# Nodal solutions for Neumann systems with gradient dependence

## Abstract

We consider the following convective Neumann systems:

$$( \mathrm{S} ) \quad \textstyle\begin{cases} -\Delta _{p_{1}}u_{1}+ \frac{ \vert \nabla u_{1} \vert ^{p_{1}}}{u_{1}+\delta _{1} }=f_{1}(x,u_{1},u_{2}, \nabla u_{1},\nabla u_{2}) & \text{in } \Omega , \\ -\Delta _{p_{2}}u_{2}+ \frac{ \vert \nabla u_{2} \vert ^{p_{2}}}{u_{2}+\delta _{2} }=f_{2}(x,u_{1},u_{2}, \nabla u_{1},\nabla u_{2}) & \text{in } \Omega , \\ \vert \nabla u_{1} \vert ^{p_{1}-2}\frac{\partial u_{1}}{\partial \eta }=0= \vert \nabla u_{2} \vert ^{p_{2}-2}\frac{\partial u_{2}}{\partial \eta } & \text{on } \partial \Omega ,\end{cases}$$

where Ω is a bounded domain in $$\mathbb{R}^{N}$$ ($$N\geq 2$$) with a smooth boundary Ω, $$\delta _{1}, \delta _{2} >0$$ are small parameters, η is the outward unit vector normal to Ω, $$f_{1}, f_{2}:\Omega \times \mathbb{R}^{2}\times \mathbb{R}^{2N} \rightarrow \mathbb{R}$$ are Carathéodory functions that satisfy certain growth conditions, and $$\Delta _{p_{i}}$$ ($$1< p_{i}< N$$, $$i=1,2$$) are the p-Laplace operators $$\Delta _{p_{i}}u_{i}=\operatorname{div}(|\nabla u_{i}|^{p_{i}-2}\nabla u_{i})$$ for $$u_{i}\in W^{1,p_{i}}(\Omega )$$. To prove the existence of solutions to such systems, we use a subsupersolution method. We also obtain nodal solutions by constructing appropriate subsolution and supersolution pairs. To the best of our knowledge, such systems have not been studied yet.

## 1 Introduction

We consider the following Neumann systems with gradient dependence:

$$( \mathrm{S} ) \quad \textstyle\begin{cases} -\Delta _{p_{1}}u_{1}+ \frac{ \vert \nabla u_{1} \vert ^{p_{1}}}{u_{1}+\delta _{1} }=f_{1}(x,u_{1},u_{2}, \nabla u_{1},\nabla u_{2}) & \text{in } \Omega , \\ -\Delta _{p_{2}}u_{2}+ \frac{ \vert \nabla u_{2} \vert ^{p_{2}}}{u_{2}+\delta _{2} }=f_{2}(x,u_{1},u_{2}, \nabla u_{1},\nabla u_{2}) & \text{in } \Omega , \\ \vert \nabla u_{1} \vert ^{p_{1}-2}\frac{\partial u_{1}}{\partial \eta }=0= \vert \nabla u_{2} \vert ^{p_{2}-2}\frac{\partial u_{2}}{\partial \eta } & \text{on } \partial \Omega ,\end{cases}$$

where Ω is a bounded domain in $$\mathbb{R}^{N}$$ ($$N\geq 2$$) with a smooth boundary Ω, $$\delta _{1}, \delta _{2} >0$$ are small parameters, η is the outward unit normal vector to Ω, $$\Delta _{p_{i}}$$ ($$1< p_{i}< N$$, $$i=1,2$$) are the p-Laplace operators $$\Delta _{p_{i}}u_{i}:=\operatorname{div}(|\nabla u_{i}|^{p_{i}-2}\nabla u_{i})$$ for $$u_{i}\in W^{1,p_{i}}(\Omega )$$.

In recent years, much has been done regarding the existence of solutions for nonlinear systems with the Dirichlet condition and the reaction term depending on the gradient using different techniques, mainly fixed point theory, variational methods, truncation methods, and subsupersolution methods. We mention for instance, Candito et al. [2], where the authors investigated a quasilinear singular Dirichlet system with gradient dependence. They combined Schauder’s fixed point theorem with subsupersolution approach to establish the existence of smooth positive solutions. For more detail, we refer the readers to some recent papers: Carl and Motreanu [5], Infante et al. [10], Miyagaki and Rodrigues [14], Kita and Otani [11], Motreanu et al. [17], Orpel [21], Ou [22], Wang et al. [24], Yang and Yang [25], and the references therein. See also the monograph by Motreanu [16].

On the other hand, the corresponding Neumann system has been much less studied. In this context, the Neumann quasilinear equation involving a connective term equation was studied by Moussaoui et al. [20]. Candito et al. [3] obtained nodal solutions for a $$(p_{1}, p_{2})$$-Laplacian Neumann system without gradient terms. Neumann systems involving variable exponent double phase operators and gradient dependence were investigated by Guarnotta et al. [9].

The main interest of the present work is the presence of the gradient term, which constitutes a serious obstacle in the investigation of system $$(\mathrm{S})$$. Note that system $$(\mathrm{S})$$ is not in the variational form. Therefore the usual critical point theory cannot be directly applied. This difficulty is overcome by using the theory of pseudomonotone operators. We first introduce an auxiliary system using truncation operators. Then we construct a subsolution $$(\underline{u}_{1},\underline{u}_{2})$$ and a supersolution $$(\overline{u}_{1},\overline{u}_{2})$$ such that $$\underline{u}_{1}\leq \overline{u}_{1}$$, $$\underline{u}_{2}\leq \overline{u}_{2}$$ (see Theorem 5.1). Finally, sub- and supersolutions and truncation techniques provide at least two solutions for system $$(\mathrm{S})$$ with precise sign properties.

We will assume that the nonlinearities $$f_{i}$$ for $$i=1,2$$ are Carathéodory functions $$f_{1}, f_{2}:\Omega \times \mathbb{R}^{2}\times \mathbb{R}^{2N} \rightarrow \mathbb{R}$$, that is, $$f_{i} (\cdot , s_{1},s_{2}, \xi _{1},\xi _{2} )$$ is measurable for every $$(s_{1},s_{2}, \xi _{1},\xi _{2} )\in \mathbb{R}^{2}\times \mathbb{R}^{2N}$$, $$f_{i} (\cdot , s_{1},s_{2}, \xi _{1},\xi _{2} )$$ is continuous for a.e. $$x\in \Omega$$, and they satisfy the following growth conditions:

$$({\mathbf{H_{1}}})$$:

There exist $$\alpha _{i} ,\beta _{i} ,M_{i}>0$$, $$i=1,2$$, such that $$\max \{\alpha _{i} ,\beta _{i} \}< p_{i}-1$$ and

\begin{aligned}& \bigl\vert f_{i}(x,s_{1},s_{2},\xi _{1},\xi _{2} ) \bigr\vert \leq M_{i} \bigl(1+ \vert s_{i} \vert ^{ \alpha _{i} }\bigr) \bigl(1+ \vert \xi _{i} \vert ^{\beta _{i} }\bigr) \\& \quad \text{for } i=1,2 \text{ and all } (x,s_{1},s_{2}, \xi _{1},\xi _{2} )\in \Omega \times \mathbb{R}^{2} \times \mathbb{R}^{2N}. \end{aligned}
$$({\mathbf{H_{2}}})$$:

With appropriate $$m_{i}>0$$, $$i=1,2$$, we have

$$\liminf_{|s_{i}|\rightarrow 0} \bigl\{ f_{i}(x,s_{1},s_{2}, \xi _{1},\xi _{2} ):(\xi _{1},\xi _{2}) \in \mathbb{R}^{2N} \bigr\} >m_{i} , \quad \text{uniformly in } x\in \Omega .$$

Our main results are the following theorems.

### Theorem 1.1

Let $$\delta _{1}, \delta _{2}>0$$ be small enough and suppose that conditions $$({\mathbf{H_{1}}})$$ and $$({\mathbf{H_{2}}})$$ are satisfied. Then system $$( \mathrm{S} )$$ has a nodal solution $$(u_{0},v_{0})\in \mathcal{C} ^{1,\gamma }(\overline{\Omega })\times \mathcal{C} ^{1,\gamma }(\overline{\Omega })$$ for some $$\gamma \in (0,1)$$ such that $$u_{0}(x)$$ and $$u'_{0}(x)$$ are negative near Ω.

### Theorem 1.2

Let $$\delta _{1}, \delta _{2}>0$$ be small enough and suppose that conditions $$({\mathbf{H_{1}}})$$ and $$({\mathbf{H_{2}}})$$ are satisfied. Then system $$( \mathrm{S} )$$ has a positive solution $$(u_{+},u^{+})\in \mathcal{C}^{1,\gamma }(\overline{\Omega })\times \mathcal{C}^{1,\gamma }(\overline{\Omega })$$ for some $$\gamma \in (0,1)$$ such that $$u_{+}(x)$$ and $$u^{+}(x)$$ are negative near Ω.

The paper is organized as follows. In Sect. 2, we collect some needed definitions and results. In Sect. 3, we study auxiliary systems. In Sect. 4, we prove Theorem 3.1. In Sect. 5, we study subsupersolutions. In Sect. 6, we study nodal solutions. In Sect. 7, we prove our main results.

## 2 Preliminaries

This part is devoted to summarizing the necessary basic definitions, notations, and function spaces. For other necessary material, we refer the reader to the comprehensive monograph by Papageorgiou et al. [23]. The Banach space $$W^{1,p}(\Omega )$$ is equipped with the usual norm

$$\Vert u \Vert _{1,p}:= \bigl( \Vert u \Vert _{p}^{p}+ \Vert \nabla u \Vert _{p}^{p} \bigr)^{1/p}\quad \text{for } u\in W^{1,p}(\Omega ),$$

where

$$\Vert v \Vert _{p}:=\textstyle\begin{cases} ( \int _{\Omega } \vert v(x) \vert ^{p}\,\mathrm{d}x ) ^{1/p}&\text{if }p< + \infty , \\ \operatorname*{ess\,sup}_{x\in \Omega } \vert v(x) \vert &\text{otherwise}.\end{cases}$$

Moreover, we denote

\begin{aligned}& \mathcal{W}=W^{1,p_{1}}(\Omega )\times W^{1,p_{2}}(\Omega ),\qquad W_{b}^{1,p_{i}}( \Omega ):=W^{1,p_{i}}(\Omega )\cap L^{\infty }(\Omega ),\\& [u_{1},u_{2}]:= \bigl\{ u\in W^{1,p}(\Omega ): u_{1}\leq u\leq u_{2} \bigr\} ,\qquad C^{1,\gamma}_{0}(\overline{\Omega}):= \bigl\{ u\in C^{1, \gamma}(\overline{\Omega }): u\backslash _{\partial \Omega}=0 \bigr\} . \end{aligned}

Now we define a weak solution of system $$(\mathrm{S})$$.

### Definition 2.1

We say that $$(u_{1}, u_{2})\in \mathcal{W}$$ is a weak solutions of system $$(\mathrm{S})$$ if

\begin{aligned} &u_{i}+\delta _{i} >0 \quad \text{a.e. in } \Omega , \qquad \frac{ \vert \nabla u_{i} \vert ^{p_{i}}}{u_{i}+\delta _{i} }\in L^{1}(\Omega )\quad \text{for } i=1,2, \\ & \int _{\Omega } \vert \nabla u_{1} \vert ^{p_{1}-2}\nabla u_{1}\nabla \varphi _{1} \, \mathrm{d}x+ \int _{\Omega }\frac{ \vert \nabla u_{1} \vert ^{p_{1}}}{u_{1}+\delta _{1} } \varphi _{1} \, \mathrm{d}x= \int _{\Omega }f_{1}(x,u_{1},u_{2}, \nabla u_{1},\nabla u_{2}) \varphi _{1} \, \mathrm{d}x, \\ & \int _{\Omega } \vert \nabla u_{2} \vert ^{p_{2}-2}\nabla u_{2}\nabla \varphi _{2} \, \mathrm{d}x+ \int _{\Omega }\frac{ \vert \nabla u_{2} \vert ^{p_{2}}}{u_{2}+\delta _{2} } \varphi _{2} \, \mathrm{d}x= \int _{\Omega }f_{2}(x,u_{1},u_{2}, \nabla u_{1},\nabla u_{2}) \varphi _{2} \, \mathrm{d}x, \end{aligned}
(2.1)

for every $$(\varphi _{1},\varphi _{2}) \in W_{b}^{1,p_{1}}(\Omega )\times W_{b}^{1,p_{2}}( \Omega )$$.

### Remark 2.2

Note that the boundedness condition for $$(\varphi _{1},\varphi _{2})$$ is necessary since $$\frac{|\nabla u_{i}|^{p}}{u_{i}+\delta _{i} }$$, $$i=1,2$$, are only in $$L^{1}(\Omega )$$.

Next, we state the definition of a sub-solution and a super-solution of system $$(\mathrm{S})$$.

