# Non-homogeneous $(p_{1},p_{2})$-fractional Laplacian systems with lack of
compactness

Debangana Mukherjee  Department of Mathematics, Indian Institute of Science
Education and Research, Dr. Homi Bhaba Road, Pune, 411008, India
<EMAIL_ADDRESS>and  Tuhina Mukherjee  Department of Mathematics, Indian
Institute of Technology Jodhpur, Rajasthan-506004, India-342037
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The present paper studies the existence of weak solutions for following type
of non-homogeneous system of equations

$(S)\left\\{\begin{aligned}
(-\Delta)^{s_{1}}_{p_{1}}u&=u|u|^{\alpha-1}|v|^{\beta+1}+f_{1}(x)\,\mbox{ in
}\,\Omega,\\\
(-\Delta)^{s_{2}}_{p_{2}}v&=|u|^{\alpha+1}v|v|^{\beta-1}+f_{2}(x)\,\mbox{ in
}\,\Omega,\\\ u=v&=0\,\mbox{ in }\,\mathbb{R}^{N}\setminus\Omega,\\\
\end{aligned}\right.$

where $\Omega\subset\mathbb{R}^{N}$ is smooth bounded domain,
$s_{1},s_{2}\in(0,1)$, $1<p_{1},p_{2}<\infty$,
$N>\max\\{p_{1}s_{1},p_{2}s_{2}\\}$, $\alpha>-1$ and $\beta>-1$. We employ the
variational techniques where the associated energy functional is minimized
over Nehari manifold set while imposing appropriate bound on dual norms of
$f_{1},f_{2}$.

###### Key words and phrases:

$(p_{1},p_{2})$\- Fractional Laplacian, Elliptic system, Critical exponent,
Non homogeneous equations.

###### 2010 Mathematics Subject Classification:

Primary 35R11, 35J20, 49J35, secondary 47G20, 45G05

##  1\. Introduction

The study of elliptic equations and systems involving fractional and
$p$-fractional Laplace operators have dynamized a lot of application in
nonlinear analysis. In recent years, enormous attention is given to the
research on very delicate reading about the existence and multiplicity of
solutions for such elliptic problems. The present article deals with non-
homogeneous system of equations involving $(p_{1},p_{2})$-type fractional
operators $(-\Delta)^{s_{i}}_{p_{i}},i=1,2$ defined as:

$(-\Delta)_{p_{i}}^{s_{i}}u(x):=\lim_{\epsilon\to
0}\int_{\mathbb{R}^{N}\setminus
B_{\epsilon}(x)}\frac{|u(x)-u(y)|^{p_{i}-2}(u(x)-u(y))}{|x-y|^{N+p_{i}s_{i}}}~{}dy,\;\;\text{for}\;\;x\in\mathbb{R}^{N}$
(1.1)

where $u\in C_{c}^{\infty}(\mathbb{R}^{N})$ and $B_{\epsilon}(x)$ denotes ball
in $\mathbb{R}^{N}$ with centre $x$ and radius $\epsilon$. When $s_{1}=s_{2}$
the equation reduces to a $(p_{1},p_{2})$-type Laplacian problem which appears
in a more general reaction-diffusion system

$u_{t}=\text{div}(a(u)\nabla u)+g(x,u)$ (1.2)

where $a(u)=|\nabla u|^{p-2}\nabla u+|\nabla u|^{q-2}\nabla u$. Such problems
have a wide range of applications in physics and related sciences such as
biophysics, plasma physics, and chemical reaction design, etc. where $u$
describes a concentration, and the first term on the right-hand side of (1.2)
corresponds to a diffusion with a diffusion coefficient $a(u)$; the term
$g(x,u)$ stands for the reaction, related to sources and energy-loss
processes. A lot of attention has been given to the study of
$(p_{1},p_{2})$-Laplace equations in the last few years, for instance, we cite
some attributing references [12, 14, 15, 20, 21, 23].

We are attentive in considering the system of non-homogeneous equations
involving $((s_{1},p_{1}),(s_{2},p_{2}))$ fractional Laplacian operators. In
case $s_{1}=s_{2}=1$, the non-homogeneous elliptic systems were initially
researched by J. Chabrowski [6] where the author has established multiple
solutions for

$\displaystyle(-\Delta)_{p_{1}}u$ $\displaystyle=\lambda
u|u|^{\alpha-1}|v|^{\beta+1}+f_{1}\,\mbox{ in }\,\Omega,$
$\displaystyle(-\Delta)_{p_{2}}v$
$\displaystyle=\lambda|u|^{\alpha+1}v|v|^{\beta-1}+f_{2}\,\mbox{ in
}\,\Omega,$ $\displaystyle u=v$ $\displaystyle=0\,\mbox{ in
}\,\mathbb{R}^{N}\setminus\Omega,$

where $\Omega\subset\mathbb{R}^{N}$ is bounded, $\lambda>0$,
$1<p_{1},p_{2}<N$, $-1<\alpha,\beta$. If we take $s_{1}=s_{2}=1$, then our
problem $(S)$ is also studied by J. Velin [22] over a regular bounded set
$\Omega$ and by Benmouloud et al. [2] over $\Omega$ any open set. The couple
of papers have adopted variational techniques with enthralling use of a
maximum norm over the product function space. Later on, a class of
$(p,q)$-Laplacian system with indefinite weights have been learned by Afrouzi
et al. in [1] via local minimization techniques. For interested readers, we
point out some striking articles [10, 13, 16, 18, 19] which deals with various
kind of elliptic systems involving the $(p,q)$-Laplace operator. In [10], the
authors have analyzed the existence of positive solutions to the non-
cooperative fractional elliptic system and the scheme heavily relies upon the
Schauder fixed point theorem. Using the mechanism of sub-super solutions, the
authors in [13] have achieved captivating results about the existence of a
weak solution.

Driven by the above research, our prospect in this article is to establish the
existence results for $(S)$. There are very few and sparse works till now
pledging with $((s_{1},p_{1}),(s_{2},p_{2}))$\- fractional Laplacian system of
equations, for instance, we cite [17]. We also present a very recent and
stirring article [3] which deals with non-homogeneous semilinear fractional
system with lack of compactness. The essence of our article lies in the fact
that we contemplate on systems of equation with fractional
$(p_{1},p_{2})$-Laplace operator, particularly, the quasilinear case over a
bounded domain which is first of its kind in literature. Moreover, we are not
assuming any fixed relation between $s_{1},s_{2}$. Since the nonlinearity is
of critical growth, we face a lack of compactness situation even in the
bounded domain case. The approach to handle our problem is variational and we
do a local minimization to establish existence results.

