Symmetry results for p-Laplacian systems involving a first-order term

Abstract

In this paper we obtain symmetry and monotonicity results for positive solutions to some p-Laplacian cooperative systems in bounded domains involving first-order terms and under zero Dirichlet boundary condition.

1 Introduction

The aim of this work is to get some symmetry and monotonicity results for nontrivial solutions $(u_1, u_2,\ldots ,u_m)\in C^{1}({\overline{\Omega }})\times C^{1}({\overline{\Omega }})\ldots \times C^{1}({\overline{\Omega }})$ to the following quasilinear elliptic system

$$\begin{aligned}{\left\{ \begin{array}{ll} -\Delta _{p_i} u_i+a_i(u_i)|\nabla u_i|^{q_i}= f_i(u_1,u_2,\ldots ,u_i,\ldots , u_m) &{} \text {in }\Omega \\ u_i>0 &{} \text {in }\Omega \\ u_i=0 &{} \text {on }\partial \Omega , \end{array}\right. }\qquad \qquad \qquad \qquad \quad \quad ({\mathcal {S}}) \end{aligned}$$

where $i=1,\ldots ,m$, $p_i>1$, $q_i=\max \{1,{p_i}-1\}$, $\Omega$ is a smooth bounded domain (connected open set) of ${\mathbb {R}}^N$, $N\ge 2$, $\Delta _{p_i} u_i:=\mathrm{div}(|\nabla u_i|^{p_i-2}\nabla u_i)$ is the p-Laplace operator and $a_i, f_i$ are problem data that obey to the set of assumptions $(hp^*)$. The solution $(u_1, u_2,\ldots ,u_m)$ has to be understood in the weak distributional meaning. Our result will be obtained by means of the moving plane method, which goes back to the papers of Alexandrov [1] and Serrin [27]. In this work we use a nice variant of this technique: in particular the one of the celebrated papers of Berestycki-Nirenberg [3] and Gidas-Ni-Nirenberg [16], where the authors used, as essential ingredient, the maximum principle by comparing the values of the solution of the equation at two different points after a suitable reflection. Such a technique can be performed in general convex domains providing partial monotonicity results near the boundary and symmetry properties when the domain is convex and symmetric. For simplicity of exposition and without loss of generality, since the system (${\mathcal {S}}$) is invariant with respect to translations and rotations, we assume directly in all the paper that $\Omega$ is a convex domain in the $x_1$-direction and symmetric with respect to the hyperplane $\{x_1=0\}$. When $m=1$ the system (${\mathcal {S}}$) is reduced to a scalar equation, that was already studied in [15] in the case of $\Omega = {\mathbb {R}}^N_+$ and $1<p<2$.

The moving plane procedure was applied to investigate symmetry properties of solutions of cooperative semilinear elliptic systems in bounded domains, firstly by Troy [28] (see also [11, 12, 26]): in this paper, the author considers the case $p_i=2$ and $a_i=0$ of (${\mathcal {S}}$). This technique is very powerful and was adapted also in the case of cooperative semilinear systems in the half-space ${\mathbb {R}}^N_+$ by Dancer [10] and in the entire space ${\mathbb {R}}^N$ by Busca and Sirakov [4]. For other results regarding semilinear elliptic systems in bounded or unbounded domains, involving also critical nonlinearities, we refer to [13].

The moving plane method for quasilinear elliptic equations in bounded domains was developed in several papers by Damascelli, Pacella and Sciunzi [7,8,9] and in [14, 18] for quasilinear elliptic equations involving the Hardy–Leray potential and other more general singular nonlinearities. For the case of quasilinear elliptic systems in bounded domains we refer to [23, 24], where the authors considered the case $m=2$ and $a_1=a_2=0$ of (${\mathcal {S}}$). Moreover, for other questions regarding existence, non-existence and Liouville type results, in the case of (pure, i.e., $a_i=0$ in (${\mathcal {S}}$)) p-Laplace systems, we refer the readers to the papers (and references therein) [2, 5, 6, 20, 21].

In this work we consider the general case of m p-Laplace equations with first-order terms.

To deal with the study of the qualitative properties of solutions to (${\mathcal {S}}$), first we point out some regularity properties of the solutions to (${\mathcal {S}}$), see Sect. 2. Indeed the fact that solutions to p-Laplace equations are not in general $C^2(\Omega )$, leads to the study of the summability properties of the second derivatives of the solutions. Thanks to these regularity results, we are able to prove a weak comparison principle in small domains, i.e., Proposition 2.5, that is a first crucial step in the proof of the main result of the paper, namely Theorem 1.1. Moreover, we also get some comparison and maximum principles that we will exploit in the Proof of Theorem 1.1.

Through all the paper, we assume that the following hypotheses (denoted by $({hp^*})$ in the sequel) hold:

$({hp^*})$:

(i):

For any $1\le i\le m$, $a_i: {\mathbb {R}}\rightarrow {\mathbb {R}}$ are locally Lipschitz continuous functions.

(ii):

For any $1\le i\le m$, $f_i: \overline{{\mathbb {R}}}_+^m\rightarrow {\mathbb {R}}$ are locally ${\mathcal {C}}^1$ functions, i.e., $f_i\in C^1_{\mathrm{loc}}(\overline{{\mathbb {R}}}^m_+)$, and assume that

$$\begin{aligned}f_i(t_1,t_2,\ldots ,t_m)>0,\end{aligned}$$

for all $t_i>0$. Moreover, the functions $f_i$ satisfy

$$\begin{aligned} \frac{\partial f_i}{\partial t_k}(t_1,t_2,\ldots ,t_m)\ge 0 \quad \text {for}\quad k\ne i, \,\,1\le i,k\le m. \end{aligned}$$

(1.1)

The monotonicity conditions (1.1) are also known as cooperativity conditions, see [10, 24, 26, 28].

Finally we have the following

Theorem 1.1

Assume that hypotheses $(hp^*)$ hold. If $\Omega$ is convex in the $x_1$-direction and symmetric with respect to the hyperplane $T_{0}=\{x \in {\mathbb {R}}^N: x_1=0 \}$, then any solution $(u_1, u_2,\ldots ,u_m)\in C^{1}({\overline{\Omega }})\times C^{1}({\overline{\Omega }})\ldots \times C^{1}({\overline{\Omega }})$ to (${\mathcal {S}}$) is symmetric with respect to the hyperplane $T_{0}$ and nondecreasing in the $x_1$-direction in the set $\Omega _0=\{x_1 <0\}$, namely

$$\begin{aligned}u_i(x_1,x_2,\ldots ,x_N)=u_i(-x_1,x_2,\ldots ,x_N)\quad \text {in }\Omega \end{aligned}$$

and

$$\begin{aligned} \frac{\partial u_i}{\partial x_1}(x) \ge 0\quad \text {in }\Omega _0, \end{aligned}$$

(1.2)

for every $i\in \{1,\ldots ,m\}.$ In particular, if $\Omega$ is a ball, then $u_i$ are radially symmetric and radially decreasing, i.e.,

$$\begin{aligned}\frac{\partial u_i}{\partial r}(r)<0 \quad \text {for }r\ne 0.\end{aligned}$$

Moreover, if $p_i>(2N+2)/(N+2)$ for every $i\in \{1,\ldots ,m\}$, then we have

$$\begin{aligned} \frac{\partial u_i}{\partial x_1}(x) > 0\quad \text {in }\Omega _0, \end{aligned}$$

(1.3)

for every $i\in \{1,\cdots ,m\}.$

The paper is organized as follows: In Section 2 we recall some preliminary results and we prove Proposition 2.5. The Proof of the Theorem 1.1 is contained in Sect. 3.

2 Preliminaries

In this section, we are going to give some results for p-Laplace equations involving a first-order term. Through all the paper, generic fixed and numerical constants will be denoted by C (with subscript or superscript in some case) and it will be allowed to vary within a single line or formula. Moreover, by ${\mathcal {L}}(\Omega )$ we will denote the Lebesgue measure of a measurable set $\Omega$.

