# Multiple solutions to Gierer-Meinhardt systems of elliptic equations

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 multiple solutions for Gierer-Meinhardt system
involving Neumann boundary conditions. The approach combines the methods of
sub-supersolution and Leray-Schauder topological degree.

###### Key words and phrases:

Neumann boundary conditions, Gierer-Meinhardt system, sub-supersolutions,
topological degree theory

###### 2010 Mathematics Subject Classification:

35J61; 35Q92; 35J75; 35J91

##  1\. Introduction

The Gierer-Meinhardt model [11] proposed in 1972 is a typical model of
reaction-diffusion systems which has been the object of extensive mathematical
treatment in recent years, see [27, 33] for a description of progress made and
references. It describes the activator-inhibitor coupled behavior for many
systems in cell biology and physiology [11, 14, 20]. An activator is a
biochemical which stimulates a change in cells or tissues so that cell
differentiation or cell division occurs at the position where the activator
concentration is high. An inhibitor is a chemical which diffuses much faster
than the activator and tempers the self-enhancing growth of the activator
concentration, thereby stabilizing the system.

The general model proposed by Gierer-Meinhardt [11] may be written as

(1.1) $\left\\{\begin{array}[]{l}u_{t}=d_{1}\Delta
u-\hat{d}_{1}u+c\rho\frac{u^{\alpha_{1}}}{v^{\beta_{1}}}+\rho_{0}\rho\text{ in
}\Omega\times\left[0,T\right]\\\ v_{t}=d_{2}\Delta
v-\hat{d}_{2}v+c^{\prime}\rho^{\prime}\frac{u^{\alpha_{2}}}{v^{\beta_{2}}}\text{
\ \ \ \ \ \ \ in }\Omega\times\left[0,T\right],\end{array}\right.$

where $\Omega$ is bounded domain of $\mathbb{R}^{N}$ ($N\geq 1$) with smooth
boundary $\partial\Omega$, under homogeneous Neumann boundary conditions

$\frac{\partial u}{\partial\eta}=\frac{\partial v}{\partial\eta}=0\text{ \
on}\;\partial\Omega,$

in which $\eta$ denotes the unit outer normal to $\partial\Omega$. The
constants $d_{1},d_{2}$ are diffusion coefficients with $d_{1}\ll d_{2}$ and
$\hat{d}_{1},\hat{d}_{2},c,c^{\prime}$ and $\rho_{0}$ are positive. The
functions $u(t,x)$ and $v(t,x)$ are the concentrations of an activator
substance and an inhibitor substance, respectively, while the exponents
$\alpha_{i},\beta_{i}\geq 0$ satisfy the relation
$\beta_{1}\alpha_{2}>\left(\alpha_{1}-1\right)\left(\beta_{2}+1\right)$.

In particular, it has been a matter of high interest to study nonconstant
positive steady states called ground states of Gierer-Meinhardt system. They
are the solutions of the elliptic system

(1.2) $\left\\{\begin{array}[]{ll}d_{1}\Delta
u-\hat{d}_{1}u+\frac{u^{\alpha_{1}}}{v^{\beta_{1}}}=0&\text{in\ }\Omega,\\\
d_{2}\Delta v-\hat{d}_{2}v+\frac{u^{\alpha_{2}}}{v^{\beta_{2}}}=0&\text{in\
}\Omega,\end{array}\right.$

subject to Neumann boundary conditions. The difficulty in dealing with problem
(1.2) is mainly due to the lack of variational structure and a priori
estimates on the solutions. An idea due to Keener [14] consists to consider
the shadow system associated to (1.2), which is obtained by dividing by
$d_{2}$ in the second equation and then letting $d_{2}\rightarrow+\infty$. It
has been shown that nonconstant solutions of the shadow system exhibit
interior or boundary spikes. Among the large number of works in this direction
we quote for instance [10, 12, 28, 29, 34]. The case of finite $d_{2}$ and
bounded domain can be found in [30], [35]-[38], where existence, stability
and/or dynamics of spike solutions are studied.

In whole space $\Omega=\mathbb{R}^{N}$, existence and uniqueness of solutions
for a class of Gierer-Meinhardt system (1.2) are shown in [25] for $N\geq 3$.
In dimension one or two ($N=1$,$2$), it has been shown that system (1.2)
exhibits single or multiple bump solutions, see [4, 5] and the references
therein. When the spatial dimension $N=3$, a smoke-ring nonradially solution
as well as a symmetric radially bound solution are constructed for system
(1.2) in [17] and [18], respectively.

The Dirichlet boundary conditions in (1.2) has received a special attention
where existence, nonexistence and uniqueness results have been shown. Relevant
contributions regarding this topic can be found in [3, 9, 16] and the
references given there.

Surprisingly enough, so far we were not able to find previous results
providing more than one solution for (1.2), whether in the Dirichlet or
Neumann boundary conditions. Motivated by this fact, our main concern is the
question of existence of multiple ground states for Gierer-Meinhardt system

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

where the exponents $\alpha_{i},\beta_{i}>0$ satisfy

(1.3) $0\leq\alpha_{i}-\beta_{i}<\alpha_{i}+\beta_{i}<1.$

It is worth noting that no loss of generality is involved by taking parameters
$d_{i},\hat{d}_{i}$ in (1.2) identically equal to $1$.

By a (weak) solution of system $(\mathrm{P})$ we mean a couple $(u,v)\in
H^{1}(\Omega)\times H^{1}(\Omega)$ such that

(1.4) $\left\\{\begin{array}[]{l}\int_{\Omega}\nabla
u\nabla\varphi\,\mathrm{d}x+\int_{\Omega}u\varphi\,\mathrm{d}x=\int_{\Omega}\frac{u^{\alpha_{1}}}{v^{\beta_{1}}}\varphi\,\mathrm{d}x,\\\
\int_{\Omega}\nabla
v\nabla\psi\,\mathrm{d}x+\int_{\Omega}v\psi\,\mathrm{d}x=\int_{\Omega}\frac{u^{\alpha_{2}}}{v^{\beta_{2}}}\psi\,\mathrm{d}x,\end{array}\right.$

for all $(\varphi,\psi)\in H^{1}(\Omega)\times H^{1}(\Omega),$ provided the
integrals in the right-hand side of the above identities exist.

Our main result is stated as follows.

######  Theorem 1.

Under assumption (1.3), problem $(\mathrm{P})$ has at least three positive
solutions in $C^{1,\tau}(\overline{\Omega})\times
C^{1,\tau}(\overline{\Omega}),$ for certain $\tau\in(0,1)$, where at least one
vanishes on $\partial\Omega$ and one is positive on $\partial\Omega$.

The proof is chiefly based on sub-supersolutions techniques and topological
degree theory. It falls naturally into two parts corresponding to the
application of each method.

In the first part, two positive solutions are obtained (cf. Theorems 3 and 4,
section 2). They are located in separate areas, identified by sub-
supersolutions pairs, where only one of them is formed by functions vanishing
at the boundary of the domain $\partial\Omega$. The other area incorporates
only positive functions in the entire domain including the boundary
$\partial\Omega$. This is achieved by a choice of suitable functions with an
adjustment of adequate constants on the basis of which sub-supersolutions
pairs are constructed. At this point, spectral properties of the Laplacian
operator have been exploited both in the case of Dirichlet and Neumann
boundary conditions. Then, Theorem 2, stated in Section $2$, ensures the
existence of two positive solutions localized in the aforementioned areas. By
the nature of the sub-supersolutions pairs constructed, these solutions do not
coincide. One is zero at the boundary of the domain while the second is
positive there. Note that Theorem 2, shown via Schauder’s fixed point Theorem
([40]) and suitable truncation, is a sub-supersolution result for elliptic
systems involving Neumann boundary conditions. It can be applied for large
classes of Neumann elliptic systems since no sign condition on the
nonlinearities is required and no specific structure is imposed, whether
cooperative or competitive. For more inquiries regarding such structures, see,
e.g., [7, 8, 15, 22, 23, 24].

The second part in this work provides a third solution of $(\mathrm{P})$
stated in Theorem 1 (cf. section 3). The proof is based on topological degree
theory. Precisely, we prove that the degree on a ball $\mathcal{B}_{L_{2}}$
comprising both solutions given by Theorem 3 is equal to $1$ while the degree
in a bigger ball $\mathcal{B}_{L_{1}}\supset\mathcal{B}_{L_{2}},$ with
$L_{2}<L_{1},$ encompassing all potential solutions of $(\mathrm{P})$ is $0$.
By the excision property of Leray-Schauder degree, this leads to the existence
of a solution for $(\mathrm{P})$ in
$\mathcal{B}_{L_{1}}\backslash\overline{\mathcal{B}}_{L_{2}},$ lying outside
the two merged areas mentioned above where the first two solutions are
located. Therefore, it is a third solution of $(\mathrm{P})$.

The rest of this article is organized as follows. Section 2 treats the
existence of two solutions for system $(\mathrm{P})$ while Section 3 provides
the third solution.

##  2\. Existence of two solutions

In the sequel, the Sobolev space $H^{1}(\Omega)$ will be equipped with the
norm

$\|w\|_{1,2}:=\left(\|w\|_{2}^{2}+\|\nabla
w\|_{2}^{2}\right)^{\frac{1}{2}},\quad w\in H^{1}(\Omega),$

where, as usual,
$\|w\|_{2}:=(\int_{\Omega}|w(x)|^{2}\mathrm{d}x)^{\frac{1}{2}}$. We denote by
$H_{+}^{1}(\Omega)=\\{w\in H^{1}(\Omega):w\geq 0$ a.e. in $\Omega\\}.$ 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 $x\in\overline{\Omega}\\}$. This
cone has a non-empty interior described as follows:

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

In what follows, we set $r^{\pm}:=\max\\{\pm r,0\\}$ and we denote by
$\gamma_{0}$ the unique continuous linear map
$\gamma_{0}:H^{1}(\Omega)\rightarrow L^{2}(\partial\Omega)$ known as the trace
map such that $\gamma_{0}(u)=u|_{\partial\Omega},$ for all $u\in
H^{1}(\Omega)\cap C(\overline{\Omega})$ and verifies the property
$\gamma_{0}(u^{+})=\gamma_{0}(u)^{+}$ for all $u\in H^{1}(\Omega)$ (see, e.g.,
[21]).

###  2.1. A sub-super-solution theorem

We investigate the existence of solutions to

$(\mathrm{P}_{f,g})\qquad\left\\{\begin{array}[]{ll}\Delta
u-u+f(x,u,v)=0&\text{in}\;\;\Omega,\\\ \Delta
v-v+g(x,u,v)=0&\text{in}\;\;\Omega,\\\ \frac{\partial
u}{\partial\eta}=\frac{\partial
v}{\partial\eta}=0&\text{on}\;\;\partial\Omega.\end{array}\right.$

where $f,g:\Omega\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ satisfy
Carathéodory’s conditions.

The following assumptions will be posited.