Organisation of the paper: The present paper is classified into the following
sections- Section 2 which is again divided into four parts- first part deals
with the suitable function spaces with which we have worked on, second part
intents to cover some notations and symbols used throughout the paper, third
part is schemed for some hypotheses needed to establish our main outcome and
the last part is designed for our main result. Section 3 embraces study of the
problem over bounded domains and the proof of our main result.

##  2\. Functional Setting and Main result

###  2.1. Functional Setting

In this section, we interpret appropriate function spaces which are imperative
for our analysis. Let $p>1,\,s\in(0,1),\,N>ps,\,p_{s}^{*}:=\frac{Np}{N-sp}.$
We denote the standard fractional Sobolev space by $W^{s,p}(\Omega)$ endowed
with the norm

$\|{u}\|_{W^{s,p}(\Omega)}:=\|{u}\|_{L^{p}(\Omega)}+\left(\int_{\Omega\times\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)^{1/p}.$

We set $Q:=\mathbb{R}^{2N}\setminus(\Omega^{c}\times\Omega^{c})$, where
$\Omega^{c}=\mathbb{R}^{N}\setminus\Omega$ and define

$X_{s,p}(\Omega):=\Big{\\{}u:\mathbb{R}^{N}\to\mathbb{R}\mbox{ measurable
}\Big{|}u|_{\Omega}\in L^{p}(\Omega)\mbox{ and
}\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy<\infty\Big{\\}}.$

The space $X_{s,p}(\Omega)$ is endowed with the norm defined as

$\|u\|_{s,p}:=|u|_{L^{p}(\Omega)}+\left(\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)^{1/p}.$

We note that in general $W^{s,p}(\Omega)$ is not same as $X_{s,p}(\Omega)$ as
$\Omega\times\Omega$ is strictly contained in $Q$. We define the space
$X_{0,s,p}(\Omega)$ as

$X_{0,s,p}(\Omega):=\Big{\\{}u\in X_{s,p}:u=0\quad\text{a.e.
in}\quad\mathbb{R}^{N}\setminus\Omega\Big{\\}}$

or equivalently as $\overline{C_{0}^{\infty}(\Omega)}^{X_{s,p}(\Omega)}$. It
is well-known that for $p>1$, $X_{0,s,p}(\Omega)$ is a uniformly convex Banach
space endowed with the norm

$[u]_{s,p}:=\|u\|_{0,s,p}=\left(\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)^{1/p}.$

Since $u=0$ in $\mathbb{R}^{N}\setminus\Omega,$ the above integral can be
extended to all of $\mathbb{R}^{N}.$ The embedding
$X_{0,s,p}(\Omega)\hookrightarrow L^{r}(\Omega)$ is continuous for any
$r\in[1,p^{*}_{s}]$ and compact for $r\in[1,p^{*}_{s}).$ Moreover, for
$1<p_{2}\leq p_{1}$, $X_{0,s_{1},p_{1}}(\Omega)\subset
X_{0,s_{2},p_{2}}(\Omega)$ (see Lemma 2.2 in Section 2 of [4]).

In the present article, we work with the product space

$X=X_{0,s_{1},p_{1}}(\Omega)\times X_{0,s_{2},p_{2}}(\Omega)$

equipped with the norm

$\|(u,v)\|=\max\\{[u]_{s_{1},p_{1}},[v]_{s_{2},p_{2}}\\}$

such that $(X,\|\cdot\|)$ forms a reflexive, separable Banach space. If we
consider the norm

$\|(u,v)\|_{0}=[u]_{s_{1},p_{1}}+[v]_{s_{2},p_{2}}$

over the space $X$ then it is easy to verify that $\|\cdot\|_{0}$ is
equivalent to the norm $\|\cdot\|$ over $X$. We represent the dual space of
$X$ as $X^{*}$ and $\left\langle\cdot,\cdot\right\rangle$ denotes the duality
pairing of $X$ and $X^{*}$.

###  2.2. Notations and symbols

In this section, we announce some notations and symbols which will be used all
through the paper.

  * •

For $i=1,2$, $s_{i}\in(0,1)$, we define

$\dot{W}^{s_{i},p_{i}}(\mathbb{R}^{N}):=\bigg{\\{}u\in
L^{p_{i}^{*}}(\mathbb{R}^{N}):\displaystyle\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+s_{i}p_{i}}}\,dxdy<\infty\bigg{\\}}$

and

$S_{p_{i}}=\inf_{u\in\dot{W}^{s_{i},p_{i}}(\mathbb{R}^{N})\setminus\\{0\\}}\frac{\displaystyle\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+s_{i}p_{i}}}dxdy}{\bigg{(}\displaystyle\int_{\mathbb{R}^{N}}|u|^{p^{*}_{i}}\bigg{)}^{\frac{p}{p^{*}_{i}}}},$
(2.1)

where $p_{i}^{*}=\frac{Np_{i}}{N-s_{i}p_{i}}$.

  * •

For $r>0,t>0$, we set

$\displaystyle
a(t)=\frac{1}{t}-\frac{1}{\alpha+\beta+2},\;\;b(r,t)=\frac{(r+1)(t-1)}{(\alpha+\beta+2)(\alpha+\beta+1)},$
$\displaystyle
c(t)=\frac{(\alpha+\beta+2-t)}{\alpha+\beta+1},\;\;d(r,t)=\frac{1}{\frac{(\alpha+1)}{p^{\prime}_{1}r^{p^{\prime}_{1}}}+\frac{(\beta+1)}{p^{\prime}_{2}t^{p^{\prime}_{2}}}},$

where $p_{i}^{\prime}=\frac{p_{i}}{p_{i}-1}$ for $i=1,2$.

  * •

We signify

$\displaystyle\varepsilon_{1}=(\alpha+1)d(\nu,\mu)\bigg{(}c(p_{1})-\frac{\nu^{p_{1}}}{p_{1}}\bigg{)}\bigg{(}b(\alpha,p_{1})\min(S_{p_{1}}^{p^{*}_{1}/p_{1}},S_{p_{2}}^{p^{*}_{2}/p_{2}})\bigg{)}^{\frac{p_{1}}{p_{1}^{*}-p_{1}}}$
$\displaystyle\varepsilon_{2}=(\beta+1)d(\nu,\mu)\bigg{(}c(p_{2})-\frac{\mu^{p_{2}}}{p_{2}}\bigg{)}\bigg{(}b(\beta,p_{2})\min(S_{p_{1}}^{p^{*}_{1}/p_{1}},S_{p_{2}}^{p^{*}_{2}/p_{2}})\bigg{)}^{\frac{p_{2}}{p_{2}^{*}-p_{2}}},$

where $\nu$ and $\mu$ are fixed numbers such that

$0<\nu<[p_{1}\,c(p_{1})]^{\frac{1}{p_{1}}},\,\,0<\mu<[p_{2}\,c(p_{2})]^{\frac{1}{p_{2}}}.$

  * •

For $i=1,2$, we set

$\left\langle f_{i},u\right\rangle_{i}=\int_{\Omega}f_{i}u_{i}~{}dx.$

###  2.3. Hypothesis

We propose the following hypotheses which will be employed in the entire
article.