### Definition 2.3

We say that the pair $$(\underline{u}_{1}, \underline{u}_{2})\in \mathcal{W}$$ is a sub-solution of system $$(\mathrm{S})$$ if

\begin{aligned} &\underline{u}_{i}+\delta _{i} >0 \quad \text{a.e. in } \Omega , \qquad \frac{ \vert \nabla \underline{u}_{i} \vert ^{p_{i}}}{\underline{u}_{i}+\delta _{i} }\in L^{1}( \Omega )\quad \text{for } i=1,2, \\ & \int _{\Omega } \vert \nabla \underline{u}_{1} \vert ^{p_{1}-2}\nabla \underline{u}_{1}\nabla \varphi _{1} \,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \underline{u}_{1} \vert ^{p_{1}}}{\underline{u}_{1}+\delta _{1} } \varphi _{1} \,\mathrm{d}x- \int _{\Omega }f_{1}(x,\underline{u}_{1},w_{2}, \nabla \underline{u}_{1},\nabla w_{2})\varphi _{1} \,\mathrm{d}x \\ &\quad{} + \int _{\Omega } \vert \nabla \underline{u}_{2} \vert ^{p_{2}-2}\nabla \underline{u}_{2}\nabla \varphi _{2} \,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \underline{u}_{2} \vert ^{p_{2}}}{\underline{u}_{2}+\delta _{2} } \varphi _{2} \, \mathrm{d}x \\ &\quad {}- \int _{\Omega }f_{2}(x,w_{1}, \underline{u}_{2},\nabla w_{1}, \nabla \underline{u}_{2})\varphi _{2} \,\mathrm{d}x\leq 0, \end{aligned}
(2.2)

and we say that the pair $$(\overline{u}_{1}, \overline{u}_{2})\in \mathcal{W}$$ is a super-solution of system $$(\mathrm{S})$$ if

\begin{aligned} &\overline{u}_{i}+\delta _{i} >0\quad \text{a.e. in } \Omega , \qquad \frac{ \vert \nabla \overline{u}_{i} \vert ^{p_{i}}}{\overline{u}_{i}+\delta _{i} }\in L^{1}(\Omega ) \quad \text{for } i=1,2, \\ & \int _{\Omega } \vert \nabla \overline{u}_{1} \vert ^{p_{1}-2}\nabla \overline{u}_{1}\nabla \varphi _{1} \,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1}+\delta _{1} } \varphi _{1} \,\mathrm{d}x- \int _{\Omega }f_{1}(x,\overline{u}_{1},w_{2}, \nabla \overline{u}_{1}, \nabla w_{2})\varphi _{1} \,\mathrm{d}x \\ &\quad{} + \int _{\Omega } \vert \nabla \overline{u}_{2} \vert ^{p_{2}-2}\nabla \overline{u}_{2}\nabla \varphi _{2} \,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \overline{u}_{2} \vert ^{p_{2}}}{\overline{u}_{2}+\delta _{2} } \varphi _{2} \, \mathrm{d}x \\ &\quad {}- \int _{\Omega }f_{2}(x,w_{1}, \overline{u}_{2},\nabla w_{1},\nabla \overline{u}_{2})\varphi _{2} \,\mathrm{d}x\geq 0, \end{aligned}
(2.3)

for all $$(\varphi _{1},\varphi _{2}) \in W_{b}^{1,p_{1}}(\Omega )\times W_{b}^{1,p_{2}}( \Omega )$$ such that $$\varphi _{1}, \varphi _{2} \geq 0$$ in Ω and for all $$(w_{1},w_{2})\in \mathcal{W}$$ such that $$\underline{u}_{i}\leq w_{i}\leq \overline{u}_{i}$$, $$i=1,2$$, with all integrals in (2.2) and (2.3) being finite.

We will use the following conditions:

$$({\mathbf{H_{3}}})$$:

Let $$0\leq q_{1}\leq p_{1}-1$$ and $$0\leq r_{1}\leq p_{2}-1$$. For every $$\rho >0$$, there exists $$M_{1}:=M_{1}(\rho )>0$$ such that

$$\bigl\vert f_{1}(x,s_{1},s_{2},\xi _{1},\xi _{2} ) \bigr\vert \leq M_{1} \bigl(1+ \vert \xi _{1} \vert ^{q_{1}}+ \vert \xi _{2} \vert ^{r_{1}}\bigr)\quad \text{in }\Omega \times {}[ -\rho ,\rho ]^{2} \times \mathbb{R}^{2N}.$$
$$({\mathbf{H_{4}}})$$:

Let $$0\leq q_{2}\leq p_{1}-1$$ and $$0\leq r_{2}\leq p_{2}-1$$. For every $$\rho >0$$, there exists $$M_{2}:=M_{2}(\rho )>0$$ such that

$$\bigl\vert f_{2}(x,s_{1},s_{2},\xi _{1},\xi _{2} ) \bigr\vert \leq M_{2} \bigl(1+ \vert \xi _{1} \vert ^{q_{2}}+ \vert \xi _{2} \vert ^{r_{2}}\bigr)\quad \text{in }\Omega \times {}[ -\rho ,\rho ]^{2} \times \mathbb{R}^{2N}.$$
$$({\mathbf{H_{5}}})$$:

There are sub- and supersolutions $$\underline{u}_{1},\overline{u}_{1}\in \mathcal{C}^{1}(\overline{\Omega })$$ of system $$(\mathrm{S})$$, respectively, satisfying

$$\overline{u}_{1}+\delta _{1} \geq \underline{u}_{1}+\delta _{1} >0\quad \text{a.e. in } \Omega .$$
(2.4)
$$({\mathbf{H_{6}}})$$:

There are sub- and supersolutions $$\underline{u}_{2},\overline{u}_{2}\in \mathcal{C}^{1}(\overline{\Omega })$$ of system $$(\mathrm{S})$$, respectively, satisfying

$$\overline{u}_{2}+\delta _{2} \geq \underline{u}_{2}+ \delta _{2} >0\quad \text{a.e. in }\Omega .$$
(2.5)

Via a standard argument, we will prove the following:

### Proposition 2.4

Suppose that conditions $$({\mathbf{H_{3}}})$$, $$({\mathbf{H_{4}}})$$, $$({\mathbf{H_{5}}})$$, and $$({\mathbf{H_{6}}})$$ are satisfied. Let $$(\underline{u}_{i},\underline{v}_{i}), (\overline{u}_{i}, \overline{v}_{i}) \in W^{1,p_{1}}_{b}(\Omega )\times W^{1,p_{2}}_{b}( \Omega )$$ be pairs of sub- and supersolutions of system $$( \mathrm{S} )$$. Set

\begin{aligned}& \overline{u}=\min \{\overline{u}_{1},\overline{u}_{2}\},\qquad \underline{u}=\max \{\underline{u}_{1},\underline{u}_{2}\},\\& \overline{v}=\min \{\overline{v}_{1},\overline{v}_{2}\},\qquad \underline{v}=\max \{\underline{v}_{1},\underline{v}_{2}\}, \end{aligned}

and assume that $$\overline{u}\leq \overline{v}$$ and $$\underline{u}\leq \underline{v}$$ Then $$(\underline{u},\underline{v})$$, $$(\overline{u},\overline{v})$$ is also a pair of sub- and supersolutions of system $$( \mathrm{S} )$$.

### Proof

The proof is inspired by the proof of Motreanu et al. [19, Lemma 3]. Fix $$\epsilon >0$$ and define the truncation function $$\xi _{\epsilon }(s)=\max \{-\epsilon ,\min \{s,\epsilon \}\}$$ for $$s\in \mathbb{R}$$. By Marcus et al. [13] we know that

\begin{aligned}& \xi _{\epsilon }\bigl((\overline{u}_{1}-\overline{u}_{2})^{-}\bigr), \xi _{\epsilon } \bigl((\underline{u}_{1}-\underline{u}_{2})^{+} \bigr)\in \mathcal{W},\\& \nabla \xi _{\epsilon }\bigl((\overline{u}_{1}- \overline{u}_{2})^{-}\bigr)=\xi _{ \epsilon }^{\prime } \bigl((\overline{u}_{1}-\overline{u}_{2})^{-} \bigr)\nabla (\overline{u}_{1}-\overline{u}_{2})^{-}, \end{aligned}

and

$$\nabla \xi _{\epsilon }\bigl((\underline{u}_{1}- \underline{u}_{2})^{+}\bigr)= \xi _{\epsilon }^{\prime } \bigl((\underline{u}_{1}-\underline{u}_{2})^{+} \bigr) \nabla (\underline{u}_{1}-\underline{u}_{2})^{+}.$$

Now letting $$\varphi \in C_{c}^{1}(\Omega )$$ be a test function such that $$\varphi \geq 0$$, we obtain

\begin{aligned}& \biggl\langle -\Delta _{{p}_{1}}\underline{u}_{1}+ \frac{ \vert \nabla \underline{u}_{1} \vert ^{p_{1}}}{\underline{u}_{1}+\delta _{1}}, \xi _{\epsilon }\bigl((\underline{u}_{1}- \underline{u}_{2})^{+}\bigr)\varphi \biggr\rangle \\& \quad \leq \int _{\Omega }f_{1}(x,\underline{u}_{1},w_{2}, \nabla \underline{u}_{1},\nabla w_{2} )\xi _{\epsilon }\bigl((\underline{u}_{1}- \underline{u}_{2})^{+} \bigr)\varphi \,\mathrm{d}x, \end{aligned}
(2.6)
\begin{aligned}& \biggl\langle -\Delta _{{p}_{1}}\overline{u}_{1}+ \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1}+\delta _{1} },\xi _{\epsilon }\bigl((\overline{u}_{1}- \overline{u}_{2})^{-}\bigr)\varphi \biggr\rangle \\& \quad \geq \int _{\Omega }f_{1}(x,\overline{u}_{1},w_{2}, \nabla \overline{u}_{1},\nabla w_{2})\xi _{\epsilon }\bigl((\overline{u}_{1}- \overline{u}_{2})^{-} \bigr)\varphi \,\mathrm{d}x \end{aligned}
(2.7)

for every $$w_{2}\in W^{1,p_{2}}(\Omega )$$ with $$\underline{u}_{2}\leq w_{2}\leq \overline{u}_{2}$$ and

\begin{aligned}& \biggl\langle -\Delta _{{p}_{1}}\underline{u}_{2}+ \frac{ \vert \nabla \underline{u}_{2} \vert ^{p_{1}}}{\underline{u}_{2}+\delta _{1} },\bigl(\epsilon -\xi _{\epsilon }\bigl(( \underline{u}_{1}-\underline{u}_{2})^{+} \bigr)\bigr)\varphi \biggr\rangle \\& \quad \leq \int _{\Omega }f_{1}(x,w_{1}, \underline{u}_{2},\nabla w_{1}, \nabla \underline{u}_{2}) \bigl( \epsilon -\xi _{\epsilon }\bigl(( \underline{u}_{1}-\underline{u}_{2})^{+} \bigr) \bigr) \varphi \,\mathrm{d}x, \end{aligned}
(2.8)
\begin{aligned}& \biggl\langle -\Delta _{{p}_{1}}\overline{u}_{2}+ \frac{ \vert \nabla \overline{u}_{2} \vert ^{p_{1}}}{\overline{u}_{2}+\delta _{1} },\bigl(\epsilon -\xi _{\epsilon }\bigl(( \overline{u}_{1}-\overline{u}_{2})^{-} \bigr)\bigr)\varphi \biggr\rangle \\& \quad \geq \int _{\Omega }f_{1}(x,w_{1}, \overline{u}_{2},\nabla w_{1}, \nabla \overline{u}_{2}) \bigl( \epsilon -\xi _{\epsilon }\bigl(( \overline{u}_{1}-\overline{u}_{2})^{-} \bigr) \bigr) \varphi \,\mathrm{d}x \end{aligned}
(2.9)

for every $$w_{1}\in W^{1,p_{1}}(\Omega )$$ with $$\underline{u}_{1}\leq w_{1}\leq \overline{u}_{1}$$. Therefore by the monotonicity of the −p-Laplacian operator we have

\begin{aligned}& \biggl\langle -\Delta _{{p}_{1}}\underline{u}_{1}+ \frac{ \vert \nabla \underline{u}_{1} \vert ^{p_{1}}}{\underline{u}_{1}+\delta _{1}}, \xi _{\epsilon }\bigl((\underline{u}_{1}- \underline{u}_{2})^{+}\bigr)\varphi \biggr\rangle + \biggl\langle -\Delta _{{p}_{1}}\underline{u}_{2}+ \frac{ \vert \nabla \underline{u}_{2} \vert ^{p_{1}}}{\underline{u}_{2}+\delta _{1} },\bigl(\epsilon -\xi _{\epsilon }\bigl(( \underline{u}_{1}-\underline{u}_{2})^{+} \bigr)\bigr)\varphi \biggr\rangle \\& \quad \geq \int _{\Omega } \vert \nabla \underline{u}_{1} \vert ^{p_{1}-2}(\nabla \underline{u}_{1},\nabla \varphi )_{ \mathbb{R} ^{N}}\xi _{\epsilon }\bigl(( \underline{u}_{1}- \underline{u}_{2})^{+}\bigr)\,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \underline{u}_{1} \vert ^{p_{1}}}{\underline{u}_{1}+\delta _{1} }\xi _{\epsilon } \bigl((\underline{u}_{1}-\underline{u}_{2})^{+} \bigr)\varphi \,\mathrm{d}x \\& \quad \quad{} + \int _{\Omega } \vert \nabla \underline{u}_{2} \vert ^{p_{1}-2}(\nabla \underline{u}_{2},\nabla \varphi )_{ \mathbb{R} ^{N}} \bigl( \epsilon -\xi _{ \epsilon }\bigl(( \underline{u}_{1}-\underline{u}_{2})^{+} \bigr) \bigr) \,\mathrm{d}x \\& \quad \quad {}+ \int _{\Omega } \frac{ \vert \nabla \underline{u}_{2} \vert ^{p_{1}}}{\underline{u}_{2}+\delta _{1} }\bigl(\epsilon -\xi _{ \epsilon }\bigl((\underline{u}_{1}- \underline{u}_{2})^{+}\bigr)\bigr)\varphi \,\mathrm{d}x \end{aligned}
(2.10)