  * $(\mathrm{H.1})$

For every $\delta>0,$ there exists $M=M(\delta)>0$ such that

$\max\\{|f(x,s_{1},s_{2})|,|g(x,s_{1},s_{2})|\\}\leq M,\text{ \ for a.e.
}x\in\Omega\text{, for all }|s_{1}|,|s_{2}|\leq\delta.$

  * $(\mathrm{H.2})$

With appropriate $(\underline{u},\underline{v}),(\overline{u},\overline{v})\in
C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega})$ one has
$\underline{u}\leq\overline{u}$, $\underline{v}\leq\overline{v}$, as well as

(2.1)
$\left\\{\begin{array}[]{l}\int_{\Omega}(\nabla\underline{u}\nabla\varphi_{1}+\underline{u}\varphi_{1})\,\mathrm{d}x-\int_{\partial\Omega}\frac{\partial\underline{u}}{\partial\eta}\gamma_{0}(\varphi_{1})-\int_{\Omega}f(\cdot,\underline{u},v)\varphi_{1}\,\mathrm{d}x\leq
0,\\\
\int_{\Omega}(\nabla\underline{v}\,\nabla\varphi_{2}+\underline{v}\varphi_{2})\,\mathrm{d}x-\int_{\partial\Omega}\frac{\partial\underline{v}}{\partial\eta}\gamma_{0}(\varphi_{2})-\int_{\Omega}g(\cdot,u,\underline{v})\varphi_{2}\,\mathrm{d}x\leq
0,\end{array}\right.$ (2.2)
$\left\\{\begin{array}[]{l}\int_{\Omega}(\nabla\overline{u}\,\nabla\varphi_{1}+\overline{u}\varphi_{1})\,\mathrm{d}x-\int_{\partial\Omega}\frac{\partial\overline{u}}{\partial\eta}\gamma_{0}(\varphi_{1})-\int_{\Omega}f(\cdot,\overline{u},v)\varphi_{1}\,\mathrm{d}x\geq
0,\\\
\int_{\Omega}(\nabla\overline{v}\,\nabla\varphi_{2}+\overline{v}\varphi_{2})\,\mathrm{d}x-\int_{\partial\Omega}\frac{\partial\overline{v}}{\partial\eta}\gamma_{0}(\varphi_{2})-\int_{\Omega}g(\cdot,u,\overline{v})\varphi_{2}\,\mathrm{d}x\geq
0\end{array}\right.$

for all $\varphi_{1},\varphi_{2}\in H_{+}^{1}(\Omega)$ and all $(u,v)\in
H^{1}(\Omega)\times H^{1}(\Omega)$ such that
$(u,v)\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$.

Under $(\mathrm{H.1})$ the above integrals involving $f$ and $g$ take sense
because $\underline{u},\underline{v},\overline{u},\overline{v}$ are bounded.

######  Theorem 2.

Suppose $(\mathrm{H.1})$-$(\mathrm{H.2})$ hold true. Then, problem
$(\mathrm{P}_{f,g})$ possesses a solution $(u,v)\in
C^{1,\tau}(\overline{\Omega})\times C^{1,\tau}(\overline{\Omega})$ with
suitable $\tau\in]0,1[$ such that

(2.3) $\underline{u}\leq
u\leq\overline{u}\quad\text{and}\quad\underline{v}\leq v\leq\overline{v}.$

Moreover, $\frac{\partial u}{\partial\eta}=\frac{\partial v}{\partial\eta}=0$
on $\partial\Omega$.

###### Proof.

Given $(z_{1},z_{2})\in C(\overline{\Omega})\times C(\overline{\Omega})$, we
define

(2.4)
$\left\\{\begin{array}[]{l}\mathrm{T}_{1}(z_{1}):=\min(\max(z_{1},\underline{u}),\overline{u})\\\
\mathrm{T}_{2}(z_{2}):=\min(\max(z_{2},\underline{v}),\overline{v}).\end{array}\right.$

If $\delta>0$ satisfies

$-\delta\leq\underline{u}\leq\overline{u}\leq\delta,\quad-\delta\leq\underline{v}\leq\overline{v}\leq\delta,$

$(\mathrm{H.1})$ enable us to deduce that

$f(x,\mathrm{T}_{1}(z_{1}),\mathrm{T}_{2}(z_{2})),\text{
}g(x,\mathrm{T}_{1}(z_{1}),\mathrm{T}_{2}(z_{2}))\in H^{-1}(\Omega).$

Then, from Minty-Browder Theorem (see, e.g., [1, Theorem V.15]), we infer that
the auxiliary problem

(2.5) $\left\\{\begin{array}[]{ll}\Delta
u-u+f(x,\mathrm{T}_{1}(z_{1}),\mathrm{T}_{2}(z_{2}))=0&\text{in }\Omega,\\\
\Delta v-v+g(x,\mathrm{T}_{1}(z_{1}),\mathrm{T}_{2}(z_{2}))=0&\text{in
}\Omega,\\\ \frac{\partial u}{\partial\eta}=\frac{\partial
v}{\partial\eta}=0&\text{on }\partial\Omega,\end{array}\right.$

admits a unique solution $(u,v)\in H^{1}(\Omega)\times H^{1}(\Omega).$

Let us introduce the operator

$\begin{array}[]{cccc}\mathcal{T}:&C(\overline{\Omega})\times
C(\overline{\Omega})&\rightarrow&C^{1}(\overline{\Omega})\times
C^{1}(\overline{\Omega})\\\ &(z_{1},z_{2})&\mapsto&(u,v).\end{array}$

We note from (2.5) that any fixed point of $\mathcal{T}$ within
$[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$ coincides
with the weak solution of $(\mathrm{P}_{f,g})$. Consequently, to achieve the
desired conclusion, we apply Schauder’s fixed point Theorem [40] to find a
fixed point of $\mathcal{T}$ in
$[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$.

By $(\mathrm{H.1}),$ Moser iteration procedure ensures that $u,v\in
L^{\infty}(\Omega)$ while the regularity theory of Lieberman [19] implies that
$(u,v)\in C^{1,\tau}(\overline{\Omega})\times C^{1,\tau}(\overline{\Omega})$
and $\left\|u\right\|_{C^{1,\tau}(\overline{\Omega})},$
$\left\|v\right\|_{C^{1,\tau}(\overline{\Omega})}\leq L_{0},$ where $L_{0}>0$
is independent of $u$ and $v$. Then, the compactness of the embedding
$C^{1,\tau}(\overline{\Omega})\subset C(\overline{\Omega})$ implies that
$\mathcal{T(}C(\overline{\Omega})\times C(\overline{\Omega}))$ is a relatively
compact subset of $C(\overline{\Omega})\times C(\overline{\Omega})$.

Next, we show that $\mathcal{T}$ is continuous with respect to the topology of
$C(\overline{\Omega})\times C(\overline{\Omega})$. Let
$(z_{1,n},z_{2,n})\rightarrow(z_{1},z_{2})$ in $C(\overline{\Omega})\times
C(\overline{\Omega})$ for all $n$. Denote
$\left(u_{n},v_{n}\right)=\mathcal{T(}z_{1,n},z_{2,n})$, which reads as

(2.6) $\begin{array}[]{c}\int_{\Omega}(\nabla
u_{n}\nabla\varphi_{1}+u_{n}\varphi_{1})\,\mathrm{d}x=\int_{\Omega}f(x,\mathrm{T}_{1}(z_{1,n}),\mathrm{T}_{2}(z_{2,n}))\varphi_{1}\,\mathrm{d}x\end{array}$

and

(2.7) $\begin{array}[]{c}\int_{\Omega}(\nabla
v_{n}\nabla\varphi_{2}+v_{n}\varphi_{2})\,\mathrm{d}x=\int_{\Omega}g(x,\mathrm{T}_{1}(z_{1,n}),\mathrm{T}_{2}(z_{2,n}))\varphi_{2}\,\mathrm{d}x\end{array}$

for all $\varphi_{1},\varphi_{2}\in H^{1}(\Omega)$. Inserting
$(\varphi_{1},\varphi_{2})=(u_{n},v_{n})$ in (2.6) and (2.7), using
$(\mathrm{H.1}),$ we get

(2.8)
$\left\|u_{n}\right\|_{1,2}^{2}=\int_{\Omega}f(x,\mathrm{T}_{1}(z_{1,n}),\mathrm{T}_{2}(z_{2,n}))u_{n}\,\mathrm{d}x\leq
M\int_{\Omega}u_{n}\,\mathrm{d}x$

and

(2.9)
$\left\|v_{n}\right\|_{1,2}^{2}=\int_{\Omega}g(x,\mathrm{T}_{1}(z_{1,n}),\mathrm{T}_{2}(z_{2,n}))v_{n}\,\mathrm{d}x\leq
M\int_{\Omega}v_{n}\,\mathrm{d}x.$

Thus, $\\{u_{n}\\}$ and $\\{v_{n}\\}$ are bounded in $H^{1}(\Omega)$. So,
passing to relabeled subsequences, we can write the weak convergence in
$H^{1}(\Omega)\times H^{1}(\Omega)$

(2.10)
$\begin{array}[]{c}(u_{n},v_{n})\rightharpoonup\left(u,v\right)\end{array}$

for some $\left(u,v\right)\in H^{1}(\Omega)\times H^{1}(\Omega)$. Setting
$\varphi_{1}=u_{n}-u$ in (2.6) and $\varphi_{2}=v_{n}-v$ in (2.7), we find
that

$\begin{array}[]{l}\int_{\Omega}(\nabla
u_{n}\nabla(u_{n}-u)+u_{n}(u_{n}-u))\,\mathrm{d}x=\int_{\Omega}f(x,\mathrm{T}_{1}(z_{1,n}),\mathrm{T}_{2}(z_{2,n}))(u_{n}-u)\
\mathrm{d}x\end{array}$

and

$\begin{array}[]{l}\int_{\Omega}(\nabla
v_{n}\nabla(v_{n}-v)+v_{n}(v_{n}-v))\text{
}\mathrm{d}x=\int_{\Omega}g(x,\mathrm{T}_{1}(z_{1,n}),\mathrm{T}_{2}(z_{2,n}))(v_{n}-v)\
\mathrm{d}x.\end{array}$

Lebesgue’s dominated convergence theorem ensures that

$\begin{array}[]{c}\underset{n\rightarrow\infty}{\lim}\left\langle-\Delta
u_{n}+u_{n},u_{n}-u\right\rangle=\underset{n\rightarrow\infty}{\lim}\left\langle-\Delta
v_{n}+v_{n},v_{n}-v\right\rangle=0.\end{array}$

The $S_{+}$-property of $-\Delta$ on $H^{1}\left(\Omega\right)$ (see, e.g.
[21, Proposition 2.72]) along with (2.10) implies

$\begin{array}[]{c}(u_{n},v_{n})\rightarrow(u,v)\text{ in }H^{1}(\Omega)\times
H^{1}(\Omega).\end{array}$

Then, through (2.6), (2.7) and the invariance of $C(\overline{\Omega})\times
C(\overline{\Omega})$ by $\mathcal{T}$, we infer that
$\left(u,v\right)=\mathcal{T(}z_{1},z_{2})$. On the other hand, from (2.6) and
(2.7) we know that the sequence $\\{\left(u_{n},v_{n}\right)\\}$ is bounded in
$C^{1,\tau}(\overline{\Omega})\times C^{1,\tau}(\overline{\Omega})$ for
certain $\tau\in(0,1)$ (see [19, Theorem 1.2]). Since the embedding
$C^{1,\tau}(\overline{\Omega})\subset C(\overline{\Omega})$ is compact, along
a relabeled subsequence there holds $(u_{n},v_{n})\rightarrow(u,v)$ in
$C(\overline{\Omega})\times C(\overline{\Omega})$. We conclude that
$\mathcal{T}$ is continuous.