Firstly, we recall the following inequalities (see, for example, [7]) that we are going to use along the paper:

For all $\mu ,\mu '\in {\mathbb {R}}^N$ with $|\mu |+|\mu '|>0$ there exist two positive constants $C,{\bar{C}}$ depending on p such that

$$\begin{aligned} \begin{aligned} {}[|\mu |^{p-2}\mu -|\mu '|^{p-2}\mu '][\mu -\mu ']&\ge C(|\mu |+|\mu '|)^{p-2}|\mu -\mu '|^2,\\ ||\mu |^{p-2}\mu -|\mu '|^{p-2}\mu '|&\le {\bar{C}}(|\mu |+|\mu '|)^{p-2}|\mu -\mu '|. \end{aligned} \end{aligned}$$

(2.1)

In the following two theorems we give some regularity results and comparison/maximum principles for the solutions to (${\mathcal {S}}$).

Theorem 2.1

(See [19, 22]). Let $\Omega$ a bounded smooth domain of ${\mathbb {R}}^N$, $N\ge 2$, $1<p<\infty$, $q\ge \max \{p-1,1\}$ and consider $u\in C^{1}( \Omega )$ a positive weak solution to

$$\begin{aligned} -\Delta _p u+a(u)|\nabla u|^q=f(x, u) \quad in \quad \Omega , \end{aligned}$$

with

(i):

$a: {\mathbb {R}}\rightarrow {\mathbb {R}}$ a locally Lipschitz continuous function;

(ii):

$f\in C^1( {{\overline{\Omega }}} \times [0,+\infty ))$.

Denoting $u_{x_i}={\partial u}/{\partial x_i}$ and setting $\nabla u_{x_i}=0$ on $Z_u$, for any $\Omega '\subset \Omega ''\subset \subset \Omega$, we have

$$\begin{aligned} \begin{aligned}&\int _{\Omega '} \frac{|\nabla u|^{p-2-\beta }|\nabla u_{x_i}|^2}{|x-y|^\gamma }\,dx\leqslant {\mathcal {C}}\qquad \forall \, i=1,\ldots ,N, \end{aligned} \end{aligned}$$

(2.2)

uniformly for any $y\in \Omega '$, with

$$\begin{aligned} {\mathcal {C}}:\,= {\mathcal {C}}\Big (a, f,p, q,\beta , \gamma , \Vert u\Vert _{L^\infty (\Omega '')},\Vert \nabla u\Vert _{L^\infty (\Omega '')}\Big ), \end{aligned}$$

for any $0\leqslant \beta <1$ and $\gamma <N-2$ if $N\ge 3$, or $\gamma =0$ if $N=2$.

Moreover, if $f(x,\cdot )$ is positive in $\Omega ''$, then it follows that

$$\begin{aligned} \begin{aligned}&\int _{\Omega '} \frac{1}{|\nabla u|^{r(p-1)}}\frac{1}{|x-y|^\gamma }dx\leqslant {{\mathcal {C}}}^*, \end{aligned} \end{aligned}$$

(2.3)

uniformly for any $y\in \Omega '$, with

$$\begin{aligned} \mathcal{C}^{*}:\,= \mathcal {C}^{*}\Big (a, f,p, q,r, \gamma , \Vert u\Vert _{L^\infty (\Omega '')},\Vert \nabla u\Vert _{L^\infty (\Omega '')}\Big ), \end{aligned}$$

for any $r<1$ and $\gamma < N-2$ if $N\ge 3$, or $\gamma =0$ if $N=2$.

In particular, these regularity results apply to the solutions $u_i$ to (${\mathcal {S}}$) with

$$\begin{aligned} f(x,u_i)=f_i(u_1,u_2,\ldots ,u_i,\ldots , u_m). \end{aligned}$$

(2.4)

Proof

The proof follows exploiting and adapting some arguments contained in [19, 22] to (2.4)-type nonlinearities. This would imply some technicalities which we rather avoid here. $\square$

For $\rho \in L^{1}(\Omega )$ and $1\le s<\infty$, the weighted space $H^{1,s}_\rho (\Omega )$ (with respect to $\rho$) is defined as the completion of $C^1({{\overline{\Omega }}})$ (or $C^{\infty }({{\overline{\Omega }}})$) with the following norm

$$\begin{aligned} \Vert v\Vert _{H^{1,s}_\rho }= \Vert v\Vert _{L^s (\Omega )}+\Vert \nabla v\Vert _{L^s (\Omega , \rho )}, \end{aligned}$$

(2.5)

where

$$\begin{aligned} \Vert \nabla v\Vert ^s_{L^s (\Omega , \rho )}:=\int _{\Omega }\rho (x)|\nabla v(x)|^s dx. \end{aligned}$$

The space $H^{1,s}_{0,\rho }(\Omega )$ is, consequently, defined as the closure of $C^1_c(\Omega )$ (or $C^{\infty }_c(\Omega )$), with respect to the norm (2.5). We refer to [9] for more details about weighted Sobolev spaces and also to [17, Chapter 1] and the references therein. Theorem 2.1 provides also the right summability of the weight $|\nabla u(x)|^{p-2}$ in order to obtain a weighted Poincaré–Sobolev type inequality that will be useful in the sequel. For the proof we refer to [9, Section 3].

Theorem 2.2

(Weighted Poincaré-Sobolev type inequality). Assume that hypotheses $(hp^*)$ hold and let $(u_1, u_2,\ldots ,u_m)\in C^{1}({\overline{\Omega }})\times C^{1}({\overline{\Omega }})\ldots \times C^{1}({\overline{\Omega }})$ be a solution to (${\mathcal {S}}$). Assume that $p_i\ge 2$ for some $i\in \{1,\ldots ,m\}$ and set $\rho _i=|\nabla u_i|^{p_i-2}$. Then, for every $w\in H^{1,2}_{0}(\Omega ,\rho _i)$, we have

$$\begin{aligned} \Vert w\Vert _{L^2(\Omega )}\leqslant C_{P}\Vert \nabla w\Vert _{L^2(\Omega , \rho _i)}=C_{P}\left( \int _{\Omega }\rho _i\, |\nabla w|^2\right) ^{\frac{1}{2}}, \end{aligned}$$

(2.6)

with $C_{P}=C_{P}(\Omega ) \rightarrow 0$ if ${\mathcal {L}}(\Omega )\rightarrow 0$.

The following theorem collects some comparison and maximum principles for solutions to the system (${\mathcal {S}}$). We have

Theorem 2.3

(See [19, 22]). Let $\Omega$ a bounded smooth domain of ${\mathbb {R}}^N$, $N\ge 2$,

$$\begin{aligned} p_i>\frac{(2N+2)}{(N+2)} \end{aligned}$$

(2.7)

and $q_i\ge \max \{p_i-1,1\}$ for $i=1,\ldots ,m$. Let $(u_1, u_2,\ldots ,u_m), (v_1, v_2,\ldots ,v_m)\in C^{1}({\overline{\Omega }})\times C^{1}({\overline{\Omega }}) \times \cdots \times C^{1}({\overline{\Omega }})$, with $(u_1, u_2,\ldots ,u_m)$ a solution to (${\mathcal {S}}$) and let us assume that assumptions $(hp^*)$ hold.

• (1) Then, for $i=1,2,\ldots ,m$, any connected domain $\Omega '\subseteq \Omega$ and for some constant $\Lambda >0$, such that

  $$\begin{aligned} -\Delta _{p_i} u_i+a_i(u_i)|\nabla u_i|^{q_i}+\Lambda u_i\le -\Delta _{p_i} v_i+a_i(v_i)|\nabla v_i|^{q_i}+\Lambda v_i,\quad u_i\le v_i \quad in\quad \Omega ' \end{aligned}$$

  in the weak distributional meaning, it follows that

  $$\begin{aligned} u_i<v_i\quad in\quad \Omega ', \end{aligned}$$

  unless $u_i\equiv v_i$ in $\Omega '$.

• (2) For any $i=1,2,\ldots ,m$, for any $j=1,2,\ldots ,N$, and for any connected domain $\Omega '\subseteq \Omega$ such that

  $$\begin{aligned} \frac{\partial u_i}{\partial x_j} \ge 0 \quad in\quad \Omega ', \end{aligned}$$

  it follows that

  $$\begin{aligned} \frac{\partial u_i}{\partial x_j} >0 \quad in\quad \Omega ', \qquad \text {unless}\qquad \frac{\partial u_i}{\partial x_j}= 0 \quad \text {in} \quad \Omega '. \end{aligned}$$

Proof

The part (1) of the statement, follows using the regularity results contained in Theorem 2.1 and then exploiting [19, Theorem 1.2].