and

\begin{aligned}& \biggl\langle -\Delta _{{p}_{1}}\overline{u}_{1}+ \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1}+\delta _{1}}, \xi _{\epsilon }\bigl((\overline{u}_{1}- \overline{u}_{2})^{-}\bigr)\varphi \biggr\rangle + \biggl\langle -\Delta _{{p}_{1}}\overline{u}_{2}+ \frac{ \vert \nabla \overline{u}_{2} \vert ^{p_{1}}}{\overline{u}_{2}+\delta _{1} },\bigl(\epsilon -\xi _{\epsilon }\bigl(( \overline{u}_{1}-\overline{u}_{2})^{-} \bigr)\bigr)\varphi \biggr\rangle \\& \quad \leq \int _{\Omega } \vert \nabla \overline{u}_{1} \vert ^{p_{1}-2}(\nabla \overline{u}_{1},\nabla \varphi )_{ \mathbb{R} ^{N}}\xi _{\epsilon }\bigl(( \overline{u}_{1}- \overline{u}_{2})^{-}\bigr)\,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1}+\delta _{1} }\xi _{\epsilon } \bigl((\overline{u}_{1}-\overline{u}_{2})^{-} \bigr)\varphi \,\mathrm{d}x \\& \quad \quad{} + \int _{\Omega } \vert \nabla \overline{u}_{2} \vert ^{p_{1}-2}(\nabla \overline{u}_{2},\nabla \varphi )_{ \mathbb{R} ^{N}} \bigl( \epsilon -\xi _{ \epsilon }\bigl(( \overline{u}_{1}-\overline{u}_{2})^{+} \bigr) \bigr) \,\mathrm{d}x \\& \quad \quad {}+ \int _{\Omega } \frac{ \vert \nabla \overline{u}_{2} \vert ^{p_{1}}}{\overline{u}_{2}+\delta _{1} }\bigl(\epsilon -\xi _{ \epsilon }\bigl((\overline{u}_{1}- \overline{u}_{2})^{-}\bigr)\bigr)\varphi \,\mathrm{d}x. \end{aligned}
(2.11)

Invoking equations (2.6), (2.8), and (2.10), we obtain

\begin{aligned} & \int _{\Omega } \vert \nabla \underline{u}_{1} \vert ^{p_{1}-2}(\nabla \underline{u}_{1},\nabla \varphi )_{ \mathbb{R} ^{N}} \frac{1}{\epsilon }\xi _{\epsilon }\bigl(( \underline{u}_{1}- \underline{u}_{2})^{+} \bigr)\,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \underline{u}_{1} \vert ^{p_{1}}}{\underline{u}_{1}+\delta _{1} }\frac{1}{\epsilon }\xi _{ \epsilon }((\underline{u}_{1}-\underline{u}_{2})^{-} \,\mathrm{d}x \\ &\quad \quad{} + \int _{\Omega } \vert \nabla \underline{u}_{2} \vert ^{p_{1}-2}(\nabla \underline{u}_{2},\nabla \varphi )_{ \mathbb{R} ^{N}} \biggl( 1-\frac{1}{\epsilon } \xi _{\epsilon } \bigl((\underline{u}_{1}-\underline{u}_{2})^{+} \bigr) \biggr) \,\mathrm{d}x \\ &\quad \quad{} + \int _{\Omega } \frac{ \vert \nabla \underline{u}_{2} \vert ^{p_{1}}}{\underline{u}_{2}+\delta _{1} }(1- \frac{1}{\epsilon }\xi _{\epsilon }\bigl((\underline{u}_{1}-\underline{u}_{2})^{+} \bigr)\,\mathrm{d}x \\ &\quad \leq \int _{\Omega }f_{1}(x,\underline{u}_{1},w_{2}, \nabla \underline{u}_{1},\nabla w_{2}) \frac{1}{\epsilon }\xi _{\epsilon }\bigl((\underline{u}_{1}- \underline{u}_{2})^{+}\bigr)\varphi \, \mathrm{d}x \\ &\quad \quad{} + \int _{\Omega }f_{1}(x,\underline{u}_{1},w_{2}, \nabla \underline{u}_{1},\nabla w_{2}) \biggl( 1- \frac{1}{\epsilon }\xi _{\epsilon }\bigl((\underline{u}_{1}- \underline{u}_{2})^{-}\bigr) \biggr) \varphi \, \mathrm{d}x. \end{aligned}

In a similar manner, invoking equations (2.7), (2.9), and (2.11), we get

\begin{aligned} & \int _{\Omega } \vert \nabla \overline{u}_{1} \vert ^{p_{1}-2}(\nabla \overline{u}_{1},\nabla \varphi )_{\mathbb{R} ^{N}}\frac{1}{\epsilon }\xi _{\epsilon }\bigl(( \overline{u}_{1}- \overline{u}_{2})^{-} \bigr)\,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1}+\delta _{1} }\frac{1}{\epsilon }\xi _{ \epsilon }((\overline{u}_{1}-\overline{u}_{2})^{-} \,\mathrm{d}x \\ &\quad \quad{} + \int _{\Omega } \vert \nabla \overline{u}_{2} \vert ^{p-2}(\nabla \overline{u}_{2},\nabla \varphi )_{ \mathbb{R} ^{N}} \biggl( 1-\frac{1}{\epsilon } \xi _{\epsilon } \bigl((\overline{u}_{1}-\overline{u}_{2})^{-} \bigr) \biggr) \,\mathrm{d}x \\ &\quad \quad{} + \int _{\Omega } \frac{ \vert \nabla \overline{u}_{2} \vert ^{p_{1}}}{\overline{u}_{2}+\delta _{1} }(1- \frac{1}{\varepsilon }\xi _{\epsilon }\bigl((\overline{u}_{1}-\overline{u}_{2})^{-} \bigr)\,\mathrm{d}x \\ &\quad \geq \int _{\Omega }f_{2}(x,w_{1}, \overline{u}_{2},\nabla w_{1}, \nabla \overline{u}_{2})\frac{1}{\epsilon }\xi _{\epsilon }\bigl(( \overline{u}_{1}-\overline{u}_{2})^{-} \bigr)\varphi \,\mathrm{d}x \\ &\quad \quad{} + \int _{\Omega }f_{2}(x,w_{1}, \overline{u}_{2},\nabla w_{1},\nabla \overline{u}_{2}) \biggl( 1- \frac{1}{\epsilon }\xi _{\epsilon }\bigl((\overline{u}_{1}-\overline{u}_{2})^{-} \bigr) \biggr) \varphi \,\mathrm{d}x. \end{aligned}

Letting $$\epsilon \rightarrow 0$$ and observing that

$$\textstyle\begin{cases} \frac{1}{\epsilon }\xi _{\epsilon }((\overline{u}_{1}-\overline{u}_{2})^{-} \rightarrow \mathrm{1}_{\{\overline{u}_{1}< \overline{u}_{2}\}}(x),& \text{a.e. in }\Omega \text{ as }\epsilon \rightarrow 0, \\ \frac{1}{\epsilon }\xi _{\epsilon }((\underline{u}_{1}- \underline{u}_{2})^{+}\rightarrow \mathrm{1}_{\{\underline{u}_{1}< \underline{u}_{2} \}}(x),& \text{a.e. in }\Omega \text{ as }\epsilon \rightarrow 0, \end{cases}$$

we see that

\begin{aligned} & \int _{\Omega } \vert \nabla \underline{u} \vert ^{p_{1}-2}\nabla \underline{u} \nabla \varphi \,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \underline{u} \vert ^{p_{1}}}{\underline{u}+\delta _{1} }\varphi \,\mathrm{d}x\leq \int _{\Omega }f_{1}(x, \underline{u}_{1},w_{2} ,\nabla \underline{u}_{1},w_{2})\varphi \,\mathrm{d}x \end{aligned}

and

\begin{aligned} & \int _{\Omega } \vert \nabla \overline{u} \vert ^{p_{1}-2}\nabla \overline{u}\nabla \varphi \,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \overline{u} \vert ^{p_{1}}}{\overline{u}+\delta _{1} }\varphi \,\mathrm{d}x\geq \int _{\Omega }f_{1}(x, \overline{u}_{1},w_{2} ,\nabla \overline{u}_{1},\nabla w_{2})\varphi \, \mathrm{d}x \end{aligned}

for every $$\varphi \in C_{c}^{1}(\Omega )$$, $$\varphi \geq 0$$ a.e. in Ω. By a similar argument we obtain

\begin{aligned} & \int _{\Omega } \vert \nabla \underline{v} \vert ^{p_{2}-2}\nabla \underline{v} \nabla \varphi \,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \underline{v} \vert ^{p_{2}}}{\underline{v}+\delta _{2} }\varphi \,\mathrm{d}x\leq \int _{\Omega }f_{2}(x,w_{1}, \underline{v}_{2},\nabla w_{1},\nabla \underline{v}_{2})\varphi \,\mathrm{d}x \end{aligned}

and

\begin{aligned} & \int _{\Omega } \vert \nabla \overline{v} \vert ^{p_{2}-2}\nabla \overline{v}\nabla \varphi \,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \overline{v} \vert ^{p_{2}}}{\overline{v}+\delta _{2} }\varphi \,\mathrm{d}x\geq \int _{\Omega }f_{2}(x,w_{1}, \overline{v}_{2},\nabla w_{1},\nabla \overline{v}_{2})\varphi \,\mathrm{d}x. \end{aligned}

Finally, in view of the denseness of $$C_{c}^{1}(\Omega )$$ in both $$W^{1,p_{1}}(\Omega )$$ and $$W^{1,p_{2}}(\Omega )$$, we deduce that $$(\underline{u},\underline{v})$$, $$(\overline{u},\overline{v})$$ is also a pair of sub- and supersolutions of system $$( \mathrm{S} )$$. □

## 3 Auxiliary systems

Let, the pairs $$(\underline{u}_{1},\underline{u}_{2}), (\overline{u}_{1}, \overline{u}_{2})\in \mathcal{C}^{1}(\overline{\Omega })\times \mathcal{C}^{1}(\overline{\Omega })$$ be sub- and supersolutions, respectively, of system $$(\mathrm{S})$$ as required in conditions $$({\mathbf{H_{5}}})$$ and $$({\mathbf{H_{6}}})$$. Now, for a given $$(u_{1},u_{2})\in \mathcal{W}$$, we define the truncation operators $$\mathcal{T}_{i}:W^{1,p_{i}}(\Omega )\rightarrow W^{1,p_{i}}(\Omega )$$ by

\begin{aligned} \mathcal{T}_{1}(u_{1}):= \textstyle\begin{cases} \underline{u}_{1} & \text{when }u_{1}\leq \underline{u}_{1}, \\ u_{1} & \text{if }\underline{u}_{1}\leq u_{1}\leq \overline{u}_{1}, \\ \overline{u}_{1} & \text{otherwise},\end{cases}\displaystyle \qquad \mathcal{T}_{2}(u_{2}):= \textstyle\begin{cases} \underline{u}_{2} & \text{when }u_{2}\leq \underline{u}_{2}, \\ u_{2} & \text{if }\underline{u}_{2}\leq u_{2}\leq \overline{u}_{2}, \\ \overline{u}_{2} & \text{otherwise}.\end{cases}\displaystyle \end{aligned}
(3.1)

Then by Carl et al. [4, Lemma 2.89], $$\mathcal{T}_{1}$$ and $$\mathcal{T}_{2}$$ are continuous, monotone, and bounded. In view of conditions $$({\mathbf{H_{3}}})$$ and $$({\mathbf{H_{4}}})$$, if $$\rho >0$$, then

$$-\rho \leq \underline{u}_{1}\leq \overline{u}_{1}\leq \rho ,\qquad {-}\rho \leq \underline{u}_{2}\leq \overline{u}_{2} \leq \rho .$$
(3.2)

We introduce the Nemitskii operators $$\mathcal{N}_{{f}_{1}}$$ and $$\mathcal{N}_{{f}_{2}}$$ generated by the Carathéodory functions $$f_{1}$$ and $$f_{2}$$, respectively, which are well defined for $$i=1,2$$ since the range of $$\mathcal{T}_{1}$$ and $$\mathcal{T}_{2}$$ lies within the region $$[\underline{u}_{i}, \overline{u}_{i}]$$. So by the Rellich–Kondrachov compactness embedding theorem the maps

\begin{aligned}& \mathcal{N}_{{f}_{1}}\circ (\mathcal{T}_{1}, \mathcal{T}_{2}):[ \underline{u}_{1}, \overline{u}_{1}]\subset \mathcal{W} \longrightarrow L^{p_{1}^{\prime }}(\Omega )\hookrightarrow W^{-1,p_{1}}( \Omega ), \end{aligned}
(3.3)
\begin{aligned}& \mathcal{N}_{{f}_{2}}\circ (\mathcal{T}_{1}, \mathcal{T}_{2}):[ \underline{u}_{2},\overline{u}_{2}] \subset \mathcal{W} \longrightarrow L^{p_{2}^{\prime }}(\Omega )\hookrightarrow W^{-1,p_{2}}( \Omega ) \end{aligned}
(3.4)

are bounded and completely continuous. Furthermore, set

$$\mathcal{F}(u)= \bigl(\mathcal{N}_{{f}_{1}}(\mathcal{T}_{1}u_{1}, \mathcal{T}_{2}u_{2},\nabla (\mathcal{T}_{1}u_{1}), \nabla ( \mathcal{T}_{2}u_{2}),\mathcal{N}_{{f}_{2}} (\mathcal{T}_{1}u_{1}, \mathcal{T}_{2}u_{2}, \nabla (\mathcal{T}_{1}u_{1}),\nabla ( \mathcal{T}_{2}u_{2}) \bigr).$$