We are thus in a position to apply Schauder’s fixed point theorem to the map
$\mathcal{T}$, which establishes the existence of $(u,v)\in
C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega})$ satisfying
$(u,v)=\mathcal{T}(u,v).$ By $(\mathrm{H.1})$ and the regularity theory of
Lieberman [19] we derive that $(u,v)\in C^{1,\tau}(\overline{\Omega})\times
C^{1,\tau}(\overline{\Omega}),$ $\tau\in(0,1)$. Moreover, due to [2, Theorem
3], one has

$\frac{\partial u}{\partial\eta}=\frac{\partial v}{\partial\eta}=0\;\;\text{on
}\partial\Omega.$

Hence, $(u,v)$ is a solution of $(\mathrm{P}_{f,g})$. Let us show that (2.3)
is fulfilled. We only prove the first inequality in (2.3) because the other
ones can justified similarly. Put $\zeta=(\underline{u}-u)^{+}$ and suppose
$\zeta\neq 0$. Then, from $(\mathrm{H.2})$, (2.5) and (2.4), we get

$\begin{array}[]{l}\int_{\\{u<\underline{u}\\}}(\nabla u\nabla\zeta+u\zeta)\
dx=\int_{\Omega}(\nabla u\nabla\zeta+u\zeta)\ dx\\\
=\int_{\Omega}f(x,\mathrm{T}_{1}(u),\mathrm{T}_{2}(v))\zeta\
dx=\int_{\\{u<\underline{u}\\}}f(x,\mathrm{T}_{1}(u),\mathrm{T}_{2}(v))\zeta\
dx\\\ =\int_{\\{u<\underline{u}\\}}f(x,\underline{u},\mathrm{T}_{2}(v))\zeta\
dx\geq\int_{\\{u<\underline{u}\\}}(\nabla\underline{u}\nabla\zeta+\underline{u}\zeta)\
dx.\end{array}$

This implies that

$\begin{array}[]{c}\int_{\\{u<\underline{u}\\}}(\nabla\underline{u}-\nabla
u)\nabla\zeta+(\underline{u}-u)\zeta)\ dx\leq 0,\end{array}$

a contradiction. Hence $u\geq\underline{u}$ in $\Omega$. A quite similar
argument provides that $v\geq\underline{v},$ $u\leq\overline{u}$ and
$v\leq\overline{v}$ in $\Omega$. This completes the proof. ∎

######  Remark 1.

The conclusion of Theorem 2 remains true if we replace Neumann boundary
conditions with Dirichlet ones.

###  2.2. Existence of a solution

Our goal is to construct sub- and supersolution pair for $(\mathrm{P})$ which,
thanks to Theorem 2, lead to solutions of $(\mathrm{P})$. With this aim,
consider the following nonlinear Neumann eigenvalue problem

(2.11) $-\Delta\phi_{1}+\phi_{1}=\lambda_{1}\phi_{1}\text{ in }\Omega,\text{ \
}\frac{\partial\phi_{1}}{\partial\eta}=0\text{ on }\partial\Omega.$

where $\phi_{1}\in int\mathcal{C}_{+}^{1}(\overline{\Omega})$ is the
eigenfunction corresponding to the first eigenvalue $\lambda_{1}>0$ which is
simple, isolate (see [26]) and characterized by

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

For a later use, set $\mu,\overline{\mu}>0$ constants such that

(2.13)
$\bar{\mu}=\max_{\overline{\Omega}}\phi_{1}(x)\geq\min_{\overline{\Omega}}\phi_{1}(x)=\underline{\mu},\text{
\ }\forall x\in\Omega.$

Consider the homogeneous Neumann problem

(2.14) $-\Delta z+z=1\text{ in }\Omega,\text{ \ }\frac{\partial
z}{\partial\eta}=0\text{ on }\partial\Omega,$

which, by Minty-Browder Theorem, admits a unique solution $z\in
H^{1}(\Omega)$. A Moser iteration procedure ensures that $z\in
L^{\infty}(\Omega)$ while the regularity theory of Lieberman [19] implies that
$z\in\mathcal{C}_{+}^{1}(\overline{\Omega}).$ Moreover, note that (2.14)
implies that $\Delta z(x)\leq z(x)$ $a.e.$ in $\Omega.$ Hence, by virtue of
the strong maximum principle of Vasquez [32] we obtain $z(x)>0$ for all
$x\in\Omega$. Suppose that for some $x_{0}\in\partial\Omega$ we have
$z(x_{0})=0$. Then, again by [32] it follows that $\frac{\partial
z}{\partial\eta}(x_{0})<0,$ which contradicts (2.14). This proves that
$z(x)>0$ for all $x\in\overline{\Omega}$, that is, $z\in
int\mathcal{C}_{+}^{1}(\overline{\Omega})$. Thus, there exists a constant
$c_{0}\in(0,1)$ such that

(2.15) $\begin{array}[]{l}z\geq c_{0}\bar{\mu}\text{ \ on
}\overline{\Omega}.\end{array}$

Furthermore, by comparison principle (see [31, Lemma 3.2]), there exists a
constant $c_{1}>0$ such that

(2.16) $z\leq c_{1}\phi_{1}\;\text{in}\;\Omega.$

Therefore, it is readily seen that $Cz\geq C^{-1}\phi_{1}$, provided $C>0$ is
big enough.

######  Theorem 3.

Assume (1.3) is fulfilled. Then, problem $(\mathrm{P})$ admits a positive
solution $(u_{1},v_{1})\in C^{1,\tau}(\overline{\Omega})\times
C^{1,\tau}(\overline{\Omega}),$ for certain $\tau\in(0,1)$, within
$[C^{-1}\phi_{1},Cz]\times[C^{-1}\phi_{1},Cz]$, provided $C>0$ is big enough.

###### Proof.

Let us show that the functions $(\overline{u},\overline{v}):=(Cz,Cz)$ satisfy
(2.2). With this aim, pick $(u,v)\in H^{1}(\Omega)\times H^{1}(\Omega)$ such
that $C^{-1}\phi_{1}\leq u\leq\overline{u}$, $C^{-1}\phi_{1}\leq
v\leq\overline{v}$. By (1.3), (2.13) and (2.16) it follows that

(2.17)
$\frac{(Cz)^{\alpha_{1}}}{v^{\beta_{1}}}\leq\frac{(Cz)^{\alpha_{1}}}{(C^{-1}\phi_{1})^{\beta_{1}}}\leq
C^{\alpha_{1}+\beta_{1}}c_{1}^{\alpha_{1}}\phi_{1}^{\alpha_{1}-\beta_{1}}\leq
C^{\alpha_{1}+\beta_{1}}c_{1}^{\alpha_{1}}\bar{\mu}^{\alpha_{1}-\beta_{1}}$

and

(2.18)
$\frac{u^{\alpha_{2}}}{(Cz)^{\beta_{2}}}\leq(Cz)^{\alpha_{2}-\beta_{2}}\leq(Cc_{1}\phi_{1})^{\alpha_{2}-\beta_{2}}\leq(Cc_{1}\bar{\mu})^{\alpha_{2}-\beta_{2}}.$

Since $\alpha_{i}+\beta_{i}<1$ ($i=1,2$) (see (1.3)), we infer that

$-\Delta(Cz)+(Cz)=C\geq\max\\{C^{\alpha_{1}+\beta_{1}}c_{1}^{\alpha_{1}}\bar{\mu}^{\alpha_{1}-\beta_{1}},(Cc_{1}\bar{\mu})^{\alpha_{2}-\beta_{2}}\\}\;\;\text{in}\;\Omega,$

provided that $C>0$ is sufficiently large. Test with
$\varphi_{1},\varphi_{2}\in H_{+}^{1}(\Omega)$ we obtain

$\int_{\Omega}(\nabla(Cz)\nabla\varphi_{1}+Cz\varphi_{1})\,\mathrm{d}x\geq\int_{\Omega}\frac{(Cz)^{\alpha_{1}}}{v^{\beta_{1}}}\varphi_{1}\,\mathrm{d}x,$
$\int_{\Omega}(\nabla(Cz)\nabla\varphi_{2}+Cz\varphi_{2})\,\mathrm{d}x\geq\int_{\Omega}\frac{u^{\alpha_{2}}}{(Cz)^{\beta_{2}}}\varphi_{2}\,\mathrm{d}x,$

as desired.

Next, we show that
$(\underline{u},\underline{v}):=(C^{-1}\phi_{1},C^{-1}\phi_{1})$ satisfy
(2.1). In view of (1.3) and after increasing $C$ when necessary, one has

$\begin{array}[]{l}C^{\alpha_{1}+\beta_{1}-1}\lambda_{1}c_{1}^{\beta_{1}}\bar{\mu}\underline{\mu}^{\beta_{1}-\alpha_{1}}\leq
1.\end{array}$

Hence, by (2.11), (2.13) and (2.16), we get

(2.19)
$\begin{array}[]{l}-\Delta(C^{-1}\phi_{1})+C^{-1}\phi_{1}=C^{-1}\lambda_{1}\phi_{1}\leq
C^{-1}\lambda_{1}\bar{\mu}\\\ \leq
C^{-(\alpha_{1}+\beta_{1})}c_{1}^{-\beta_{1}}\underline{\mu}^{\alpha_{1}-\beta_{1}}\leq
C^{-(\alpha_{1}+\beta_{1})}c_{1}^{-\beta_{1}}\phi_{1}^{\alpha_{1}-\beta_{1}}\\\
\leq\frac{(C^{-1}\phi_{1})^{\alpha_{1}}}{(Cz)^{\beta_{1}}}\leq\frac{(C^{-1}\phi_{1})^{\alpha_{1}}}{v^{\beta_{1}}}\text{
\ in }\Omega,\end{array}$

for all $v\in[C^{-1}\phi_{1},Cz]$. A quite similar argument based on (2.13)
and (1.3) furnishes

(2.20)
$\begin{array}[]{l}-\Delta(C^{-1}\phi_{1})+C^{-1}\phi_{1}=C^{-1}\lambda_{1}\phi_{1}\leq
C^{-1}\lambda_{1}\bar{\mu}\\\
\leq(C^{-1}\underline{\mu})^{\alpha_{2}-\beta_{2}}\leq(C^{-1}\phi_{1})^{\alpha_{2}-\beta_{2}}\\\
\leq\frac{u^{\alpha_{2}}}{(C^{-1}\phi_{1})^{\beta_{2}}}\text{\ \ in
}\Omega,\text{ for all }u\in[C^{-1}\phi_{1},Cz].\end{array}$

Finally, test (2.19)–(2.20) with $\varphi\in H_{+}^{1}(\Omega)$ we derive that

$\int_{\Omega}(\nabla(C^{-1}\phi_{1})\nabla\varphi+C^{-1}\phi_{1}\varphi)\,\mathrm{d}x\leq\int_{\Omega}\min\\{\frac{(C^{-1}\phi_{1})^{\alpha_{1}}}{v^{\beta_{1}}},\frac{u^{\alpha_{2}}}{(C^{-1}\phi_{1})^{\beta_{2}}}\\}\varphi\,\mathrm{d}x.$