  * $\textit{(}H_{1})$

$(\alpha+1)\frac{(N-s_{1}p_{1})}{Np_{1}}+(\beta+1)\frac{(N-s_{2}p_{2})}{Np_{2}}=1$.

  * $\textit{(}H_{2})$

$\max\\{p_{1},p_{2}\\}<\alpha+\beta+2$

  * $\textit{(}H_{3})$

$0<\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}(\Omega)}+\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}(\Omega)}<\min\\{\varepsilon_{1},\varepsilon_{2},1\\}$.

###  2.4. Main result

Towards the primary objective, we are involved in analyzing the following
problem

$(\mathcal{P}_{f_{1},f_{2}})\left\\{\begin{aligned}
(-\Delta)^{s_{1}}_{p_{1}}u&=u|u|^{\alpha-1}|v|^{\beta+1}+f_{1}(x)\,\mbox{ in
}\,\Omega,\\\
(-\Delta)^{s_{2}}_{p_{2}}v&=|u|^{\alpha+1}v|v|^{\beta-1}+f_{2}(x)\,\mbox{ in
}\,\Omega,\\\ u=v&=0\,\mbox{ in }\,\mathbb{R}^{N}\setminus\Omega,\\\
\end{aligned}\right.$

where $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain,
$s_{1},s_{2}\in(0,1)$; $f_{1}$ and $f_{2}$ are not identically zero on
$\Omega$. Furthermore, $\alpha>-1$, $\beta>-1$.

Our goal is to exercise on the existence of solutions for the above model of
elliptic system associated with the critical Sobolev exponents in the nonlocal
framework. We fetch this result when $f_{1}$ and $f_{2}$ are chosen small in
the sense of dual norm. The outcome of our research is planted in the
sequential steps:

  * •

some fundamental results are obtained;

  * •

the associated energy functional has a negative infimum on a suitable
manifold;

  * •

a minimization solution to the problem $(\mathcal{P}_{f_{1},f_{2}})$ is
achieved;

  * •

some property of this minimizing solution is also retrieved;

  * •

the existence of a weak solution for the problem $(P_{f_{1},f_{2}})$ is
accomplished.

Our main result in this manuscript concerning $(\mathcal{P}_{f_{1},f_{2}})$ is
demonstrated below:

######  Theorem 2.1.

Let $f_{i}\in X_{0,s_{i},p_{i}}(\Omega)^{*}$ for $i=1,2$ and $\Omega$ be
sufficiently smooth bounded domain in $\mathbb{R}^{N}$. We furthermore assume
that $\textit{(}H_{1}),\textit{(}H_{2}),\textit{(}H_{3})$ holds true. Then
there exists $(u^{*},v^{*})\in\Theta$ which solves
$(\mathcal{P}_{f_{1},f_{2}})$ and satisfies $\mathcal{I}(u^{*},v^{*})<0$.

In the upcoming section, we meticulously take care of the some elementary
lemmas, followed by some pivotal results to study our problem
$(\mathcal{P}_{f_{1},f_{2}})$ in bounded domain designed in the nonlocal
pattern.

##  3\.  System over bounded domain

In this section, we move our attention in proving the existence of solutions
for the elliptic system $(\mathcal{P}_{f_{1},f_{2}})$ involving critical
Sobolev exponents. Owing to the nonlocal picture, we design the methodology in
a very delicate manner using suitable ingredients by a local minimization for
an adapted variational problem in the $(s_{i},p_{i})$-fractional scheme for
$i=1,2$. We consider the corresponding energy functional
$\mathcal{I}:X\to\mathbb{R}$ as

$\displaystyle\mathcal{I}(u_{1},u_{2})$
$\displaystyle=\frac{\alpha+1}{p_{1}}[u_{1}]^{p_{1}}_{s_{1},p_{1}}+\frac{\beta+1}{p_{2}}[u_{2}]^{p_{2}}_{s_{2},p_{2}}-\int_{\Omega}|u_{1}|^{\alpha+1}|u_{2}|^{\beta+1}\,dx$
$\displaystyle\quad-(\alpha+1)\left\langle
f_{1},u_{1}\right\rangle_{1}-(\beta+1)\left\langle
f_{2},u_{2}\right\rangle_{2}$

for $(u_{1},u_{2})\in X$. Let $\Theta\subset X$ be defined as

$\displaystyle\Theta:=\big{\\{}(u,v)\in
X\setminus\\{0\\}:\left\langle\mathcal{I}^{\prime}(u,v),(u,v)\right\rangle=0\big{\\}}$
(3.1)

which gives us that

$\displaystyle\mathcal{I}_{|\Theta}(u,v)$
$\displaystyle=(\alpha+1)a(p_{1})[u]_{s_{1},p_{1}}^{p_{1}}+(\beta+1)a(p_{2})[v]_{s_{2},p_{2}}^{p_{2}}$
$\displaystyle\qquad-(\alpha+1)a(1)\left\langle
f_{1},u\right\rangle_{1}-(\beta+1)a(1)\left\langle f_{2},v\right\rangle_{2}.$

###  3.1. Some Preliminary Lemmas

To obtain our main result, we need the following preliminary results by a
local minimization.

######  Lemma 3.1.

Let $\Theta$ be defined as in (3.1) then $\Theta\neq\emptyset.$

###### Proof.

Let $u_{f_{1}}\in X_{0,s_{1},p_{1}}(\Omega)$ be the unique non-trivial
solution of the problem

$\begin{cases}\begin{aligned} (-\Delta)^{s_{1}}_{p_{1}}u&=f_{1}\,\text{ in
}\,\Omega,\\\ u&=0\,\text{ in
}\,\mathbb{R}^{N}\setminus\Omega.\end{aligned}\end{cases}$

Then, $[u]^{p_{1}}_{0,s_{1},p_{1}}=\left\langle
f_{1},u_{f_{1}}\right\rangle_{1}$. Therefore, by the definition of
$\mathcal{I}(\cdot,\cdot)$ we have

$\left\langle\mathcal{I}^{\prime}(u_{f_{1}},0),(u_{f_{1}},0)\right\rangle=(\alpha+1)[u_{f_{1}}]_{s_{1},p_{1}}^{p_{1}}-(\alpha+1)\left\langle
f_{1},u_{f_{1}}\right\rangle_{1}=0.$

Hence, $(u_{f_{1}},0)$ is non-trivial and $(u_{f_{1}},0)\in\Theta$. ∎

######  Lemma 3.2.

Let $f_{i}\in X^{*}_{0,s_{i},p_{i}}$ and $u_{f_{i}}$ for $i=1,2$ be the non
solution of

$\begin{cases}\begin{aligned} (-\Delta)^{s_{i}}_{p_{i}}u&=f_{i}\,\text{ in
}\,\Omega,\\\ u&=0\,\text{ in
}\,\mathbb{R}^{N}\setminus\Omega.\end{aligned}\end{cases}$ (3.2)

Then, we have

$[u_{f_{i}}]_{s_{i},p_{i}}^{p_{i}}=\|f_{i}\|_{X^{*}_{0,s_{i},p_{i}}}^{p^{\prime}_{i}}$

where $p_{i}^{\prime}=\frac{p_{i}}{p_{i}-1}$ is the conjugate of $p_{i}$.

