# nodal solutions with synchronous sign changing components and Constant sign
solutions for singular Gierer-Meinhardt type system

Abdelkrim Moussaoui  Applied Mathematics Laboratory, Faculty of Exact
Sciences,
and Biology Department, Faculty of Natural and Life Sciences
A. Mira Bejaia University, Targa Ouzemour, 06000 Bejaia, Algeria
<EMAIL_ADDRESS>
###### Abstract.

We establish the existence of three solutions for singular semilinear elliptic
system, two of which are of opposite constant-sign. Under a strong singularity
effect, the third solution is nodal with synchronous sign components. The
approach combines sub-supersolutions method and Leray-Schauder topological
degree involving perturbation argument.

###### Key words and phrases:

Singularity, Gierer-Meinhardt system, perturbation, sub-supersolutions,
topological degree theory.

###### 1991 Mathematics Subject Classification:

35J75, 35J62, 35J92.

##  1\. Introduction

Let $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ $\left(N\geq 2\right)$
with a smooth boundary $\partial\Omega$. We consider the following system of
semilinear elliptic equations

$(\mathrm{P})\qquad\left\\{\begin{array}[]{ll}-\Delta
u+u=sgn(v)\frac{|u|^{\alpha_{1}}}{|v|^{\beta_{1}}}&\text{in }\Omega\\\ -\Delta
v+v=sgn(u)\frac{|u|^{\alpha_{2}}}{|v|^{\beta_{2}}}&\text{in }\Omega\\\
u,v=0\text{ }&\text{on }\partial\Omega,\end{array}\right.$

where $\Delta$ stands for the Laplacian differential operator on
$\mathcal{H}_{0}^{1}(\Omega),$ $sgn(\cdot)$ denotes the sign function and the
exponents $\alpha_{i}\in(-1,1)$ and $\beta_{i}\in(0,1)$ satisfy the following
condition

(1.1) $\alpha_{i}+\beta_{i}<1\text{ \ and \ }0>\alpha_{i}-\beta_{i}>-1,\text{
for }i=1,2\text{.}$

System $(\mathrm{P})$ exhibits a singularity that, without loss of generality,
is located at zero. This make difficult any study attendant to sign properties
of solutions for $(\mathrm{P}),$ especially those that change sign since they
inevitably pass through the singularity. From a structural perspective,
$(\mathrm{P})$ is closely related to Gierer-Meinhardt system which is
originally arose in studies of biological pattern formation, describing the
activator-inhibitor coupled behavior for many systems in cell biology and
physiology [11, 14, 19]. It is characterized by

(1.2) $\left\\{\begin{array}[]{ll}-d_{1}\Delta
u+a_{1}u=\frac{u^{\alpha_{1}}}{v^{\beta_{1}}}&\text{in }\Omega\\\ -d_{2}\Delta
v+a_{2}v=\frac{u^{\alpha_{2}}}{v^{\beta_{2}}}&\text{in
}\Omega\end{array}\right.$

subject to Neumann boundary conditions $\frac{\partial
u}{\partial\eta}=\frac{\partial v}{\partial\eta}=0$ on$\;\partial\Omega,$
where $u$ and $v$ represent the scaled activator and inhibitor concentrations,
respectively, $d_{1},d_{2}$ are diffusion coefficients with $d_{1}\ll d_{2}$
and the exponents $\alpha_{i},\beta_{i}\in\mathbb{R}$ satisfy the relations

$\beta_{1}\alpha_{2}<\left(1-\alpha_{1}\right)\left(1-\beta_{2}\right)\text{ \
with \ }\alpha_{i}\geq 0\geq\beta_{i},\text{ }i=1,2.$

Depending on boundedness of the diffusive coefficient $d_{2}$, existence,
stability and/or dynamics of spike solutions are widely studied for system
(1.2). We refer to [9, 12, 29, 30, 33] when $d_{2}\rightarrow+\infty$ while in
the bounded case $d_{2}<+\infty,$ we quote [13, 32, 31, 34, 35]. The case
$d_{1}=d_{2}=1$ for Neumann system (1.2) is examined recently in [26]
establishing existence of at least three positive solutions. In whole space
$\Omega=\mathbb{R}^{N}$, existence, uniqueness as well as structural
properties of solutions for Gierer-Meinhardt type systems are studied in [27]
for $N\geq 3$, in [4, 5] for $N=1$,$2$, and in [16, 17], when $N=3$.

Gierer-Meinhardt system (1.2) has interesting and challenging mathematical
properties, especially with Dirichlet boundary conditions when the nonlinear
terms become singular near the boundary. In this context, numerous papers are
devoted to study quantitative and qualitative properties of solutions for
system $(\mathrm{P})$ subject to Dirichlet boundary conditions $u,v=0$ on
$\partial\Omega$. A long this direction, among many other interesting works,
we refer interested reader to [3, 8, 28] and the references therein.

The passage from (1.2) to $(\mathrm{P})$ summoned up a dependence of the
nonlinearities on sign of components of solutions. Admittedly, this further
complicates our study since the structure of system $(\mathrm{P})$ will switch
from the cooperative case to the competitive one, depending on the sign of
solutions considered (see, e.g., [1, 7, 10, 15, 21, 22, 23]). However, the
involvement of the sign of solutions components promotes the emergence of
nonpositive solutions for $(\mathrm{P})$, such as nodal solutions, that have
been rarely studied for singular systems. By definition, a nodal solution is
neither positive nor negative. Thus, in the scalar case, it should necessarily
be a sign changing function. However, in the context of system $(\mathrm{P})$,
the concept of a nodal solution is more nuanced since it incorporates several
types of solutions depending on the sign of their components. Actually, [25]
is the only paper that has considered this issue for singular systems. The
existence of nodal solutions is established for a class of semilinear singular
system by means of the trapping region formed by specific sign changing sub-
supersolutions pairs. Exploiting spectral properties of Laplacian operator as
well as adequate truncation, in [25], it is established that nodal solutions
vanish on negligible sets. This is an essential point enabling nodal solutions
investigation for singular problems. Hence, by a solution of problem
$(\mathrm{P})$ we mean
$(u,v)\in\mathcal{H}_{0}^{1}(\Omega)\times\mathcal{H}_{0}^{1}(\Omega)$ such
that $u$ and/or $v$ vanish on zero measure sets and

$\displaystyle\int_{\Omega}(\nabla u\nabla\varphi_{1}+u\varphi_{1})\
\mathrm{d}x$ $\displaystyle=$
$\displaystyle\int_{\Omega}sgn(v)\frac{|u|^{\alpha_{1}}}{|v|^{\beta_{1}}}\varphi_{1}\
\mathrm{d}x,$ $\displaystyle\int_{\Omega}(\nabla
v\nabla\varphi_{2}+v\varphi_{2})\ \mathrm{d}x$ $\displaystyle=$
$\displaystyle\int_{\Omega}sgn(u)\frac{|u|^{\alpha_{2}}}{|v|^{\beta_{2}}}\varphi_{2}\
\mathrm{d}x,$

for all $\varphi_{i}\in\mathcal{H}_{0}^{1}(\Omega),$ for $i=1,2,$ provided the
integrals in the right-hand side of the above identities exist.

The aim of this work is to establish multiplicity result for singular system
$(\mathrm{P})$ with a precise sign information. We provide three solutions for
system $(\mathrm{P}),$ two of which are of opposite constant-sign. The sign
property of the third solution is closely related to the structure of system
$(\mathrm{P})$ which, in turn, depends both on sign of exponents
$\alpha_{i},\beta_{i}$ and on sign of components of solutions of
$(\mathrm{P})$. Specifically, when $\alpha_{i}\leq 0$, beside the uniqueness
of the positive and negative solutions, we show that the third solution is
nodal with synchronous sign changing components. According to our knowledge,
this topic is a novelty. Nodal solutions with such property have never been
discussed for systems, whether singular or regular (without singularities),
and even for those with a variational structure.

The main result is stated as follows.

######  Theorem 1.

Under assumption (1.1), problem $(\mathrm{P})$ has at least three nontrivial
solutions: $(u_{+},v_{+})\in int\mathcal{C}_{+}^{1}(\overline{\Omega}))\times
int\mathcal{C}_{+}^{1}(\overline{\Omega}),$ $(u_{-},v_{-})\in-
int\mathcal{C}_{+}^{1}(\overline{\Omega})\times-
int\mathcal{C}_{+}^{1}(\overline{\Omega})$ and
$(u_{\ast},v_{\ast})\in\mathcal{H}_{0}^{1}(\Omega)\times\mathcal{H}_{0}^{1}(\Omega)$.
If $\alpha_{i}\leq 0$ then the opposite constant sign solutions
$(u_{+},v_{+})$ and $(u_{-},v_{-})$ are unique and the third solution
$(u_{\ast},v_{\ast})$ is nodal with synchronous sign components, that is,

(1.3) $u_{\ast}v_{\ast}>0\text{ \ a.e. in }\Omega\text{.}$

The proof of Theorem 1 combines sub-supersolution method and topological
degree theory. By a choice of suitable functions with an adjustment of
adequate constants, we construct two opposite constant sign sub-supersolutions
pairs on the basis of which positive and negative rectangles are formed. The
latter provide a localisation of a positive and negative solutions
$(u_{+},v_{+})$ and $(u_{-},v_{-})$ of $(\mathrm{P})$, whose existence is
derived from the sub-supersolutions Theorem for singular systems in [15,
Theorem 2]. When $\alpha_{i}\leq 0$ in (1.1), the uniqueness of solutions
$(u_{+},v_{+})$ and $(u_{-},v_{-})$ is established by monotonicity argument.