Therefore, the couples
$(\underline{u},\overline{u}):=(\underline{v},\overline{v}):=(C^{-1}\phi_{1},Cz)$
satisfy assumption $(\mathrm{H.2})$, whence Theorem 2 applies leading to
existence of a solution $(u_{1},v_{1})\in C^{1,\tau}(\overline{\Omega})\times
C^{1,\tau}(\overline{\Omega}),$ $\tau\in(0,1)$, for problem $(\mathrm{P})$
such that

(2.21) $C^{-1}\phi_{1}\leq u_{1},v_{1}\leq Cz.$

This ends the proof. ∎

###  2.3. Existence of a second solution

Consider the homogeneous Dirichlet problem

(2.22) $-\Delta y+y=1\text{ in }\Omega,\text{ }y=0\text{ on }\partial\Omega,$

which admits a unique solution $y\in C_{0}^{1,\tau}(\overline{\Omega})$ for
suitable $\tau\in(0,1)$. By a quite similar argument as in the proof of [6,
Lemma 3.1], we show that there exist constants $c>1$ such that

(2.23) $\frac{d}{c}\leq y\leq cd\;\;\text{in}\;\Omega,\;\;\frac{\partial
y}{\partial\eta}<0\;\;\text{on}\;\partial\Omega.$

Now, given $\delta>0$, denote by $y_{\delta}\in
C_{0}^{1,\tau}(\overline{\Omega})$ the solution of the Dirichlet problem

(2.24) $-\Delta u+u=\left\\{\begin{array}[]{ll}1&\text{if
}x\in\Omega\backslash\overline{\Omega}_{\delta},\\\
-1&\text{otherwise},\end{array}\right.\quad u=0\text{ on }\partial\Omega.$

where

$\Omega_{\delta}:=\\{x\in\overline{\Omega}:d(x,\partial\Omega)<\delta\\}.$

Existence and uniqueness of $y_{\delta}$ directly stem from Minty-Browder’s
Theorem. Moreover, according to [13, Corollary 3.1], if $\delta>0$ is small
enough then

(2.25) $y_{\delta}\geq\frac{y}{2}\text{ in }\Omega\text{ \ \ and \
}\frac{\partial y_{\delta}}{\partial\eta}<\frac{1}{2}\frac{\partial
y}{\partial\eta}<0\text{ \ on }\partial\Omega,$

Obviously, for a large $C>0,$ one has

$Cy\geq C^{-1}y_{\delta}\text{ in }\overline{\Omega}.$

######  Theorem 4.

Let (1.3) be satisfied. Then, for $C>0$ big enough, problem $(\mathrm{P})$
admits a positive solution $(u_{2},v_{2})\in
C^{1,\tau}(\overline{\Omega})\times C^{1,\tau}(\overline{\Omega}),$
$\tau\in(0,1)$, within $[C^{-1}y_{\delta},Cy]\times[C^{-1}y_{\delta},Cy]$,
such that

$u_{2}\neq u_{1}\text{ \ and \ }v_{2}\neq v_{1}.$

###### Proof.

We shall prove that functions $(\overline{u},\overline{v})=(Cy,Cy)$ and
$(\underline{u},\underline{v})=(C^{-1}y_{\delta},C^{-1}y_{\delta})$ satisfy
(2.2) and (2.1), respectively. With this aim, let $(u,v)\in
H^{1}(\Omega)\times H^{1}(\Omega)$ such that

$C^{-1}y_{\delta}\leq u,v\leq Cy.$

Then, from (2.23), (2.25) and (1.3), we get

(2.26)
$\begin{array}[]{l}\frac{v^{\beta_{1}}}{(Cy)^{\alpha_{1}}}\leq(Cy)^{\beta_{1}-\alpha_{1}}\leq(Ccd(x))^{\beta_{1}-\alpha_{1}}\leq(Cc|\Omega|)^{\beta_{1}-\alpha_{1}}\text{
in }\Omega\end{array}$

and

(2.27)
$\begin{array}[]{l}\frac{(Cy_{\delta})^{\beta_{2}}}{u^{\alpha_{2}}}\leq\frac{(Cy)^{\beta_{2}}}{(C^{-1}y_{\delta})^{\alpha_{2}}}\leq\frac{(Ccd(x))^{\beta_{2}}}{(C^{-1}\frac{y}{2})^{\alpha_{2}}}\leq\frac{(Ccd(x))^{\beta_{2}}}{(C^{-1}\frac{d(x)}{2c})^{\alpha_{2}}}\\\
\leq(Cc)^{\beta_{2}+\alpha_{2}}2^{\alpha_{2}}d(x)^{\beta_{2}-\alpha_{2}}\leq(Cc)^{\beta_{2}+\alpha_{2}}2^{\alpha_{2}}|\Omega|^{\beta_{2}-\alpha_{2}}\text{
\ in }\Omega.\end{array}$

On the basis of (1.3) and for $C>0$ sufficiently large we may write

$C>\max\\{(Cc|\Omega|)^{\beta_{1}-\alpha_{1}},(Cc)^{\beta_{2}+\alpha_{2}}2^{\alpha_{2}}|\Omega|^{\beta_{2}-\alpha_{2}}\\}.$

Now, from (2.22), a direct computation gives

(2.28) $-\Delta(Cy)+Cy=C.$

Then, test with $\varphi\in H_{+}^{1}(\Omega),$ Green’s formula [2] together
with (2.26)-(2.28) imply that

$\begin{array}[]{c}\int_{\Omega}(\nabla(Cy)\nabla\varphi+Cy\varphi)\,\mathrm{d}x-\int_{\Omega}\frac{\partial(Cy)}{\partial\eta}\gamma_{0}(\varphi)\text{
{d}}\sigma=\int_{\Omega}C\varphi\text{ }\mathrm{d}x\\\
\geq\int_{\Omega}\max\\{\frac{v^{\beta_{1}}}{(Cy)^{\alpha_{1}}},\frac{(Cy)^{\beta_{2}}}{u^{\alpha_{2}}}\\}\varphi\text{
}\mathrm{d}x,\end{array}$

as desired.

Next, we show that
$(\underline{u},\underline{v})=(C^{-1}y_{\delta},C^{-1}y_{\delta})$ satisfy
(2.1). From (2.24) a direct computation shows that

(2.29)
$-\Delta(C^{-1}y_{\delta})+C^{-1}y_{\delta}=\left\\{\begin{array}[]{ll}C^{-1}&\text{in
}\Omega\backslash\overline{\Omega}_{\delta},\\\
-C^{-1}&\text{otherwise.}\end{array}\right.$

By (2.23), (2.25) and (1.3), we get

(2.30)
$\begin{array}[]{l}\frac{v^{\beta_{1}}}{(C^{-1}y_{\delta})^{\alpha_{1}}}\geq(C^{-1}y_{\delta})^{\beta_{1}-\alpha_{1}}\geq(C^{-1}\frac{y}{2})^{\beta_{1}-\alpha_{1}}\geq(\frac{d(x)}{2cC})^{\beta_{1}-\alpha_{1}}\text{
in }\Omega\end{array}$

and

(2.31)
$\begin{array}[]{l}\frac{(C^{-1}y_{\delta})^{\beta_{2}}}{u^{\alpha_{2}}}\geq\frac{(C^{-1}y_{\delta})^{\beta_{2}}}{(Cy)^{\alpha_{2}}}\geq\frac{(C^{-1}\frac{y}{2})^{\beta_{2}}}{(Ccd(x))^{\alpha_{2}}}\geq\frac{(C^{-1}\frac{d(x)}{2c})^{\beta_{2}}}{(Ccd(x))^{\alpha_{2}}}=\frac{d(x)^{\beta_{2}-\alpha_{2}}}{2^{\beta_{2}}(Cc)^{\beta_{2}+\alpha_{2}}}\text{
in }\Omega.\end{array}$

In $\Omega\backslash\overline{\Omega}_{\delta},$ it holds

$(\frac{d(x)}{2cC})^{\beta_{1}-\alpha_{1}}>(\frac{\delta}{2cC})^{\beta_{1}-\alpha_{1}},$
$\frac{d(x)^{\beta_{2}-\alpha_{2}}}{2^{\beta_{2}}(Cc)^{\beta_{2}+\alpha_{2}}}>\frac{\delta^{\beta_{2}-\alpha_{2}}}{2^{\beta_{2}}(Cc)^{\beta_{2}+\alpha_{2}}},$

because $d(x)>\delta$. Therefore, in view of (1.3) and for $C>0$ large, we
infer that

(2.32)
$\min\\{(\frac{\delta}{2cC})^{\beta_{1}-\alpha_{1}},\frac{\delta^{\beta_{2}-\alpha_{2}}}{2^{\beta_{2}}(Cc)^{\beta_{2}+\alpha_{2}}}\\}>C^{-1}\text{
\ in }\Omega\backslash\overline{\Omega}_{\delta}\text{.}$

In $\Omega_{\delta},$ it is easily seen that

(2.33)
$\min\\{(\frac{d(x)}{2cC})^{\beta_{1}-\alpha_{1}},\frac{d(x)^{\beta_{2}-\alpha_{2}}}{2^{\beta_{2}}(Cc)^{\beta_{2}+\alpha_{2}}}\\}\geq
0>-C^{-1}.$

Test with $\varphi\in H_{+}^{1}(\Omega)$ in (2.29), gathering (2.32), (2.33)
together and using Green’s formula [2], we get

$\begin{array}[]{c}\int_{\Omega}(\nabla(C^{-1}y_{\delta})\nabla\varphi+C^{-1}y_{\delta}\varphi)\,\mathrm{d}x-\int_{\Omega}\frac{\partial(C^{-1}y_{\delta})}{\partial\eta}\gamma_{0}(\varphi)\text{
{d}}\sigma\\\
=\int_{\Omega\backslash\overline{\Omega}_{\delta}}C^{-1}\varphi\text{
}\mathrm{d}x-\int_{\Omega_{\delta}}C^{-1}\varphi\text{ }\mathrm{d}x\\\
\leq\int_{\Omega}\min\\{\frac{v^{\beta_{1}}}{(C^{-1}y_{\delta})^{\alpha_{1}}},\frac{(C^{-1}y_{\delta})^{\beta_{2}}}{u^{\alpha_{2}}}\\}\varphi\text{
}\mathrm{d}x.\end{array}$

showing that (2.1) is fulfilled for
$(\underline{u},\underline{v}):=(C^{-1}y_{\delta},C^{-1}y_{\delta})$.

Consequently, owing to Theorem 2 we conclude that there exists a solution
$(u_{2},v_{2})\in C^{1,\tau}(\overline{\Omega})\times
C^{1,\tau}(\overline{\Omega}),$ $\tau\in(0,1)$, of problem $(\mathrm{P})$ such
that

(2.34) $C^{-1}y_{\delta}\leq u_{2},v_{2}\leq Cy.$

On the basis of (2.22), (2.24) and (2.34), we have $u_{2},v_{2}=0$ on
$\partial\Omega$ while, in view of (2.13) and (2.21), the solution
$(u_{1},v_{1})$ in Theorem 3 satisfies $u_{1},v_{1}>0$ on $\partial\Omega$.
This shows that $(u_{2},v_{2})$ is a second positive solution for
$(\mathrm{P})$ with $u_{2}\neq u_{1}$ and $v_{2}\neq v_{1}$. The proof is
completed. ∎

##  3\. A third solution

In what follows, for any constant $\Lambda>0,$ we denote by
$\mathcal{B}_{\Lambda}$ the ball in $C^{1}(\overline{\Omega})\times
C^{1}(\overline{\Omega})$ defined by

$\mathcal{B}_{\Lambda}(0):=\left\\{(u,v)\in C^{1}(\overline{\Omega})\times
C^{1}(\overline{\Omega}):\|u\|_{C^{1}(\overline{\Omega})}+\|v\|_{C^{1}(\overline{\Omega})}<\Lambda\right\\}.$

Recall from [21, Definition 1.60] that nodal domain of $w\in H^{1}(\Omega)$ is
a connected component of $\Omega\backslash\\{x\in\Omega:w=0\\}$. By [21,
Proposition 1.61], if $\Omega_{1}$ is a nodal domain of $w\in
H^{1}(\Omega)\cap C(\Omega)$ then

(3.1) $\chi_{\Omega_{1}}w\in H^{1}(\Omega),$

where the symbol $\chi_{\Omega_{1}}$ stands for the characteristic function of
the set $\Omega_{1}$.