Next, define the cut-off functions $$b_{i}:\Omega \times \mathbb{R}\longrightarrow \mathbb{R}$$, $$i=1,2$$, by

\begin{aligned} b_{1}(x,s):=-\bigl(\underline{u}_{1}(x)-s \bigr)_{+}^{p_{1}-1}+\bigl(s-\overline{u}_{1}(x) \bigr)_{+}^{p_{1}-1} \quad \text{for } (x,s)\in \Omega \times \mathbb{R}, \end{aligned}
(3.5)
\begin{aligned} b_{2}(x,s):=-\bigl(\underline{u}_{2}(x)-s \bigr)_{+}^{p_{2}-1}+\bigl(s-\overline{u}_{2}(x) \bigr)_{+}^{p_{2}-1} \quad \text{for } (x,s)\in \Omega \times \mathbb{R}. \end{aligned}
(3.6)

It is easy to see that $$b_{i}$$, $$i=1,2$$, are Carathéodory functions fulfilling the following growth condition:

\begin{aligned}& \bigl\vert b_{1}(x,s) \bigr\vert \leq \varphi _{1}(x)+c_{1} \vert s \vert ^{p_{1}-1}\quad \text{for a.e. }x \in \Omega \text{ and every }s\in \mathbb{R}, \end{aligned}
(3.7)
\begin{aligned}& \bigl\vert b_{2}(x,s) \bigr\vert \leq \varphi _{2}(x)+c_{2} \vert s \vert ^{p_{2}-1}\quad \text{for a.e. }x \in \Omega \text{ and every }s\in \mathbb{R}, \end{aligned}
(3.8)

with $$\varphi _{1}$$, $$\varphi _{2}$$ $$\in L^{\infty }(\Omega )$$ and $$c_{1}, c_{2}> 0$$. Moreover, we have the following estimates:

\begin{aligned} \int _{\Omega }b_{1}(\cdot ,u_{1})u_{1} \,\mathrm{d}x\geq C_{1} \Vert u_{1} \Vert _{p_{1}}^{p}-C_{2}\quad \text{for every }u_{1} \in W^{1,p_{1}}(\Omega ), \end{aligned}
(3.9)
\begin{aligned} \int _{\Omega }b_{2}(\cdot ,u_{2})u_{2} \,\mathrm{d}x\geq C'_{1} \Vert u_{1} \Vert _{p_{1}}^{p}-C'_{2}\quad \text{for every }u_{1}\in W^{1,p_{2}}(\Omega ), \end{aligned}
(3.10)

where $$C_{1}$$, $$C_{2}$$, $$C'_{1}$$, $$C'_{2}$$ are some positive constants (for more detail, see, e.g., Carl et al. [4, pp. 95–96]). Let $$\mu > 0$$ and set

$$\mu \mathcal{B}(u)= \bigl(\mu \mathcal{B}_{1}(u_{1}), \mu \mathcal{B}_{2}(u_{2}) \bigr).$$

Now we introduce the following auxiliary problem:

$$( \mathrm{S}_{\mu} ) \quad \textstyle\begin{cases} -\Delta _{p_{1}}u_{1}+ \frac{ \vert \nabla (\mathcal{T} u_{1}) \vert ^{p_{1}}}{\mathcal{T}u_{1}+\delta _{1} }=f_{1}(x, \mathcal{T}u_{1},\mathcal{T}u_{2},\nabla (\mathcal{T}u_{1}),\nabla ( \mathcal{T}u_{2}))-\mu b_{1}(x,u) & \text{in } \Omega , \\ -\Delta _{p_{2}}u_{2}+ \frac{ \vert \nabla (\mathcal{T}u_{2}) \vert ^{p_{2}}}{\mathcal{T}u_{2}+\delta _{2} }=f_{2}(x, \mathcal{T}u_{1},\mathcal{T}u_{2},\nabla (\mathcal{T}u_{1}),\nabla ( \mathcal{T}u_{2}))-\mu b_{2}(x,u) & \text{in } \Omega , \\ \vert \nabla u_{1} \vert ^{p_{1}-2}\frac{\partial u_{1}}{\partial \eta }=0= \vert \nabla u_{2} \vert ^{p_{2}-2}\frac{\partial u_{2}}{\partial \eta } & \text{on } \partial \Omega ,\end{cases}$$

where $$(u_{1},u_{2})\in \mathcal{W}$$. Our main result in this section concerning system $$(\mathrm{S}_{\mu })$$ is as follows.

### Theorem 3.1

Suppose that conditions $$({\mathbf{H_{3}}})$$, $$({\mathbf{H_{4}}})$$, $$({\mathbf{H_{5}}})$$, and $$({\mathbf{H_{6}}})$$ are satisfied. Then system $$(\mathrm{S}_{\mu })$$ has a pair of weak solutions $$(u_{1}, u_{2})\in \mathcal{W}$$.

The following estimates will be a key for the proof of Theorem 3.1 in the next section.

### Lemma 3.2

Suppose that conditions $$({\mathbf{H_{3}}})$$ and $$({\mathbf{H_{4}}})$$ are satisfied. Then there exist constants $$k_{0}, k'_{0}>0$$, depending only on $$p_{1}$$, $$p_{2}$$, and Ω, such that

\begin{aligned} \int _{\Omega } \bigl\vert f_{1} \bigl(x , \mathcal{T}u_{1},\mathcal{T}u_{2}, \nabla ( \mathcal{T}u_{1}),\nabla (\mathcal{T}u_{2}) \bigr) \bigr\vert \vert u_{1} \vert \,\mathrm{d}x& \leq \frac{1}{2} \bigl( \Vert \nabla u_{1} \Vert _{p_{1}}^{p_{1}}+ \Vert \nabla u_{2} \Vert _{p_{2}}^{p_{2}} \bigr) \\ &\quad{} + k_{0} \bigl(1+ \Vert u_{1} \Vert _{p_{1}}+ \Vert u_{1} \Vert _{p_{1}}^{p_{1}}+ \Vert u_{1} \Vert _{p_{2}}^{p_{2}} \bigr) \end{aligned}

and

\begin{aligned} \int _{\Omega } \bigl\vert f_{2} \bigl(x , \mathcal{T}u_{1},\mathcal{T}u_{2}, \nabla ( \mathcal{T}u_{1}),\nabla (\mathcal{T}u_{2}) \bigr) \bigr\vert \vert u_{2} \vert \,\mathrm{d}x& \leq \frac{1}{2} \bigl( \Vert \nabla u_{1} \Vert _{p_{1}}^{p_{1}}+ \Vert \nabla u_{2} \Vert _{p_{2}}^{p_{2}} \bigr) \\ &\quad{} + k'_{0} \bigl(1+ \Vert u_{2} \Vert _{p_{2}}+ \Vert u_{2} \Vert _{p_{2}}^{p_{2}}+ \Vert u_{2} \Vert _{p_{1}}^{p_{1}} \bigr) \end{aligned}

for every $$(u_{1},u_{2})\in \mathcal{W}$$.

### Proof

We will only prove the first inequality. The second inequality can be verified similarly. First, by condition $$({\mathbf{H_{3}}})$$ we have

\begin{aligned} & \int _{\Omega } \bigl\vert f_{1} \bigl(x , \mathcal{T}u_{1}, \mathcal{T}u_{2},\nabla ( \mathcal{T}u_{1}),\nabla (\mathcal{T}u_{2}) \bigr) \bigr\vert \vert u_{1} \vert \,\mathrm{d}x \\ &\quad \leq M_{1} \int _{\Omega} \bigl(1+ \bigl\vert \nabla ( \mathcal{T}u_{1}) \bigr\vert ^{q_{1}}+ \bigl\vert \nabla (\mathcal{T}u_{2}) \bigr\vert ^{r_{1}} \bigr) \vert u_{1} \vert \,\mathrm{d}x. \end{aligned}
(3.11)

Using Young’s inequality, we get that for any fixed $$\varepsilon \in ]0,\frac{1}{2M}[$$ and every $$u_{1}\in W^{1,p_{1}}(\Omega )$$,

$$\bigl\vert \nabla (\mathcal{T}u_{1}) \bigr\vert ^{q_{1}} \vert u_{1} \vert \leq \varepsilon \bigl\vert \nabla ( \mathcal{T}u_{1}) \bigr\vert ^{\frac{q_{1}p_{1}}{p_{1}-1}}+c_{\varepsilon } \vert u_{1} \vert ^{p_{1}}\leq \varepsilon \bigl(1+ \bigl\vert \nabla ( \mathcal{T}u_{1}) \bigr\vert ^{p_{1}} \bigr)+c_{ \varepsilon } \vert u_{1} \vert ^{p_{1}}.$$
(3.12)

Similarly, for every $$u_{2}\in W^{1,p_{2}}(\Omega )$$, we have

$$\bigl\vert \nabla (\mathcal{T}u_{2}) \bigr\vert ^{r_{1}} \vert u_{1} \vert \leq \varepsilon \bigl\vert \nabla ( \mathcal{T}u_{2}) \bigr\vert ^{\frac{r_{1}p_{2}}{p_{2}-1}}+c'_{\varepsilon } \vert u_{1} \vert ^{p_{2}}\leq \varepsilon \bigl(1+ \bigl\vert \nabla (\mathcal{T}u_{2}) \bigr\vert ^{p_{2}} \bigr)+c'_{ \varepsilon } \vert u_{1} \vert ^{p_{2}}.$$
(3.13)

On the other hand, using equation (3.1), we can see that

\begin{aligned} \int _{\Omega } \bigl\vert \nabla (\mathcal{T}u_{1}) \bigr\vert ^{p_{1}} \,\mathrm{d}x& = \int _{\{\underline{u}_{1}\leq u_{1}\leq \overline{u}_{1}\}} \vert \nabla u_{1} \vert ^{p_{1}} \,\mathrm{d}x+ \int _{\{u_{1}\geq \overline{u}_{1}\}} \vert \nabla \overline{u}_{1} \vert ^{p_{1}} \,\mathrm{d}x+ \int _{\{u_{1}\leq \underline{u}_{1}\}} \vert \nabla \underline{u}_{1} \vert ^{p_{1}} \,\mathrm{d}x \\ & \leq \int _{\Omega } \vert \nabla u_{1} \vert ^{p_{1}} \,\mathrm{d}x+ \int _{ \Omega } \vert \nabla \underline{u}_{1} \vert ^{p_{1}} \,\mathrm{d}x+ \int _{\Omega } \vert \nabla \overline{u}_{1} \vert ^{p_{1}} \,\mathrm{d}x \\ & \leq \Vert \nabla u_{1} \Vert _{p_{1}}^{p_{1}}+ \bigl( \Vert \nabla \underline{u}_{1} \Vert _{\infty }^{p_{1}}+ \Vert \nabla \overline{u}_{1} \Vert _{\infty }^{p_{1}} \bigr) \vert \Omega \vert . \end{aligned}
(3.14)

Using the same techniques, we get

\begin{aligned} \int _{\Omega } \bigl\vert \nabla (\mathcal{T}u_{2}) \bigr\vert ^{p_{2}} \,\mathrm{d}x& \leq \Vert \nabla u_{2} \Vert _{p_{2}}^{p_{2}}+ \bigl( \Vert \nabla \underline{u}_{2} \Vert _{\infty }^{p_{2}}+ \Vert \nabla \overline{u}_{2} \Vert _{\infty }^{p_{2}} \bigr) \vert \Omega \vert . \end{aligned}
(3.15)

Consequently, using equations (3.12)–(3.15), we get

\begin{aligned} & \int _{\Omega } \bigl\vert f_{1} \bigl(x , \mathcal{T}u_{1}, \mathcal{T}u_{2},\nabla ( \mathcal{T}u_{1}),\nabla (\mathcal{T}u_{2}) \bigr) \bigr\vert \vert u_{1} \vert \,\mathrm{d}x _{p_{1}} \\ &\quad \leq M_{1} \bigl( \vert \Omega \vert ^{ \frac{p_{1}-1}{p_{1}}} \Vert u_{1} \Vert + \epsilon \vert \Omega \vert \bigl(1+ \Vert \nabla \underline{u}_{1} \Vert _{\infty }^{p_{1}}+ \Vert \nabla \overline{u}_{1} \Vert _{\infty }^{p_{1}} \bigr)+\epsilon \Vert \nabla u_{1} \Vert _{p_{1}}^{p_{1}}+c_{\epsilon } \Vert u_{1} \Vert _{p_{1}}^{p_{1}} \\ &\quad \quad{} + \epsilon \vert \Omega \vert \bigl(1+ \Vert \nabla \underline{u}_{2} \Vert _{\infty }^{p_{2}}+ \Vert \nabla \overline{u}_{2} \Vert _{\infty }^{p_{2}} \bigr)+\epsilon \Vert \nabla u_{2} \Vert _{p_{2}}^{p_{2}}+c_{ \epsilon } \Vert u_{1} \Vert _{p_{2}}^{p_{2}} \bigr) \\ &\quad \leq \frac{1}{2} \bigl( \Vert \nabla u_{1} \Vert _{p_{1}}^{p_{1}}+ \Vert \nabla u_{2} \Vert _{p_{2}}^{p_{2}} \bigr)+k_{0} \bigl(1+ \Vert u_{1} \Vert _{p_{1}}+ \Vert u_{1} \Vert _{p_{1}}^{p_{1}}+ \Vert u_{1} \Vert _{p_{2}}^{p_{2}} \bigr) \end{aligned}
(3.16)

for a suitable $$k_{0}>0$$. The proof of the lemma is thus completed. □

The following useful estimates can be verified in a similar way as in Moussaoui et al. [20, Lemma 2.2].