###### Proof.

Recall $u_{f_{i}}$ be the unique solution of (3.2) which gives that for
$\Phi_{i}\in X_{0,s_{i},p_{i}}(\Omega)\setminus\\{0\\}$,

$\int_{\mathbb{R}^{2N}}\frac{|u_{f_{i}}(x)-u_{f_{i}}(y)|^{p_{i}-2}(u_{f_{i}(x)}-u_{f_{i}(y)})(\Phi_{i}(x)-\Phi_{i}(y))}{|x-y|^{N+s_{i}p_{i}}}\,dxdy=\left\langle
f_{i},\Phi_{i}\right\rangle_{i}.$ (3.3)

Using Hölder’s inequality, we note that

$\displaystyle\left\langle f_{i},\Phi_{i}\right\rangle_{i}$
$\displaystyle\leq\int_{\mathbb{R}^{2N}}\frac{|u_{f_{i}}(x)-u_{f_{i}}(y)|^{p_{i}-1}|\Phi_{i}(x)-\Phi_{i}(y)|}{|x-y|^{N+s_{i}p_{i}}}\,dxdy$
$\displaystyle\leq\left(\int_{\mathbb{R}^{2N}}\frac{|u_{f_{i}}(x)-u_{f_{i}}(y)|^{p_{i}}}{|x-y|^{N+s_{i}p_{i}}}\,dxdy\right)^{\frac{p_{i}-1}{p_{i}}}\left(\int_{\mathbb{R}^{2N}}\frac{|\Phi_{i}(x)-\Phi_{i}(y)|^{p_{i}}}{|x-y|^{N+s_{i}p_{i}}}\,dxdy\right)^{\frac{1}{p_{i}}}$
$\displaystyle=[u_{f_{i}}]_{s_{i},p_{i}}^{p_{i}-1}[\Phi_{i}]_{s_{i},p_{i}}.$

This implies,

$\displaystyle\Bigg{|}\frac{\left\langle
f_{i},\Phi_{i}\right\rangle_{i}}{[\Phi]_{s_{i},p_{i}}}\Bigg{|}\leq[u_{f_{i}}]_{s_{i},p_{i}}^{p_{i}-1},$

which immediately yields us

$\displaystyle\|f_{i}\|_{X^{*}_{0,s_{i},p_{i}}}^{p^{\prime}_{i}}\leq[u_{f_{i}}]_{s_{i},p_{i}}^{p_{i}}.$
(3.4)

On the other hand, taking $\Phi_{i}=u_{f_{i}}$ in (3.3) we get,

$\displaystyle[u_{f_{i}}]_{s_{i},p_{i}}^{p_{i}}\leq\|f_{i}\|^{p^{\prime}_{i}}_{X^{*}_{0,s_{i},p_{i}}}.$
(3.5)

Combining (3.4) and (3.5) we come by the following conclusion,

$\displaystyle[u_{f_{i}}]_{s_{i},p_{i}}^{p_{i}}=\|f_{i}\|_{X^{*}_{0,s_{i},p_{i}}}^{p^{\prime}_{i}}.$

This winds-up the proof of the Lemma. ∎

The subsequent result is an easy consequence of the Ekeland Variational
Principle.

######  Lemma 3.3.

Let $\alpha,\beta$ satisfy $\alpha+\beta+2>\max\\{p_{1},p_{2}\\}$. Then there
exists a sequence $\\{(u_{k},v_{k})\\}_{k\in\mathbb{N}}$ such that

$\displaystyle\inf_{(u,v)\in\Theta}\mathcal{I}(u,v)<\mathcal{I}(u_{k},v_{k})<\inf_{(u,v)\in\Theta}\mathcal{I}(u,v)+\frac{1}{k},$
(3.6)
$\displaystyle\|\mathcal{I}^{\prime}_{|_{\Theta}}(u_{k},v_{k})\|_{X^{*}}\leq\frac{1}{k}\,\text{
for all }\,k\in\mathbb{N}.$ (3.7)

###### Proof.

Using Hölder’s inequality and Young’s inequality we get,

$\displaystyle\|\mathcal{I}^{\prime}_{|_{\Theta}}(u,v)\|$
$\displaystyle\geq(\alpha+1)a(p_{1})[u]_{s_{1},p_{1}}^{p_{1}}+(\beta+1)a(p_{2})[v]_{s_{2},p_{2}}^{p_{2}}$
$\displaystyle\qquad-(\alpha+1)\theta_{1}^{p_{1}}[u]_{s_{1},p_{1}}^{p_{1}}-(\alpha+1)[a(1){\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}^{p_{1}}}]^{p^{\prime}_{1}}\theta_{1}^{-p^{\prime}_{1}}$
$\displaystyle\qquad-(\beta+1)\theta_{2}^{p_{2}}[v]_{s_{2},p_{2}}^{p_{2}}-(\beta+1)[a(1){\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}}]^{p^{\prime}_{2}}$
$\displaystyle\geq(\alpha+1)[u]_{s_{1},p_{1}}^{p_{1}}\left(a(p_{1})-\theta_{1}^{p_{1}}\right)+(\beta+1)[v]_{s_{2},p_{2}}^{p_{2}}\left(a(p_{2})-\theta_{2}^{p_{2}}\right)$
$\displaystyle\qquad-\theta_{1}^{-p^{\prime}_{1}}(\alpha+1)\left(a(1)\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}\right)^{p^{\prime}_{1}}-\theta_{2}^{-p^{\prime}_{2}}(\beta+1)\left(a(1)\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}\right)^{p^{\prime}_{2}}$
$\displaystyle\geq-\theta_{1}^{-p^{\prime}_{1}}(\alpha+1)\left(a(1)\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}\right)^{p^{\prime}_{1}}-\theta_{2}^{-p^{\prime}_{2}}(\beta+1)\left(a(1)\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}\right)^{p^{\prime}_{2}}$

by choosing $\theta_{1},\theta_{2}>0$ small enough. Thus
$\mathcal{I}(\cdot,\cdot)$ is bounded below on $\Theta$. Now applying Ekeland
Variational principle, we assert that there exists a sequence
$\\{(u_{k},v_{k})\\}_{k\in\mathbb{N}}\subset\Theta$ such that (3.6) and (3.7)
hold. This concludes the proof of the Lemma. ∎