To prove the part (2) we need to define the linearized equations to the system (${\mathcal {S}}$). In order to do this, since $(u_1, u_2,\ldots ,u_m)\in C^{1}({\overline{\Omega }})\times C^{1}({\overline{\Omega }}) \times \cdots \times C^{1}({\overline{\Omega }})$ is a weak solution of (${\mathcal {S}}$), then we set

$$\begin{aligned}&L_{(u_1,\ldots ,u_m)}\big ((\partial _{x_j} u_1,\ldots ,\partial _{x_j} u_i,\ldots , \partial _{x_j} u_m), (\varphi _1,\ldots , \varphi _m)\big ) \\&\quad =\Big (L^1_{(u_1,\ldots ,u_m)}\big ((\partial _{x_j} u_1,\ldots ,\partial _{x_j} u_i,\ldots , \partial _{x_j} u_m), \varphi _1\big ),\ldots ,\\&L^i_{(u_1,\ldots ,u_m)}\big ((\partial _{x_j} u_1,\ldots ,\partial _{x_j} u_i,\ldots , \partial _{x_j} u_m), \varphi _i\big ), \ldots , \\&L^m_{(u_1,\ldots ,u_m)}\big ((\partial _{x_j} u_1,\ldots ,\partial _{x_j} u_i,\ldots , \partial _{x_j} u_m), \varphi _m\big ) \Big ), \end{aligned}$$

where for $p_i>1$,

$$\begin{aligned}&L^i_{(u_1,\ldots ,u_m)}\big ((\partial _{x_j} u_1,\ldots ,\partial _{x_j} u_i,\ldots , \partial _{x_j} u_m), \varphi _i\big ) \\&\quad = \int _{\Omega }|\nabla u_i|^{p_i-2}(\nabla \partial _{x_j}u_i,\nabla \varphi _i) +(p_i-2)\int _{\Omega }|\nabla u_i|^{p_i-4}(\nabla u_i,\nabla \partial _{x_j}u_i)(\nabla u_i,\nabla \varphi _i)\\&\qquad +\int _{\Omega }a_i'(u_i)|\nabla u_i|^{q_i}\partial _{x_j}u_i\, \varphi _i+q_i\int _{\Omega }a_i(u_i)|\nabla u_i|^{q_i-2}(\nabla u_i,\nabla \partial _{x_j}u_i)\varphi _i \\&\qquad -\int _{\Omega }\sum _{k=1}^m\frac{\partial f_i}{\partial u_k}(u_1,\ldots ,u_i,\ldots ,u_m)\partial _{x_j}u_k\,\varphi _i, \end{aligned}$$

for any $\varphi _1,\ldots , \varphi _m \in C^1_0 (\Omega )$. Moreover, using the regularity results contained in Theorem 2.1 (see [22]), the following equation holds

$$\begin{aligned} L_{(u_1,\ldots ,u_m)}\big ((\partial _{x_j} u_1,\ldots ,\partial _{x_j} u_i,\ldots , \partial _{x_j} u_m), (\varphi _1,\ldots , \varphi _m)\big ) =0, \end{aligned}$$

(2.8)

for all $(\varphi _1,\ldots ,\varphi _i,\ldots ,\varphi _m)$ in $H^{1,2}_{0, \rho _{u_1}}(\Omega )\times \cdots \times H^{1,2}_{0, \rho _{u_i}}(\Omega )\times \cdots \times H^{1,2}_{0, \rho _{u_m}}(\Omega )$ where

$$\begin{aligned} \rho _{u_i}(x):= |\nabla u_i(x)|^{p_i-2},\qquad i=1,\ldots ,m. \end{aligned}$$

Since $f_i$ are locally ${\mathcal {C}}^1$ functions and $\Vert u_i\Vert _{L^{\infty }(\Omega )}\le C$ for any $i\in \{1,\ldots , m\}$, there exists a positive constant $\Theta$ such that

$$\begin{aligned} \frac{\partial f_i}{\partial u_i}+\Theta \ge 0\,\,\text {for all }u_1,u_2,\dots ,u_m>0. \end{aligned}$$

(2.9)

Moreover, in light of (1.1) we have

$$\begin{aligned} \frac{\partial f_i}{\partial u_k}(u_1,\ldots ,u_i,\ldots ,u_m)\ge 0 \end{aligned}$$

(2.10)

for $i\ne k$. Therefore, using (2.9) and (2.10) and taking into account (2.8), it follows, for all $j=1,\ldots ,N$ and for all $i=1,\ldots ,m$, that $\partial _{x_j} u_i$ are nonnegative functions solving the inequalities

$$\begin{aligned}&\int _{\Omega }|\nabla u_i|^{p_i-2}(\nabla \partial _{x_j}u_i,\nabla \varphi _i) +(p_i-2)\int _{\Omega }|\nabla u_i|^{p_i-4}(\nabla u_i,\nabla \partial _{x_j}u_i)(\nabla u_i,\nabla \varphi _i)\\&\quad +\int _{\Omega }a_i'(u_i)|\nabla u_i|^{q_i}\partial _{x_j}u_i\, \varphi _i+q_i\int _{\Omega }a_i(u_i)|\nabla u_i|^{q_i-2}(\nabla u_i,\nabla \partial _{x_j}u_i)\varphi _i \\&\quad +\Theta \int _{\Omega } \partial _{x_j}u_i \, \varphi _i\ge 0 \end{aligned}$$

for all nonnegative test functions $\varphi _i\ge 0$.

Therefore, we can apply [22, Theorem 3.1] to each $\partial _{x_j} u_i$ separately obtaining that, for every $s>1$ sufficiently close to 1 and some positive $\delta$ sufficiently small, there exists a positive constant C such that

$$\begin{aligned} \Vert \partial _{x_j} u_i\Vert _{L^s (B(x, 2\delta ))}\le C_1\inf _{B(x, \delta )}\partial _{x_j} u_i. \end{aligned}$$

(2.11)

Then, the sets $\{x\in \Omega ': \partial _{x_j} u_i=0\}$ are both closed (by continuity) and open (via inequality (2.11)) in the domain $\Omega '$. This yields the assertion. $\square$

Remark 2.4

We point out that Theorem 2.3 holds without any a priori assumption on the critical set of the solution $(u_1, u_2,\ldots ,u_m)$, that is, the set where the gradients $\nabla u_i$ vanish. On the other hand, though, condition (2.7) can be removed when we work in connected domain $\Omega '$ such that $\nabla u_i\ne 0$ for all $x\in \Omega '$ and for all $i\in \{1,\ldots , m\}$. Indeed, the statements (1) and (2) of Theorem 2.3 hold in the whole range $p_i>1$.

Note that the positivity of $f(x,\cdot )$, is actually needed to obtain (2.3). Furthermore, by (2.3) it follows that the critical set $\{x\in \Omega \, : \,\nabla u(x)=0\}$ has zero Lebesgue measure.

An essential tool in the Proof of Theorem 1.1 is Proposition 2.5, i.e., a weak comparison principle in small domains. To prove it, we start giving the following assumptions:

$(*)$:

We suppose that $(u_1, u_2,\ldots ,u_m) \in C^{1}({\overline{\Omega }}_1)\times C^{1}({\overline{\Omega }}_1)\times \cdots \times C^{1}({\overline{\Omega }}_1)$ is a solution to (${\mathcal {S}}$) in the smooth bounded domain $\Omega _1\subset {\mathbb {R}}^N$ and $({{\tilde{u}}}_1, {{\tilde{u}}}_2,\ldots ,{{\tilde{u}}}_m) \in C^{1}({\overline{\Omega }}_2)\times C^{1}({\overline{\Omega }}_2)\times \cdots \times C^{1}({\overline{\Omega }}_2)$ is a solution to (${\mathcal {S}}$) in the smooth bounded domain $\Omega _2 \subset {\mathbb {R}}^N$, with

$$\begin{aligned} \Omega _1\cap \Omega _2\ne \emptyset . \end{aligned}$$

Proposition 2.5

Assume that $(*)$ holds, $p_i>1$, $q_i=\max \{1,{p_i}-1\}$ for every $i\in \{1,2,\ldots , m\}$ and let $\Omega \subset \Omega _1\cap \Omega _2$ be a connected set. Then, there exists a positive number $\delta$, depending upon $m, p_i,q_i,a_i, f_i,\Vert u_i\Vert _{L^\infty (\Omega )},\Vert \nabla u_i\Vert _{L^{\infty }(\Omega )}, \Vert \nabla {\tilde{u}}_i\Vert _{L^{\infty }(\Omega )}$, $i=1,2, \ldots , m$, such that if $\Omega _0\subset \Omega$ with

$$\begin{aligned} {\mathcal {L}}(\Omega _0) \le \delta \quad \text {and} \quad u_i\le {{\tilde{u}}}_i\,\,\text {on }\partial \Omega _0\text { for every }i\in \{1,\ldots ,m\}, \end{aligned}$$

then

$$\begin{aligned} u_i\le {{\tilde{u}}}_i\,\,\text {in }\Omega _0, \end{aligned}$$

for every $i\in \{1,\ldots ,m\}$.