###  3.1. An auxiliary result

The next lemma is crucial in finding a third solution for problem
$(\mathrm{P})$.

######  Lemma 1.

For any constant $\lambda\in(0,\lambda_{1})$ the Neumann problem

(3.2) $-\Delta u+u=\lambda|u^{+}-\phi_{1}|\text{ in }\Omega,\text{ \ \
}\frac{\partial u}{\partial\eta}=0\text{ \ on }\partial\Omega,$

does not admit solutions $u\in H^{1}(\Omega)$.

###### Proof.

By contradiction, let $u\in H^{1}(\Omega)$ be a solution of (3.2) from which
it is easily seen that $0$ and $\phi_{1}$ do not fulfill (3.2). Thus,

(3.3) $u\neq 0\text{ and }u\neq\phi_{1}.$

First, we show that $u\in L^{\infty}(\Omega)$. To this end, let
$\theta:\mathbb{R}\rightarrow[0,1]$ be a $C^{1}$ cut-off function such that

$\theta(s)=\left\\{\begin{array}[]{l}0\text{ if }s\leq 0,\\\ 1\text{ if }s\geq
1\end{array}\right.\text{ \ and }\phi^{\prime}(s)\geq 0\text{ in }[0,1].$

Fix a constant $L>0$. Here, it does not involve any loss of generality by
assuming that $L>\left\|\phi_{1}\right\|_{\infty}$. Given $\delta>0,$ we
define $\theta_{\delta}(t)=\theta(\frac{t-L}{\delta})$ for all
$t\in\mathbb{R}$. It follows that

(3.4) $\begin{array}[]{l}\theta_{\delta}\circ z\in H^{1}(\Omega)\text{ \ and \
}\nabla(\theta_{\delta}\circ z)=(\theta_{\delta}^{\prime}\circ z)\nabla
z,\text{ \ for }z\in H^{1}(\Omega)\text{.}\end{array}$

Testing in (3.2) with $(\theta_{\delta}\circ u)\varphi$, for $\varphi\in
H^{1}(\Omega)$ and $\varphi\geq 0$ in $\Omega$, we get

(3.5) $\begin{array}[]{l}\int_{\Omega}(\nabla u\nabla((\theta_{\delta}\circ
u)\varphi)+u(\theta_{\delta}\circ
u)\varphi)\,\mathrm{d}x=\lambda\int_{\Omega}|u^{+}-\phi_{1}|(\theta_{\delta}\circ
u)\varphi\,\mathrm{d}x.\end{array}$

From (3.4) we have

$\int_{\Omega}\nabla u\nabla((\theta_{\delta}\circ
u)\varphi)\,\mathrm{d}x=\int_{\Omega}(\theta_{\delta}^{\prime}\circ
u)\varphi\,\mathrm{d}x+\int_{\Omega}\nabla u\nabla\varphi\text{
}(\theta_{\delta}\circ u)\,\mathrm{d}x.$

Since $\theta_{\delta}^{\prime}\circ u\geq 0,$ (3.5) becomes

$\begin{array}[]{l}\int_{\Omega}(\nabla u\nabla\varphi(\theta_{\delta}\circ
u)+u(\theta_{\delta}\circ
u)\varphi)\,\mathrm{d}x\leq\lambda\int_{\Omega}|u^{+}-\phi_{1}|(\theta_{\delta}\circ
u)\varphi\,\mathrm{d}x.\end{array}$

Letting $\delta\rightarrow 0$ we achieve

$\begin{array}[]{l}\int_{\\{{u>L}\\}}(\nabla{u}\nabla{\varphi}+u\varphi)\text{
}\mathrm{d}x\leq\lambda\int_{\\{{u>L}\\}}|u^{+}-\phi_{1}|\varphi\,\mathrm{d}x,\end{array}$

for all $\varphi\in H^{1}(\Omega)$ with $\varphi\geq 0$ in $\Omega$. Noting
that $|u^{+}-\phi_{1}|<u$ for $u>L>0$, it follows that

(3.6)
$\begin{array}[]{l}\int_{\\{{u>L}\\}}(\nabla{u}\nabla{\varphi}+u\varphi)\text{
}\mathrm{d}x\text{ }\mathrm{d}x\leq\lambda\int_{\\{{u>L}\\}}u\varphi\
\mathrm{d}x.\end{array}$

Set $\varphi=u$ in (3.6) it follows that

(3.7)
$\begin{array}[]{l}\int_{\\{{u>L}\\}}(\left|\nabla{u}\right|^{2}+u^{2})\text{
}\mathrm{d}x\leq\lambda\int_{\\{{u>L}\\}}u^{2}\ \mathrm{d}x,\end{array}$

that is,

(3.8)
$\begin{array}[]{l}\int_{\Omega}(\left|\nabla({u}\chi_{\\{u>L\\}})\right|^{2}+({u}\chi_{\\{u>L\\}})^{2})\text{
}\mathrm{d}x\leq\lambda\int_{\Omega}(u\chi_{\\{u>L\\}})^{2}\
\mathrm{d}x.\end{array}$

By the first eigenvalue’s characterization in (2.12) one has

$\begin{array}[]{l}\lambda_{1}\int_{\Omega}(u\chi_{\\{u>L\\}})^{2}\text{
}\mathrm{d}x\leq\int_{\Omega}(\left|\nabla({u}\chi_{\\{u>L\\}})\right|^{2}+({u}\chi_{\\{u>L\\}})^{2})\text{
}\mathrm{d}x,\end{array}$

while, combined with (3.8), entails

$(\lambda_{1}-\lambda)\int_{\Omega}(\chi_{\\{u>L\\}}u)^{2}\text{
}\mathrm{d}x\leq 0.$

Since $\lambda<\lambda_{1,p},$ we conclude that

$\chi_{\\{u>L\\}}u=0.$

whereas the weak comparison principle implies that

(3.9) $u\geq 0\text{ \ in }\Omega.$

Thus

(3.10) $0\leq u\leq L\text{ \ in }\Omega,$

showing that $u\in L^{\infty}(\Omega)$ as desired. Consequently, the
regularity result in [19, Theorem 1.2] implies that $u\in
C^{1}(\overline{\Omega})$. Moreover, due to (3.10), the proof of Lemma 1 is
completed by showing that the interval $[0,L]$ does not contain any solution
of (3.2). To this end, let us begin by analyzing the case when $u>\phi_{1}$.
By (3.1) and due to $C^{1}(\overline{\Omega})$-regularity of $u$, we can
assert that $\chi_{\\{u>\phi_{1}\\}}u\in H^{1}(\Omega)$. Testing by
$\chi_{\\{u>\phi_{1,p}\\}}u$ in (3.2) we get

(3.11)
$\begin{array}[]{l}\int_{\Omega}(|\nabla(\chi_{\\{u>\phi_{1}\\}}u)|^{2}+(\chi_{\\{u>\phi_{1}\\}}u)^{2})\text{
}\mathrm{d}x\\\ =\int_{\Omega}(\nabla
u\nabla(\chi_{\\{u>\phi_{1}\\}}u)+u\chi_{\\{u>\phi_{1}\\}}u)\text{
}\mathrm{d}x\\\
=\lambda\int_{\Omega}|u^{+}-\phi_{1}|\chi_{\\{u>\phi_{1}\\}}u\text{
}\mathrm{d}x\leq\lambda\int_{\Omega}|u-\phi_{1}|\chi_{\\{u>\phi_{1}\\}}u\text{
}\mathrm{d}x\\\ \leq\lambda\int_{\Omega}\chi_{\\{u>\phi_{1}\\}}u^{2}\text{
}\mathrm{d}x=\lambda\int_{\Omega}(\chi_{\\{u>\phi_{1,p}\\}}u)^{2}\text{
}\mathrm{d}x.\end{array}$

Recalling the first eigenvalue’s characterization in (2.12) it follows that

$\begin{array}[]{l}\lambda_{1}\int_{\Omega}(\chi_{\\{u>\phi_{1}\\}}u)^{2}\text{
}\mathrm{d}x\leq\int_{\Omega}(|\nabla(\chi_{\\{u>\phi_{1}\\}}u)|^{2}+(\chi_{\\{u>\phi_{1}\\}}u)^{2})\text{
}\mathrm{d}x,\end{array}$

which, combined with (3.11) entails

$(\lambda_{1}-\lambda)\int_{\Omega}(\chi_{\\{u>\phi_{1,p}\\}}u)^{2}\text{
}\mathrm{d}x\leq 0.$

Since $\lambda<\lambda_{1},$ we infer that

(3.12) $\chi_{\\{u>\phi_{1}\\}}u=0.$

Next, we examine the situation when $u<\phi_{1}$ by processing separately the
cases $u\geq\frac{\phi_{1}}{2}$ and $u\leq\frac{\phi_{1}}{2}$. Notice that

(3.13) $\left\\{\begin{array}[]{l}|u-\phi_{1}|\leq u\text{ if
}u\in[\frac{\phi_{1}}{2},\phi_{1}]\\\ \text{and \ }|u-\phi_{1}|\geq u\text{ if
}u\in[0,\frac{\phi_{1}}{2}].\end{array}\right.$

Testing (3.2) with $\chi_{[\frac{\phi_{1}}{2},\phi_{1}]}u\in H^{1}(\Omega),$
by (3.13), we get

(3.14)
$\begin{array}[]{l}\int_{\Omega}|\nabla(\chi_{[\frac{\phi_{1}}{2},\phi_{1}]}u)|^{2}\text{
}\mathrm{d}x=\int_{\Omega}(\nabla
u\nabla(\chi_{[\frac{\phi_{1}}{2},\phi_{1}]}u)+u\chi_{[\frac{\phi_{1}}{2},\phi_{1}]}u)\text{
}\mathrm{d}x\\\
=\lambda\int_{\Omega}|u^{+}-\phi_{1}|\chi_{{}_{[\frac{\phi_{1}}{2},\phi_{1}]}}u\text{
}\mathrm{d}x\leq\lambda\int_{\Omega}(\chi_{{}_{[\frac{\phi_{1}}{2},\phi_{1}]}}u)^{2}\text{
}\mathrm{d}x.\end{array}$