###  3.2. Few technical lemmas

In this section, we are dealing with some technical lemmas in order to obtain
our main result. Let us denote

$m:=\inf_{(u,v)\in\Theta}\mathcal{I}(u,v).$

######  Lemma 3.4.

It holds that

$\displaystyle
m<\min\left\\{-\frac{\alpha+1}{p^{\prime}_{1}}\|f_{1}\|^{p^{\prime}_{1}}_{X^{*}_{0,s_{1},p_{1}}},\,-\frac{\beta+1}{p^{\prime}_{2}}\|f_{2}\|^{p^{\prime}_{2}}_{X^{*}_{0,s_{2},p_{2}}}\right\\}.$

###### Proof.

Let $u_{f_{i}}$ be unique solution of (3.2) given in Lemma 3.2. Then, by Lemma
3.2 we have

$\displaystyle\mathcal{I}(u_{f_{1}},0)$
$\displaystyle=(\alpha+1)\left(\frac{1}{p_{1}}[u_{f_{1}}]_{s_{1},p_{1}}^{p_{1}}-\left\langle
f_{1},u_{f_{1}}\right\rangle_{1}\right)=-\frac{(\alpha+1)}{p^{\prime}_{1}}\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}^{p^{\prime}_{1}}.$

In a similar fashion, we have

$\mathcal{I}(0,u_{f_{2}})=-\frac{(\beta+1)}{p^{\prime}_{2}}\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}^{p^{\prime}_{2}}.$

Since $m<\mathcal{I}(u_{f_{1}},0)$ and $m<\mathcal{I}(0,u_{f_{2}})$, so we
accomplish the result. ∎

Let us define $\mathcal{K}:X\to\mathbb{R}$ defined by

$\displaystyle\mathcal{K}(u,v)$
$\displaystyle=\left\langle\mathcal{I}^{\prime}(u,v),(u,v)\right\rangle$ (3.8)
$\displaystyle=(\alpha+1)[u]_{s_{1},p_{1}}^{p_{1}}+(\beta+1)[v]_{s_{2},p_{2}}-(\alpha+\beta+2)\int_{\Omega}|u|^{\alpha+1}|v|^{\beta+1}\,dx$
$\displaystyle\qquad-(\alpha+1)\left\langle
f_{1},u\right\rangle_{1}-(\beta+1)\left\langle f_{2},v\right\rangle_{2},$

for $(u,v)\in X$. Following result ensures that every critical point of
$\mathcal{K}$ is non trivial.

######  Lemma 3.5.

Let $\mathcal{K}$ be defined in (3.8) above. Then,
$\mathcal{K}^{\prime}(u,v)=0$ for $(u,v)\in\Theta$ implies that $u\not\equiv
0\not\equiv v$.

###### Proof.

We will use the method by contradiction for proof. So we assume
$(u_{1},v_{1})\in\Theta$ be such that $\mathcal{K^{\prime}}(u_{1},v_{1})=0$
and either $u_{1}\equiv 0$ or $v_{1}\equiv 0$. Then $(u_{1},v_{1})\in\Theta$
implies that $(u_{1},v_{1})\neq(0,0)$, by definition of $\Theta$. Without loss
of generality, let $v_{1}\equiv 0$. As $(u_{1},v_{1})\in\Theta$, so
$\left\langle\mathcal{I}^{\prime}(u_{1},v_{1}),(u_{1},v_{1})\right\rangle=0$.
This implies $\mathcal{K}(u_{1},v_{1})=0$ which asserts that

$\displaystyle(\alpha+1)[u_{1}]_{s_{1},p_{1}}^{p_{1}}-(\alpha+1)\left\langle
f_{1},u_{1}\right\rangle_{1}=0$ (3.9)

since $v_{1}\equiv 0$. Again as $\mathcal{K^{\prime}}(u_{1},v_{1})=0$, so we
get
$\left\langle\mathcal{K^{\prime}}(u_{1},v_{1}),(u_{1},v_{1})\right\rangle=0$
which yields us

$\displaystyle
p_{1}(\alpha+1)[u_{1}]_{s_{1},p_{1}}^{p_{1}}-(\alpha+1)\left\langle
f_{1},u_{1}\right\rangle_{1}=0.$ (3.10)

Equations (3.9) and (3.10) together imply

$\displaystyle(p_{1}-1)(\alpha+1)[u_{1}]_{s_{1},p_{1}}^{p_{1}}=0,$

that is $u_{1}\equiv 0$ a.e. in $\Omega$. Hence,
$(u_{1},v_{1})=(0,0)\in\Theta$ which is a contradiction to the definition of
$\Theta$. This completes the proof. ∎

The next outcome is an important consequence of Lemma 3.5.

######  Lemma 3.6.

Let $(f_{1},f_{2})\in X^{*}\setminus\\{(0,0)\\}$ be such that

$\displaystyle
0<\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}+\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}<\min\\{\varepsilon_{1},\varepsilon_{2},1\\}.$
(3.11)

Then, $\mathcal{K^{\prime}}(u,v)\neq 0$ for every $(u,v)\in\Theta$.

###### Proof.

We will prove this result by the method of contradiction. Let us suppose that

$\displaystyle\mathcal{K}^{\prime}(\tilde{u},\tilde{v})=0\,\text{ for some
}\,(\tilde{u},\tilde{v})\in\Theta.$

By Lemma 3.5, we know that $\tilde{u}\not\equiv 0\not\equiv\tilde{v}$. Whereas
$\left\langle\mathcal{K}^{\prime}(\tilde{u},\tilde{v}),(\tilde{u},\tilde{v})\right\rangle=0$
implies

$\displaystyle(\alpha+1)p_{1}[\tilde{u}]_{s_{1},p_{1}}^{p_{1}}+(\beta+1)p_{2}[\tilde{v}]_{s_{2},p_{2}}^{p_{2}}$
(3.12)
$\displaystyle-(\alpha+\beta+2)^{2}\int_{\Omega}|\tilde{u}|^{\alpha+1}|\tilde{v}|^{\beta+1}\,dx-(\alpha+1)\left\langle
f_{1},\tilde{u}\right\rangle_{1}-(\beta+1)\left\langle
f_{2},\tilde{v}\right\rangle_{2}=0.$