Proof

Let us set

$$\begin{aligned} U_i=(u_i- {\tilde{u}}_i)^+. \end{aligned}$$

We will prove the result by showing that

$$\begin{aligned} (u_i-{\tilde{u}}_i)^+\equiv 0, \end{aligned}$$

for every $i\in \{1,2, \ldots , m\}$. Since $u_i \le \tilde{u_i}$ on $\partial \Omega _0$, then the functions $(u_i-{\tilde{u}}_i)^+$ belong to $W^{1,p_i}_0(\Omega _0)$. Therefore, since $u_i, {{\tilde{u}}}_i$ are both weak solutions to (${\mathcal {S}}$) in $\Omega$, for all $\varphi \in C^\infty _c(\Omega )$ we have

$$\begin{aligned}&\int _\Omega |\nabla u_i|^{p_i-2}(\nabla u_i,\nabla \varphi ) dx+\int _\Omega a_i(u_i)|\nabla u_i|^{q_i}\varphi dx= \int _\Omega f_i(u_1,u_2,\ldots ,u_m) \varphi dx \end{aligned}$$

(2.12)

$$\begin{aligned}&\text {and}\nonumber \\&\int _\Omega |\nabla {\tilde{u}}_i|^{p_i-2}(\nabla {\tilde{u}}_i,\nabla \varphi ) dx+\int _\Omega a_i({\tilde{u}}_i)|\nabla {\tilde{u}}_i|^{q_i}\varphi dx= \int _\Omega f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots ,{\tilde{u}}_m) \varphi dx, \end{aligned}$$

(2.13)

for $i=1,2, \ldots , m$. By a density argument, we can put, respectively, $\varphi =(u_i-{\tilde{u}}_i)^+$ in Eqs. (2.12) and (2.13). Subtracting, we get for any i

$$\begin{aligned}&\int _{\Omega _0}\big ( |\nabla u_i|^{p_i-2}\nabla u_i -|\nabla {\tilde{u}}_i|^{p_i-2}\nabla {\tilde{u}}_i, \nabla (u_i-{\tilde{u}}_i)^+ \big ) \, dx\nonumber \\&\qquad +\int _{\Omega _0}\big (a_i(u_i)|\nabla u_i|^{q_i}-a_i({\tilde{u}}_i)|\nabla {\tilde{u}}_i|^{q_i} \big ) (u_i-{\tilde{u}}_i)^+ \, dx \nonumber \\&\quad =\int _{\Omega _0}[f_i(u_1,u_2,\ldots ,u_m)-f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots ,{\tilde{u}}_m)](u_i-{\tilde{u}}_i)^+ \, dx. \end{aligned}$$

(2.14)

The second term on the left-hand side of (2.14) can be estimated as follows

$$\begin{aligned} \begin{aligned}&\left|\int _{\Omega _0} \big (a_i(u_i)|\nabla u_i|^{q_i}-a_i({\tilde{u}}_i)|\nabla {\tilde{u}}_i|^{q_i}\big )(u_i-{\tilde{u}}_i)^+ \, dx \right|\\&\quad =\left|\int _{\Omega _0}\big (a_i(u_i)|\nabla u_i|^{q_i}-a_i(u_i)|\nabla {\tilde{u}}_i|^{q_i}+a_i(u_i)|\nabla {\tilde{u}}_i|^{q_i}-a_i({\tilde{u}}_i)|\nabla {\tilde{u}}_i|^{q_i}\big )(u_i-{\tilde{u}}_i)^+ \, dx\right|\\&\quad \le \int _{\Omega _0}|a_i(u_i)|\big ||\nabla u_i|^{q_i}-|\nabla {\tilde{u}}_i|^{q_i}\big | (u_i-{\tilde{u}}_i)^+ \, dx\\&\qquad +\int _{\Omega _0}|\nabla {\tilde{u}}_i|^{q_i}(a_i(u_i)-a_i({\tilde{u}}_i)) (u_i-{\tilde{u}}_i)^+ \, dx. \end{aligned} \end{aligned}$$

Since $a_i$ is a locally Lipschitz continuous function (see $({hp^*})$), it follows that there exists a positive constant $K_{a_i}=K_{a_i}(\Vert u_i\Vert _{L^{\infty }(\Omega )})$ such that for every $u_i \in [0,\Vert u_i\Vert _{L^{\infty }(\Omega )}]$

$$\begin{aligned} \vert a_i(u_i) \vert \le K_{a_i}. \end{aligned}$$

Moreover, denoting by $L_{a_i}=L_{a_i}(\Vert u_i\Vert _{L^{\infty }(\Omega )})$ the Lipschitz constant of $a_i$, we obtain

$$\begin{aligned} \begin{aligned}&\left|\int _{\Omega _0}\big (a_i(u_i)|\nabla u_i|^{q_i}-a_i({\tilde{u}}_i)|\nabla {\tilde{u}}_i|^{q_i}\big ) (u_i-{\tilde{u}}_i)^+ \, dx \right|\\&\quad \le K_{a_i}\int _{\Omega _0}\big | |\nabla u_i|^{q_i}-|\nabla {\tilde{u}}_i|^{q_i}\big | (u_i-{\tilde{u}}_i)^+ \, dx \\&\qquad +C(q_i, L_{a_i}, \Vert \nabla {{\tilde{u}}}_i\Vert _{L^{\infty }(\Omega )})\int _{\Omega _0}[(u_i-{\tilde{u}}_i)^+]^2 \,dx. \end{aligned} \end{aligned}$$

(2.15)

By the mean value’s theorem and taking into account that $q_i\ge 1$, it follows that

$$\begin{aligned} \begin{aligned}&K_{a_i}\int _{\Omega _0}\big | |\nabla u_i|^{q_i}-|\nabla {\tilde{u}}_i|^{q_i}\big | (u_i-{\tilde{u}}_i)^+ \, dx \\&\le C(q_i,K_{a_i})\int _{\Omega _0}(|\nabla u_i|+|\nabla {\tilde{u}}_i|)^{q_{i}-1}|\nabla (u_i-{\tilde{u}}_i)^+| (u_i-{\tilde{u}}_i)^+ \, dx. \end{aligned} \end{aligned}$$

The last term (recall that $q_i\ge \max \{1,p_i-1\}$) can be written as follows,

$$\begin{aligned}&C\int _{\Omega _0}(|\nabla u_i|+|\nabla {\tilde{u}}_i|)^{q_{i}-1}|\nabla (u_i-{\tilde{u}}_i)^+| (u_i-{\tilde{u}}_i)^+\, dx\nonumber \\&\quad = C\int _{\Omega _0}\frac{(|\nabla u_i|+|\nabla {\tilde{u}}_i|)^{q_i-1}}{(|\nabla u_i|+|\nabla {\tilde{u}}_i|)^\frac{p_i-2}{2}}(|\nabla u_i|+|\nabla {\tilde{u}}_i|)^\frac{p_i-2}{2}|\nabla (u_i-{\tilde{u}}_i)^+| (u_i-{\tilde{u}}_i)^+\,dx\nonumber \\&\quad \le C\int _{\Omega _0}(|\nabla u_i|+|\nabla {\tilde{u}}_i|)^\frac{p_i-2}{2}|\nabla (u_i-{\tilde{u}}_i)^+|(u_i-{\tilde{u}}_i)^+ \,dx, \end{aligned}$$