Then, on account of (2.12), (3.14) becomes

$\lambda_{1}\int_{\Omega}(\chi_{{}_{[\frac{\phi_{1}}{2},\phi_{1}]}}u)^{2}\text{
}\mathrm{d}x\leq\lambda\int_{\Omega}(\chi_{{}_{[\frac{\phi_{1}}{2},\phi_{1}]}}u)^{2}\text{
}\mathrm{d}x,$

leading to

(3.15) $\chi_{{}_{[\frac{\phi_{1}}{2},\phi_{1}]}}u=0.$

Consequently, on the basis of (3.3), (3.10), (3.12) and (3.15), we inevitably
must have

(3.16) $\chi_{[0,\frac{\phi_{1}}{2}]}u\neq 0.$

Testing (3.2) with $\chi_{[0,\frac{\phi_{1}}{2}]}u\in H^{1}(\Omega),$ by
(3.13) and (2.12), we obtain

$\begin{array}[]{l}\int_{\Omega}|\nabla(\chi_{[0,\frac{\phi_{1}}{2}]}u)|^{2}\text{
}\mathrm{d}x=\int_{\Omega}|\nabla
u\nabla(\chi_{[0,\frac{\phi_{1}}{2}]}u)+u\chi_{[0,\frac{\phi_{1}}{2}]}u)\text{
}\mathrm{d}x\\\
=\lambda\int_{\Omega}|u-\phi_{1}|\chi_{[0,\frac{\phi_{1}}{2}]}u\text{
}\mathrm{d}x\geq\lambda\int_{\Omega}(\chi_{[0,\frac{\phi_{1}}{2}]}u)^{2}\text{
}\mathrm{d}x,\end{array}$

a contradiction with (3.16) because $\lambda<\lambda_{1}$. Hence,

(3.17) $\chi_{[0,\frac{\phi_{1}}{2}]}u=0.$

Finally, according to (3.3), (3.10), (3.12), (3.15) and (3.17), the desired
conclusion follows. ∎

###  3.2. Topological degree results

####  3.2.1. Topological degree on $\mathcal{B}_{L_{1}}(0)$

For $t\in[0,1]$ and for $\lambda\in(0,\lambda_{1})$, we shall study the
homotopy class of problem

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

The next result shows that solutions of problem $(\mathrm{P}_{t})$ cannot
occur outside the ball $\mathcal{B}_{L_{1}}(0)$ for a large constant
$L_{1}>0$.

######  Lemma 2.

Assume (1.3) holds. Then, there is a large constant $L_{1}>0$ such that every
solution $(u,v)$ of $(\mathrm{P}_{t})$ satisfies
$\left\|u\right\|_{C^{1}(\overline{\Omega})},\left\|v\right\|_{C^{1}(\overline{\Omega})}<L_{1},$
for all $t\in[0,1]$. Moreover, if $t=0$ problem $(\mathrm{P}_{0})$ does not
admit solutions.

###### Proof.

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 $(\mathrm{P}_{t_{n}})$ such
that

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

Thus, $(\mathrm{P}_{t_{n}})$ is reads as

$\left\\{\begin{array}[]{l}-\Delta
u_{n}+u_{n}=t_{n}\frac{u_{n}^{\alpha_{1}}}{v_{n}^{\beta_{1}}}+(1-t_{n})\lambda|u_{n}^{+}-\phi_{1}|\text{
in }\Omega,\\\ -\Delta
v_{n}+v_{n}=t_{n}\frac{u_{n}^{\alpha_{2}}}{v_{n}^{\beta_{2}}}+(1-t_{n})\lambda|v_{n}^{+}-\phi_{1}|\text{
in }\Omega,\\\ \frac{\partial u_{n}}{\partial\eta}=\frac{\partial
v_{n}}{\partial\eta}=0\text{ on }\partial\Omega,\text{ \ for all
}n\in\mathbb{N}.\end{array}\right.$

Set

(3.18)
$\begin{array}[]{c}(\theta_{n},\hat{\theta}_{n}):=(\|u_{n}\|_{C^{1}(\overline{\Omega})},\|v_{n}\|_{C^{1}(\overline{\Omega})})\rightarrow\infty\text{
as }n\rightarrow\infty\end{array}$

and denote

(3.19)
$\left\\{\begin{array}[]{l}(\mathcal{U}_{n},\mathcal{V}_{n}):=(\frac{u_{n}}{\theta_{n}},\frac{v_{n}}{\hat{\theta}_{n}})\in
C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega}),\\\ \text{with
}\|\mathcal{U}_{n}\|_{C^{1}(\overline{\Omega})}=\|\mathcal{V}_{n}\|_{C^{1}(\overline{\Omega})}=1\text{
for all }n\in\mathbb{N}.\end{array}\right.$

$(\mathrm{P}_{t_{n}})$ results in

(3.20)
$\left\\{\begin{array}[]{l}-\Delta\mathcal{U}_{n}+\mathcal{U}_{n}=\frac{1}{\theta_{n}}\left(t_{n}\frac{u_{n}^{\alpha_{1}}}{v_{n}^{\beta_{1}}}+(1-t_{n})\lambda|u_{n}^{+}-\phi_{1}|\right)\text{
in }\Omega,\\\
-\Delta\mathcal{V}_{n}+\mathcal{V}_{n}=\frac{1}{\hat{\theta}_{n}}\left(t_{n}\frac{u_{n}^{\alpha_{2}}}{v_{n}^{\beta_{2}}}+(1-t_{n})\lambda|v_{n}^{+}-\phi_{1}|\right)\text{
in }\Omega,\\\
\frac{\partial\mathcal{U}_{n}}{\partial\eta}=\frac{\partial\mathcal{V}_{n}}{\partial\eta}=0\text{
on }\partial\Omega.\end{array}\right.$

Note that (3.20) implies that

$\Delta\mathcal{U}_{n}(x)\leq\mathcal{U}_{n}(x)\text{ \ and \
}\Delta\mathcal{V}_{n}(x)\leq\mathcal{V}_{n}(x)\text{ for a.e. }x\in\Omega.$

Hence, by virtue of the strong maximum principle in [32], we obtain
$\mathcal{U}_{n}(x),\mathcal{V}_{n}(x)>0$ for all $x\in\Omega$. Suppose that
for some $x_{0},x_{0}^{\prime}\in\partial\Omega$ we have
$\mathcal{U}_{n}(x_{0})=\mathcal{V}_{n}(x_{0}^{\prime})=0$. Then, again by
[32] it follows that
$\frac{\partial\mathcal{U}_{n}}{\partial\eta}(x_{0}),\frac{\partial\mathcal{V}_{n}}{\partial\eta}(x_{0}^{\prime})<0,$
which contradicts (2.13). This proves that
$\mathcal{U}_{n}(x),\mathcal{V}_{n}(x)>0$ for all $x\in\overline{\Omega}$,
that is, $\mathcal{U}_{n},\mathcal{V}_{n}\in
int\mathcal{C}_{+}^{1}(\overline{\Omega})$. Thus, there exists a constant
$\rho>0$ such that

(3.21) $\begin{array}[]{l}\mathcal{U}_{n},\mathcal{V}_{n}>\rho\text{ \ on
}\overline{\Omega}.\end{array}$

By (3.21), (1.3), (3.19) and in view of (3.18), one has

(3.22)
$\begin{array}[]{c}\left|\frac{1}{\theta_{n}}\left(t_{n}\frac{u_{n}^{\alpha_{1}}}{v_{n}^{\beta_{1}}}+(1-t_{n})\lambda|u_{n}^{+}-\phi_{1}|\right)\right|\\\
=\frac{t_{n}}{\theta_{n}^{1+\beta_{1}-\alpha_{1}}}\frac{\mathcal{U}_{n}^{\alpha_{1}}}{\mathcal{V}_{n}^{\beta_{1}}}+(1-t_{n})\lambda|\mathcal{U}_{n}-\frac{\phi_{1}}{\theta_{n}}|\\\
\leq\frac{t_{n}}{\theta_{n}^{1+\beta_{1}-\alpha_{1}}\rho^{\beta_{1}}}\mathcal{U}_{n}^{\alpha_{1}}+(1-t_{n})\lambda\mathcal{U}_{n}\leq\mathcal{U}_{n}^{\alpha_{1}}+\lambda\mathcal{U}_{n}\\\
\leq\|\mathcal{U}_{n}\|_{C^{1}(\overline{\Omega})}^{\alpha_{1}}+\lambda\|\mathcal{U}_{n}\|_{C^{1}(\overline{\Omega})}=1+\lambda\text{
\ in }\Omega,\end{array}$

and

(3.23)
$\begin{array}[]{l}\left|\frac{1}{\hat{\theta}_{n}}\left(t_{n}\frac{u_{n}^{\alpha_{2}}}{v_{n}^{\beta_{2}}}+(1-t_{n})\lambda|v_{n}^{+}-\phi_{1}|\right)\right|\\\
=\frac{t_{n}}{\hat{\theta}_{n}^{1+\beta_{2}-\alpha_{2}}}\frac{\mathcal{U}_{n}^{\alpha_{2}}}{\mathcal{V}_{n}^{\beta_{2}}}+(1-t_{n})\lambda|\mathcal{V}_{n}-\frac{\phi_{1}}{\hat{\theta}_{n}}|\\\
\leq\frac{t_{n}}{\hat{\theta}_{n}^{1+\beta_{2}-\alpha_{2}}\rho^{\beta_{2}}}\mathcal{U}_{n}^{\alpha_{2}}+(1-t_{n})\lambda\mathcal{V}_{n}\leq\mathcal{U}_{n}^{\alpha_{2}}+\lambda\mathcal{V}_{n}\\\
\leq\|\mathcal{U}_{n}\|_{C^{1}(\overline{\Omega})}^{\alpha_{2}}+\lambda\|\mathcal{V}_{n}\|_{C^{1}(\overline{\Omega})}=1+\lambda\text{
\ in }\Omega.\end{array}$

Then, owing to [19] together with (3.19), one derives that
$\mathcal{U}{\normalsize{}_{n}}$ and $\mathcal{V}_{n}$ are bounded in
$C^{1,\gamma}(\overline{\Omega})$ for certain $\gamma\in(0,1)$. The
compactness of the embedding $C^{1,\gamma}(\overline{\Omega})\subset
C^{1}(\overline{\Omega})$ implies

$\mathcal{U}{\normalsize{}_{n}}\rightarrow\mathcal{U}\ \ \text{and \
}\mathcal{V}{\normalsize{}_{n}}\rightarrow\mathcal{V}\text{ \ in
}C^{1}(\overline{\Omega}),$

with

(3.24) $\begin{array}[]{l}\mathcal{U},\mathcal{V}>\rho\text{ \ on
}\overline{\Omega}.\end{array}$

Passing to the limit in (3.20) as $n\rightarrow\infty$ one gets

(3.25)
$\left\\{\begin{array}[]{ll}-\Delta\mathcal{U}+\mathcal{U}=(1-t)\lambda\mathcal{U}&\text{in
}\Omega\\\ -\Delta\mathcal{V}+\mathcal{V}=(1-t)\lambda\mathcal{V}&\text{in
}\Omega\\\
\frac{\partial\mathcal{U}}{\partial\eta}=\frac{\partial\mathcal{V}}{\partial\eta}=0\text{
}&\text{on }\partial\Omega\end{array}\right.,\text{ \ for }t\in[0,1].$

If $t=1$ then $\mathcal{U}=\mathcal{V}=0$ which contradicts (3.24). Assume
$t\in[0,1).$ Since $(1-t)\lambda<\lambda<\lambda_{1},$ the unique solution of
(3.25) is $(\mathcal{U},\mathcal{V})=0$, which is impossible in view of
(3.24). The claim is thus proved. ∎

######  Proposition 1.

Assume (1.3) holds. With a constant $L_{1}>0$, let the homotopy $\mathcal{H}$
on $\left[0,1\right]\times\overline{\mathcal{B}}_{L_{1}}(0)$ defined by

$\mathcal{H}(t,u,v)=(\mathcal{H}_{1}(t,u,v),\mathcal{H}_{2}(t,u,v)),$

with

$\mathcal{H}_{1}(t,u,v)=u-(-\Delta+I)^{-1}(t\frac{u^{\alpha_{1}}}{v^{\beta_{1}}}+(1-t)\lambda|u^{+}-\phi_{1}|),$

and

$\mathcal{H}_{2}(t,u,v)=v-(-\Delta+I)^{-1}(t\frac{u^{\alpha_{2}}}{v^{\beta_{2}}}+(1-t)\lambda|v^{+}-\phi_{1}|),$

where $\overline{\mathcal{B}}_{L_{1}}(0)$ is the closure of
$\mathcal{B}_{L_{1}}(0)$ in $C^{1}(\overline{\Omega})\times
C^{1}(\overline{\Omega})$.