The third solution of $(\mathrm{P})$ is obtained via topological degree
theory. It is located in the area between the positive and the negative
rectangles. This is achieved by introducing first a parameter $\varepsilon>0$
in $(\mathrm{P}),$ thus producing a regularized system
$(\mathrm{P}_{\varepsilon})$ whose study is relevant for problem
$(\mathrm{P})$. Then, we prove that the degree on a ball
$\mathcal{B}_{R_{\varepsilon}}$, encompassing all potential solutions of
$(\mathrm{P}_{\varepsilon}),$ is $0$ while the degree in
$\mathcal{B}_{R_{\varepsilon}},$ but excluding the area located between the
aforementioned positive and negative rectangles, is equal to $1$. By the
excision property of Leray-Schauder degree, this leads to the existence of a
nontrivial solution $(u_{\varepsilon},v_{\varepsilon})$ for
$(\mathrm{P}_{\varepsilon}).$ Here, it is important to note that, unlike [7,
24, 26], the independence of the radius $R_{\varepsilon}$ on $\varepsilon$ is
not required. Then, through a priori estimates, dominated convergence Theorem
as well as $S_{+}$-property of the negative Laplacian, we may pass to the
limit as $\varepsilon\rightarrow 0$ in $(\mathrm{P}_{\varepsilon})$. This
leads to a solution $(u_{\ast},v_{\ast})$ of $(\mathrm{P})$ which, according
to its localization, does not coincide with the above mentioned solutions
$(u_{+},v_{+})$ and $(u_{-},v_{-})$. Thus, $(u_{\ast},v_{\ast})$ is a third
solution of $(\mathrm{P})$. Furthermore, when $\alpha_{i}\leq 0$ in (1.1), the
uniqueness of the aforementioned constant-sign solutions forces
$(u_{\ast},v_{\ast})$ to be nodal in the sens that both components $u_{\ast}$
and $v_{\ast}$ are nontrivial and at least are not of the same constant sign.
However, exploiting the sign coupled property of system $(\mathrm{P})$, we
conclude that $u_{\ast}$ and $v_{\ast}$ are both of synchronous sign changing.

The rest of this article is organized as follows. Section 2 deals with
existence of solutions for regularized system $(\mathrm{P}_{\varepsilon})$
while Section 3 provides multiplicity result for system $(\mathrm{P})$.

##  2\. An auxiliary system

In the sequel, the Banach spaces $\mathcal{H}_{0}^{1}(\Omega)$ and
$L^{2}(\Omega)$ are equipped with the usual norms $\|\cdot\|_{1,2}$ and
$\|\cdot\|_{2}$, respectively. We also utilize the Hölder spaces
$C^{1}(\overline{\Omega})$ and $C^{1,\tau}(\overline{\Omega}),$ $\tau\in(0,1)$
as well as the order cone $\mathcal{C}_{+}^{1}(\overline{\Omega})=\\{w\in
C^{1}(\overline{\Omega}):w(x)\geq 0$ for all $x\in\overline{\Omega}\\},$ which
has a non-empty interior described as follows:

$int\mathcal{C}_{+}^{1}(\overline{\Omega})=\\{w\in\mathcal{C}_{+}^{1}(\overline{\Omega}):w(x)>0\text{
for all }x\in\overline{\Omega}\\}.$

Hereafter, we denote by $d(x)$ the distance from a point
$x\in\overline{\Omega}$ to the boundary $\partial\Omega$, where
$\overline{\Omega}=\Omega\cup\partial\Omega$ is the closure of
$\Omega\subset\mathbb{R}^{N}$. For
$w_{1},w_{2}\in\mathcal{C}^{1}(\overline{\Omega})$, the notation $w_{1}\ll
w_{2}$ means that

$\begin{array}[]{c}w_{1}(x)<w_{2}(x)\,\,\,\forall
x\in\Omega\,\,\,\mbox{and}\,\,\,\frac{\partial
w_{2}}{\partial\eta}<\frac{\partial
w_{1}}{\partial\eta}\,\,\,\mbox{on}\,\,\,\partial\Omega,\end{array}$

$\eta$ is the outward normal to $\partial\Omega$.

Let $y_{i},z_{i}\in int\mathcal{C}_{+}^{1}(\overline{\Omega})$ ($i=1,2$) be
the unique solutions of the Dirichlet problems

(2.1) $-\Delta y_{i}(x)+y_{i}(x)=d(x)^{\alpha_{i}-\beta_{i}}\text{ in
}\Omega,\text{ \ }y_{i}=0\text{ on }\partial\Omega,$ (2.2) $-\Delta
z_{i}(x)+z_{i}(x)=\left\\{\begin{array}[]{ll}d(x)^{\alpha_{i}-\beta_{i}}&\text{in
\ }\Omega\backslash\overline{\Omega}_{\delta},\\\ -1&\text{in \
}\Omega_{\delta},\end{array}\right.,\text{ \ }z_{i}=0\text{ \ on
}\partial\Omega,$

which is known to satisfy

(2.3) $c^{-1}d(x)\leq z_{i}(x)\leq y_{i}(x)\leq cd(x)\text{ in }\Omega,$

where $c>1$ is a constant and

$\Omega_{\delta}=\left\\{x\in\Omega:d(x)<\delta\right\\},$

with a fixed $\delta>0$ sufficiently small (see, e.g., [6]).

Let $\phi_{1}$ be the positive eigenfunction defined by

$-\Delta\phi_{1}+\phi_{1}=\lambda_{1}\phi_{1}\text{ \ in }\Omega,\text{ \
}\phi_{1}=0\text{ \ on }\partial\Omega,$

where $\lambda_{1}$ is the principle eigenvalue characterized by

(2.4)
$\lambda_{1}=\inf_{w\in\mathcal{H}_{0}^{1}(\Omega)\backslash\\{0\\}}\frac{\int_{\Omega}(|\nabla
w|^{2}+|w|^{2})\,\mathrm{d}x}{\int_{\Omega}|w|^{2}\,\mathrm{d}x}.$

We will make use the topological degree theory to get a third solution for
system $(\mathrm{P})$. However, the singular terms in system $(\mathrm{P})$
prevent the degree calculation to be well defined. This is mainly due to the
difficulty in getting estimates from below for solutions of $(\mathrm{P})$. To
this challenge, we disturb system $(\mathrm{P})$ by introducing a parameter
$\varepsilon>0$. This gives rise to a regularized system for $(\mathrm{P})$
whose study is relevant to our initial problem.

For $\varepsilon\in(0,1),$ we state the regularized system

$(\mathrm{P}_{\varepsilon})\qquad\left\\{\begin{array}[]{ll}-\Delta
u+u=sgn(v)\frac{(|u|+\varepsilon)^{\alpha_{1}}}{(|v|+\varepsilon)^{\beta_{1}}}&\text{in
}\Omega\\\ -\Delta
v+v=sgn(u)\frac{(|u|+\varepsilon)^{\alpha_{2}}}{(|v|+\varepsilon)^{\beta_{2}}}&\text{in
}\Omega\\\ u,v=0&\text{on }\partial\Omega.\end{array}\right.$

The existence result regarding problem $(\mathrm{P}_{\varepsilon})$ is stated
as follows.

######  Theorem 2.

Under assumption (1.1), system $(\mathrm{P}_{\varepsilon})$ admits nontrivial
solutions $(u_{\varepsilon},v_{\varepsilon})$ in
$\mathcal{C}^{1}(\overline{\Omega})\times\mathcal{C}^{1}(\overline{\Omega})$
satisfying

(2.5) $-C^{-1}z_{1}(x)\leq u_{\varepsilon}(x)\leq C^{-1}z_{1}(x),\text{
}\forall x\in\Omega,$ (2.6) $-C^{-1}z_{2}(x)\leq v_{\varepsilon}(x)\leq
C^{-1}z_{2}(x),\text{ }\forall x\in\Omega,$

for all $\varepsilon\in(0,1)$ and all $C>1$. Moreover, there exists
$(u_{\ast},v_{\ast})\in\mathcal{H}_{0}^{1}(\Omega)\times\mathcal{H}_{0}^{1}(\Omega)$,
solution of problem $(\mathrm{P}),$ within

$(u_{\ast},v_{\ast})\in[-C^{-1}z_{1},C^{-1}z_{1}]\times[-C^{-1}z_{2},C^{-1}z_{2}],$

such that

(2.7) $u_{\varepsilon}\rightarrow u_{\ast}\text{ and
}v_{\varepsilon}\rightarrow v_{\ast}\text{ in
}\mathcal{H}_{0}^{1}(\Omega)\text{ as }\varepsilon\rightarrow 0.$

