dx\ge m_{1}C^{p-1}\int _{\Omega } \xi dx\ . \end{aligned}$$
(2.5)

Define \( \bar{f}_{1} (s):= \max _{t\in [0,s]}f_{1}(t)\). Then \( \bar{f}_{1} (s) \) is nondecreasing and \( \bar{f}_{1} (s)\ge f_{1}(s) \) for all \( s \ge 0 \). It follows from (H2), (H3) and (H4) that there exists \(C\ge 1\) sufficiently large such that

$$\begin{aligned} m_{1}\ge \frac{{\lambda _{1}\bar{f}_{1}(C\Vert e_p \Vert _\infty ) +\mu _{1}\tilde{g}_{1}([\frac{(\lambda _{2}+\mu _{2})}{m_{2}}\tilde{f}_{2} (C\Vert e_p \Vert \infty )]^{\frac{1}{q-1}}\Vert e_q \Vert _{\infty })}}{C^{p-1}}. \end{aligned}$$

Now since \( \bar{f}_{1} \) and \( \tilde{g}_{1} \) are nondecreasing, we have

$$\begin{aligned} C^{p-1}\ge&\lambda _{1}\bar{f}_{1}(Ce_p)+\mu _{1}\tilde{g}_{1}\left( [\frac{(\lambda _{2} +\mu _{2})}{m_{2}}\tilde{f}_{2}(C\Vert e_p \Vert \infty )]^{\frac{1}{q-1}}e_q\right) \nonumber \\ &=\lambda _{1}\bar{f}_{1}(z_{1})+\mu _{1}\tilde{g}_{1}(z_{2})\ge \lambda _{1} f_{1}(z_{1}) +\mu _{1}\tilde{g}_{1}(z_{2})\ . \end{aligned}$$
(2.6)

Combining (2.5) and (2.6) yields

$$\begin{aligned}&M_{1}\Big (\int _{\Omega }|\nabla z_{1}|^{p}dx\Big )\int _{\Omega }\ |\nabla z_{1}|^{p-2}\nabla z_{1}\cdot \nabla w\ dx\nonumber \\&\ge \lambda _{1} \int _{\Omega } f_{1}(z_{1}) \xi dx +\mu _{1} \int _{\Omega }\tilde{g}_{1}(z_{2})\xi dx\ . \end{aligned}$$
(2.7)

Next, it is easy to see that \( z_{2}=\left[ \frac{(\lambda _{2}+\mu _{2})}{m_{2}} \tilde{f}_{2}(C\Vert e_p \Vert _\infty )\right] ^{\frac{1}{q-1}}e_p \) satisfies

$$\begin{aligned}&M_{2}\Big (\int _{\Omega }|\nabla z_{2}|^{q}dx\Big ) \int _{\Omega }\ |\nabla z_{2}|^{q-2}\nabla z_{2}\cdot \nabla w\ dx \nonumber \\ &=\frac{(\lambda _{2}+\mu _{2})}{m_{2}}\tilde{f}_{2}(C\Vert e_p \Vert _\infty )M_{2}\Big (\int _{\Omega }|\nabla z_{2}|^{q}dx\Big ) \int _{\Omega }\vert \nabla e_p \vert ^{q-2} \nabla e_p \cdot \nabla \xi dx \nonumber \\ &\ge(\lambda _{2}+\mu _{2})\tilde{f}_{2}(C\Vert e_p \Vert _\infty ) \int _{\Omega }\xi dx \nonumber \\ &=[\lambda _{2} \tilde{f}_{2}(C\Vert e_p \Vert _\infty )+\mu _{2} \tilde{f}_{2}(C\Vert e_p \Vert _\infty )]\int _{\Omega }\xi dx\ . \end{aligned}$$
(2.8)

By (H2) and (H3), for \( C > 0 \) sufficiently large, we have

$$\begin{aligned} \tilde{f}_{2}(C\Vert e_p \Vert _\infty )\ge&\tilde{g}_{2}([\frac{(\lambda _{2}+\mu _{2})}{m_{2}}\tilde{f}_{2} (C\Vert e_p \Vert _\infty )]^{\frac{1}{q-1}}\Vert e_q \Vert _\infty )\\ \ge&\tilde{g}_{2}([\frac{(\lambda _{2}+\mu _{2})}{m_{2}}\tilde{f}_{2} (C\Vert e_p \Vert _\infty )]^{\frac{1}{q-1}}e_p \end{aligned}$$

where \( \bar{g}_{2}(s):=\max _{t\in [0,s]}g_{2}(t) \) is a nondecreasing function. Then since \( \bar{g}_{2}(s)\ge g_{2}(s) \) for \( s\ge 0 \), (2.8) yields

$$\begin{aligned}&M_{2}\Big (\int _{\Omega }|\nabla z_{2}|^{q}dx\Big ) \int _{\Omega }\ |\nabla z_{2}|^{q-2}\nabla z_{2}\cdot \nabla w\ dx\\ \ge&\lambda _{2} \int _{\Omega }\tilde{f}_{2} (Ce_p)\xi dx+\mu _{2} \int _{\Omega }g_{2}([\frac{(\lambda _{2}+\mu _{2})}{m_{2}}\tilde{f}_{2} (C\Vert e_p \Vert _\infty )]^{\frac{1}{q-1}}e_p)\xi dx\\ &=\lambda _{2} \int _{\Omega }\tilde{f}_{2} (z_{1})\xi dx+\mu _{2} \int _{\Omega }g_{2}(z_{2})\xi dx \end{aligned}$$

and hence \( (z_{1},z_{2}) \) is a supersolution. Clearly for \( C \gg 1 \), \( (\psi _{1},\psi _{2})\le (z_{1},z_{2}) \) and this completes the proof of Theorem 2.2. \(\square \)

Examples

If \( p = 2 \), \( q = 3 \) and \( \alpha _{1}=\frac{2}{3} \) and \( \alpha _{2}=\frac{1}{3} \), it is easy to verify that \( f_{1}(s)=s^{1/2}-\epsilon _{1} \); \( g_{1}(s)=s-\epsilon _{2} \); \( f_{2}(s)=s^{10/3}-\epsilon _{3} \); \( g_{2}(s)=s^{4/3}-\epsilon _4 \) satisfy the hypotheses of Theorem 2.2, when \( \epsilon _{1} , \epsilon _4 \in \mathbb {R}\) and \( \epsilon _{2}, \epsilon _{3} >0. \)

3 Conclusion

This article concerns the existence of positive solutions for elliptic (p, q)-Kirchhoff type systems with multiple parameters. we establish our abstract existence result via the method of sub- and super-solutions.Our results in this note improve the previous one [19] in which \(M_{1}(t)\equiv M_{2}(t)\equiv 1.\)