### Lemma 3.3

Suppose that conditions $$({\mathbf{H_{3}}})$$, $$({\mathbf{H_{4}}})$$, $$({\mathbf{H_{5}}})$$, and $$({\mathbf{H_{6}}})$$ are satisfied. Then for every $$u=(u_{1},u_{2})\in \mathcal{W}$$, there exist constants $$k_{1}$$ and $$k_{2}$$, independent of $$u_{1}$$ and $$u_{2}$$, respectively, such that

\begin{aligned} &\frac{ \vert \nabla (\mathcal{T}u_{1}) \vert ^{p_{1}}}{\mathcal{T}u_{1}+\delta _{1} } \vert u_{1} \vert \in L^{1}(\Omega ) \quad \textit{and}\quad \int _{\Omega } \frac{ \vert \nabla (\mathcal{T}u_{1}) \vert ^{p_{1}}}{\mathcal{T}u_{1}+\delta _{1} } \vert u_{1} \vert \,\mathrm{d}x\leq k_{1} \bigl(1+ \Vert u_{1} \Vert _{p_{1}} \bigr), \end{aligned}
(3.17)
\begin{aligned} &\frac{ \vert \nabla (\mathcal{T}u_{2}) \vert ^{p_{2}}}{\mathcal{T}u_{2}+\delta _{2} } \vert u_{2} \vert \in L^{1}(\Omega ) \quad \textit{and}\quad \int _{\Omega } \frac{ \vert \nabla (\mathcal{T}u_{2}) \vert ^{p_{2}}}{\mathcal{T}u_{2}+\delta _{1} } \vert u_{2} \vert \,\mathrm{d}x\leq k_{2} \bigl(1+ \Vert u_{2} \Vert _{p_{2}} \bigr). \end{aligned}
(3.18)

## 4 Proof of Theorem 3.1

First, by the growth conditions (3.7) and (3.8) we know that the Nemytskii operators $$\mathcal{B}_{i}:W^{1,p_{i}}(\Omega )\longrightarrow W^{-1,p^{\prime }_{i}}( \Omega )$$ given by $$\mathcal{B}_{i}u_{i}(x)=b(\cdot ,u_{i})$$ are well defined, continuous, and bounded for $$i=1,2$$. Also, the operator $$\mathcal{B}(u)=(\mathcal{B}_{1}(u_{1}),\mathcal{B}_{2}(u_{2}))$$ is well defined. Moreover, using the compact embedding $$W^{1,p_{i}}(\Omega )\hookrightarrow L^{p_{i}}(\Omega )$$, we have that the operator $$\mathcal{B}$$ is completely continuous. Next, using conditions $$(\mathrm{H.}3)$$ and $$(\mathrm{H.}4)$$, we can introduce the functions $$\pi _{p_{i},\delta _{i} }:(-\delta _{i} ,+\infty )\times \mathbb{R}^{N} \longrightarrow \mathbb{R}$$, $$i=1,2$$, defined by

$$\pi _{p_{i},\delta _{i} }(s_{i},\xi _{i} )= \frac{ \vert \xi _{i} \vert ^{p_{i}}}{s_{i}+\delta _{i} }$$

having the growth

$$\bigl\vert \pi _{p_{i},\delta _{i} }(s_{i},\xi _{i} ) \bigr\vert \leq \delta _{0} \vert \xi _{i} \vert ^{p_{i}}$$

for all $$s_{i}>-\delta _{i}$$ and $$\xi _{i} \in \mathbb{R}^{N}$$, where $$\delta _{0}>0$$ is a constant such that $$\overline{u}_{i}+\delta _{i} \geq \underline{u}_{i}+\delta _{i} > \delta _{0}$$ a.e. in Ω for $$i=1,2$$.

By Motreanu et al. [18, Theorem 2.76] and Gasinski et al. [8, Theorem 3.4.4]) we know that the corresponding Nemytskii operators

$$\Pi _{p_{i},\delta _{i} }:[\underline{u}_{i},\overline{u}_{i}] \subset W^{1,p_{i}}(\Omega )\longrightarrow L^{1}(\Omega ) \subset W^{-1,p_{i}^{ \prime }}(\Omega )$$

are bounded and continuous for $$i=1,2$$. By virtue of the compact embedding of $$W^{1,p}(\Omega )$$ into $$L^{p}(\Omega )$$, we know that $$\Pi _{p,\delta }(u) =(\Pi _{p_{1},\delta _{1} }(u_{1}),\Pi _{p_{2}, \delta _{2} }(u_{2}) )$$ is completely continuous. Finally, $$\mathcal{A}(u)=(A_{1}(u_{1}), A_{2}(u_{2}))$$, where $$\mathcal{A}: \mathcal{W} \rightarrow \mathcal{W}^{*}$$ is defined in equation (2.1), is well defined, bounded, continuous, strictly monotone, and of type $$(S_{+})$$. Therefore, for every u and $$\varphi \in \mathcal{W}$$, we have the following representations:

\begin{aligned}& \bigl\langle \mathcal{A}(u), \varphi \bigr\rangle _{\mathcal{W}} = \sum _{i=1}^{2} \int _{\Omega } \vert \nabla u_{i} \vert ^{p_{i}-2}\nabla u_{i} \nabla \varphi _{i} \, \mathrm{d}x, \\& \bigl\langle \Pi _{p,\delta }(u),\varphi \bigr\rangle _{\mathcal{W}} = \sum_{i=1}^{2} \int _{\Omega } \Pi _{p_{i},\delta _{i} }(u_{1},u_{2}, \nabla u_{1}, \nabla u_{2})\varphi _{i} \, \mathrm{d}x, \\& \bigl\langle \mathcal{B}(u),\varphi \bigr\rangle _{\mathcal{W}} = \sum _{i=1}^{2} \int _{\Omega } \mathcal{B}_{i}(u_{1},u_{2}, \nabla u_{1},\nabla u_{2}) \varphi _{i} \, \mathrm{d}x, \\& \bigl\langle \mathcal{F}(u),\varphi \bigr\rangle _{\mathcal{W}} = \sum _{i=1}^{2} \int _{\Omega }\mathcal{N}_{{f}_{i}}( \mathcal{T}_{1}u_{1}, \mathcal{T}_{2}u_{2}, \nabla (\mathcal{T}_{1}u_{1}),\nabla ( \mathcal{T}_{2}u_{2})\varphi _{i} \, \mathrm{d}x. \end{aligned}

Now for every u and $$\varphi \in \mathcal{W}$$, system $$(\mathrm{P}_{\mu })$$ can be given in the form

$$\bigl\langle \mathcal{A}(u)+\mu \mathcal{B}u+\Pi _{p,\delta }(u),\varphi \bigr\rangle _{\mathcal{W}}=\bigl\langle \mathcal{F}(u),\varphi \bigr\rangle _{ \mathcal{W}}.$$
(4.1)

Set

$$\mathcal{\chi}_{\mu}:=\mathcal{A}(u)+\mu \mathcal{B}u+\Pi _{p,\delta }(u)- \mathcal{F}(u).$$

First, by conditions $$(\mathrm{H.}1)$$ and $$(\mathrm{H.}2)$$, $$\mathcal{\chi}_{\mu}$$ is well defined, continuous, and bounded. The next step in the proof is showing that the operator $$\mathcal{\chi}_{\mu }$$ is pseudo-monotone. To this end, using the $$(\mathrm{S})_{+}$$-property of $$\mathcal{A}$$, in view of the compactness of the operators $$\Pi _{p,\delta }$$, $$\mathcal{B}$$, $$\mathcal{F}$$, we can use Gambera et al. [7, Lemma 2.2] to deduce that the operator $$\mathcal{\chi}_{\mu}$$ also has the $$(\mathrm{S})_{+}$$-property. Furthermore, we can apply Zeidler [26, Proposition 26.2] to see that the operator $$\mathcal{\chi}_{\mu}$$ is pseudo-monotone.

Let us show that the operator $$\mathcal{\chi}_{\mu }:\mathcal{W}\rightarrow \mathcal{W}^{*}$$ is coercive. To this end, using equation (4.1), we get

\begin{aligned} \bigl\langle \mathcal{\chi}_{\mu }(u),u \bigr\rangle & = \sum_{i=1}^{2} \int _{\Omega } \vert \nabla u_{i} \vert ^{p_{i}} \,\mathrm{d}x+\mu \sum_{i=1}^{2} \int _{\Omega }b_{i}(x,u_{i})u_{i} \,\mathrm{d} x+\sum_{i=1}^{2} \int _{\Omega } \frac{ \vert \nabla (\mathcal{T}u_{i}) \vert ^{p_{i}}}{\mathcal{T}u_{i}+\delta _{i} }u_{i} \, \mathrm{d}x \\ &\quad{} - \sum_{i=1}^{2} \int _{\Omega }f_{i}\bigl(x ,\mathcal{T}u_{1}, \mathcal{T}u_{2},\nabla (\mathcal{T}u_{1}),\nabla ( \mathcal{T}u_{2})\bigr)u_{i} \,\mathrm{d}x \\ & \geq \sum_{i=1}^{2} \int _{\Omega } \vert \nabla u_{i} \vert ^{p_{i}} \,\mathrm{d}x+\mu \sum_{i=1}^{2} \int _{\Omega }b_{i}(x,u_{i})u_{i} \,\mathrm{d} x-\sum_{i=1}^{2} \int _{\Omega } \frac{ \vert \nabla (\mathcal{T}u_{i}) \vert ^{p_{i}}}{\mathcal{T}u_{i}+\delta _{i} }u_{i} \, \mathrm{d}x \\ &\quad{} - \sum_{i=1}^{2} \int _{\Omega }f_{i}\bigl(x ,\mathcal{T}u_{1}, \mathcal{T}u_{2},\nabla (\mathcal{T}u_{1}),\nabla ( \mathcal{T}u_{2})\bigr)u_{i} \,\mathrm{d}x. \end{aligned}
(4.2)

Now using equation (4.2) and combining equations (3.9) and (3.10) with Lemmas 3.2 and 3.3, we obtain

\begin{aligned} \bigl\langle \mathcal{\chi}_{\mu }(u),u \bigr\rangle & \geq \sum_{i=1}^{2} \Vert \nabla u_{i} \Vert _{p_{i}}^{p_{i}}+\mu \bigl( C_{1} \Vert u_{1} \Vert _{p_{1}}^{p_{1}}-C_{2} \bigr)+\mu \bigl( C'_{1} \Vert u_{2} \Vert _{p_{2}}^{p_{2}}-C'_{2} \bigr) \\ &\quad{} - \sum_{i=1}^{2} k_{i} \bigl(1+ \Vert u_{i} \Vert _{p_{i}} \bigr)- \bigl( \Vert \nabla u_{1} \Vert _{p_{1}}^{p_{1}}+ \Vert \nabla u_{2} \Vert _{p_{2}}^{p_{2}} \bigr) \\ &\quad{} - k_{0} \bigl(1+ \Vert u_{1} \Vert _{p_{1}}+ \Vert u_{1} \Vert _{p_{1}}^{p_{1}}+ \Vert u_{1} \Vert _{p_{2}}^{p_{2}} \bigr) \\ &\quad{} - k'_{0} \bigl(1+ \Vert u_{2} \Vert _{p_{2}}+ \Vert u_{2} \Vert _{p_{2}}^{p_{2}}+ \Vert u_{2} \Vert _{p_{1}}^{p_{1}} \bigr) \\ & \geq \bigl( \Vert \nabla u_{1} \Vert _{p_{1}}^{p_{1}}+ \Vert \nabla u_{2} \Vert _{p_{2}}^{p_{2}} \bigr)+ \mu C^{*}_{1} \bigl( \Vert u_{1} \Vert _{p_{1}}^{p_{1}}+ \Vert u_{2} \Vert _{p_{2}}^{p_{2}} \bigr)-\mu \bigl( C_{2}+C'_{2} \bigr) \\ &\quad{} - \sum_{i=1}^{2} k_{i} \bigl(1+ \Vert u_{i} \Vert _{p_{i}} \bigr)-k^{*}_{0} \bigl(1+ \Vert u_{1} \Vert _{p_{1}}+ \Vert u_{1} \Vert _{p_{1}}^{p_{1}}+ \Vert u_{1} \Vert _{p_{2}}^{p_{2}} \bigr), \end{aligned}
(4.3)

where $$k^{*}_{0}:=\max \{k_{0}, k'_{0}\}$$ and $$C^{*}_{1}:=\min \{C_{1}, C'_{1}\}$$. Then invoking Moussaoui and Saoudi [20, Lemma 2.2], we can deduce that

$$\bigl\Vert \nabla (u_{i}\mathbbm{1}_{\{\underline{u}_{i}< u_{i}< \overline{u}_{i}\}}) \bigr\Vert _{p_{i}}\leq \tilde{C}_{i} \quad \text{for some } \tilde{C}_{i}>0 \text{ independent of } u_{i}, i=1,2.$$

Furthermore, for sufficiently large $$\mu >0$$ such that $$\mu C^{*}_{1}-k^{*}_{0}>0$$ and for every sequence $$(u_{n})_{n}$$ in $$\mathcal{W}$$, inequality (4.3) implies

$$\bigl\langle \mathcal{\chi}_{\mu}(u_{n}),u_{n} \bigr\rangle \rightarrow +\infty \quad \text{as } \Vert u_{n} \Vert _{\mathcal{W}}\rightarrow +\infty .$$

Therefore, since $$\mathcal{\chi}_{\mu}$$ is continuous, bounded, coercive, and pseudomonotone, invoking the pseudo-monotone operator theorem (see, e.g., Carl et al. [4, Theorem 2.99]), we get the existence of $$u\in \mathcal{W}$$ such that

$$\bigl\langle \mathcal{\chi}_{\mu }(u_{1},u_{2}),( \varphi _{1}, \varphi _{2}) \bigr\rangle =0\quad \text{for every } (\varphi _{1}, \varphi _{2}) \in \mathcal{W}.$$
(4.4)

Moreover, using Casas et al. [6, Theorem 3], we have

$$\vert \nabla u_{1} \vert ^{p_{1}-2}\frac{\partial u_{1}}{\partial \eta }=0= \vert \nabla u_{2} \vert ^{p_{2}-2}\frac{\partial u_{2}}{\partial \eta }=0 \quad \text{on }\partial \Omega .$$

Therefore we conclude that $$u=(u_{1},u_{2})\in \mathcal{W}$$ is a weak solution of $$(\mathrm{S}_{\mu })$$. This completes the proof of the theorem.