###  2.1. Topological degree results

We shall study the homotopy class of problem

$(\mathrm{P}_{\varepsilon,\theta}^{t})\qquad\left\\{\begin{array}[]{l}-\Delta
u+u=\mathrm{F}_{1,\theta}^{t}({x,}u,v)\text{ in }\Omega,\\\ -\Delta
v+v=\mathrm{F}_{2,\theta}^{t}({x,}u,v)\text{ in }\Omega,\\\ u,v=0\text{ \ on
}\partial\Omega,\end{array}\right.$

with

(2.8) $\mathrm{F}_{1,\theta}^{t}({x,}u,v)=t\text{
}sgn(v)\frac{(|u|+\varepsilon)^{\alpha_{1}}}{(|v|+\varepsilon)^{\beta_{1}}}+(1-t)sgn((1-\theta)v)\text{
}(1+\theta\lambda_{1}u^{+}),$ (2.9)
$\mathrm{F}_{2,\theta}^{t}({x,}u,v)=t\text{
}sgn(u)\frac{(|u|+\varepsilon)^{\alpha_{2}}}{(|v|+\varepsilon)^{\beta_{2}}}+(1-t)sgn((1-\theta)u)\text{
}(1+\theta\lambda_{1}v^{+}),$

for $\varepsilon\in(0,1),$ for $t\in[0,1],$ where $\theta$ is a constant such
that $\theta\in\\{0,1\\}$ and $s^{+}:=\max\\{0,s\\},$ for $s\in\mathbb{R}.$

It is worth noting that all solutions
$(u,v)\in\mathcal{H}_{0}^{1}(\Omega)\times\mathcal{H}_{0}^{1}(\Omega)$ of
$(\mathrm{P}_{\varepsilon,\theta}^{t})$ satisfy

(2.10) $u(x),v(x)\neq 0\text{ \ for a.e. }x\in\Omega,$

for all $t\in[0,1],$ all $\varepsilon\in(0,1)$ and for $\theta\in\\{0,1\\}$.
This is due to the fact that $(0,0)$ cannot be a solution of
$(\mathrm{P}_{\varepsilon,\theta}^{t})$ because
$\mathrm{F}_{i,\theta}^{t}({x,}0,0)\neq 0$ as well as the fact that ”a.e. in
$\Omega$” is an equivalence relation in $L^{1}(\Omega)$.

######  Remark 1.

The decoupled system $(\mathrm{P}_{\varepsilon,1}^{0})$ (that is
$(\mathrm{P}_{\varepsilon,\theta}^{t})$ for $t=0$ and $\theta=1$) which reads
as

$(\mathrm{P}_{\varepsilon,1}^{0})\qquad\left\\{\begin{array}[]{l}-\Delta
u+u=\mathrm{F}_{1,1}^{0}({x,}u,v)=1+\lambda_{1}u^{+}\text{ in }\Omega\\\
-\Delta v+v=\mathrm{F}_{2,1}^{0}({x,}u,v)=1+\lambda_{1}v^{+}\text{ in
}\Omega\\\ u,v=0\text{ \ on }\partial\Omega,\end{array}\right.$

does not admit solutions
$(u,v)\in\mathcal{H}_{0}^{1}(\Omega)\times\mathcal{H}_{0}^{1}(\Omega)$, for
all $\varepsilon\in(0,1)$. This is due to [20, Proposition 9.64] with $p=2$
and $\beta(x),\xi(x),h(x)\equiv 1$.

In the sequel, we denote by $\mathcal{B}_{R_{\varepsilon}}$ and
$\mathcal{B}_{z}$ the balls in
$\mathcal{C}^{1}(\overline{\Omega})\times\mathcal{C}^{1}(\overline{\Omega}),$
centered at the origin, defined by

$\mathcal{B}_{R_{\varepsilon}}:=\left\\{(u,v)\in\mathcal{C}^{1}(\overline{\Omega})\times\mathcal{C}^{1}(\overline{\Omega}):\|u\|_{C^{1}(\overline{\Omega})}+\|v\|_{C^{1}(\overline{\Omega})}<R_{\varepsilon}\right\\},$
$\mathcal{B}_{z}:=\left\\{(u,v)\in\mathcal{B}_{R_{\varepsilon}}:-C^{-1}z_{1}\leq
u\leq C^{-1}z_{1},\text{ \ }-C^{-1}z_{2}\leq v\leq C^{-1}z_{2}\right\\},$

where, without loss of generality, we assumed that
$R_{\varepsilon}>\max_{i=1,2}\left\|z_{i}\right\|_{\infty},$ for all
$\varepsilon\in(0,1).$ It is readily seen that $\mathcal{B}_{z}$ is an open
set in
$\mathcal{C}^{1}(\overline{\Omega})\times\mathcal{C}^{1}(\overline{\Omega})$.

The next result shows that solutions of problem
$(\mathrm{P}_{\varepsilon,\theta}^{t})$ cannot occur outside the ball
$\mathcal{B}_{R_{\varepsilon}}$.

######  Proposition 1.

Assume (1.1) holds. Then, there is a constant $R_{\varepsilon}>0$ such that
every solution $(u,v)$ of $(\mathrm{P}_{\varepsilon,\theta}^{t})$ belongs to
$\mathcal{C}^{1}(\overline{\Omega})\times\mathcal{C}^{1}(\overline{\Omega})$
and satisfy

(2.11)
$\left\|u\right\|_{\mathcal{C}^{1}(\overline{\Omega})},\left\|v\right\|_{\mathcal{C}^{1}(\overline{\Omega})}<R_{\varepsilon},$

for all $t\in(0,1],$ all $\varepsilon\in(0,1)$ and for $\theta\in\\{0,1\\}$.
Moreover, if $\theta=0$, then all positive solutions $(u_{+},v_{+})$ and all
negative solutions $(u_{-},v_{-})$ of $(\mathrm{P}_{\varepsilon,0}^{t})$
($(\mathrm{P}_{\varepsilon,\theta}^{t})$ with $\theta=0$) satisfy

(2.12) $\begin{array}[]{c}C^{-1}z_{1}(x)\ll u_{+}(x)\text{ \ and \
}u_{-}(x)\ll-C^{-1}z_{1}(x)\text{,}\quad\forall x\in\Omega,\end{array}$ (2.13)
$\begin{array}[]{c}C^{-1}z_{2}(x)\ll v_{+}(x)\text{ \ and \
}v_{-}(x)\ll-C^{-1}z_{2}(x)\text{,}\quad\forall x\in\Omega,\end{array}$

for a constant $C>1$ large.

###### Proof.

We begin by proving (2.11). If $\alpha_{i}\leq 0$ in (1.1), by (2.8) and (2.9)
we have

$\begin{array}[]{l}|\mathrm{F}_{1,\theta}^{t}({x,}u,v)|\leq\varepsilon^{\alpha_{1}+\beta_{1}}+\lambda_{1}|u|\text{
\ and \
}|\mathrm{F}_{2,\theta}^{t}({x,}u,v)|\leq\varepsilon^{\alpha_{2}+\beta_{2}}+\lambda_{1}|v|.\end{array}$

Then, the regularity result [20, Corollary 8.13] together with the compact
embedding
$\mathcal{C}^{1,\tau}(\overline{\Omega})\subset\mathcal{C}^{1}(\overline{\Omega})$
show that all solutions $(u,v)$ of $(\mathrm{P}_{\varepsilon,\theta}^{t})$ are
bounded in
$\mathcal{C}^{1}(\overline{\Omega})\times\mathcal{C}^{1}(\overline{\Omega})$
and satisfy (2.11).

Let us examine the case when $\alpha_{i}>0$ in (1.1), $i=1,2$. By
contradiction suppose that for every $n\in\mathbb{N},$ there exist
$t_{n}\in(0,1]$ and a solution $(u_{n},v_{n})$ of
$(\mathrm{P}_{\varepsilon,\theta}^{t_{n}})$ such that

$t_{n}\rightarrow t\in(0,1]\text{ \ and \
}\|u_{n}\|_{\mathcal{C}^{1}(\overline{\Omega})},\|v_{n}\|_{\mathcal{C}^{1}(\overline{\Omega})}\rightarrow\infty\text{
\ as }n\rightarrow\infty.$

Without loss of generality we may admit that

(2.14)
$\begin{array}[]{c}\gamma_{n}:=\|u_{n}\|_{\mathcal{C}^{1}(\overline{\Omega})}\rightarrow\infty\text{
as }n\rightarrow\infty.\end{array}$

Denote

(2.15)
${\normalsize\tilde{u}}_{n}:=\frac{u_{n}}{\gamma_{n}}\in\mathcal{C}^{1}(\overline{\Omega})\text{
with
}\|{\normalsize\tilde{u}}_{n}\|_{\mathcal{C}^{1}(\overline{\Omega})}=1,\text{
for all }n\in\mathbb{N}.$

Problem $(\mathrm{P}_{\varepsilon,\theta}^{t_{n}})$ results in

(2.16)
$\begin{array}[]{l}-\Delta{\normalsize\tilde{u}}_{n}+{\normalsize\tilde{u}}_{n}=\frac{1}{\gamma_{n}}\mathrm{F}_{1,\theta}^{t_{n}}({x,}u_{n},v_{n})\\\
=\frac{t_{n}}{\gamma_{n}}\text{
}sgn(v_{n})\frac{(|u_{n}|+\varepsilon)^{\alpha_{1}}}{(|v_{n}|+\varepsilon)^{\beta_{1}}}+\frac{1-t_{n}}{\gamma_{n}}\text{
}sgn((1-\theta)v_{n})\text{ }(1+\theta\lambda_{1}u_{n}^{+}).\end{array}$