(2.16)

with $C=C(p_i,q_i,K_{a_i},\Vert \nabla u_i\Vert _{L^{\infty }(\Omega )}, \Vert \nabla {\tilde{u}}_i\Vert _{L^{\infty }(\Omega )})$ is a positive constant. Exploiting Young’s inequality in the right-hand side of (2.16) we finally obtain

$$\begin{aligned}&C\int _{\Omega _0}(|\nabla u_i|+|\nabla {\tilde{u}}_i|)^{q_{i}-1}|\nabla (u_i-{\tilde{u}}_i)^+||(u_i-{\tilde{u}}_i)^+|\,dx\nonumber \\&\quad \leqslant \varepsilon C\int _{\Omega _0}(|\nabla u_i|+|\nabla {\tilde{u}}_i|)^{p_i-2}|\nabla (u_i-{\tilde{u}}_i)^+|^2\,dx \nonumber \\&\qquad +\frac{C}{\varepsilon } \int _{\Omega _0}[(u_i-{\tilde{u}}_i)^+]^2\,dx. \end{aligned}$$

Therefore, collecting the previous estimates, from (2.15), we obtain

$$\begin{aligned} \begin{aligned}&\left|\int _{\Omega _0} \big (a_i(u_i)|\nabla u_i|^{q_i}-a_i({\tilde{u}}_i)|\nabla {\tilde{u}}_i|^{q_i}\big )(u_i-{\tilde{u}}_i)^+ \, dx \right|\\&\quad \leqslant \varepsilon C\int _{\Omega _0}(|\nabla u_i|+|\nabla {\tilde{u}}_i|)^{p_i-2}|\nabla (u_i-{\tilde{u}}_i)^+|^2\,dx \\&\qquad +\frac{C}{\varepsilon } \int _{\Omega _0}[(u_i-{\tilde{u}}_i)^+]^2\,dx. \end{aligned} \end{aligned}$$

Finally, using (2.1) and fixing $\varepsilon$ sufficiently small, from (2.14) we get

$$\begin{aligned} \begin{aligned}&\int _{\Omega _0}(|\nabla u_i|+|\nabla {\tilde{u}}_i|)^{p_i-2}|\nabla (u_i-{\tilde{u}}_i)^+|^2\,dx\\&\quad \le \int _{\Omega _0}\big ( |\nabla u_i|^{p_i-2}\nabla u_i -|\nabla {\tilde{u}}_i|^{p_i-2}\nabla {\tilde{u}}_i, \nabla (u_i-{\tilde{u}}_i)^+ \big ) \, dx\\&\quad \le C\int _{\Omega _0}[f_i(u_1,u_2,\ldots ,u_m)-f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots ,{\tilde{u}}_m)](u_i-{\tilde{u}}_i)^+ \, dx \\&\qquad +C \int _{\Omega _0}[(u_i-{\tilde{u}}_i)^+]^2\,dx, \end{aligned} \end{aligned}$$

(2.17)

where $C=C(p_i,q_i,K_{a_i}, L_{a_i},\Vert \nabla u_i\Vert _{L^{\infty }(\Omega )}, \Vert \nabla {\tilde{u}}_i\Vert _{L^{\infty }(\Omega )})$ is a positive constant.

The first term on the right-hand side of (2.17) can be arranged as follows

$$\begin{aligned} \begin{aligned}&\int _{\Omega _0}[f_i(u_1,u_2,\ldots ,u_m)-f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots ,{\tilde{u}}_m)] (u_i-{\tilde{u}}_i)^+ \, dx\\&\quad =\int _{\Omega _0}[f_i(u_1,u_2,\ldots ,u_m)-f_i({\tilde{u}}_1,u_2,\ldots ,u_m)+f_i({\tilde{u}}_1,u_2,\ldots ,u_m)\\&\qquad -f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots ,{\tilde{u}}_m)] (u_i-{\tilde{u}}_i)^+ \,dx\\&\quad =\int _{\Omega _0}[f_i(u_1, u_2,\ldots ,u_m)-f_i({\tilde{u}}_1,u_2,\ldots ,u_m) \\&\qquad +f_i({\tilde{u}}_1,u_2,\ldots ,u_m)-f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots ,u_m)\\&\qquad +\ldots + f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots , u_i, \ldots ,u_m)-f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots , {{\tilde{u}}}_i, \ldots ,u_m) \\&\qquad +f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots , {{\tilde{u}}}_i, \ldots ,u_m) \\&\qquad \vdots \\&\qquad \ldots +f_i({\tilde{u}}_1,{\tilde{u}}_2, \ldots ,{u}_m)-f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots ,{\tilde{u}}_m)] (u_i-{\tilde{u}}_i)^+ \,dx. \end{aligned} \end{aligned}$$

(2.18)

Using the fact that $f_i$ are ${\mathcal {C}}^1_{loc}$ functions satisfying (1.1), see $({hp^*})$, by (2.18) we have

$$\begin{aligned} \begin{aligned}&\int _{\Omega _0}[f_i(u_1,u_2,\ldots ,u_m)-f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots ,{\tilde{u}}_m)] (u_i-{\tilde{u}}_i)^+ \, dx\\&\quad \le \int _{\Omega _0}\frac{f_i(u_1,u_2,\ldots ,u_m)-f_i({\tilde{u}}_1,u_2,\ldots ,u_m)}{(u_1-{\tilde{u}}_1)^+}(u_1-{\tilde{u}}_1)^+(u_i-{\tilde{u}}_i)^+\,dx\\&\qquad +\int _{\Omega _0}\frac{f_i({{\tilde{u}}}_1,u_2,\ldots ,u_m)-f_i({\tilde{u}}_1,{{\tilde{u}}}_2,\ldots ,u_m)}{(u_2-{\tilde{u}}_2)^+}(u_2-{\tilde{u}}_2)^+(u_i-{\tilde{u}}_i)^+\, dx\\&\qquad \vdots \\&\qquad + \int _{\Omega _0}\frac{f_i({{\tilde{u}}}_1,{{\tilde{u}}}_2,\ldots , u_i, \ldots , u_m)-f_i({\tilde{u}}_1,{{\tilde{u}}}_2,\ldots ,{{\tilde{u}}}_i, \ldots , u_m)}{(u_i-{\tilde{u}}_i)}[(u_i-{\tilde{u}}_i)^+]^2\, dx\\&\qquad \vdots \\&\qquad +\int _{\Omega _0}\frac{f_i({\tilde{u}}_1,{\tilde{u}}_2, {\tilde{u}}_3\ldots ,{u}_m)-f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots ,{\tilde{u}}_m)}{(u_m-{{\tilde{u}}}_m)^+}](u_m-{{\tilde{u}}}_m)^+ (u_i-{\tilde{u}}_i)^+ \,dx\\&\quad \le L_{f_i} \sum _{j=1}^{m}\int _{\Omega _0} (u_j-{\tilde{u}}_j)^+(u_i-{\tilde{u}}_i)^+ \, dx, \end{aligned} \end{aligned}$$

(2.19)

where $L_{f_i}$ is the Lipschitz constant of $f_i$ that depends on the $\displaystyle \max _{1\le j\le m }\{\Vert u_j\Vert _{L^\infty (\Omega )}\}$. Exploiting Young’s inequality on the right-hand side of (2.19), we get

$$\begin{aligned}&\int _{\Omega _0}[f_i(u_1,u_2,\ldots ,u_m)-f_i({\tilde{u}}_1,{\tilde{u}}_2,\ldots ,{\tilde{u}}_m)] (u_i-{\tilde{u}}_i)^+ \, dx\nonumber \\&\quad \le C \sum _{j=1}^m\int _{\Omega _0} [(u_j-{\tilde{u}}_j)^+]^2 \, dx, \end{aligned}$$

(2.20)

where $\displaystyle C=C\big (m, L_{f_i}\big )$ is a positive constant. Finally, from (2.17) and (2.20) we infer for $i=1,\ldots , m$

$$\begin{aligned} \int _{\Omega _0} (|\nabla u_i|+|\nabla {\tilde{u}}_i|)^{p_i-2}|\nabla (u_i-{\tilde{u}}_i)^+|^2 dx\le C_i \sum _{j=1}^m \int _{\Omega _0} [(u_j-{\tilde{u}}_j)^+]^2 \, dx, \end{aligned}$$

(2.21)

where $\displaystyle C_i=C_i(m,p_i,q_i,K_{a_i},L_{a_i}, L_{f_i},\Vert \nabla u_i\Vert _{L^{\infty }(\Omega )}, \Vert \nabla {\tilde{u}}_i\Vert _{L^{\infty }(\Omega )})$ is a positive constant.