Again as $(\tilde{u},\tilde{v})\in\Theta$, so we have
$\left\langle\mathcal{I}^{\prime}(\tilde{u},\tilde{v}),(\tilde{u},\tilde{v})\right\rangle=0$,
that is,

$\displaystyle(\alpha+1)[\tilde{u}]_{s_{1},p_{1}}^{p_{1}}+(\beta+1)[\tilde{v}]_{s_{2},p_{2}}^{p_{2}}$
(3.13)
$\displaystyle-(\alpha+\beta+2)\int_{\Omega}|\tilde{u}|^{\alpha+1}|\tilde{v}|^{\beta+1}\,dx-(\alpha+1)\left\langle
f_{1},\tilde{u}\right\rangle_{1}-(\beta+1)\left\langle
f_{2},\tilde{v}\right\rangle_{2}=0.$

Equations (3.12) and (3.13) together yields,

$\displaystyle(\alpha+1)(p_{1}-1)[\tilde{u}]_{s_{1},p_{1}}^{p_{1}}+(\beta+1)(p_{2}-1)[\tilde{v}]_{s_{2},p_{2}}^{p_{2}}=(\alpha+\beta+2)(\alpha+\beta+1)\int_{\Omega}|\tilde{u}|^{\alpha+1}|\tilde{v}|^{\beta+1}\,dx.$

Using Young’s and Sobolev inequality we have,

$\displaystyle\frac{(\alpha+1)(p_{1}-1)}{(\alpha+\beta+2)(\alpha+\beta+1)}[\tilde{u}]_{s_{1},p_{1}}^{p_{1}}+\frac{(\beta+1)(p_{2}-1)}{(\alpha+\beta+2)(\alpha+\beta+1)}[\tilde{v}]_{s_{2},p_{2}}^{p_{2}}$
$\displaystyle=\int_{\Omega}|\tilde{u}|^{\alpha+1}|\tilde{v}|^{\beta+1}\,dx\leq\frac{(\alpha+1)}{p_{1}^{*}}\big{(}\int_{\Omega}|\tilde{u}|^{p_{1}^{*}}\,dx\big{)}^{\frac{1}{p_{1}^{*}}}+\frac{(\beta+1)}{p_{2}^{*}}\big{(}\int_{\Omega}|\tilde{v}|^{p_{2}^{*}}\,dx\big{)}^{\frac{1}{p_{2}^{*}}}$
$\displaystyle=\frac{(\alpha+1)}{p_{1}^{*}}|\tilde{u}|_{L^{p_{1}^{*}}(\Omega)}+\frac{(\beta+1)}{p_{2}^{*}}|\tilde{v}|_{L^{p_{2}^{*}}(\Omega)}$
$\displaystyle\leq\frac{(\alpha+1)}{p_{1}^{*}}\frac{1}{S_{p_{1}}^{\frac{1}{p_{1}}}}[\tilde{u}]_{s_{1},p_{1}}+\frac{(\beta+1)}{p_{2}^{*}}\frac{1}{S_{p_{2}}^{\frac{1}{p_{2}}}}[\tilde{v}]_{s_{2},p_{2}},$

where we have used

$\frac{\alpha+1}{p_{1}^{*}}+\frac{\beta+1}{p_{2}^{*}}=1\,\text{ and
}\,S_{p_{i}}^{\frac{1}{p_{i}}}|u|_{L^{p_{i}}}\leq[u]_{s_{i},p_{i}}.$

Now without loss of generality, we may assume that

$[\tilde{u}]_{s_{1},p_{1}}^{p_{1}^{*}}\leq[\tilde{v}]_{s_{2},p_{2}}^{p_{2}^{*}}.$

Hence we get,

$\displaystyle
b(\alpha,p_{1})[\tilde{u}]_{s_{1},p_{1}}^{p_{1}}+b(\beta,p_{2})[\tilde{v}]_{s_{2},p_{2}}^{p_{2}}\leq\frac{(\alpha+1)}{p_{1}^{*}}\frac{1}{S_{p_{1}}^{p_{1}^{*}/p_{1}}}[\tilde{u}]_{s_{1},p_{1}}^{p_{1}^{*}}+\frac{(\beta+1)}{p_{2}^{*}}\frac{1}{S_{p_{2}}^{p_{2}^{*}/p_{2}}}[\tilde{v}]_{s_{2},p_{2}}^{p_{2}^{*}},$

which implies that

$\displaystyle
b(\beta,p_{2})[\tilde{v}]_{s_{2},p_{2}}^{p_{2}}\leq\left(\frac{\alpha+1}{p_{1}^{*}}+\frac{\beta+1}{p_{2}^{*}}\right)\frac{1}{\min\\{S_{p_{1}}^{p_{1}^{*}/p_{1}},S_{p_{2}}^{p_{2}^{*}/p_{2}}\\}}[\tilde{v}]_{s_{2},p_{2}}^{p_{2}^{*}}$

This yields us,

$M:=b(\beta,p_{2})\big{(}\min\\{S_{p_{1}}^{p_{1}^{*}/p_{1}}S_{p_{2}}^{p_{2}^{*}/p_{2}}\\}\big{)}^{\frac{1}{p_{2}^{*}-p_{2}}}\leq[\tilde{v}]_{s_{2},p_{2}}.$
(3.14)

On the other hand, if we have assumed

$[\tilde{u}]_{s_{1},p_{1}}^{p_{1}^{*}}\geq[\tilde{v}]_{s_{2},p_{2}}^{p_{2}^{*}},$

then we would get

$M\leq[\tilde{u}]_{s_{1},p_{1}}.$ (3.15)

Multiplying (3.13) by $(\alpha+\beta+2)$ and then subtracting from (3.12), we
get using Hölder and Young’s inequality,

$\displaystyle(\alpha+1)c(p_{1})[\tilde{u}]_{s_{1},p_{1}}^{p_{1}}+(\beta+1)c(p_{2})[\tilde{v}]_{s_{2},p_{2}}^{p_{2}}$
$\displaystyle=(\alpha+1)\left\langle
f_{1},\tilde{u}\right\rangle+(\beta+1)\left\langle
f_{2},\tilde{v}\right\rangle$
$\displaystyle\leq(\alpha+1)\frac{\delta_{1}^{p_{1}}}{p_{1}}[\tilde{u}]_{s_{1},p_{1}}^{p_{1}}+\frac{(\alpha+1)}{\delta_{1}^{p^{\prime}_{1}}p^{\prime}_{1}}\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}^{p^{\prime}_{1}}+(\beta+1)\frac{\delta_{2}^{p_{2}}}{p_{2}}[\tilde{v}]_{s_{2},p_{2}}^{p_{2}}+\frac{(\beta+1)}{\delta_{2}^{p^{\prime}_{2}}p^{\prime}_{2}}\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}^{p^{\prime}_{2}},$

for some $\delta_{1},\delta_{2}>0$. Choosing
$0<\delta_{1}<\big{(}p_{1}\,c(p_{1})\big{)}^{\frac{1}{p_{1}}}$ and
$0<\delta_{2}<\big{(}p_{2}\,c(p_{2})\big{)}^{\frac{1}{p_{2}}}$, we get