By (2.14) and since $1>\alpha_{1}>0$, one has

$\begin{array}[]{l}\left|\frac{t_{n}}{\gamma_{n}}\text{
}sgn(v_{n})\frac{(|u_{n}|+\varepsilon)^{\alpha_{1}}}{(|v_{n}|+\varepsilon)^{\beta_{1}}}+\frac{1-t_{n}}{\gamma_{n}}\text{
}sgn((1-\theta)v_{n})\text{ }(1+\theta\lambda_{1}u_{n}^{+})\right|\\\ \\\
\leq\frac{1}{\gamma_{n}}\frac{(|u_{n}|+\varepsilon)^{\alpha_{1}}}{\varepsilon^{\beta_{1}}}+1+\lambda_{1}\tilde{u}_{n}^{+}=\gamma_{n}^{\alpha_{1}-1}\frac{(|\tilde{u}_{n}|+\frac{\varepsilon}{\gamma_{n}})^{\alpha_{1}}}{\varepsilon^{\beta_{1}}}+1+\lambda_{1}\tilde{u}_{n}^{+}\\\
\\\
\leq\frac{(|\tilde{u}_{n}|+1)^{\alpha_{1}}}{\varepsilon^{\beta_{1}}}+1+\lambda_{1}\tilde{u}_{n}^{+}\leq
C_{\varepsilon}(1+\|\tilde{u}_{n}\|_{C^{1}(\overline{\Omega})}^{\alpha_{1}}+\|\tilde{u}_{n}\|_{C^{1}(\overline{\Omega})})\text{
in }\Omega,\end{array}$

for some constant $C_{\varepsilon}>0$ independent of $n$. Then, thanks to the
regularity up to the boundary in [18], we derive that $\tilde{u}_{n}$ is
bounded in $\mathcal{C}^{1,\tau}(\overline{\Omega})$ for certain
$\tau\in(0,1)$. The compactness of the embedding
$\mathcal{C}^{1,\tau}(\overline{\Omega})\subset\mathcal{C}^{1}(\overline{\Omega})$
implies

$\tilde{u}_{n}\rightarrow\tilde{u}\text{ \ in
}\mathcal{C}^{1}(\overline{\Omega})\text{.}$

Taking $\theta=0$ and passing to the limit in (2.16) as $n\rightarrow\infty,$
we obtain

$-\Delta{\normalsize\tilde{u}}+{\normalsize\tilde{u}}=0\text{ in
}\Omega,\text{ }{\normalsize\tilde{u}}=0\text{ on }\partial\Omega.$

Therefore ${\normalsize\tilde{u}=0}$ which contradicts (2.15). If $\theta=1,$
in the limit results in

(2.17)
$\left\\{\begin{array}[]{ll}-\Delta{\normalsize\tilde{u}}+{\normalsize\tilde{u}}=(1-t)\lambda_{1}{\normalsize\tilde{u}}^{+}&\text{
in }\Omega,\\\ {\normalsize\tilde{u}}=0,&\text{on
}\partial\Omega\end{array}\right.,\text{ for }t\in(0,1].$

Test with $-{\normalsize\tilde{u}}$ in (2.17) leads to
${\normalsize\tilde{u}}={\normalsize\tilde{u}^{+}}$, which is nonzero because
of (2.15). Acting with ${\normalsize\tilde{u}}$ in (2.17) and integrating over
$\Omega,$ we get

$\int_{\Omega}(|\nabla{\normalsize\tilde{u}}|^{2}+{\normalsize\tilde{u}}^{2})\,\mathrm{d}x=(1-t)\lambda_{1}\int_{\Omega}{\normalsize\tilde{u}^{2}}\,\mathrm{d}x,\text{
for }t\in(0,1].$

Put $t=1$ implies ${\normalsize\tilde{u}}=0$ which contradicts the fact that
${\normalsize\tilde{u}}\neq 0.$ If $t\in(0,1)$, (2.4) implies

$\lambda_{1}\int_{\Omega}{\normalsize\tilde{u}^{2}}\,\mathrm{d}x\leq(1-t)\lambda_{1}\int_{\Omega}{\normalsize\tilde{u}^{2}}\,\mathrm{d}x,$

which is absurd because ${\normalsize\tilde{u}}>0$ and $t<1.$ Consequently,
this shows that there exists a constant $R_{\varepsilon}>0$ such that (2.11)
holds true.

We proceed to show (2.12) and (2.13). Let $(u_{+},v_{+})$ be a positive
solution of $(\mathrm{P}_{\varepsilon,0}^{t})$. Then
$sgn(u_{+}),sgn(v_{+})\equiv 1.$ On account of (1.1), (2.3) and (2.11), we
have

$\begin{array}[]{l}\left\\{\begin{array}[]{ll}C^{-1}d(x)^{\alpha_{i}-\beta_{i}}&\text{in\
}\Omega\backslash\overline{\Omega}_{\delta}\\\ -C^{-1}&\text{in
}\Omega_{\delta}\end{array}\right.\leq\left\\{\begin{array}[]{ll}C^{-1}\delta^{\alpha_{i}-\beta_{i}}&\text{in\
}\Omega\backslash\overline{\Omega}_{\delta}\\\ -C^{-1}&\text{in
}\Omega_{\delta}\end{array}\right.\\\ \\\
<\left\\{\begin{array}[]{ll}t\frac{\varepsilon^{\alpha_{i}}}{(R_{\varepsilon}+1)^{\beta_{i}}}+(1-t)&\text{if
}\alpha_{i}\geq 0\\\
t(R_{\varepsilon}+1)^{\alpha_{i}-\beta_{i}}+(1-t)&\text{if }\alpha_{i}\leq
0\end{array}\right.\\\ \\\
\leq\left\\{\begin{array}[]{ll}t\frac{\varepsilon^{\alpha_{i}}}{(|v_{+}|+1)^{\beta_{i}}}+(1-t)&\text{if
}\alpha_{i}\geq 0\\\
t\frac{(|u_{+}|+1)^{\alpha_{i}}}{(|v_{+}|+1)^{\beta_{i}}}+(1-t)&\text{if
}\alpha_{i}\leq
0\end{array}\right.\leq\mathrm{F}_{i,0}^{t}({x,}u_{+},v_{+})\text{ \ in\
}\Omega,\text{ }i=1,2,\end{array}$

for all $t\in[0,1]$ and all $\varepsilon\in(0,1),$ provided $C>1$ is large.
Thus, for each compact set $\mathrm{K}\subset\subset\Omega$, there is a
constant $\sigma=\sigma(\mathrm{K})>0$ such that

$\sigma+C^{-1}\left\\{\begin{array}[]{ll}d(x)^{\alpha_{i}-\beta_{i}}&\text{in
\ }\Omega\backslash\overline{\Omega}_{\delta},\\\ -1&\text{in \
}\Omega_{\delta},\end{array}\right.<\mathrm{F}_{i,0}^{t}({x,}u_{+},v_{+})\text{
a.e. in }\Omega\cap\mathrm{K},i=1,2.$

Then, by (2.2) and the strong comparison principle [2, Proposition $2.6$], we
infer that $C^{-1}z_{1}(x)\ll u_{+}(x)$ and $C^{-1}z_{2}(x)\ll v_{+}(x)$, for
a.a. $x\in\overline{\Omega}$. In the same manner we can show that
$-C^{-1}z_{1}(x)\gg u_{-}(x)$ and $-C^{-1}z_{2}(x)\gg v_{-}(x)$, for a.a.
$x\in\overline{\Omega}$. This ends the proof. ∎

On account of (2.10), $\mathrm{F}_{i,\theta}^{t}{(x},\cdot,\cdot)$ is
continuous for a.e. $x\in\Omega,$ for $i=1,2$. Thus, the homotopy
$\mathcal{H}_{\varepsilon,\theta}:[0,1]\times\mathcal{C}^{1}(\overline{\Omega})\times\mathcal{C}^{1}(\overline{\Omega})\rightarrow\mathcal{C}(\overline{\Omega})$
given by

$\mathcal{H}_{\varepsilon,\theta}(t,u,v)=I(u,v)-\left(\begin{array}[]{cc}(-\Delta+I)^{-1}&0\\\
0&(-\Delta+I)^{-1}\end{array}\right)\left(\begin{array}[]{l}\mathrm{F}_{1,\theta}^{t}({x,}u,v)\\\
\lx@intercol\hfil\mathrm{F}_{2,\theta}^{t}({x,}u,v)\hfil\lx@intercol\end{array}\right)$

is well defined for all $t\in[0,1],$ all $\varepsilon\in(0,1)$, and for
$\theta=0,1$. Moreover, the compactness of the operator
$(-\Delta+I)^{-1}:\mathcal{C}(\overline{\Omega})\rightarrow\mathcal{C}^{1}(\overline{\Omega})$
implies that $\mathcal{H}_{\varepsilon,\theta}$ is completely continuous.

The next result provides the values of the the topological degree of
$\mathcal{H}_{\varepsilon,\theta}$ in certain specific sets for $\theta=0$ and
$\theta=1$.

######  Proposition 2.

Assume that (1.1) is satisfied. Then, the Leray-Schauder topological degrees
$\deg(\mathcal{H}_{\varepsilon,1}(1,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}},0)$
(with $\theta=1$) and
$\deg(\mathcal{H}_{\varepsilon,0}(1,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}}\backslash\overline{\mathcal{B}}_{z},0)$
(with $\theta=0$) are well defined for all $t\in[0,1]$ and all
$\varepsilon\in(0,1)$. Moreover, it holds

(2.18)
$\deg(\mathcal{H}_{\varepsilon,1}(1,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}},0)=\deg(\mathcal{H}_{\varepsilon,1}(0,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}},0)=0$

and

(2.19)
$\deg(\mathcal{H}_{\varepsilon,0}(1,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}}\backslash\overline{\mathcal{B}}_{z},0)=\deg(\mathcal{H}_{\varepsilon,0}(0,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}}\backslash\overline{\mathcal{B}}_{z},0)\neq
0,$

where $\overline{\mathcal{B}}_{z}$ is the closure of $\mathcal{B}_{z}$ in
$\mathcal{C}^{1}(\overline{\Omega})\times\mathcal{C}^{1}(\overline{\Omega})$.