In the case $p_j\ge 2$, a weighted Poincaré inequality holds true on the right-hand side of (2.21), see Theorem 2.2. Indeed, Eq. (2.6) yields

$$\begin{aligned} \int _{\Omega _0} [(u_j-{\tilde{u}}_j)^+]^2dx\le C_{P, j}(\Omega _0)\int _{\Omega _0} (|\nabla u_j|+|\nabla {{\tilde{u}}}_j|)^{p_j-2}|\nabla (u_j-{\tilde{u}}_j)^+|^2dx,\quad \text {if } p_j \ge 2, \end{aligned}$$

(2.22)

where the Poincaré constant $C_{P, j}(\Omega _0)\rightarrow 0$, when the Lebesgue measure ${\mathcal {L}}(\Omega _0)\rightarrow 0$. Actually, we used the fact that, since $p_j\ge 2$,

$$\begin{aligned}|\nabla u_j|^{p_j-2}\le (|\nabla u_j|+|\nabla {{\tilde{u}}}_j|)^{p_j-2}.\end{aligned}$$

In the case $p_j <2$, we use the standard Poincaré inequality on the right-hand side of (2.21), namely

$$\begin{aligned} \int _{\Omega _0}[(u_j-{\tilde{u}}_j)^+]^2 \, dx\le C_{P, j}(\Omega _0)\int _{\Omega _0} |\nabla (u_j-{\tilde{u}}_j)^+|^2 \, dx,\quad \text {if } p_j < 2, \end{aligned}$$

and $C_{P, j}(\Omega _0)\rightarrow 0$ if ${\mathcal {L}}(\Omega _0)\rightarrow 0$. Moreover, in the case $p_j< 2$ since $u_j, {{\tilde{u}}}_j \in C^1({{\overline{\Omega }}})$, we deduce also

$$\begin{aligned}&\int _{\Omega _0} |\nabla (u_j-{\tilde{u}}_j)^+|^2 dx \nonumber \\&\quad \le C(p_j, \Vert \nabla u_j\Vert _{L^{\infty }(\Omega )}, \Vert \nabla {\tilde{u}}_j\Vert _{L^{\infty }(\Omega )})\int _{\Omega _0} (|\nabla u_j|+|\nabla {\tilde{u}}_j|)^{p_j-2}|\nabla (u_j-{\tilde{u}}_j)^+|^2 dx. \end{aligned}$$

(2.23)

Using (2.23), up to redefine the Poincaré constant in this case, we obtain

$$\begin{aligned}\int _{\Omega _0}[(u_j-{\tilde{u}}_j)^+]^2 \, dx\le C_{P, j}(\Omega _0)\int _{\Omega _0} (|\nabla u_j|+|\nabla {\tilde{u}}_j|)^{p_j-2} |\nabla (u_j-{\tilde{u}}_j)^+|^2 \, dx,\quad \text {if } p_j \nonumber < 2, \end{aligned}$$

(2.24)

and $C_{P, j}(\Omega _0)\rightarrow 0$ if ${\mathcal {L}}(\Omega _0)\rightarrow 0$. Let us set now

$$\begin{aligned} C_{P}(\Omega _0)=\max _{1\le j\le m}\{C_{P, j}(\Omega _0)\}. \end{aligned}$$

(2.25)

Furthermore, by combining (2.21) with (2.22), (2.24) and (2.25), we obtain for $i=1,\ldots ,m$

$$\begin{aligned}&\int _{\Omega _0} (|\nabla u_i|+|\nabla {\tilde{u}}_i|)^{p_i-2}|\nabla (u_i-{\tilde{u}}_i)^+|^2 dx\nonumber \\&\quad \le C_iC_{P}(\Omega _0)\sum _{j=1}^m\int _{\Omega _0} (|\nabla u_j|+|\nabla {\tilde{u}}_j|)^{p_j-2}|\nabla (u_j-{\tilde{u}}_j)^+|^2 dx. \end{aligned}$$

(2.26)

Let us define $\displaystyle {{\hat{C}}}= m\cdot \max _{1\le i\le m} \{C_i\}$. By adding Eq. (2.26) and setting

$$\begin{aligned} I(\Omega _0)=\sum _{i=1}^m\int _{\Omega _0} (|\nabla u_i|+|\nabla {\tilde{u}}_i|)^{p_i-2}|\nabla (u_i-{\tilde{u}}_i)^+|^2 dx, \end{aligned}$$

we obtain

$$\begin{aligned} I(\Omega _0)\le {{\hat{C}}}C_{P}(\Omega _0) I(\Omega _0). \end{aligned}$$

(2.27)

Now, we choose $\delta >0$ sufficiently small such that the condition ${\mathcal {L}}(\Omega _0)\le \delta$ implies

$$\begin{aligned} {{\hat{C}}}C_{P}(\Omega _0)<1. \end{aligned}$$

Therefore, from (2.27) we get the desired contradiction, namely

$$\begin{aligned} U_i=(u_i-{\tilde{u}}_i)^+ \equiv 0, \end{aligned}$$

for all $i=1,\ldots ,m.$ $\square$

3 Symmetry results for solutions to (${\mathcal {S}}$): Proof of Theorem 1.1

In this section, we prove our main result. As we said in the introduction, without loss of generality and for the sake of simplicity, since the problem is invariant with respect to translations, reflections and rotations, we suppose that $\Omega$ is a bounded smooth domain which is convex in the $x_1$-direction and symmetric with respect to $\{x_1=0\}$. Let us now recall the main ingredients of the moving plane method. We set

$$\begin{aligned} T_\lambda :=\{x \in {\mathbb {R}}^N: x_1= \lambda \}. \end{aligned}$$

Given $x\in {\mathbb {R}}^N$ and $\lambda <0$, we define

$$\begin{aligned} x_\lambda =R_\lambda (x):= (2\lambda -x_1, x_2,\ldots ,x_N) \end{aligned}$$

and the reflected functions

$$\begin{aligned} u_{i,{\lambda }}(x):=u_i(x_\lambda ),\qquad i=1,2,\ldots , m. \end{aligned}$$

We also set

$$\begin{aligned}&\Omega _\lambda :=\{x\in \Omega \,:\, x_1<\lambda \},\nonumber \\&a:=\inf _{x\in \Omega } x_1, \end{aligned}$$

(3.1)

$$\begin{aligned}&\Lambda :=\Big \{a<\lambda <0 \,\, : u_i\le u_{i,t}\,\, \text {in}\,\, \Omega _t, \,\,\text {for all}\,\, t\in (a,\lambda ] \,\, \text {and}\,\, \text {for all}\,\,i=1,2,\ldots, m \Big\} \end{aligned}$$

(3.2)

and (if $\Lambda \ne \emptyset$)

$$\begin{aligned} {{\bar{\lambda }}}=\sup \Lambda . \end{aligned}$$

Finally, for $i=1,\ldots ,m$, we define the critical sets

$$\begin{aligned} Z_{u_i} := \{x\in \Omega : \nabla u_i(x)=0\}. \end{aligned}$$

Proof of Theorem 1.1

For $a<\lambda <0$ [see (3.1)] and $\lambda$ sufficiently close to a, we assume that ${\mathcal {L}}(\Omega _\lambda )$ is as small as we need. In particular, we may assume that Proposition 2.5 works with $\Omega _1=\Omega,\, \Omega _2= R_\lambda (\Omega ),\, \Omega _0=\Omega _\lambda$ and ${{\tilde{u}}}_i=u_{i,\lambda }$. Therefore, we set

$$\begin{aligned} W_{i,\lambda } :=u_i-u_{i,\lambda },\quad i=1,2,\ldots , m \end{aligned}$$

and we observe that, by construction, we have

$$\begin{aligned} W_{i,\lambda }\le 0\,\,\text {on }\partial \Omega _\lambda ,\quad i=1,2,\ldots , m. \end{aligned}$$

By Proposition 2.5, it follows that

$$\begin{aligned} W_{i,\lambda } \le 0\,\,\text {in }\Omega _\lambda , \quad i=1,2,\ldots , m. \end{aligned}$$

Hence, the set $\Lambda$ [see (3.2)] is not empty and ${{\bar{\lambda }}} \in (a,0]$. Note that, by continuity, it follows $u_i\le u_{i,{{\bar{\lambda }}}}$. We have to show that, actually ${{\bar{\lambda }}} =0$. Hence, we assume by contradiction that ${{\bar{\lambda }}} < 0$ and we argue as follows.