## 5 Subsupersolutions

The aim of this section is to construct pairs of sub- and supersolutions of system $$(\mathrm{S})$$.

### Theorem 5.1

Assume that conditions $$({\mathbf{H_{3}}})$$, $$({\mathbf{H_{4}}})$$, $$({\mathbf{H_{5}}})$$, and $$({\mathbf{H_{6}}})$$ are satisfied. Then system $$(\mathrm{S})$$ has a solution $$u=(u_{1},u_{2})\in \mathcal{C}^{1,\gamma }( \overline{\Omega }) \times \mathcal{C}^{1,\gamma }(\overline{\Omega })\cap [\underline{u}_{1},\overline{u}_{1}]\times [ \underline{u}_{2},\overline{u}_{2}]$$ for some $$\gamma \in (0,1)$$.

### Proof of Theorem 5.1

First, using Theorem 3.1, we can fix $$\mu >0$$ sufficiently large such that system $$(\mathrm{S}_{\mu })$$ admits a pair of weak solutions $$u=(u_{1},u_{2})\in \mathcal{W}$$. It remains to verify that $$u=(u_{1},u_{2})\in [\underline{u}_{1},\overline{u}_{1}]\times [ \underline{u}_{2},\overline{u}_{2}]$$. Here we give just the proof for $$u_{1}\in [\underline{u}_{1},\overline{u}_{1}]$$. A similar reasoning yields the second inequality. To this end, we set $$(\varphi _{1},\varphi _{2})=((u_{1}-\overline{u}_{1})_{+},0)$$. By Lemma 3.3 and condition $$({\mathbf{H_{5}}})$$, combined with equation (4.4), we obtain

\begin{aligned} & \int _{\Omega } \vert \nabla u_{1} \vert ^{p_{1}-2}\nabla u_{1} \nabla (u_{1}- \overline{u}_{1})_{+} \,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla (\mathcal{T}u_{1}) \vert ^{p_{1}}}{\mathcal{T}u_{1}+\delta _{1} }(u_{1}- \overline{u}_{1})_{+} \,\mathrm{d}x \\ &\quad = \int _{\Omega }f\bigl(x ,\mathcal{T}u_{1}, \mathcal{T}u_{2},\nabla ( \mathcal{T}u_{1}),\nabla ( \mathcal{T}u_{2})\bigr) (u_{1}-\overline{u}_{1})_{+} \,\mathrm{d}x-\mu \int _{\Omega }b(x ,u_{1}) (u_{1}- \overline{u}_{1})_{+} \,\mathrm{d}x \\ &\quad = \int _{\Omega }f\bigl(x ,\overline{u}_{1},, \mathcal{T}u_{2},\nabla \overline{u}_{1},\nabla ( \mathcal{T}u_{2})\bigr) (u_{1}-\overline{u}_{1})_{+} \,\mathrm{d}x-\mu \int _{\Omega }(u_{1}-\overline{u}_{1})_{+}^{p_{1}} \,\mathrm{d}x \\ &\quad \leq \int _{\Omega } \vert \nabla \overline{u}_{1} \vert ^{p_{1}-2}\nabla \overline{u}_{1} \nabla (u_{1}-\overline{u}_{1})_{+} \, \mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1}+\delta _{1} }(u_{1}- \overline{u}_{1})_{+} \,\mathrm{d}x-\mu \int _{\Omega }(u_{1}-\overline{u}_{1})_{+}^{p_{1}} \,\mathrm{d}x. \end{aligned}

Now, according to equation (3.1),

$$\int _{\Omega } \frac{ \vert \nabla (\mathcal{T}u_{1}) \vert ^{p_{1}}}{\mathcal{T}u_{1}+\delta _{1} }(u_{1}- \overline{u}_{1})_{+} \,\mathrm{d}x= \int _{\Omega } \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1}+\delta _{1} }(u_{1}- \overline{u}_{1})_{+} \,\mathrm{d}x,$$

so it follows that

$$\int _{\Omega } \bigl( \vert \nabla u_{1} \vert ^{p_{1}-2}\nabla u_{1}- \vert \nabla \overline{u}_{1} \vert ^{p_{1}-2}\nabla \overline{u}_{1} \bigr) \nabla (u_{1}- \overline{u}_{1})_{+} \, \mathrm{d}x\leq -\mu \int _{\Omega }(u_{1}-\overline{u}_{1})_{+}^{p_{1}} \,\mathrm{d}x\leq 0.$$
(5.1)

Hence it follows from equation (5.1), combined with the monotonicity of $$A_{1}$$, that $$u_{1}\leq \overline{u}_{1}$$. In the same way, to see that $$\underline{u}_{1}\leq u_{1}$$, we set $$(\varphi _{1},\varphi _{2})=((\underline{u}_{1}-u_{1})_{+},0)$$. So, $$u=(u_{1},u_{2})\in [\underline{u}_{1},\overline{u}_{1}]\times [ \underline{u}_{2},\overline{u}_{2}]$$. Moreover, according to Miyajima et al. [15, Remark 8], we obtain that $$u=(u_{1},u_{2})\in \mathcal{C}^{1,\gamma }( \overline{\Omega }) \times \mathcal{C}^{1,\gamma }(\overline{\Omega })$$ for some $$\gamma \in (0,1)$$ and $$\frac{\partial u_{1}}{\partial \eta }=\frac{\partial u_{2}}{\partial \eta }=0$$ on Ω. Therefore we have shown that $$u=(u_{1},u_{2})\in \mathcal{C}^{1,\gamma }( \overline{\Omega }) \times \mathcal{C}^{1,\gamma }(\overline{\Omega })$$ is a solution of the system $$(\mathrm{S})$$ within $$[\underline{u}_{1},\overline{u}_{1}]\times [\underline{u}_{2}, \overline{u}_{2}]$$. □

## 6 Nodal solutions

The objective of this section is to show the existence of nodal solutions of system $$(\mathrm{S} )$$. The proof is mostly based on finding pairs of sub- and supersolutions of system $$(\mathrm{S} )$$. To this end, first, recall from Candito et al. [3, Lemma 2] that $$z_{i}\in \mathcal{C}^{1,\gamma }(\overline{\Omega })$$, $$i=1,2$$, for some $$\gamma \in (0,1)$$ are the unique solutions of the homogeneous Dirichlet problem

$$\textstyle\begin{cases} -\Delta _{p_{i}}u=1 & \text{in }\Omega , \\ u=0& \text{on }\partial \Omega , \end{cases}$$
(6.1)

which satisfies

\begin{aligned}& \Vert z_{i} \Vert _{\mathcal{C}^{1,\gamma }(\overline{\Omega })}\leq L \quad \text{and}\quad \Vert \nabla z_{i} \Vert _{\infty}\leq \hat{L}, \end{aligned}
(6.2)
\begin{aligned}& ld(x)\leq z_{i}\leq Ld(x)\quad \text{in } \Omega ,\qquad \frac{\partial z_{i}}{\partial \eta }< 0\quad \text{on }\partial \Omega , \end{aligned}
(6.3)

for certain constants , l, and L. Moreover, by the Minty–Browder theorem (see Brezis [1]), combined with the Lieberman regularity Theorem [12], we know that the Dirichlet problem

$$-\Delta _{{p}_{i}}u= \textstyle\begin{cases} 1 & \text{if }x\in \Omega \backslash \overline{\Omega }_{\tau }, \\ -1 & \text{otherwise},\end{cases}\displaystyle \qquad u=0\quad \text{on }\partial \Omega ,$$
(6.4)

has a unique solution, denoted by $$z_{i,\tau }\in \mathcal{C}^{1,\gamma }(\overline{\Omega })$$ for a given $$0<\tau <\operatorname{diam}(\Omega )$$, satisfying

\begin{aligned}& z_{i,\tau }\leq z_{i}\quad \text{in }\Omega , \end{aligned}
(6.5)
\begin{aligned}& \frac{\partial z_{i,\tau }}{\partial \eta }< \frac{1}{2} \frac{\partial z_{i}}{\partial \eta }< 0\quad \text{on }\partial \Omega , \quad \text{and}\quad z_{i, \tau }\geq \frac{1}{2} z_{i} \quad \text{in }\Omega . \end{aligned}
(6.6)

Now for a given $$\tau >0$$, we define

\begin{aligned}& \underline{u}_{1}:=\tau ^{\frac{1}{p_{1}}}z_{1,\tau }^{\omega _{1} }- \tau ,\qquad \underline{u}_{2}:=\tau ^{\frac{1}{p_{2}}}z_{2, \tau }^{\omega _{2} }- \tau , \end{aligned}
(6.7)
\begin{aligned}& \overline{u}_{1}:=\tau ^{-p_{1}}z_{1}^{\overline{\omega}_{1}}- \tau , \qquad \overline{u}_{2}:=\tau ^{-p_{2}}z_{2}^{ \overline{\omega}_{2}}- \tau , \end{aligned}
(6.8)

where

$$\frac{\omega _{i} -1}{\omega _{i} }>\frac{1}{p_{i}-1}> \frac{\overline{\omega}_{i}-1}{\overline{\omega}_{i}}\quad \text{with }\omega _{i} >\overline{ \omega}_{i}>1$$
(6.9)

and

$$\overline{\omega _{i} }< 1+p_{i} \biggl(1- \frac{\max \{\alpha _{i} ,\beta _{i} \}}{p_{i}-1} \biggr).$$
(6.10)

According to equations (6.2)–(6.3), we have

\begin{aligned}& \overline{u}_{1}\leq \tau ^{-p_{1}}(Ld)^{\overline{\omega}_{1}}, \qquad \overline{u}_{2}\leq \tau ^{-p_{2}}(Ld)^{ \overline{\omega}_{2}} \end{aligned}
(6.11)
\begin{aligned}& \Vert \nabla \overline{u}_{1} \Vert _{\infty }\leq \tau ^{-p_{1}}\hat{L}_{1}, \qquad \Vert \nabla \overline{u}_{2} \Vert _{\infty }\leq \tau ^{-p_{2}} \hat{L}_{2}, \end{aligned}
(6.12)

with $$\hat{L}_{i}:=\overline{\omega}_{i}L^{\overline{\omega}_{i}}$$ for $$i=1,2$$. Furthermore, on Ω, we have

$$\textstyle\begin{cases} \frac{\partial \overline{u}_{1}}{\partial \eta }=\tau ^{-p_{1}} \frac{\partial (z_{1}^{\overline{\omega}_{1}})}{\partial \eta }=\tau ^{-p_{1}} \overline{\omega}_{1}z_{1}^{\overline{\omega}_{1}-1} \frac{\partial z_{1}}{\partial \eta }=0, \\ \frac{\partial \overline{u}_{2}}{\partial \eta }=\tau ^{-p_{2}} \frac{\partial (z_{2}^{\overline{\omega}_{2}})}{\partial \eta }=\tau ^{-p_{2}} \overline{\omega}_{2}z_{2}^{\overline{\omega}_{2}-1} \frac{\partial z_{2}}{\partial \eta }=0, \end{cases}$$
(6.13)

since $$z_{i}$$ is a solution of the Dirichlet problem (6.1) for $$\omega _{i} ,\overline{\omega}_{i}>1$$, $$i=1,2$$.

Now we will prove the following result.

### Lemma 6.1

For a sufficiently small $$\tau >0$$, we have $$\underline{u}_{1}\leq \overline{u}_{1}$$ and $$\underline{u}_{2}\leq \overline{u}_{2}$$.