###### Proof.

Proposition 1 expressly establishes that solutions of
$(\mathrm{P}_{\varepsilon,\theta}^{t})$ must lie in
$\mathcal{B}_{R_{\varepsilon}}$ and, if $\theta=0$, positive and negative
solutions of $(\mathrm{P}_{\varepsilon,0}^{t})$ are located in
$\mathcal{B}_{R_{\varepsilon}}\backslash\overline{\mathcal{B}}_{z}$. Hence,
the degrees
$\deg(\mathcal{H}_{\varepsilon,1}(1,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}},0)$
and
$\deg(\mathcal{H}_{\varepsilon,0}(1,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}}\backslash\overline{\mathcal{B}}_{z},0)$
are well defined for all $\varepsilon\in(0,1)$. Moreover, the homotopy
invariance property of the degree ensures that the first equality in (2.18)
and (2.19) is fulfilled. On the other hand, according to Remark 1, problem
$(\mathrm{P}_{\varepsilon,1}^{0})$ ($(\mathrm{P}_{\varepsilon,\theta}^{t})$
with $t=0$ and $\theta=1$) has no solutions whereas
$(\mathrm{P}_{\varepsilon,0}^{1})$ ($(\mathrm{P}_{\varepsilon,\theta}^{t})$
with $t=1$ and $\theta=0$), that is reduced to the following decoupled torsion
problems

$\left\\{\begin{array}[]{l}-\Delta u+u=1\text{ in }\Omega\\\ -\Delta
v+v=1\text{ in }\Omega\end{array}\right.,\text{ }u,v=0\text{ on
}\partial\Omega,$

admits a unique solution. Thence

$\deg\left(\mathcal{H}_{\varepsilon,1}(0,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}},0\right)=0,$

while

$\deg(\mathcal{H}_{\varepsilon,0}(0,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}}\backslash\overline{\mathcal{B}}_{z},0)\neq
0,$

for all $\varepsilon\in(0,1).$ Consequently, we deduce that

$\deg\left(\mathcal{H}_{\varepsilon,1}(1,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}},0\right)=0\text{
\ and \
}\deg(\mathcal{H}_{\varepsilon,0}(1,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}}\backslash\overline{\mathcal{B}}_{z},0)\neq
0,$

for all $\varepsilon\in(0,1).$ This completes the proof. ∎

###  2.2. Proof of Theorem 2.

By the definition of the homotopy $\mathcal{H}_{\varepsilon,\theta}$, observe
that

$\mathcal{H}_{\varepsilon,0}(1,\cdot,\cdot)=\mathcal{H}_{\varepsilon,1}(1,\cdot,\cdot),\text{
for all }\varepsilon\in(0,1).$

Moreover, $(u,v)$ is a solution for $(\mathrm{P}_{\varepsilon})$ if, and only
if,

$\begin{array}[]{c}(u,v)\in\mathcal{B}_{R_{\varepsilon}}(0)\,\,\,\mbox{and}\,\,\,\mathcal{H}_{\varepsilon,1}(1,u,v)=0.\end{array}$

By virtue of the domain additivity property of Leray-Schauder degree we have

$\displaystyle\deg(\mathcal{H}_{\varepsilon,1}(1,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}},0)$
$\displaystyle=$
$\displaystyle\deg(\mathcal{H}_{\varepsilon,1}(1,\cdot,\cdot),\mathcal{B}_{R_{\varepsilon}}\backslash\overline{\mathcal{B}}_{z},0)+\deg(\mathcal{H}_{\varepsilon,1}(1,\cdot,\cdot),\mathcal{B}_{z},0).$

Then, on the basis of Proposition 2, we infer that

$\deg(\mathcal{H}_{\varepsilon,1}(1,\cdot,\cdot),\mathcal{B}_{z},0)\neq 0,$

showing that problem $(\mathrm{P}_{\varepsilon})$ has a solution
$(u_{\varepsilon},v_{\varepsilon})$ in $\mathcal{B}_{z}$. Consequently,
$(u_{\varepsilon},v_{\varepsilon})\in\mathcal{C}^{1}(\overline{\Omega})\times\mathcal{C}^{1}(\overline{\Omega})$
fulfills (2.5)-(2.6) while (2.10) forces $u_{\varepsilon},v_{\varepsilon}\neq
0$ a.e in $\Omega,$ for all $\varepsilon\in(0,1)$. This proves the first part
of the theorem.

We proceed to show the limit in (2.7). Set $\varepsilon=\frac{1}{n}$ in
$(\mathrm{P}_{\varepsilon})$ with any positive integer $n\geq 1.$ From above,
there exists
$(u_{n},v_{n}):=(u_{\frac{1}{n}},v_{\frac{1}{n}})\in\mathcal{C}^{1}(\overline{\Omega})\times\mathcal{C}^{1}(\overline{\Omega})$
solution of $(\mathrm{P}_{n})$ ($(\mathrm{P}_{\varepsilon})$ with
$\varepsilon=\frac{1}{n}$) such that

(2.20)
$(u_{n},v_{n})\in[-C^{-1}z_{1},C^{-1}z_{1}]\times[-C^{-1}z_{2},C^{-1}z_{2}]$

and

(2.21) $\left\\{\begin{array}[]{l}\int_{\Omega}(\nabla
u_{n}\nabla\varphi_{1}+u_{n}\text{\thinspace}\varphi_{1})\text{
}\mathrm{d}x=\int_{\Omega}sgn(v_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{1}}}{(|v_{n}|+\frac{1}{n})^{\beta_{1}}}\varphi_{1}\
\mathrm{d}x,\\\ \int_{\Omega}(\nabla
v_{n}\nabla\varphi_{2}+v_{n}\text{\thinspace}\varphi_{2})\text{
}\mathrm{d}x=\int_{\Omega}sgn(u_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{2}}}{(|v_{n}|+\frac{1}{n})^{\beta_{2}}}\varphi_{2}\
\mathrm{d}x,\end{array}\right.$

for all $\varphi_{1},\varphi_{2}\in\mathcal{H}_{0}^{1}(\Omega)$.

Taking $t=1$ and $\varepsilon=\frac{1}{n}$ in (2.8), by (2.10), we infer that

(2.22) $u_{n}(x),v_{n}(x)\neq 0\text{ for a.e. }x\in\Omega,\text{ for all }n.$

Acting with $(\varphi_{1},\varphi_{2})=(u_{n},v_{n})$ in (2.21), by (1.1),
(2.20) and (2.22), we get

$\begin{array}[]{l}\int_{\Omega}(|\nabla u_{n}|^{2}+|u_{n}|^{2})\text{
}\mathrm{d}x=\int_{\Omega}sgn(v_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{1}}}{(|v_{n}|+\frac{1}{n})^{\beta_{1}}}u_{n}\
\mathrm{d}x\\\
\leq\int_{\Omega}\left|sgn(v_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{1}}}{(|v_{n}|+\frac{1}{n})^{\beta_{1}}}u_{n}\right|\
\mathrm{d}x\\\
\leq\left\\{\begin{array}[]{ll}\int_{\Omega}\frac{|u_{n}|^{\alpha_{1}+1}}{|v_{n}|^{\beta_{1}}}\
\mathrm{d}x&\text{if }\alpha_{1}\leq 0\\\
\int_{\Omega}\frac{(|u_{n}|+1)^{\alpha_{1}+1}}{|v_{n}|^{\beta_{1}}}\
\mathrm{d}x&\text{if }\alpha_{1}>0\end{array}\right.\\\
\leq\left\\{\begin{array}[]{ll}\int_{\Omega}\frac{(C^{-1}z_{1})^{\alpha_{1}+1}}{|v_{n}|^{\beta_{1}}}\
\mathrm{d}x&\text{if }\alpha_{1}\leq 0\\\
\int_{\Omega}\frac{(C^{-1}z_{1}+1)^{\alpha_{1}+1}}{|v_{n}|^{\beta_{1}}}\
\mathrm{d}x&\text{if }\alpha_{1}>0\end{array}\right.<\infty\end{array}$

and

$\begin{array}[]{l}\int_{\Omega}(|\nabla v_{n}|^{2}+|v_{n}|^{2})\text{
}\mathrm{d}x=\int_{\Omega}sgn(u_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{2}}}{(|v_{n}|+\frac{1}{n})^{\beta_{2}}}v_{n}\
\mathrm{d}x\\\
\leq\int_{\Omega}\left|sgn(u_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{2}}}{(|v_{n}|+\frac{1}{n})^{\beta_{2}}}v_{n}\right|\
\mathrm{d}x\\\
\leq\left\\{\begin{array}[]{ll}\int_{\Omega}|u_{n}|^{\alpha_{1}}|v_{n}|^{1-\beta_{1}}\
\mathrm{d}x&\text{if }\alpha_{1}\leq 0\\\
\int_{\Omega}(|u_{n}|+1)^{\alpha_{1}}|v_{n}|^{1-\beta_{1}}\
\mathrm{d}x&\text{if }\alpha_{1}>0\end{array}\right.\\\
\leq\left\\{\begin{array}[]{ll}\int_{\Omega}|u_{n}|^{\alpha_{1}}|v_{n}|^{1-\beta_{1}}\
\mathrm{d}x&\text{if }\alpha_{1}\leq 0\\\
\int_{\Omega}(C^{-1}z_{1}+1)^{\alpha_{1}}|v_{n}|^{1-\beta_{1}}\
\mathrm{d}x&\text{if }\alpha_{1}>0\end{array}\right.<\infty,\end{array}$

showing that $\\{u_{n}\\}_{n}$ and $\\{v_{n}\\}_{n}$ are bounded in
$\mathcal{H}_{0}^{1}(\Omega)$. We are thus allowed to extract a subsequence
(still denoted by $\\{u_{n}\\}_{n},\\{v_{n}\\}_{n}$) such that