First of all, we point out that ${\mathcal {L}}(Z_{u_i})=0$ for all i. Indeed, if we apply Theorem 2.1, for $u_i$ with $f(x,u_i)=f_i(u_1,u_2,\ldots ,u_i,\ldots , u_m)$, from (2.3) the conclusion follows. Hence, let A be an open set such that for $i=1,\ldots ,m$

$$\begin{aligned} Z_{u_i}\cap \Omega _{{{\bar{\lambda }}}} \subset A \subset \Omega _{{{\bar{\lambda }}}}, \end{aligned}$$

with the Lebesgue measure ${\mathcal {L}}(A)$ small as we like. Notice now that, since $f_i$ are locally ${\mathcal {C}}^1$ functions and $\Vert u_i\Vert _{L^{\infty }(\Omega )}\le C$ for any $i\in \{1,\ldots , m\}$, there exists a positive constant $\Theta$ such that

$$\begin{aligned} \frac{\partial f_i}{\partial u_i}+\Theta \ge 0\,\,\text {for all }u_1,u_2,\dots ,u_m>0. \end{aligned}$$

(3.3)

Furthermore, using (1.1) we obtain

$$\begin{aligned}&-\Delta _{p_i} u_i +a_i(u_i)|\nabla u_i|^{q_i}+ \Theta u_i =f_i (u_1,u_2,\ldots ,u_m)+ \Theta u_i \nonumber \\&\quad \le f_i (u_{1,\lambda },u_{2,\lambda },\ldots ,u_{m,\lambda })+\Theta u_{i,\lambda } = -\Delta _{p_i} u_{i,\lambda } +a_i(u_{i,\lambda })|\nabla u_{i,\lambda }|^{q_i}+ \Theta u_{i,\lambda } \end{aligned}$$

(3.4)

for any $a< \lambda \le {{\bar{\lambda }}}$. In light of (3.4) we have

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _{p_i} u_i +a_i(u_i)|\nabla u_i|^{q_i}+ \Theta u_i \le -\Delta _{p_i} u_{i,\lambda } +a_i(u_{i,\lambda })|\nabla u_{i,\lambda }|^{q_i}+ \Theta u_{i,\lambda } &{} \text {in }\Omega _\lambda ,\\ u_i \le u_{i,\lambda } &{} \text {in }\Omega _\lambda . \end{array}\right. } \end{aligned}$$

(3.5)

Then, by (3.5) and the strong comparison principle, see statement (1) of Theorem 2.3, for any $i=1,2,\ldots ,m$ such that $p_i\ge 2$, we have

$$\begin{aligned} u_i < u_{i,{{\bar{\lambda }}}} \qquad \text {or}\qquad u_i \equiv u_{i,{{\bar{\lambda }}}} , \end{aligned}$$

in $\Omega _{{{\bar{\lambda }}}}$.

In the case $1<p_i<2$, we prove first the following

Claim: The case $u_i\equiv u_{i,{{\bar{\lambda }}}}$ in some connected component ${\mathcal {C}}$ of $\Omega _{{{\bar{\lambda }}}}{\setminus } Z_{u_i}$, such that $\overline{{\mathcal {C}}}\subset \Omega$, is not possible.

We proceed by contradiction. Let us assume that such component exists, namely

$$\begin{aligned} {\mathcal {C}}\subset \Omega \quad \text {such that} \quad \partial {\mathcal {C}}\subset Z_{u_i}. \end{aligned}$$

For all $\varepsilon >0$, let us define $G_\varepsilon :{\mathbb {R}}^+_0\rightarrow {\mathbb {R}}$ by setting

$$\begin{aligned} G_\varepsilon (t)= {\left\{ \begin{array}{ll} 0 &{} \text {if }0\le t\le \varepsilon \\ 2t-2\varepsilon &{} \text {if }\varepsilon \le t\le 2\varepsilon \\ t &{} \text {if }t\ge 2\varepsilon . \end{array}\right. } \end{aligned}$$

(3.6)

Let $\chi _{{\mathcal {A}}}$ be the characteristic function of a set ${\mathcal {A}}$. We define

$$\begin{aligned} \Psi _{\varepsilon }\,:=\,e^{-s_i(u_i)}\frac{G_\varepsilon (|\nabla u_i|)}{|\nabla u_i|}\chi _{({\mathcal {C}}\cup {\mathcal {C}}^\lambda )}, \end{aligned}$$

(3.7)

where ${\mathcal {C}}^\lambda$ is the reflected set of ${\mathcal {C}}$ with respect to the hyperplane $T_{{{\bar{\lambda }}}}$ and

$$\begin{aligned} s_i(t)={{\hat{C}}}_i\cdot \int _0^t \, a_i^+(t')dt', \end{aligned}$$

(3.8)

where $a_i^+:=\max \{0, a_i\}$ ($a_i^-:=-\min \{0, a_i\}$) and ${{\hat{C}}}_i$ denotes some positive constant to be chosen later.

We point out that supp$\Psi _\varepsilon \subset {\mathcal {C}}\cup {\mathcal {C}}^\lambda$, which implies $\Psi _\varepsilon \in W^{1,p}_0( {\mathcal {C}}\cup {\mathcal {C}}^\lambda )$. Indeed by definition of ${\mathcal {C}}$ we have that $\nabla u_i=0$ on $\partial ({\mathcal {C}}\cup {\mathcal {C}}^\lambda )$. Moreover, using the test function $\Psi _{\varepsilon }$ defined in (3.7), we are able to integrate on the boundary $\partial ({\mathcal {C}}\cup {\mathcal {C}}^\lambda )$ which could be not regular.

Hence, we obtain

$$\begin{aligned}&\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }\, |\nabla u_i|^{p_i-2} (\nabla u_i ,\nabla \Psi _{\varepsilon }) dx+ \int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }a_i^+(u_i)|\nabla u_i|^{q_i}\Psi _{\varepsilon } dx\nonumber \\&\quad =\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }a_i^-(u_i)|\nabla u_i|^{q_i}\Psi _{\varepsilon } dx +\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }f_i(u_1,u_2,\ldots ,u_m)\Psi _{\varepsilon } dx. \end{aligned}$$

(3.9)

It is easy to see that for every $x\in [0, M]$ and for every $l,q\ge 1$ and $\sigma >0$, there exists a positive constant $C=C(l,q,\sigma , M)$ such that

$$\begin{aligned} x^{q}\le C \cdot x^l+\sigma ,\quad x\in [0,M]. \end{aligned}$$

(3.10)

Therefore, (3.9) and (3.10) imply:

$$\begin{aligned}&\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }\, |\nabla u_i|^{p_i-2} (\nabla u_i ,\nabla \Psi _\varepsilon ) dx\nonumber \\&\qquad+C_i(\sigma _i, p_i,q_i,\Vert \nabla u_i\Vert _{L^{\infty }(\Omega )})\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }a_i^+(u_i)|\nabla u_i|^{p_i}\Psi _\varepsilon dx\nonumber \\&\qquad +\sigma _i\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }a_i^+(u_i)\Psi _\varepsilon dx \nonumber \\&\quad \ge \int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }a_i^-(u_i)|\nabla u_i|^{q_i}\Psi _\varepsilon dx+\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }f_i(u_1,u_2,\ldots ,u_m)\Psi _\varepsilon dx\nonumber \\&\quad \ge \int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }f_i(u_1,u_2,\ldots ,u_m)\Psi _\varepsilon dx. \end{aligned}$$

(3.11)

By $({hp^*})-(ii)$, since $\overline{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }\subset \Omega$ we have that there exists $\gamma _i>0$ such that

$$\begin{aligned}f_i(u_1,u_2,\ldots ,u_m)\ge \gamma _i.\end{aligned}$$

Hence, we can choose $\sigma _i$ in (3.10), say ${{\bar{\sigma }}}_i$, small enough such that

$$\begin{aligned} \gamma _i-{{\bar{\sigma }}}_i \,\Vert a_i^+(u_i)\Vert _{\infty }={{\tilde{C}}}_i>0\,, \end{aligned}$$

so that

$$\begin{aligned}&\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }\, |\nabla u_i|^{p_i-2} (\nabla u_i ,\nabla \Psi _\varepsilon ) dx\nonumber \\&\quad +C_i({{\bar{\sigma }}}_i, p_i,q_i,\Vert \nabla u_i\Vert _{L^{\infty }(\Omega )})\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }a_i^+(u_i)|\nabla u_i|^{p_i}\Psi _\varepsilon dx\nonumber \\&\quad \ge {{\tilde{C}}}_i\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }\Psi _\varepsilon dx. \end{aligned}$$