$\displaystyle(\alpha+1)\left(c(p_{1})-\frac{\delta_{1}^{p_{1}}}{p_{1}}\right)[\tilde{u}]_{s_{1},p_{1}}^{p_{1}}+(\beta+1)\left(c(p_{2})-\frac{\delta_{2}^{p_{2}}}{p_{2}}\right)[\tilde{v}]_{s_{2},p_{2}}^{p_{2}}$
(3.16)
$\displaystyle\qquad\leq\frac{(\alpha+1)}{\delta_{1}^{p^{\prime}_{1}}p^{\prime}_{1}}\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}^{p^{\prime}_{1}}+\frac{(\beta+1)}{\delta_{2}^{p^{\prime}_{2}}p^{\prime}_{2}}\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}^{p^{\prime}_{2}}.$

Hence, (3.16) yields us

$(\alpha+1)\left(c(p_{1})-\frac{\delta_{1}^{p_{1}}}{p_{1}}\right)[\tilde{u}]_{s_{1},p_{1}}^{p_{1}}\leq\frac{(\alpha+1)}{\delta_{1}^{p^{\prime}_{1}}p^{\prime}_{1}}\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}^{p^{\prime}_{1}}+\frac{(\beta+1)}{\delta_{2}^{p^{\prime}_{2}}p^{\prime}_{2}}\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}^{p^{\prime}_{2}},$
(3.17)

and

$(\beta+1)\left(c(p_{2})-\frac{\delta_{2}^{p_{2}}}{p_{2}}\right)[\tilde{v}]_{s_{2},p_{2}}^{p_{2}}\leq\frac{(\alpha+1)}{\delta_{1}^{p^{\prime}_{1}}p^{\prime}_{1}}\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}^{p^{\prime}_{1}}+\frac{(\beta+1)}{\delta_{2}^{p^{\prime}_{2}}p^{\prime}_{2}}\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}^{p^{\prime}_{2}}.$
(3.18)

Using (3.14), we get from (3.18) that,

$(\beta+1)d(\delta_{1},\delta_{2})\left(c(p_{2})-\frac{\delta_{2}^{p_{2}}}{p_{2}}\right)b(\beta,p_{2})\Big{(}\min\\{S_{p_{1}}^{p_{1}^{*}/p_{1}},S_{p_{2}}^{p_{2}^{*}/p_{2}}\\}\Big{)}^{\frac{p_{2}}{p_{2}^{*}-p_{2}}}<\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}^{p^{\prime}_{1}}+\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}^{p^{\prime}_{2}}.$

By similar analysis, using (3.15), we would obtain,

$(\alpha+1)d(\delta_{1},\delta_{2})\left(c(p_{1})-\frac{\delta_{1}^{p_{1}}}{p_{1}}\right)b(\alpha,p_{1})\Big{(}\min\\{S_{p_{1}}^{p_{1}^{*}/p_{1}},S_{p_{2}}^{p_{2}^{*}/p_{2}}\\}\Big{)}^{\frac{p_{1}}{p_{1}^{*}-p_{1}}}<\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}^{p^{\prime}_{1}}+\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}^{p^{\prime}_{2}}.$

Consequently, we get

$\displaystyle\min\\{\varepsilon_{1},\varepsilon_{2}\\}<\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}^{p^{\prime}_{1}}+\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}^{p^{\prime}_{2}}$

which is a contradiction to (3.11), bearing out the proof of the lemma. ∎

The ensuing results below treat some property of the minimizing sequence.

######  Lemma 3.7.

Let $\frac{\alpha+1}{p_{1}^{*}}+\frac{\beta+1}{p_{2}^{*}}=1$ and $m$ be
defined as in Lemma 3.4. Let
$\\{(u_{k},v_{k})\\}_{k\in\mathbb{N}}\subset\Theta$ be such that
$\lim_{k\to\infty}\mathcal{I}(u_{k},v_{k})=m$. Then, there exists
$(u^{*},v^{*})\in X$ such that $(u_{k},v_{k})\rightharpoonup(u^{*},v^{*})$
weakly in $X$.

###### Proof.

Using Lemma 3.3, we have,

$\displaystyle-m-\frac{1}{k}\|(u_{k},v_{k})\|_{X}$
$\displaystyle\leq\left\langle\mathcal{I}^{\prime}_{\Theta}(u_{k},v_{k}),(u_{k},v_{k})\right\rangle-\mathcal{I}_{\Theta}(u_{k},v_{k})$
$\displaystyle\qquad\leq m+\frac{1}{k}\|(u_{k},v_{k})\|_{X}.$

This gives

$\displaystyle-m-\frac{1}{k}\|(u_{k},v_{k})\|_{X}$
$\displaystyle\leq\frac{\alpha+1}{p_{1}^{*}}\left(1-\frac{p_{1}}{\alpha+\beta+1}\right)[u_{k}]_{s_{1},p_{1}}^{p_{1}}+\frac{\beta+1}{p_{2}^{*}}\left(1-\frac{p_{2}}{\alpha+\beta+1}\right)[v_{k}]_{s_{2},p_{2}}^{p_{2}}$
$\displaystyle\leq m+\frac{1}{k}\|(u_{k},v_{k})\|_{X}$

which gives a contradiction if $\|(u_{k},v_{k})\|_{X}\to\infty$ as
$k\to\infty$. Therefore $\\{(u_{k},v_{k})\\}_{k\in\mathbb{N}}$ is bounded in
$X$. Since $X$ is a reflexive Banach space, there exists $(u^{*},v^{*})\in X$
such that up to a subsequence, $(u_{k},v_{k})\rightharpoonup(u^{*},v^{*})$
weakly in $X$ as $k\to\infty$, thereby completing the proof. ∎

######  Proposition 3.1.

Let $\delta_{1},\delta_{2}\in\mathbb{R}$ be such that
$0<\delta_{i}<\big{(}p_{i}\,c(p_{i})\big{)}^{\frac{1}{p_{i}}}$ and $f_{i}\in
X^{*}_{0,s_{i},p_{i}}$ for $i=1,2$ satisfying

$0<\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}+\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}<\min\\{\varepsilon_{1},\varepsilon_{2},1\\}.$

Then there exists $\delta_{0}>0$ such that

$\Big{|}\left\langle\mathcal{K}^{\prime}(u_{k},v_{k}),(u_{k},v_{k})\right\rangle\Big{|}\geq\delta_{0}>0,$

for any minimizing sequence $\\{(u_{k},v_{k})\\}_{k\in\mathbb{N}}$ of
$\mathcal{I}$ in $\Theta$.