(2.23) $\begin{array}[]{c}u_{n}\rightharpoonup u_{\ast}\text{ \ and \
}v_{n}\rightharpoonup v_{\ast}\text{ \ in
}\mathcal{H}_{0}^{1}(\Omega).\end{array}$

Moreover, on account of (2.20) and (2.23), we have

$-C^{-1}z_{1}\leq u_{\ast}\leq C^{-1}z_{1}\ \text{ and \ }-C^{-1}z_{2}\leq
v_{\ast}\leq C^{-1}z_{2}\text{ \ in }\Omega.$

Inserting $(\varphi_{1},\varphi_{2})=(u_{n}-u_{\ast},v_{n}-v_{\ast})$ in
(2.21) yields

$\int_{\Omega}(\nabla
u_{n}\text{\thinspace}\nabla(u_{n}-u_{\ast})+u_{n}(u_{n}-u_{\ast}))\text{
}\mathrm{d}x=\int_{\Omega}sgn(v_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{1}}}{(|v_{n}|+\frac{1}{n})^{\beta_{1}}}(u_{n}-u_{\ast})\
\mathrm{d}x$ $\int_{\Omega}(\nabla
v_{n}\text{\thinspace}\nabla(v_{n}-v_{\ast})+v_{n}(v_{n}-v_{\ast}))\text{
}\mathrm{d}x=\int_{\Omega}sgn(u_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{2}}}{(|v_{n}|+\frac{1}{n})^{\beta_{2}}}(v_{n}-v_{\ast})\
\mathrm{d}x$

By (1.1), (2.20) and for $C>1$, we have

$\begin{array}[]{l}\left|sgn(v_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{1}}}{(|v_{n}|+\frac{1}{n})^{\beta_{1}}}(u_{n}-u_{\ast})\right|\leq\left\\{\begin{array}[]{ll}2C^{-1}z_{1}\frac{|u_{n}|^{\alpha_{1}}}{|v_{n}|^{\beta_{1}}}&\text{if
}\alpha_{1}<0\\\
2C^{-1}z_{1}\frac{(C^{-1}z_{1}+1)^{\alpha_{1}}}{|v_{n}|^{\beta_{1}}}&\text{if
}\alpha_{1}\geq 0\end{array}\right.\\\
\leq\left\\{\begin{array}[]{ll}2\left\|z_{1}\right\|_{\infty}\frac{|u_{n}|^{\alpha_{1}}}{|v_{n}|^{\beta_{1}}}&\text{if
}\alpha_{1}<0\\\
2\left\|z_{1}\right\|_{\infty}\frac{(\left\|z_{1}\right\|_{\infty}+1)^{\alpha_{1}}}{|v_{n}|^{\beta_{1}}}&\text{if
}\alpha_{1}\geq 0,\end{array}\right.\end{array}$

while, by (2.22), we infer that

(2.24)
$sgn(v_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{1}}}{(|v_{n}|+\frac{1}{n})^{\beta_{1}}}(u_{n}-u_{\ast})\in
L^{1}(\Omega).$

Then, using (2.23), (2.24) and applying Fatou’s Lemma, we derive that

$\underset{n\rightarrow\infty}{\lim}\int_{\Omega}(\nabla
u_{n}\text{\thinspace}\nabla(u_{n}-u_{\ast})+u_{n}(u_{n}-u_{\ast}))\text{
}\mathrm{d}x\leq 0.$

Therefore, the $\mathcal{S}_{+}$-property of $-\Delta+I$ on
$\mathcal{H}_{0}^{1}(\Omega)$ (see, e.g., [20, Proposition 3.5]) guarantees
that

(2.25) $\begin{array}[]{c}u_{n}\rightarrow u_{\ast}\text{ in
}\mathcal{H}_{0}^{1}(\Omega).\end{array}$

In the same manner, we show that

(2.26) $\begin{array}[]{c}v_{n}\rightarrow v_{\ast}\text{ in
}\mathcal{H}_{0}^{1}(\Omega).\end{array}$

On the other hand, by (1.1), (2.25), (2.20) and (2.26), it holds

$\displaystyle|sgn(v_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{1}}}{(|v_{n}|+\frac{1}{n})^{\beta_{1}}}\varphi_{1}|$
$\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{|u_{n}|^{\alpha_{1}}}{|v_{n}|^{\beta_{1}}}|\varphi_{1}|&\text{if
}\alpha_{1}<0\\\
\frac{(|u_{n}|+1)^{\alpha_{1}}}{|v_{n}|^{\beta_{1}}}|\varphi_{1}|&\text{if
}\alpha_{1}\geq 0\end{array}\right.$ $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{|u_{n}|^{\alpha_{1}}}{|v_{n}|^{\beta_{1}}}|\varphi_{1}|&\text{if
}\alpha_{1}<0\\\
\frac{(\left\|z_{1}\right\|_{\infty}+1)^{\alpha_{1}}}{|v_{n}|^{\beta_{1}}}|\varphi_{1}|&\text{if
}\alpha_{1}\geq 0\end{array}\right.$

and

$\displaystyle|sgn(u_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{2}}}{(|v_{n}|+\frac{1}{n})^{\beta_{2}}}\varphi_{2}|$
$\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{|u_{n}|^{\alpha_{2}}}{|v_{n}|^{\beta_{2}}}|\varphi_{2}|&\text{if
}\alpha_{2}<0\\\
\frac{(|u_{n}|+1)^{\alpha_{2}}}{|v_{n}|^{\beta_{2}}}|\varphi_{2}|&\text{if
}\alpha_{2}\geq 0\end{array}\right.$ $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{|u_{n}|^{\alpha_{2}}}{|v_{n}|^{\beta_{2}}}|\varphi_{2}|&\text{if
}\alpha_{2}<0\\\
\frac{(\left\|z_{1}\right\|_{\infty}+1)^{\alpha_{2}}}{|v_{n}|^{\beta_{2}}}|\varphi_{2}|&\text{if
}\alpha_{2}\geq 0\end{array}\right.,$

for all $\varphi_{1},\varphi_{2}\in\mathcal{H}_{0}^{1}(\Omega).$ Then, on the
basis of (2.22), Lebesgue’s dominated convergence theorem entails

$\displaystyle\lim_{n\rightarrow\infty}\int_{\Omega}sgn(v_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{1}}}{(|v_{n}|+\frac{1}{n})^{\beta_{1}}}\varphi_{1}\
\mathrm{d}x$ $\displaystyle=$
$\displaystyle\int_{\Omega}sgn(v_{\ast})\frac{|u_{\ast}|^{\alpha_{1}}}{|v_{\ast}|^{\beta_{1}}}\varphi_{1}\
\mathrm{d}x,$
$\displaystyle\lim_{n\rightarrow\infty}\int_{\Omega}sgn(u_{n})\frac{(|u_{n}|+\frac{1}{n})^{\alpha_{2}}}{(|v_{n}|+\frac{1}{n})^{\beta_{2}}}\varphi_{2}\
\mathrm{d}x$ $\displaystyle=$
$\displaystyle\int_{\Omega}sgn(u_{\ast})\frac{|u_{\ast}|^{\alpha_{2}}}{|v_{\ast}|^{\beta_{2}}}\varphi_{2}\
\mathrm{d}x,$

for all $\varphi_{1},\varphi_{2}\in\mathcal{H}_{0}^{1}(\Omega)$. Hence, we may
pass to the limit in (2.21) to conclude that $(u_{\ast},v_{\ast})$ is a
solution of problem $(\mathrm{P})$ within
$[-C^{-1}z_{1},C^{-1}z_{1}]\times[-C^{-1}z_{2},C^{-1}z_{2}]$.

##  3\. Proof of Theorem 1

This section is devoted to the proof of the main result Theorem 1. It is
performed into two steps, distinguishing the study of constant-sign solutions
from that of nodal solutions.

######  Remark 2.

All solutions
$(u,v)\in\mathcal{H}_{0}^{1}(\Omega)\times\mathcal{H}_{0}^{1}(\Omega)$ of
$(\mathrm{P})$ satisfy $u(x),v(x)\neq 0$ for a.e. $x\in\Omega$. This is due to
the singular character at the origin of right hand-side of the equations in
$(\mathrm{P})$ together with the fact that ”a.e. in $\Omega$” is an
equivalence relation in $L^{1}(\Omega)$.

###  3.1. Constant sign solutions

######  Proposition 3.

Assume (1.1) is fulfilled with $\alpha_{i}\leq 0,$ $i=1,2$. Then, problem
$(\mathrm{P})$ does not admit more than two opposite constant-sign solutions
$(u_{+},v_{+})$ and $(u_{-},v_{-})$ in $(\mathcal{H}_{0}^{1}(\Omega)\cap
L^{\infty}(\Omega))^{2}$.