(3.12)

Choosing ${{\hat{C}}}_i$ in (3.8) equal to $C_i({{\bar{\sigma }}}_i, p_i,q_i,\Vert \nabla u_i\Vert _{L^{\infty }(\Omega )})$ in (3.12) we obtain

$$\begin{aligned}&\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }e^{-s_i(u_i)}|\nabla u_i|^{p_i-2} \left( \nabla u_i, \nabla \frac{G_\varepsilon (|\nabla u_i|)}{|\nabla u_i|}\right) dx\nonumber \\&\quad \ge {{\tilde{C}}}_i\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }e^{-s_i(u_i)}\displaystyle \frac{G_\varepsilon (|\nabla u_i|)}{|\nabla u_i|} dx. \end{aligned}$$

(3.13)

We set ${\displaystyle h_\varepsilon (t)=\frac{G_\varepsilon (t)}{t}}$, meaning that $h_\varepsilon (t)=0$ for $0\le t\le \varepsilon$. We have:

$$\begin{aligned}&\left| \int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }e^{-s_i(u_i)}|\nabla u_i|^{p_i-2}\left( \nabla u_i, \nabla \frac{G_\varepsilon (|\nabla u_i|)}{|\nabla u_i|}\right) dx\right| \nonumber \\&\quad \le \int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }|\nabla u_i|^{p_i-1}|h_\varepsilon '(|\nabla u_i|)||\nabla (|\nabla u_i|)|dx\nonumber \\&\quad \le C_i\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }|\nabla u_i|^{p_i-2}\Big (|\nabla u_i|h_\varepsilon '(|\nabla u_i|)\Big )\Vert D^2 u_i\Vert dx, \end{aligned}$$

(3.14)

where $\Vert D^2 u_i\Vert$ denotes the Hessian norm and $C_i$ a positive constant.

We let $\varepsilon \rightarrow 0$. To this aim, let us first show that

(i):

$|\nabla u_i|^{p_i-2}\Vert D^2 u_i\Vert \in L^1({\mathcal {C}}\cup {\mathcal {C}}^\lambda );$

(ii):

$|\nabla u_i|h_\varepsilon '(|\nabla u_i|)\rightarrow 0$ a.e. in ${\mathcal {C}}\cup {\mathcal {C}}^\lambda$ as $\varepsilon \rightarrow 0$ and $|\nabla u_i|h_\varepsilon '(|\nabla u_i|)\le C$ with C not depending on $\varepsilon$.

Let us prove (i). By Hölder’s inequality it follows

$$\begin{aligned}&\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }|\nabla u_i|^{p_i-2}\Vert D^2u_i\Vert dx\le \sqrt{{\mathcal {L}}({\mathcal {C}}\cup {\mathcal {C}}^\lambda )}\left( \int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }|\nabla u_i|^{2(p_i-2)}\Vert D^2u_i\Vert ^2 dx \right) ^{\frac{1}{2}}\nonumber \\&\quad \le C_i\left( \int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }|\nabla u_i|^{p_i-2-\beta _i}\Vert D^2u_i\Vert ^2|\nabla u_i|^{p_i-2+\beta _i}dx \right) ^{\frac{1}{2}}\nonumber \\&\quad \le C_i \Vert \nabla u_i\Vert ^{(p_i-2+\beta _i)/2}_{L^{\infty }(\Omega )} \left( \int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }|\nabla u_i|^{p_i-2-\beta _i}\Vert D^2 u_i\Vert ^2 dx \right) ^{\frac{1}{2}}, \end{aligned}$$

(3.15)

with $0\le \beta _i<1$ and $C_i$ a positive constant.

Using (2.2) of Theorem 2.1, we infer that

$$\begin{aligned} \left( \int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }|\nabla u_i|^{p_i-2-\beta _i}\Vert D^2 u_i\Vert ^2 dx \right) ^{\frac{1}{2}}\le C. \end{aligned}$$

Then, by (3.15) we obtain

$$\begin{aligned} \int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }|\nabla u_i|^{p_i-2}\Vert D^2u_i\Vert dx\le C. \end{aligned}$$

Let us prove (ii). Recalling (3.6), we obtain

$$\begin{aligned} h'_\varepsilon (t)= {\left\{ \begin{array}{ll} 0 &{} \text {if }0< t \le \varepsilon \\ \frac{2\varepsilon }{t^2}&{} \text {if }\varepsilon< t<2\varepsilon \\ 0 &{} \text {if }t\ge 2\varepsilon , \end{array}\right. } \end{aligned}$$

and, then, $|\nabla u_i|h_\varepsilon '(|\nabla u_i|)$ tends to 0 almost everywhere in ${\mathcal {C}}\cup {\mathcal {C}}^\lambda$ as $\varepsilon$ goes to 0 and $|\nabla u_i|h_\varepsilon '(|\nabla u_i|)\le 2$.

Finally, by the Lebesgue’s dominate convergence theorem, passing to the limit for $\varepsilon \rightarrow 0$ in (3.13) we obtain

$$\begin{aligned} 0 \ge {{\tilde{C}}}_i\int _{{\mathcal {C}}\cup {\mathcal {C}}^\lambda }e^{-s_i(u_i)}dx >0. \end{aligned}$$

This gives a contradiction, hence the Claim holds.

Then, using also Hopf’s boundary lemma (see [25, Theorem 5.5.1]) for

$$\begin{aligned}-\Delta _{p_i} u_i+a_i(u_i)|\nabla u_i|^{q_i}= f_i(u_1,u_2,\ldots ,u_i,\ldots , u_m)\ge 0, \end{aligned}$$

$u_i> 0$ in $\Omega$ and $u_i= 0$ on $\partial \Omega$, we deduce that the set $\Omega _{{{\bar{\lambda }}}}{\setminus } Z_{u_i}$ is connected. Indeed, thanks to Hopf’s lemma, $Z_{u_i}$ lies far from the boundary $\partial \Omega$. Moreover, we also remark that since $\Omega$ is convex in the $x_1$-direction, we have that the boundary $\partial \Omega$ is connected. Consequently, for any $i=1,2,\ldots ,m$ we get

$$\begin{aligned} u_i < u_{i,{{\bar{\lambda }}}} \end{aligned}$$

(3.16)

in $\Omega _{{{\bar{\lambda }}}}{\setminus } Z_{u_i}$.

Consider now a compact set K in $\Omega _{{{\bar{\lambda }}}}$ such that ${{\mathcal {L}}}(\Omega _{{{\bar{\lambda }}}}{\setminus } K)$ is sufficiently small so that Proposition 2.5 can be applied. By what we proved before, for any $i\in \{1,\ldots , m\}$, it holds that $u_i < u_{i,{{\bar{\lambda }}}}$ in $K {\setminus } A$, which is compact. Then, by (uniform) continuity, we find $\epsilon >0$ such that, ${{\bar{\lambda }}}+\epsilon < 0$ and for ${{\bar{\lambda }}}<\lambda <{{\bar{\lambda }}}+\epsilon$ we have that ${{\mathcal {L}}}(\Omega _{\lambda }{\setminus } (K{\setminus } A))$ is small enough as before, and $u_{i,\lambda }-u_i >0$ in $K {\setminus } A$ for any i. In particular, $u_{i,\lambda }-u_i >0$ on $\partial (K {\setminus } A)$. Consequently, $u_i \le u_{i,\lambda }$ on $\partial (\Omega _{\lambda }{\setminus }(K{\setminus } A))$. By Proposition 2.5 it follows $u_i \le u_{i,\lambda }$ in $\Omega _{\lambda }{\setminus }(K{\setminus } A)$ and, consequently in $\Omega _{\lambda }$, which contradicts the assumption ${{\bar{\lambda }}}<0$. Therefore ${{\bar{\lambda }}} =0$ and the thesis is proved. Finally, (1.2) follows by the monotonicity of the solution that is implicit in the moving plane method.

Finally, if $\Omega$ is a ball, repeating this argument along any direction, it follows that $u_i$, $i=1,\ldots ,m$, are radially symmetric. The fact that $\displaystyle \frac{\partial u_i}{\partial r}(r)<0$ for $r\ne 0$, follows by the Hopf’s boundary lemma which works in this case since the level sets are balls and, therefore, fulfill the interior sphere condition.

Finally (1.3) follows by (1.2) using Theorem 2.3 (see the statement (2)) and the Dirichlet boundary condition of (${\mathcal {S}}$). $\square$