### Proof

First, we show that $$\underline{u}_{1}\leq \overline{u}_{1}$$ in Ω. By a direct computation we obtain

\begin{aligned} \overline{u}_{1}(x)- \underline{u}_{1}(x)&= \bigl( \tau ^{-p_{1}}z_{1}^{ \bar{\omega _{1}}}- \tau \bigr) - \bigl( \tau ^{\frac{1}{p_{1}}}z_{i, \tau }^{\omega _{i} }- \tau \bigr) \\ &\geq \tau ^{-p_{1}}z_{1}^{\bar{\omega _{1}}}-\tau ^{\frac{1}{p_{1}}}z_{1}^{ \omega _{1} }=z^{\omega _{1} }\bigl(\tau ^{-p_{1}}z_{1}^{\bar{\omega}_{1}- \omega _{1} }-\tau ^{\frac{1}{p_{1}}}\bigr) \\ &\geq z_{1}^{\omega _{1} }\bigl(\tau ^{-p_{1}}\bigl(cd(x) \bigr)^{\bar{\omega}_{1}- \omega _{1} }-\tau ^{\frac{1}{p_{1}}}\bigr)\geq 0, \end{aligned}

since $$\omega _{1} >\overline{\omega }_{1}$$ and $$z_{1,\tau }\leq z_{1}$$ for every small enough $$\tau <\operatorname{diam}(\Omega )$$. Therefore $$\underline{u}_{1}\leq \overline{u}_{1}$$ in Ω. Finally, using a similar argument, we can obtain that $$\underline{u}_{2}\leq \overline{u}_{2}$$ in Ω. □

## 7 Proofs of main results

### Proof of Theorem 1.1

First, we claim that equation (2.3) is satisfied by the pair of functions $$(\overline{u}_{1},\overline{u}_{2})$$ given by equation (6.8). To see this, pick $$(u_{1},u_{2})\in W^{1,p_{1}}(\Omega )\times W^{1,p_{2}}(\Omega )$$ within $$[\underline{u}_{1},\overline{u}_{1}]\times [\underline{u}_{2}, \overline{u}_{2}]$$ such that $$\underline{ u}_{2}\leq u_{2}\leq \overline{u}_{2}$$, $$\underline{ u}_{1}\leq u_{1}\leq \overline{u}_{1}$$. Now, in view of condition $$({\mathbf{H_{1}}})$$, combined with equations (6.11) and (6.12), we have

\begin{aligned}[b] \bigl\vert f_{1}(\cdot , \overline{u}_{1},u_{2},\nabla \overline{u}_{1}, \nabla u_{2}) \bigr\vert & \leq M_{1}\bigl(1+ \vert \overline{u}_{1} \vert ^{\alpha _{1} }\bigr) \bigl(1+ \vert \nabla \overline{u}_{1} \vert ^{\beta _{1} }\bigr) \\ &\leq M_{1}\bigl(1+\bigl(\tau ^{-p_{1}}(Ld)^{\overline{\omega }_{1}} \bigr)^{\alpha _{1} }\bigr) \bigl(1+\bigl(\tau ^{-p_{1}} \hat{L} \bigr)^{\beta _{1} }\bigr) \\ &\leq 2M_{1}(C_{1}C_{2})^{\alpha _{1} \beta _{1} } \tau ^{-p_{1}\max \{\alpha _{1} ,\beta _{1} \}} \\ &\leq C\tau ^{-p_{1}\max \{\alpha _{1} ,\beta _{1} \}}, \end{aligned}
(7.1)

where $$C:=2M_{1}(C_{1}C_{2})^{\alpha _{1} \beta _{1} }$$, and $$\tau >0$$ is small enough. Using the same argument as in equation (7.1), we obtain

$$\bigl\vert f_{2}(\cdot ,u_{1}, \overline{u}_{2},\nabla u_{1},\nabla \overline{u}_{2}) \bigr\vert \leq C'\tau ^{-p_{2}\max \{\alpha _{2} ,\beta _{2} \}}$$
(7.2)

for some constant $$C'>0$$ and for $$\tau >0$$ small enough. Now, in view of equations (6.8) and (6.9), we have

$$-\Delta _{p_{1}}\overline{u}_{1}+ \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1}+\delta _{1} }=\tau ^{-p_{1}(p_{1}-1)} \biggl(-\Delta _{p_{1}}z_{1}^{ \bar{\omega}_{1}}+ \frac{ \vert \nabla z_{1}^{\bar{\omega}_{1}} \vert ^{p}_{1}}{z_{1}^{\bar{\omega}_{1}}} \biggr).$$
(7.3)

On the other hand, by a direct computation we get

\begin{aligned} -\Delta _{p_{1}}z_{1}^{\bar{\omega}_{1}}+ \frac{ \vert \nabla z_{1}^{\bar{\omega}_{1}} \vert ^{p_{1}}}{z_{1}^{\bar{\omega}_{1}}}& = \bar{\omega}_{1}^{p_{1}-1} \biggl( 1-(\bar{\omega}_{1}-1) ( p_{1}-1) \frac{ \vert \nabla z_{1} \vert ^{p_{1}}}{z_{1}} \biggr) z_{1}^{( \bar{\omega _{1}}-1)(p_{1}-1)} \\ &\quad{} + \bar{\omega _{1}}^{p_{1}}\frac{z_{1}^{(\bar{\omega}_{1}-1)p_{1}} \vert \nabla z_{1} \vert ^{p_{1}}}{z_{1}^{\bar{\omega}_{1}}} \\ & = \bar{\omega}^{p_{1}-1} \biggl( 1-(\bar{\omega}_{1}-1) ( p_{1}-1) \frac{ \vert \nabla z_{1} \vert ^{p_{1}}}{z_{1}} \biggr) z_{1}^{(\bar{\omega}_{1}-1)(p_{1}-1)} \\ &\quad{} + \bar{\omega}_{1}^{p_{1}}z_{1}^{(\bar{\omega}_{1}-1)(p_{1}-1)} \frac{z_{1}^{\bar{\omega}_{1}-1} \vert \nabla z_{1} \vert ^{p_{1}}}{z_{1}^{\bar{\omega}_{1}}} \\ & = \bar{\omega}_{1}^{p_{1}-1} \biggl[ 1+\bar{ \omega}_{1}\biggl(1- \frac{(\bar{\omega}_{1}-1)(p_{1}-1) }{\bar{\omega}_{1}}\biggr) \frac{ \vert \nabla z_{1} \vert ^{p_{1}}}{z_{1}} \biggr] z_{1}^{(\bar{\omega}_{1}-1)(p_{1}-1)}. \end{aligned}
(7.4)

Invoking equations (7.3) and (7.4), it follows that

\begin{aligned}& -\Delta _{p_{1}}\overline{u}_{1}+ \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1}+\delta _{1} } \\& \quad =\tau ^{-p_{1}(p_{1}-1)}\bar{\omega}_{1}^{p_{1}-1} \biggl[ 1+\bar{\omega}_{1}\biggl(1- \frac{(\bar{\omega}_{1}-1)( p_{1}-1) }{\bar{\omega}_{1}}\biggr) \frac{ \vert \nabla z_{1} \vert ^{p_{1}}}{z_{1}} \biggr] z_{1}^{(\bar{\omega}_{1}-1)(p_{1}-1)} \\& \quad \geq \tau ^{-p_{1}(p_{1}-1)}\bar{\omega}_{1}^{p_{1}-1} \textstyle\begin{cases} z_{1}^{(\bar{\omega}_{1}-1)(p_{1}-1)} & \text{in } \Omega \backslash \overline{\Omega }_{\tau }, \\ \bar{\omega}_{1}(1- \frac{(\bar{\omega}_{1}-1)( p_{1}-1) }{\bar{\omega}_{1}})z_{1}^{(\bar{\omega}_{1}-1)(p_{1}-1)-1} \vert \nabla z_{1} \vert ^{p_{1}} & \text{in } \Omega _{\tau}. \end{cases}\displaystyle \end{aligned}

Moreover, using equation (6.10) and decreasing τ if necessary, we have

\begin{aligned} \tau ^{-p_{1}(p_{1}-1)}\bar{\omega}_{1}^{p_{1}-1}z_{1}^{( \bar{\omega}_{1}-1)(p_{1}-1)} & \geq \tau ^{-p_{1}(p_{1}-1)} \bar{\omega}^{p_{1}-1}\bigl(c^{-1}d(x) \bigr)^{(\bar{\omega}_{1}-1)(p_{1}-1)} \\ & \geq \tau ^{-p_{1}(p_{1}-1)}\bar{\omega}_{1}^{p_{1}-1} \bigl(c^{-1}\tau \bigr)^{( \bar{\omega}-1)(p_{1}-1)} \\ & = \tau ^{(\bar{\omega}_{1}-1-p_{1})(p_{1}-1)}\bar{\omega}_{1}^{p_{1}-1}c^{-( \bar{\omega}_{1} -1)(p_{1}-1)} \\ & \geq \tau ^{-p_{1}\max \{\alpha _{1} ,\beta _{1} \}}\quad \text{in } \Omega \backslash \overline{\Omega }_{\tau }. \end{aligned}
(7.5)

Finally, combining equations (7.1) and (7.5), we obtain

$$-\Delta _{p_{1}}\overline{u}_{1}+ \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1} +\delta _{1} }\geq f_{1}(\cdot ,\overline{u}_{1},u_{2}, \nabla \overline{u}_{1},\nabla u_{2}) \quad\text{in }\Omega \backslash \overline{\Omega }_{\tau }.$$
(7.6)

Now pick any $$x\in {\Omega }_{\tau }$$. By equations (6.3) and (6.9) we can find a constant $$\beta >0$$ such that

$$\biggl(1- \frac{(\bar{\omega}_{1}-1) ( p_{1}-1 ) }{\bar{\omega}_{1}}\biggr) \vert \nabla z_{1} \vert >\beta\quad \text{in }{\Omega }_{\tau }.$$

By equations (6.3) and (6.9) we have

\begin{aligned} &\tau ^{-p_{1}(p_{1}-1)}\bar{\omega}_{1}^{p_{1}} \biggl(1- \frac{(\bar{\omega}_{1}-1) ( p_{1}-1 ) }{\bar{\omega}_{1}} \biggr)z_{1}^{(\bar{\omega}_{1}-1)(p_{1}-1)-1}| \nabla z_{1}|^{p_{1}} \\ &\quad \geq \tau ^{-p_{1}(p_{1}-1)}\bar{\omega}_{1}^{p_{1}} \bigl(Ld(x)\bigr)^{( \bar{\omega}_{1}-1)(p_{1}-1)-1}\bar{\mu}^{p} \\ &\quad \geq \tau ^{-p_{1}(p_{1}-1)}\bar{\omega}_{1}^{p_{1}}(L\tau )^{( \bar{\omega}_{1}-1)(p_{1}-1)-1}\beta ^{p_{1}} \\ & \quad \geq \tau ^{-p_{1}\max \{\alpha _{1} ,\beta _{1} \}}\quad \text{in } \Omega _{\tau }. \end{aligned}

Therefore, for $$\tau >0$$ sufficiently small, we obtain

$$-\Delta _{p_{1}}\overline{u}_{1}+ \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1} +\delta _{1} }\geq f_{1}(\cdot ,\overline{u}_{1},u_{2}, \nabla \overline{u}_{1},\nabla u_{2}) \quad\text{in }\Omega _{\tau}.$$
(7.7)

Combining equations (7.6) and (7.7), we get

$$-\Delta _{p_{1}}\overline{u}_{1}+ \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1} +\delta _{1} }\geq f_{1}(\cdot ,\overline{u}_{1},u_{2}, \nabla \overline{u}_{1},\nabla u_{2}) \quad\text{in }\Omega .$$
(7.8)

A similar argument yields

$$-\Delta _{p_{2}}\overline{u}_{2}+ \frac{ \vert \nabla \overline{u}_{2} \vert ^{p_{2}}}{\overline{u}_{2} +\delta _{2} }\geq f_{2}(\cdot ,u_{1}, \overline{u}_{2},\nabla u_{1}, \nabla \overline{u}_{2})\quad\text{in }\Omega .$$
(7.9)

Now test equation (7.8), equation (7.9) with $$(\varphi _{1},\varphi _{2}) \in W_{b}^{1,p_{1}}(\Omega )\times W_{b}^{1,p_{2}}( \Omega )$$, $$\varphi \geq 0$$ a.e. in Ω, and equation (6.13) yield

\begin{aligned}& \int _{\Omega } \vert \nabla \overline{u}_{1} \vert ^{p_{1}-2}\nabla \overline{u}_{1} \nabla \varphi _{1} \,\mathrm{d}x + \int _{\Omega } \frac{ \vert \nabla \overline{u}_{1} \vert ^{p_{1}}}{\overline{u}_{1} +\delta _{1}}\varphi _{1} \, \mathrm{d}x- \biggl\langle \frac{\partial \overline{u}}{\partial \eta _{p_{1}}},\gamma _{0}(\varphi _{1} ) \biggr\rangle _{ \partial \Omega } \\& \quad \geq \int _{\Omega }f_{1}(\cdot ,\overline{u}_{1},u_{2}, \nabla \overline{u}_{1},\nabla u_{2})\varphi _{1} \,\mathrm{d}x, \\& \int _{\Omega } \vert \nabla \overline{u}_{2} \vert ^{p_{2}-2}\nabla \overline{u}_{2} \nabla \varphi _{2} \,\mathrm{d}x+ \int _{\Omega } \frac{ \vert \nabla \overline{u}_{2} \vert ^{p_{2}}}{\overline{u}_{2} +\delta _{2}}\varphi _{2} \, \mathrm{d}x- \biggl\langle \frac{\partial \overline{u}}{\partial \eta _{p_{2}}},\gamma _{0}(\varphi _{2}) \biggr\rangle _{ \partial \Omega } \\& \quad \geq \int _{\Omega }f_{2}(\cdot ,u_{1}, \overline{u}_{2}, \nabla u_{1},\nabla \overline{u}_{2})\varphi _{2} \,\mathrm{d}x, \end{aligned}

where $$\gamma _{0}$$ is the trace operator on Ω,

$$\frac{\partial w}{\partial \eta _{p_{i}}}:= \vert \nabla w \vert ^{p_{i}-2} \frac{\partial w}{\partial \eta }\quad \text{for every } w\in W^{1,p_{i}}(\Omega )\cap C^{1}( \overline{\Omega }),$$
(7.10)

and $$\langle \cdot ,\cdot \rangle _{\partial \Omega }$$ is the duality brackets for the pair

$$\bigl(W^{1/p_{i}^{\prime },p_{i}}(\partial \Omega ),W^{-1/p_{i}^{\prime },p_{i}^{ \prime }}(\partial \Omega ) \bigr).$$

The proof of the claim is now completed.