###### Proof.

We only show the existence of a unique positive solution $(u_{+},v_{+})$ for
problem $(\mathrm{P})$ because the existence of the negative solution
$(u_{1,-},u_{2,-})$ can be justified similarly. By contradiction, let
$(u_{+},v_{+})$ and $(\hat{u}_{+},\hat{v}_{+})$ be two distinct positive
solutions of $(\mathrm{P})$ in $(\mathcal{H}_{0}^{1}(\Omega)\cap
L^{\infty}(\Omega))^{2}$ satisfying

$\left\\{\begin{array}[]{ll}-\Delta
u_{+}+u_{+}=\frac{|u_{+}|^{\alpha_{1}}}{|v_{+}|^{\beta_{1}}}&\text{in
}\Omega\\\ -\Delta
v_{+}+v_{+}=\frac{|u_{+}|^{\alpha_{2}}}{|v_{+}|^{\beta_{2}}}&\text{in
}\Omega\\\ u_{+},v_{+}=0&\text{on }\partial\Omega,\end{array}\right.$
$\left\\{\begin{array}[]{ll}-\Delta\hat{u}_{+}+\hat{u}_{+}=\frac{|\hat{u}_{+}|^{\alpha_{1}}}{|\hat{v}_{+}|^{\beta_{1}}}&\text{in
}\Omega\\\
-\Delta\hat{v}_{+}+\hat{v}_{+}=\frac{|\hat{u}_{+}|^{\alpha_{2}}}{|\hat{v}_{+}|^{\beta_{2}}}&\text{in
}\Omega\\\ \hat{u}_{+},\hat{v}_{+}=0&\text{on
}\partial\Omega,\end{array}\right.$

where $sgn(u_{+}),sgn(v_{+}),sgn(\hat{u}_{+}),sgn(\hat{u}_{+})\equiv 1$. Thus

(3.1) $\left\\{\begin{array}[]{l}-\Delta
u_{+}+\Delta\hat{u}_{+}+u_{+}-\hat{u}_{+}=\frac{|u_{+}|^{\alpha_{1}}}{|v_{+}|^{\beta_{1}}}-\frac{|\hat{u}_{+}|^{\alpha_{1}}}{|\hat{v}_{+}|^{\beta_{1}}}\text{
in }\Omega\\\ u_{+}-\hat{u}_{+}=0\text{ \ on
}\partial\Omega.\end{array}\right.$

Multiply (3.1) by $(u_{+}-\hat{u}_{+})$ and integrate over $\Omega,$ the
assumption $\alpha_{i}\leq 0$ yields

$\begin{array}[]{l}0\leq\int_{\Omega}|\nabla(u_{+}-\hat{u}_{+})|^{2}\
dx+\int_{\Omega}|u_{+}-\hat{u}_{+}|^{2}\ dx\\\
=\int_{\Omega}\left(\frac{|u_{+}|^{\alpha_{1}}}{|v_{+}|^{\beta_{1}}}-\frac{|\hat{u}_{+}|^{\alpha_{1}}}{|\hat{v}_{+}|^{\beta_{1}}}\right)(u_{+}-\hat{u}_{+})\
dx\leq 0,\end{array}$

showing that $u_{+}=\hat{u}_{+}$ in $\Omega$. A similar argument produces
$v_{+}=\hat{v}_{+}.$ ∎

######  Theorem 3.

Under assumption (1.1), problem $(\mathrm{P})$ admits at least two opposite
constant-sign solutions $(u_{+},v_{+})$ and $(u_{-},v_{-})$ in
$\mathcal{C}^{1,\tau}(\overline{\Omega})\times\mathcal{C}^{1,\tau}(\overline{\Omega}),$
for certain $\tau\in(0,1).$ Moreover, for a constant $C>0$ large, it holds

(3.2) $C^{-1}z_{1}(x)\ll u_{+}(x)\ll Cy_{1}(x),\text{ }\forall x\in\Omega,$
(3.3) $C^{-1}z_{2}(x)\ll v_{+}(x)\ll Cy_{2}(x),\text{ }\forall x\in\Omega,$
(3.4) $-Cy_{1}(x)\ll u_{-}(x)\ll-C^{-1}z_{1}(x),\text{ }\forall x\in\Omega$

and

(3.5) $-Cy_{2}(x)\ll v_{-}(x)\ll-C^{-1}z_{2}(x),\text{ }\forall x\in\Omega.$

If $\alpha_{i}\leq 0,i=1,2,$ then the constant-sign solutions $(u_{+},v_{+})$
and $(u_{-},v_{-})$ are unique.

###### Proof.

By using sub-supersolutions method, we first prove the existence of a positive
solution $(u_{+},v_{+})$ within
$[C^{-1}z_{1},Cy_{1}]\times[C^{-1}z_{2},Cy_{2}]$. Given that $z_{i},y_{i}\geq
0$ in $\overline{\Omega}$, we have $sgn(u_{+}),sgn(v_{+})\equiv 1$ in
$[C^{-1}z_{1},Cy_{1}]$ and $[C^{-1}z_{2},Cy_{2}],$ respectively. Obviousely,
we have

$-C^{-1}<0\leq\min\\{\frac{(C^{-1}z_{1})^{\alpha_{i}}}{(Cy_{2})^{\beta_{i}}},\frac{(Cy_{1})^{\alpha_{i}}}{(Cy_{2})^{\beta_{i}}}\\},\
\text{for all }x\in\Omega_{\delta}.$

Moreover, by (1.1), (2.3) and for $C>1$, for $\alpha_{i}>0,$ we obtain

$\displaystyle C^{-1}d(x)^{\alpha_{i}-\beta_{i}}$ $\displaystyle\leq$
$\displaystyle(Cc)^{-(\alpha_{i}+\beta_{i})}d(x)^{\alpha_{i}-\beta_{i}}$
$\displaystyle\leq$
$\displaystyle\frac{(C^{-1}c^{-1}d(x))^{\alpha_{i}}}{(Ccd(x))^{\beta_{i}}}\leq\frac{(C^{-1}z_{1})^{\alpha_{i}}}{(Cy_{2})^{\beta_{i}}},\
\ \text{for all }x\in\Omega\backslash\overline{\Omega}_{\delta},$
$\displaystyle Cd(x)^{\alpha_{i}-\beta_{i}}$ $\displaystyle\geq$
$\displaystyle c^{\alpha_{i}+\beta_{i}}(Cd(x))^{\alpha_{i}-\beta_{i}}$
$\displaystyle\geq$
$\displaystyle\frac{(Ccd(x))^{\alpha_{i}}}{(Cc^{-1}d(x))^{\beta_{i}}}\geq\frac{(Cy_{1})^{\alpha_{i}}}{(Cy_{2})^{\beta_{i}}},\text{
\ for all }x\in\overline{\Omega},$

while, if $\alpha_{i}\leq 0,$ it holds

$C^{-1}d(x)^{\alpha_{i}-\beta_{i}}\leq(Ccd(x))^{\alpha_{i}-\beta_{i}}\leq\frac{(Cy_{1})^{\alpha_{i}}}{(Cy_{2})^{\beta_{i}}},\
\ \text{for all }x\in\Omega\backslash\overline{\Omega}_{\delta},$
$Cd(x)^{\alpha_{i}-\beta_{i}}\geq(C^{-1}c^{-1}d(x))^{\alpha_{i}-\beta_{i}}\geq\frac{(C^{-1}z_{1})^{\alpha_{i}}}{(C^{-1}z_{2})^{\beta_{i}}},\text{
\ for all }x\in\overline{\Omega},$

provided that $C>1$ is sufficiently large. Then, in view of (2.1) and (2.2),
it follows that

$\begin{array}[]{l}-\Delta(C^{-1}z_{i})+C^{-1}z_{i}=C^{-1}\left\\{\begin{array}[]{ll}d(x)^{\alpha_{i}-\beta_{i}}&\text{in
\ }\Omega\backslash\overline{\Omega}_{\delta},\\\ -1&\text{in \
}\Omega_{\delta},\end{array}\right.\\\ \\\
\leq\left\\{\begin{array}[]{cc}\frac{(C^{-1}z_{1})^{\alpha_{i}}}{(Cy_{2})^{\beta_{i}}}&\text{if
}\alpha_{i}\geq 0\\\
\frac{(Cy_{1})^{\alpha_{i}}}{(Cy_{2})^{\beta_{i}}}&\text{if }\alpha_{i}\leq
0\end{array}\right.\leq\frac{|u|^{\alpha_{i}}}{|v|^{\beta_{i}}}\text{ in
}\Omega,\end{array}$

and

$\begin{array}[]{l}-\Delta(Cy_{i})+Cy_{i}=Cd(x)^{\alpha_{i}-\beta_{i}}\\\ \\\
\geq\left\\{\begin{array}[]{cc}\frac{(Cy_{1})^{\alpha_{i}}}{(Cy_{2})^{\beta_{i}}}&\text{if
}\alpha_{i}\geq 0\\\
\frac{(C^{-1}z_{1})^{\alpha_{i}}}{(C^{-1}z_{2})^{\beta_{i}}}&\text{if
}\alpha_{i}\leq
0\end{array}\right.\geq\frac{|u|^{\alpha_{i}}}{|v|^{\beta_{i}}}\text{ in
}\Omega,\end{array}$

for all $(u_{1},u_{2})\in[C^{-1}z_{1},Cy_{1}]\times[C^{-1}z_{2},Cy_{2}].$
Then, [15, Theorem 2] ensures the existence a solution
$(u_{+},v_{+})\in\mathcal{C}^{1,\tau}(\overline{\Omega})\times\mathcal{C}^{1,\tau}(\overline{\Omega}),$
for certain $\tau\in(0,1),$ for problem $(\mathrm{P})$ within
$[C^{-1}z_{1},Cy_{1}]\times[C^{-1}z_{2},Cy_{2}]$. In view of Proposition 3,
$(u_{+},v_{+})$ is a unique positive solution of $(\mathrm{P})$.