If $L_{1}>0$ is sufficiently large, then the Leray-Schauder topological degree
$\deg(\mathcal{H}(t,\cdot,\cdot),\mathcal{B}_{L_{1}}(0),0)$ is well defined
for every $t\in[0,1],$ and it holds

(3.26)
$\deg\left(\mathcal{H}(1,\cdot,\cdot),\mathcal{B}_{L_{1}}(0),0\right)=0.$

###### Proof.

According to Lemma 2, an analysis similar to the one showing (3.21) implies
that there exists a constant $\hat{\rho}>0$ such that all solutions $(u,v)\in
C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega})$ of $(\mathrm{P}_{t})$
satisfy $u,v>\hat{\rho}$ on $\overline{\Omega}$, for all $t\in[0,1]$. Hence,
the right hand-side of the equations in $(\mathrm{P}_{t})$ are continuous
almost everywhere in $\Omega$ and therefore, homotopies
$\mathcal{H}_{1}$,$\mathcal{H}_{2}$ are well defined. Moreover,
$\mathcal{H}_{1},\mathcal{H}_{2}:\left[0,1\right]\times
C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega})\rightarrow
C(\overline{\Omega})$ are completely continuous. This is due to the
compactness of the operators $(-\Delta+I)^{-1}:C(\overline{\Omega})\rightarrow
C^{1}(\overline{\Omega})$. Hence, $(u,v)\in\mathcal{B}_{L_{1}}(0)$ is a
solution for $(\mathrm{P})$ if, and only if,

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

Clearly, the previous Lemma 2 ensures that solutions of $(\mathrm{P}_{t})$ lie
in $\mathcal{B}_{L_{1}}(0)$. Furthermore, since $(\mathrm{P}_{t})$ has no
solutions for $t=0$, one can deduce that

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

Consequently, through the homotopy invariance property, we conclude that
(3.26) holds true. ∎

####  3.2.2. Topological degree on $\mathcal{B}_{L_{2}}(0)$.

Let us define the problem

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

for $t\in[0,1].$

######  Lemma 3.

Under assumption (1.3), every solution $(u,v)$ of $(\mathrm{\tilde{P}}_{t}),$
for $t\in[0,1]$, is bounded in $C^{1}(\overline{\Omega})\times
C^{1}(\overline{\Omega})$ and there exists a constant $L_{2}>0$ such that

(3.27)
$\left\|u\right\|_{C^{1}(\overline{\Omega})},\left\|v\right\|_{C^{1}(\overline{\Omega})}<L_{2}.$

Moreover, problem $(\mathrm{\tilde{P}}_{0})$ admits only a trivial solution.

###### Proof.

Arguing as in the proof of Lemma 2 we show that the solution set of problem
$(\mathrm{\tilde{P}}_{t})$ is bounded in $C^{1}(\overline{\Omega})\times
C^{1}(\overline{\Omega})$ uniformly with respect to $t\in[0,1]$. Indeed, by
contradiction, suppose that for every positive integer $n$, there exist
$t_{n}\in[0,1]$ and a solution $(u_{n},v_{n})$ of
$(\mathrm{\tilde{P}}_{t_{n}})$ such that $t_{n}\rightarrow t\in[0,1]$ and
$\|u_{n}\|_{C^{1}(\overline{\Omega})},\|v_{n}\|_{C^{1}(\overline{\Omega})}\rightarrow\infty$
as $n\rightarrow\infty.$ Observing that

$(u_{n}-\phi_{1})^{+}\leq|u_{n}^{+}-\phi_{1}|\text{ \ and \
}(v_{n}-\phi_{1})^{+}\leq|v_{n}^{+}-\phi_{1}|,$

the reasoning developed in the proof of Lemma 2 based on assumptions (1.3)
leads to the existence of a positive function
$(\mathcal{\bar{U}},\mathcal{\bar{V}})\in C^{1}(\overline{\Omega})\times
C^{1}(\overline{\Omega})$ solving the decoupled system (3.25). This leads to
the same contradiction as in the proof of Lemma 2, thereby completing the
proof of (3.27).

Finally, for $t=0$, $(\mathrm{\tilde{P}}_{0})$ is expressed as a decoupled
system:

$\left\\{\begin{array}[]{ll}-\Delta u+u=\lambda(u-\phi_{1})^{+}&\text{in
}\Omega,\\\ -\Delta v+v=\lambda(v-\phi_{1})^{+}&\text{in }\Omega,\\\
\frac{\partial u}{\partial\eta}=\frac{\partial v}{\partial\eta}=0\text{
}&\text{on }\partial\Omega,\end{array}\right.$

which, since $\lambda\in(0,\lambda_{1})$, admits only the trivial solution
$(u,v)=(0,0)$. ∎

######  Proposition 2.

Assume (1.3) holds. With a constant $L_{2}>0$, let the homotopy $\mathcal{N}$
on $\left[0,1\right]\times\overline{\mathcal{B}}_{L_{2}}(0)$ defined by

$\mathcal{N}(t,u,v)=(\mathcal{N}_{1}(t,u,v),\mathcal{N}_{2}(t,u,v)),$

with

$\mathcal{N}_{1}(t,u,v)=u-(-\Delta+I)^{-1}(t\frac{u^{\alpha_{1}}}{v^{\beta_{1}}}+(1-t)\lambda(u-\phi_{1})^{+}),$

and

$\mathcal{N}_{2}(t,u,v)=v-(-\Delta+I)^{-1}(t\frac{u^{\alpha_{2}}}{v^{\beta_{2}}}+(1-t)\lambda(v-\phi_{1})^{+}),$

If $L_{2}>0$ is sufficiently large, then the Leray-Schauder topological degree
$\deg(\mathcal{N}(t,\cdot,\cdot),\mathcal{B}_{L_{2}}(0),0)$ is well defined
for every $t\in[0,1],$ and it holds

(3.28)
$\deg\left(\mathcal{N}(1,\cdot,\cdot),\mathcal{B}_{L_{2}}(0),0\right)=1.$

###### Proof.

By a similar argument to the one showing (3.21), we infer that all solutions
$(u,v)\in C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega})$ of
$(\mathrm{\tilde{P}}_{t})$ satisfy $u,v>\hat{\rho}$ on $\overline{\Omega}$,
for certain constant $\hat{\rho}>0,$ for all $t\in[0,1]$. Hence, due to the
continuity of the right hand-side of the equations in $(\mathrm{P}_{t})$ for
almost everywhere in $\Omega$, the homotopies
$\mathcal{N}_{1}$,$\mathcal{N}_{2}$ are well defined. Moreover, from the
compactness of the operators $(-\Delta+I)^{-1}:C(\overline{\Omega})\rightarrow
C^{1}(\overline{\Omega})$ we derive that
$\mathcal{N}_{1},\mathcal{N}_{2}:\left[0,1\right]\times
C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega})\rightarrow
C(\overline{\Omega})$ are completely continuous.

By the definition of $\mathcal{N}$, we infer that
$(u,v)\in\mathcal{B}_{L_{2}}(0)$ is a solution of system $(\mathrm{P})$ if,
and only if,

$\begin{array}[]{c}(u,v)\in\mathcal{B}_{L_{2}}(0)\,\,\,\text{and}\,\,\,\mathcal{N}(1,u,v)=0.\end{array}$

In view of Lemma 3 solutions of $(\mathrm{\tilde{P}}_{t})$ lie in
$\mathcal{B}_{L_{2}}(0)$ and

$\deg\left(\mathcal{N}(0,\cdot,\cdot),\mathcal{B}_{L_{2}}(0),0\right)=1.$

Consequently, the homotopy invariance property leads to (3.28). ∎

####  3.2.3. The degree on
$\mathcal{B}_{L_{1}}(0)\backslash\overline{\mathcal{B}}_{L_{2}}(0)$.

######  Proposition 3.

Assume that (1.3) is satisfied. Let the map
$\mathcal{M}:C^{1}(\overline{\Omega})\times
C^{1}(\overline{\Omega})\rightarrow C(\overline{\Omega})$ be given by the
following:

$\mathcal{M}(u,v)=\left(u-(-\Delta+I)^{-1}(\frac{u^{\alpha_{1}}}{v^{\beta_{1}}}),v-(-\Delta+I)^{-1}(\frac{u^{\alpha_{2}}}{v^{\beta_{2}}})\right).$

For constants $L_{1},L_{2}>0$ with $L_{1}>L_{2}$, if the Leray-Schauder
topological degree $\deg(\mathcal{M}(\cdot,\cdot),\mathcal{B}_{L_{i}}(0),0)$
is well defined for $i=1,2$, then

(3.29)
$\deg(\mathcal{M}(\cdot,\cdot),\mathcal{B}_{L_{1}}(0)\backslash\overline{\mathcal{B}}_{L_{2}}(0),0)\neq
0.$

###### Proof.

Fix $L_{2}>0$ such that the conclusion of Proposition 2 be satisfied and
choose $L_{1}>L_{2}$ so large to fulfil the conclusion of Proposition 1. The
weak comparison principle ([31, Lemma 3.2]) applied to problems
$(\mathrm{P}_{t})$ and $(\mathrm{\tilde{P}}_{t})$ by making use of the
inequality $(s-\phi_{1})^{+}\leq|s^{+}-\phi_{1}|,$ for all $s\in\mathbb{R}$,
implies that the inclusion
$\overline{\mathcal{B}}_{L_{2}}(0)\subset\mathcal{B}_{L_{1}}(0)$ holds. In
view of the expressions of $\mathcal{M}$ and the homotopies $\mathcal{H}$,
$\mathcal{N}$, it is seen that

$\mathcal{H}(1,\cdot,\cdot)=\mathcal{N}(1,\cdot,\cdot)\,\,\,\text{in}\,\,\,\overline{\mathcal{B}}_{L_{2}}(0)$

and

(3.30)
$\mathcal{M}(\cdot,\cdot)=\mathcal{H}(1,\cdot,\cdot)\,\,\,\text{in}\,\,\,\mathcal{B}_{L_{1}}(0).$