Next, we show that equation (2.2) is satisfied by the pair of functions $$(\underline{u}_{1},\underline{u}_{2})$$ given by equation (6.7). A direct computation yields

\begin{aligned} -\Delta _{p_{1}}z_{1,\tau }^{\omega _{1} }+ \frac{ \vert \nabla z{1,\tau }^{\omega _{1} } \vert ^{p_{1}}}{z_{1,\tau }^{\omega _{1} }}& = \omega _{1}^{p_{1}-1} \biggl( 1-( \omega _{1} -1) ( p_{1}-1 ) \frac{ \vert \nabla z_{1,\tau } \vert ^{p_{1}}}{z_{1,\tau }} \biggr) z_{1,\tau }^{( \omega _{1} -1)(p_{1}-1)} \\ &\quad{} + \omega _{1}^{p_{1}} \frac{z_{1,\tau }^{(\omega _{1} -1)p_{1}} \vert \nabla z_{1,\tau } \vert ^{p_{1}}}{z_{1,\tau }^{\omega _{1}}} \\ & = \omega _{1}^{p_{1}-1} \biggl[ 1+\omega _{1} \biggl(1- \frac{(\omega _{1} -1) ( p_{1}-1 ) }{\omega _{1} }\biggr)\frac{ \vert \nabla z_{1,\tau } \vert ^{p_{1}}}{z_{1,\tau }} \biggr] z_{1,\tau }^{( \omega _{1} -1)(p_{1}-1)} \end{aligned}

in $$\Omega \backslash \overline{\Omega }_{\tau }$$. Similarly, it follows that

\begin{aligned} -\Delta _{p_{1}}z_{1,\tau}^{\omega _{1} }+ \frac{ \vert \nabla z_{1,\tau }^{\omega _{1} } \vert ^{p_{1}}}{z_{1,\tau }^{\omega _{1} }}=\omega _{1}^{p_{1}-1} \biggl[ -1+\omega _{1} \biggl(1- \frac{(\omega _{1} -1) ( p_{1}-1 ) }{\omega }\biggr) \frac{ \vert \nabla z_{1,\tau } \vert ^{p_{1}}}{z_{1,\tau}} \biggr] z_{1,\tau }^{(\omega _{1} -1)(p_{1}-1)} \end{aligned}

in $$\Omega _{\tau }$$. In fact, by equations (6.7) and (6.9) we have

\begin{aligned}[b] -\Delta _{p_{1}}\underline{u}_{1}+ \frac{ \vert \nabla \underline{u}_{1} \vert ^{p_{1}}}{\underline{u}_{1} +\delta _{1} }&=\tau ^{\frac{1}{p_{1}^{\prime }}}\biggl(-\Delta _{p_{1}}z_{1, \tau }^{\omega _{1}}+\frac{ \vert \nabla z_{1,\tau}^{\omega _{1} } \vert ^{p_{1}}}{z_{1,\tau }^{\omega _{1} }}\biggr) \\ &\leq \textstyle\begin{cases} \tau ^{\frac{1}{p_{1}^{\prime }}}\omega _{1}^{p_{1}-1}z_{1,\tau}^{( \omega _{1} -1)(p_{1}-1)} & \text{in }\Omega \backslash \overline{\Omega }_{\tau}, \\ 0 & \text{in }\Omega _{\tau }.\end{cases}\displaystyle \end{aligned}
(7.11)

In view of equations (6.2)–(6.5), choosing an appropriate constant $$m_{1}$$ in $$({\mathbf{H_{2}}})$$, we have

$$m_{1}>\tau ^{\frac{1}{p_{1}^{\prime }}}\omega _{1}^{p_{1}-1}L^{( \omega _{1} -1)(p_{1}-1)}\quad \text{for }\tau >0\text{ small enough}.$$
(7.12)

Combining equations (7.11) and (7.12), we arrive at

$$-\Delta _{p_{1}}\underline{u}_{1}+ \frac{ \vert \nabla \underline{u}_{1} \vert ^{p_{1}}}{\underline{u}_{1} +\delta _{1} }\leq f_{1}(\cdot ,\underline{u}_{1},u_{2}, \nabla \underline{u}_{1},\nabla u_{2}).$$
(7.13)

Using a similar argument, we obtain

$$-\Delta _{p_{2}}\underline{u}_{2}+ \frac{ \vert \nabla \underline{u}_{2} \vert ^{p_{2}}}{\underline{u}_{2} +\delta _{2} }\leq f_{2}(\cdot ,u_{1}, \underline{u}_{2},\nabla u_{1}, \underline{u}_{2}).$$
(7.14)

Finally, by test equations (7.13) and (7.14) with $$(\varphi _{1}, \varphi _{2}) \in W_{b}^{1,p_{1}}(\Omega )\times W_{b}^{1,p_{2}}( \Omega )$$, where $$\varphi _{1}, \varphi _{2} \geq 0$$ a.e. in Ω, equation (6.13), and the Green formula [6] we obtain

\begin{aligned}& \int _{\Omega } \vert \nabla \underline{u}_{1} \vert ^{p_{1}-2}\nabla \underline{u}_{1}\nabla \varphi _{1} \,\mathrm{d}x + \int _{\Omega } \frac{ \vert \nabla \underline{u}_{1} \vert ^{p_{1}}}{\underline{u}_{1} +\delta _{1} }\varphi _{1} \, \mathrm{d}x \\& \quad \leq \int _{\Omega } \vert \nabla \underline{u}_{1} \vert ^{p_{1}-2}\nabla \underline{u}_{1}\nabla \varphi _{1} \,\mathrm{d}x- \biggl\langle \frac{\partial \underline{u}_{1}}{\partial \eta _{p_{1}}},\gamma _{0}( \varphi _{1} ) \biggr\rangle _{\partial \Omega }+ \int _{\Omega } \frac{ \vert \nabla \underline{u}_{1} \vert ^{p_{1}}}{\underline{u}_{1}+\delta _{1} } \varphi _{1} \,\mathrm{d}x \\& \quad = \int _{\Omega }\biggl(-\Delta _{p_{1}} \underline{u}_{1} + \frac{ \vert \nabla \underline{u}_{1} \vert ^{p_{1}}}{\underline{u}_{1}+\delta _{1} }\biggr)\varphi _{1} \,\mathrm{d}x \\& \quad \leq \int _{\Omega }f_{1}(\cdot ,\underline{u}_{1},u_{2},\nabla \underline{u}_{1}, \nabla u_{2}) \varphi _{1} \,\mathrm{d}x,\\& \int _{\Omega } \vert \nabla \underline{u}_{2} \vert ^{p_{2}-2}\nabla \underline{u}_{2}\nabla \varphi _{2} \,\mathrm{d}x + \int _{\Omega } \frac{ \vert \nabla \underline{u}_{2} \vert ^{p_{2}}}{\underline{u}_{2} +\delta _{2} }\varphi _{2} \, \mathrm{d}x \\& \quad \leq \int _{\Omega } \vert \nabla \underline{u}_{2} \vert ^{p_{2}-2}\nabla \underline{u}_{2}\nabla \varphi _{2} \,\mathrm{d}x- \biggl\langle \frac{\partial \underline{u}_{2}}{\partial \eta _{p_{2}}},\gamma _{0}( \varphi _{2} ) \biggr\rangle _{\partial \Omega }+ \int _{\Omega } \frac{ \vert \nabla \underline{u}_{2} \vert ^{p_{2}}}{\underline{u}_{2}+\delta _{2} } \varphi _{2} \, \mathrm{d}x \\& \quad = \int _{\Omega }\biggl(-\Delta _{p_{2}} \underline{u}_{2} + \frac{ \vert \nabla \underline{u}_{2} \vert ^{p_{2}}}{\underline{u}_{2}+\delta _{2}}\biggr)\varphi _{2} \,\mathrm{d}x \\& \quad \leq \int _{\Omega }f_{2}(\cdot , u_{1} \underline{u}_{2},\nabla u_{1}, \nabla \underline{u}_{2})\varphi _{2} \,\mathrm{d}x, \end{aligned}

since $$\gamma _{0}(\varphi _{1} ), \gamma _{0}(\varphi _{2} )\geq 0$$ whenever $$(\varphi _{1},\varphi _{2}) \in W^{1,p_{1}}(\Omega )\times W^{1,p_{2}}( \Omega )$$ (for more detail, see Carl et al. [4, p. 35]).

Consequently, $$(\underline{u}_{1},\underline{u}_{2})$$ and $$(\overline{u}_{1},\overline{u}_{2})$$ satisfy equations (2.4) and (2.5). Therefore we can apply Theorem 5.1 to obtain the existence of a solution $$(u_{0},u'_{0})\in \mathcal{C}^{1,\gamma }(\overline{\Omega })\times \mathcal{C}^{1,\gamma }(\overline{\Omega })$$ of system $$( \mathrm{S} )$$ satisfying

$$\underline{u}_{1}\leq u_{0}\leq \overline{u}_{1},\qquad \underline{u}_{2}\leq u'_{0}\leq \overline{u}_{2}.$$
(7.15)

Furthermore, $$(u_{0},u'_{0})$$ is a nodal solution. Indeed, combining equations (6.3), (6.7), and (6.8), we arrive at

\begin{aligned}& \overline{u}_{1}=\tau ^{-p_{1}}z^{\overline{\omega }_{1}}-\tau \leq \tau ^{-p_{1}}\bigl(Ld(x)\bigr)^{\overline{\omega }_{1}}-\tau ,\\& \overline{u}_{2}=\tau ^{-p_{2}}z^{\overline{\omega }_{2}}-\tau \leq \tau ^{-p_{2}}\bigl(Ld(x)\bigr)^{\overline{\omega }_{2}}-\tau , \end{aligned}

which implies that

$$\max \bigl\{ \overline{u}_{1}(x), \overline{u}_{2}(x)\bigr\} < 0,\quad \text{provided that } d(x)< L^{-1}\tau ^{ \frac{p_{i}+1}{\overline{\omega }_{i}}}$$
(7.16)

for $$i=1,2$$. Combining equations (6.3), (6.7), and (6.8) yields

\begin{aligned}& \underline{u}_{1}=\tau ^{\frac{1}{p_{1}}}z_{1,\tau}^{\omega _{1} }- \tau \geq \tau ^{\frac{1}{p_{1}}}\bigl(ld(x)\bigr)^{\omega _{1}}-\tau ,\\& \underline{u}_{2}=\tau ^{\frac{1}{p_{2}}}z_{2,\tau}^{\omega _{2} }- \tau \geq \tau ^{\frac{1}{p_{2}}}\bigl(ld(x)\bigr)^{\omega _{2}}-\tau , \end{aligned}

and hence

$$\min \bigl\{ \underline{u}_{1}(x), \underline{u}_{2}(x)\bigr\} >0\quad \text{when } d(x)>l\tau ^{\frac{1}{\omega _{i} p_{i}^{\prime }}}$$
(7.17)

for $$i=1,2$$. The conclusion now follows from equations (7.16) and (7.17). This completes the proof of Theorem 1.1. □

### Proof of Theorem 1.2

First, using the same notation as in equations (6.7) and (6.8) and applying the same argument as in the proof of Theorem 1.1, we can ensure that $$(\underline{u}_{+},\underline{u}^{+})$$ and $$(\overline{u}_{+},\overline{u}^{+})$$ satisfy equations (2.4) and (2.5). Therefore, invoking Theorem 5.1, we obtain the existence of a solution $$(u_{+},u^{+})\in \mathcal{C}^{1,\gamma }(\overline{\Omega })\times \mathcal{C}^{1,\gamma }(\overline{\Omega })$$ with the following properties:

\begin{aligned}& u_{0}\leq u_{+}\leq \overline{u}_{+} \quad \text{and}\quad u_{+}\geq 0 \quad \text{on } \Omega ,\\& u'_{0}\leq u^{+}\leq \overline{u}^{+}\quad \text{and}\quad u^{+}\geq 0 \quad \text{on } \Omega . \end{aligned}

Finally, using equation (6.8), we can easily deduce that $$u_{+}(x)$$ and $$u^{+}(x)$$ are zero as $$d(x)\rightarrow 0$$. This completes the proof of Theorem 1.2. □

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## Acknowledgements

The authors gratefully acknowledge comments and suggestions by the reviewers.

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Repovš was supported by the Slovenian Research and Innovation Agency grants P1-0292, J1-4031, J1-4001, N1-0278, N1-0114, and N1-0083.

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Saoudi, K., Alzahrani, E. & Repovš, D.D. Nodal solutions for Neumann systems with gradient dependence. Bound Value Probl 2024, 4 (2024). https://doi.org/10.1186/s13661-023-01814-2