On the other hand, by (2.3) and (1.1), for each compact set
$\mathrm{K}\subset\Omega$, there is a constant $\eta=\eta(\mathrm{K})>0$ such
that

$\begin{array}[]{l}\eta+\underline{\mathrm{X}}_{i}(x):=\eta+C^{-1}\left\\{\begin{array}[]{ll}d(x)^{\alpha_{i}-\beta_{i}}&\text{in
\ }\Omega\backslash\overline{\Omega}_{\delta},\\\ -1&\text{in \
}\Omega_{\delta},\end{array}\right.\\\ \\\
<\min\\{(Cc)^{-(\alpha_{i}+\beta_{i})},(Cc)^{\alpha_{i}-\beta_{i}}\\}\text{
}d(x)^{\alpha_{i}-\beta_{i}}\\\ \\\
\leq\left\\{\begin{array}[]{cc}\frac{(C^{-1}c^{-1}d(x))^{\alpha_{i}}}{(Ccd(x))^{\beta_{i}}}&\text{if
}\alpha_{i}\geq 0\\\
\frac{(Ccd(x))^{\alpha_{i}}}{(Ccd(x))^{\beta_{i}}}&\text{if }\alpha_{i}\leq
0\end{array}\right.\leq\left\\{\begin{array}[]{cc}\frac{(C^{-1}z_{1})^{\alpha_{i}}}{(Cy_{2})^{\beta_{i}}}&\text{if
}\alpha_{i}\geq 0\\\
\frac{(Cy_{1})^{\alpha_{i}}}{(Cy_{2})^{\beta_{i}}}&\text{if }\alpha_{i}\leq
0\end{array}\right.\\\ \\\
\leq\frac{u_{+}^{\alpha_{i}}}{v_{+}^{\beta_{i}}}:=\mathrm{X}_{i}(x)\text{ a.e.
in }\Omega\cap\mathrm{K}\end{array}$

and

$\begin{array}[]{l}\overline{\mathrm{X}}_{i}(x):=Cd(x)^{\alpha_{i}-\beta_{i}}>\eta+\max\\{(Cc)^{\alpha_{i}+\beta_{i}},(C^{-1}c^{-1})^{\alpha_{i}-\beta_{i}}\\}\text{
}d(x)^{\alpha_{i}-\beta_{i}}\\\ \\\
\geq\eta+\left\\{\begin{array}[]{cc}\frac{(Ccd(x))^{\alpha_{i}}}{(C^{-1}c^{-1}d(x))^{\beta_{i}}}&\text{if
}\alpha_{i}\geq 0\\\
\frac{(C^{-1}z_{1})^{\alpha_{i}}}{(C^{-1}z_{2})^{\beta_{i}}}&\text{if
}\alpha_{i}\leq
0\end{array}\right.\geq\eta+\left\\{\begin{array}[]{cc}\frac{(Cy_{1})^{\alpha_{i}}}{(C^{-1}z_{2})^{\beta_{i}}}&\text{if
}\alpha_{i}\geq 0\\\
\frac{(C^{-1}z_{1})^{\alpha_{i}}}{(C^{-1}z_{2})^{\beta_{i}}}&\text{if
}\alpha_{i}\leq 0\end{array}\right.\\\ \\\
\geq\eta+\frac{u_{+}^{\alpha_{i}}}{v_{+}^{\beta_{i}}}:=\eta+\mathrm{X}_{i}(x)\text{
a.e. in }\Omega\cap\mathrm{K},\end{array}$

with
$\underline{\mathrm{X}}_{i},\mathrm{X}_{i}\mathrm{,}\overline{\mathrm{X}}_{i}\in
L_{loc}^{\infty}(\Omega)$. By the strong comparison principle [2, Proposition
$2.6$], we infer that (3.2) holds true.

Following a quite similar argument as above we obtain the existence a solution
$(u_{-},v_{-})\in\mathcal{C}^{1,\tau}(\overline{\Omega})\times\mathcal{C}^{1,\tau}(\overline{\Omega}),$
for certain $\tau\in(0,1),$ for problem $(\mathrm{P})$ satisfying (3.5), which
actually is the unique negative solution of $(\mathrm{P})$. ∎

###  3.2. Nodal solutions

######  Theorem 4.

Under assumption (1.1), problem $(\mathrm{P})$ admits at least a solution
$(u_{\ast},v_{\ast})$ in
$\mathcal{H}_{0}^{1}(\Omega)\times\mathcal{H}_{0}^{1}(\Omega)$ within
$[-C^{-1}z_{1},C^{-1}z_{1}]\times[-C^{-1}z_{2},C^{-1}z_{2}].$ Moreover, if
$\alpha_{i}\leq 0$, then $(u_{\ast},v_{\ast})$ is nodal with synchronous sign
components.

###### Proof.

According to Theorem 2 and Remark 2, system $(\mathrm{P})$ admits a nontrivial
solution
$(u_{\ast},v_{\ast})\in\mathcal{H}_{0}^{1}(\Omega)\times\mathcal{H}_{0}^{1}(\Omega)$
located in $[-C^{-1}z_{1},C^{-1}z_{1}]\times[-C^{-1}z_{2},C^{-1}z_{2}]$. In
view of (3.2)-(3.5), we infer that $(u_{\ast},v_{\ast})$ is a third solution
of $(\mathrm{P})$ while Theorem 3 together with Remark 2 force that
$(u_{\ast},v_{\ast})$ is nodal in the sens that the components $u_{\ast}$ and
$v_{\ast}$ are nontrivial and at least are not of the same constant sign.

Assume that $u_{\ast}<0<v_{\ast}$. Test the first equation in $(\mathrm{P})$
by $-u_{\ast}^{-}$ we get

$\displaystyle\int_{\Omega}(|\nabla
u_{\ast}^{-}|^{2}+|u_{\ast}^{-}|^{2})\text{ }\mathrm{d}x$ $\displaystyle=$
$\displaystyle-\int_{\Omega}sgn(v_{\ast})\frac{|u_{\ast}|^{\alpha_{1}}}{|v_{\ast}|^{\beta_{1}}}u_{\ast}^{-}\text{
}\mathrm{d}x$ $\displaystyle=$
$\displaystyle-\int_{\Omega}\frac{|u_{\ast}|^{\alpha_{1}}}{|v_{\ast}|^{\beta_{1}}}u_{\ast}^{-}\text{
}\mathrm{d}x<0,$

which forces $u_{\ast}^{-}=0$, a contradiction. The same conclusion can be
drawn if we assume $v_{\ast}<0<u_{\ast}$. Hence, $u_{\ast}$ and $v_{\ast}$
cannot be of opposite constant sign, therefore, at least $u_{\ast}$ or
$v_{\ast}$ change sign.

Assume that $u_{\ast}$ change sign and $v_{\ast}>0$. Let
$\Omega_{\ast}\subset\Omega$ be a subset such that $u_{\ast}<0$ in
$\Omega_{\ast}$. In view of [20, Proposition 1.61],
$u_{\ast}^{-}\mathbbm{1}_{\Omega_{\ast}}\in\mathcal{H}_{0}^{1}(\Omega)$, where
$\mathbbm{1}_{\Omega_{\ast}}$ denotes the characteristic function of
$\Omega_{\ast}$. Test the first equation in $(\mathrm{P})$ by
$-u_{\ast}^{-}\mathbbm{1}_{\Omega_{\ast}}\in\mathcal{H}_{0}^{1}(\Omega)$ we
get

$\displaystyle\int_{\Omega_{\ast}}(|\nabla
u_{\ast}^{-}|^{2}+|u_{\ast}^{-}|^{2})\text{ }\mathrm{d}x$ $\displaystyle=$
$\displaystyle-\int_{\Omega_{\ast}}sgn(v_{\ast})\frac{|u_{\ast}|^{\alpha_{1}}}{|v_{\ast}|^{\beta_{1}}}u_{\ast}^{-}\text{
}\mathrm{d}x$ $\displaystyle=$
$\displaystyle-\int_{\Omega_{\ast}}\frac{|u_{\ast}|^{\alpha_{1}}}{|v_{\ast}|^{\beta_{1}}}\hat{u}_{1}^{-}\text{
}\mathrm{d}x\leq 0,$

which forces $u_{\ast}^{-}\mathbbm{1}_{\Omega_{\ast}}=0$, a contradiction. The
same conclusion can be drawn if we assume $u_{\ast}$ change sign and
$v_{\ast}<0$ or $v_{\ast}$ change sign and $u_{\ast}>0$ or $u_{\ast}<0$.
Hence, $u^{\ast}$ and $v^{\ast}$ are necessarily synchronous sign changing
satisfying (1.3). This completes the proof. ∎

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