Moreover, since $\mathcal{H}(1,\cdot,\cdot)$ and $\mathcal{N}(1,\cdot,\cdot)$
do not vanish on $\partial\mathcal{B}_{L_{1}}(0)$ and
$\partial\mathcal{B}_{L_{2}}(0)$, respectively, the Leray-Schauder topological
degree of $\mathcal{H}(1,\cdot,\cdot)$ on
$\mathcal{B}_{L_{1}}\backslash\overline{\mathcal{B}}_{L_{2}}$ makes sens. By
the excision property of the degree, we get

$\begin{array}[]{l}\deg(\mathcal{H}(1,\cdot,\cdot),\mathcal{B}_{L_{1}}(0),0)=\deg(\mathcal{H}(1,\cdot,\cdot),\mathcal{B}_{L_{1}}(0)\backslash\partial\mathcal{B}_{L_{2}}(0),0).\end{array}$

By virtue of the domain additivity property of the degree it follows that

$\begin{array}[]{c}\deg(\mathcal{H}(1,\cdot,\cdot),\mathcal{B}_{L_{1}}(0)\backslash\overline{\mathcal{B}}_{L_{2}}(0),0)+\deg(\mathcal{H}(1,\cdot,\cdot),\mathcal{B}_{L_{2}}(0),0)\\\
=\deg(\mathcal{H}(1,\cdot,\cdot),\mathcal{B}_{L_{1}}(0),0).\end{array}$

Hence by (3.26), (3.28) and bearing in mind (3.30), we deduce that

$\begin{array}[]{l}\deg(\mathcal{M}(\cdot,\cdot),\mathcal{B}_{L_{1}}(0)\backslash\overline{\mathcal{B}}_{L_{2}}(0),0)\\\
=\deg(\mathcal{H}(1,\cdot,\cdot),\mathcal{B}_{L_{1}}(0)\backslash\overline{\mathcal{B}}_{L_{2}}(0),0)=-1.\end{array}$

This ends the proof. ∎

###  3.3. Proof of the main result (Existence of a third solution)

###### Proof of Theorem 1.

Two distinct positive solutions
$(u_{1},v_{1})\in[C^{-1}\phi_{1},Cz]\times[C^{-1}\phi_{1},Cz]$ and
$(u_{2},v_{2})\in[C^{-1}y_{\delta},Cy]\times[C^{-1}y_{\delta},Cy],$ belonging
to $C^{1,\tau}(\overline{\Omega})\times C^{1,\tau}(\overline{\Omega})$ for
some $\tau\in(0,1),$ are obtained for system $(\mathrm{P})$ by Theorems 3 and
4. The proof of Theorem 1 is completed by showing that $(\mathrm{P})$ admits a
positive solution $(u_{3},v_{3})\in C^{1,\tau}(\overline{\Omega})\times
C^{1,\tau}(\overline{\Omega}),$ for certain $\tau\in(0,1)$, such that

(3.31) $u_{3}\neq u_{i}\text{ \ and \ }v_{3}\neq v_{i}\text{, }i=1,2.$

Fix

$L_{2}>C\max\\{\left\|z\right\|_{C^{1,\tau}(\overline{\Omega})},\left\|y\right\|_{C^{1,\tau}(\overline{\Omega})}\\}$

such that the conclusion of Proposition 2 be satisfied. As a result, both
positive solutions $(u_{1},v_{1})$ and $(u_{2},v_{2})$ lie in
$\mathcal{B}_{L_{2}}(0)$. Now, we take $L_{1}>L_{2}$ such that the conclusion
of Proposition 1 be fulfilled. Here, without any loss of generality, we may
assume that the constant $L_{1}$ is sufficiently large so that the ball
$\mathcal{B}_{L_{1}}(0)$ contains all $C^{1}$-bound solutions of
$(\mathrm{P})$. Otherwise, there would be an infinite number of solutions with
$C^{1}$-regularity and the proof of Theorem 1 is therefore completed.

Thus, in view of Proposition 3 there exists $(u_{3},v_{3})\in
C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega})$ satisfying
$\mathcal{M}(u_{3},v_{3})=0$. This implies that the pair $(u_{3},v_{3})$ is a
solution of $(\mathrm{P})$. Since
$(u_{3},v_{3})\in\mathcal{B}_{L_{1}}(0)\backslash\overline{\mathcal{B}}_{L_{2}}(0)$
and the ordered rectangles $[C^{-1}\phi_{1},Cz]\times[C^{-1}\phi_{1},Cz]$ and
$[C^{-1}y_{\delta},Cy]\times[C^{-1}y_{\delta},Cy]$ are contained in the ball
$\mathcal{B}_{L_{2}}(0)$, the assertion (3.31) holds true and therefore
$(u_{3},v_{3})$ is a third nontrivial positive solution of $(\mathrm{P})$. The
regularity theory (see [19, Theorem 1.2]) ensures that $(u_{3},v_{3})\in
C^{1,\tau}(\overline{\Omega})\times C^{1,\tau}(\overline{\Omega})$ for some
$\tau\in(0,1)$. ∎

## References

  * [1] H. Brézis, _Analyse fonctionnelle theorie et applications_ , Masson, Paris, 1983.
  * [2] E. Casas and L.A. Fernandez, _A Green’s formula for quasilinear elliptic operators_ , J. Math. Anal. Appl. 142 (1989), 62–73.
  * [3] Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt system of elliptic equations: the classical case, Nonlinear Anal. 55 (2003), 521-541.
  * [4] M. Del Pino, M. Kowalczyk and X. Chen, The Gierer-Meinhardt system: the breaking of homoclinics and multi-bump ground states, Commun. Contemp. Math. 3 (2001), 419-439.
  * [5] M. Del Pino, M. Kowalczyk and J. Wei, Multi-bump ground states of the Gierer-Meinhardt system in $\mathbb{R}^{2}$, Ann. Inst. H. Poincaré, Anal. Non Linéaire 20 (2003), 53-85.
  * [6] H. Dellouche and A. Moussaoui, _Singular quasilinear elliptic systems with gradient dependence_ , Positivity 26 (2022), doi:10.1007/s11117-022-00868-3.
  * [7] H. Didi & A. Moussaoui, _Multiple positive solutions for a class of quasilinear singular elliptic systems_ , Rend. Circ. Mat. Palermo, II. Ser 69 (2020), 977-994.
  * [8] H. Didi, B. Khodja & A. Moussaoui, _Singular Quasilinear Elliptic Systems With (super-) Homogeneous Condition_ , J. Sibe. Fede. Univ. Math. Phys. 13(2 ) (2020), 1-9.
  * [9] M. Ghergu and V. Radulescu, On a class of Gierer-Meinhardt systems arising in morphogenesis, C. R.Acad.Sci. Paris, Ser.I 344 (2007).
  * [10] C. Ghoussoub and C. Gui, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z. 229 (1998), 443-474.
  * [11] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972), 30–39.
  * [12] C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Lin´eaire 17 (2000), 47-82.
  * [13] D. D. Hai, _On a class of singular p-Laplacian boundary value problems_ , J. Math. Anal. Appl. 383 (2011), 619-626.
  * [14] J.P. Keener, Activators and inhibitors in pattern formation, Stud. Appl. Math. 59 (1978), 1–23.
  * [15] B. Khodja & A. Moussaoui, _Positive solutions for infinite semipositone/positone quasilinear elliptic systems with singular and superlinear terms_ , Diff. Eqts. App. 8(4) (2016), 535-546.
  * [16] E.H. Kim, _Singular Gierer-Meinhardt systems of elliptic boundary value problems_ , J. Math. Anal. Appl. (1) 308 (2005), 1–10.
  * [17] T. Kolokolonikov and X. Ren, Smoke-ring solutions of Gierer-Meinhardt System in R3 , SIAM Journal on Applied Dynamical Systems, 10(1):251-277, 2011.
  * [18] T. Kolokolonikov, J. Wei and W. Yang, On large ring solutions for Gierer–Meinhardt system in $\mathbb{R}^{3}$, J. Diff. Eqts. 255 (2013), 1408-1436.
  * [19] G. M. Lieberman, _Boundary regularity for solutions of degenerate elliptic equations_ , Nonl. Anal. 12 (1988), 1203-1219.
  * [20] H. Meinhardt, Models of biological pattern formation, Acad. Press, London, 1982.
  * [21] D. Motreanu, V.V. Motreanu & N. Papageorgiou, _Topological and variational methods with applications to nonlinear boundary value problems_. Springer, New York, 2014.
  * [22] D. Motreanu & A. Moussaoui, _An existence result for a class of quasilinear singular competitive elliptic systems_ , Appl. Math. Lett.__ 38 (2014), 33-37.
  * [23] D. Motreanu & A. Moussaoui, A quasilinear singular elliptic system without cooperative structure, Acta Math. Sci.__ 34 (B) (2014), 905-916.
  * [24] D. Motreanu & A. Moussaoui, _Existence and boundedness of solutions for a singular cooperative quasilinear elliptic system_ , Complex Var. Elliptic Equ. 59 (2014), 285-296.
  * [25] A. Moussaoui, B. Khodja and S. Tas, A singular Gierer Meinhardt system of elliptic equations in $\mathbb{R}^{N},$ Nonlinear Analysis 71 (2009), 708-716.
  * [26] D. Mugnai and N. S. Papageorgiou. Resonant nonlinear Neumann problems with indefinite weight. Ann. Scuola Norm. Super. Pisa Cl. Sci., 11 (2012), 729-788.
  * [27] W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices of Amer. Math. Soc. 45 (1998), 9-18.
  * [28] W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), 819–851.
  * [29] W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247–281.
  * [30] W. M. Ni, I. Takagi and E. Yanagida, Stability of least energy patterns of the shadow system for an activator-inhibitor model, Japan J.Industrial Appl. Math. 18(2)(2001), 259-272.
  * [31] K. Sreenadh and S. Tiwari, Global multiplicity results for $p(x)$-Laplacian equation with nonlinear Neumann boundary condition, Diff. Integral Eqts. (7-8) 26 (2013), 815-836.
  * [32] J.L. Vázquez, _A strong maximum principle for some quasilinear elliptic equations_ , Appl. Math. Optimization 12 (1984), 191–202.
  * [33] J. Wei, Existence and stability of spikes for the Gierer-Meinhardt system, Handbook of Differential Equations-stationary partial differential equations, Vol. 5, Elsevier, 489-581.
  * [34] J. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem, J. Diff. Eqns 134 (1997), 104-133.
  * [35] J. Wei and M. Winter, On the two-dimensional Gierer-Meinhardt system with strong coupling, SIAM J.Math.Anal. 30 (1999), 1241-1263.
  * [36] J. Wei and M. Winter, Spikes for the Gierer-Meinhardt system in two dimensions: the strong coupling case, J. Diff. Eqts 178 (2004), 478-518.
  * [37] J. Wei and M. Winter, Existence and stability analysis of asymmetric for the Gierer-Meinhardt system, J. Math. Pures Appl. 83 (2004), 433-476.
  * [38] J. Wei, On single interior spike solutions of Gierer-Meinhardt system: uniqueness and spectrum estimates, Europ. J. Appl. Math. 10 (1999), 353-378.
  * [39] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonlinear Science 6(2001), 415-458.
  * [40] E. Zeidler, Nonlinear functional analysis and its applications. I. Fixed-point theorems. Springer-Verlag New York, 1986.