###### Proof.

By contradiction, suppose the assertion is not true. Then, there exists a
minimizing sequence $\\{(u_{k},v_{k})\\}_{k\in\mathbb{N}}\subset\Theta$ of
$\mathcal{I}$ such that

$\Big{|}\left\langle\mathcal{K^{\prime}}(u_{k},v_{k}),(u_{k},v_{k})\right\rangle\Big{|}\to
0\text{ as }\,k\to\infty.$ (3.19)

Let
$t_{k}:=\left\langle\mathcal{K^{\prime}}(u_{k},v_{k}),(u_{k},v_{k})\right\rangle\,\text{for
all}\,k\in\mathbb{N}$. Then by (3.19), we have

$\lim_{k\to\infty}|t_{k}|=0,\,\text{ that is }\,\lim_{k\to\infty}t_{k}=0.$
(3.20)

Since $(u_{k},v_{k})\subset\Theta$, we have

$\displaystyle(\alpha+1)[u_{k}]_{s_{1},p_{1}}^{p_{1}}+(\beta+1)[v_{k}]_{s_{2},p_{2}}^{p_{2}}-(\alpha+\beta+2)\int_{\Omega}|u_{k}|^{\alpha+1}|v_{k}|^{\beta+1}\,dx$
(3.21) $\displaystyle\qquad-(\alpha+1)\left\langle
f_{1},u_{k}\right\rangle_{1}-(\beta+1)\left\langle
f_{2},v_{k}\right\rangle_{2}=0.$

Combining definition of $t_{k}$ with (3.21) we have,

$\displaystyle(\alpha+1)(p_{1}-1)[u_{k}]_{s_{1},p_{1}}^{p_{1}}+(\beta+1)(p_{2}-1)[v_{k}]_{s_{2},p_{2}}^{p_{2}}$
(3.22)
$\displaystyle\qquad=(\alpha+\beta+2)(\alpha+\beta+1)\int_{\Omega}|u_{k}|^{\alpha+1}|v_{k}|^{\beta+1}\,dx+t_{k}.$

By similar argument as in the proof of Lemma 3.6, we obtain assuming
$[u_{k}]_{s_{1},p_{1}}^{p_{1}^{*}}\leq[v_{k}]_{s_{2},p_{2}}^{p_{2}^{*}}$,

$\min\\{S_{p_{1}}^{p_{1}^{*}/p_{1}},S_{p_{2}}^{p_{2}^{*}/p_{2}}\\}\left(b(\beta,p_{2})-\frac{t_{k}}{[v_{k}]_{s_{2},p_{2}}^{p_{2}}}\right)\leq[v_{k}]_{s_{2},p_{2}}^{p_{2}^{*}-p_{2}}.$

Therefore, we conclude using Lemma 3.4 that

$\frac{1}{[v_{k}]_{s_{2},p_{2}}}\leq M_{0}\,\text{ for some }\,M_{0}>0.$

Thus, we have

$\frac{t_{k}}{[v_{k}]_{s_{2},p_{2}}^{p_{2}}}<b(\beta,p_{2})\,\text{ for some
}\,k\,\text{ large enough. }\,$

This yields

$\displaystyle\Big{(}b(\beta,p_{2})\min\\{S_{p_{1}}^{p_{1}^{*}/p_{1}},S_{p_{2}}^{p_{2}^{*}/p_{2}}\\}\Big{)}^{\frac{1}{p_{2}^{*}-p_{2}}}-At_{k}\leq[v_{k}]_{s_{2},p_{2}},$

where $A$ is a constant depending on $\alpha,\beta,p_{i},p_{i}^{*},S_{p_{i}}$,
$i=1,2$. As a result, we have for
$0<\delta_{i}<\big{(}p_{i}\,c(p_{i})\big{)}^{\frac{1}{p_{i}}}$

$\displaystyle(\alpha+1)d(\delta_{1},\delta_{2})\left(c(p_{1})-\frac{\delta_{1}^{p_{1}}}{p_{1}}\right)\Big{(}b(\alpha,p_{1})\min\\{S_{p_{1}}^{p_{1}^{*}/p_{1}},S_{p_{2}}^{p_{2}^{*}/p_{2}}\\}\Big{)}^{\frac{p_{1}}{p_{1}^{*}-p_{1}}}-At_{k}$
$\displaystyle\leq\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}+\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}+t_{k},$

and

$\displaystyle(\beta+1)d(\delta_{1},\delta_{2})\left(c(p_{2})-\frac{\delta_{2}^{p_{2}}}{p_{1}}\right)\Big{(}b(\beta,p_{2})\min\\{S_{p_{1}}^{p_{1}^{*}/p_{1}},S_{p_{2}}^{p_{2}^{*}/p_{2}}\\}\Big{)}^{\frac{p_{2}}{p_{2}^{*}-p_{2}}}-Bt_{k}$
$\displaystyle\leq\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}+\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}+t_{k}.$

Letting $k\to\infty$ we get,

$(\alpha+1)d(\delta_{1},\delta_{2})\left(c(p_{1})-\frac{\delta_{1}^{p_{1}}}{p_{1}}\right)\Big{(}b(\alpha,p_{1})\,\min\\{S_{p_{1}}^{p_{1}^{*}/p_{1}},S_{p_{2}}^{p_{2}^{*}/p_{2}}\\}\Big{)}^{\frac{p_{1}}{p_{1}^{*}-p_{1}}}\leq\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}+\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}$

and

$(\beta+1)d(\delta_{1},\delta_{2})\left(c(p_{2})-\frac{\delta_{2}^{p_{2}}}{p_{2}}\right)\Big{(}b(\beta,p_{2})\,\min\\{S_{p_{1}}^{p_{1}^{*}/p_{1}},S_{p_{2}}^{p_{2}^{*}/p_{2}}\\}\Big{)}^{\frac{p_{2}}{p_{2}^{*}-p_{2}}}\leq\|f_{1}\|_{X^{*}_{0,s_{1},p_{1}}}+\|f_{2}\|_{X^{*}_{0,s_{2},p_{2}}}.$

This contradicts the hypothesis of the proposition, thereby winding-up the
proof. ∎

###  3.3. Proof of Theorem 2.1

Taking into account the minimizing sequence
$\\{(u_{k},v_{k})\\}_{k\in\mathbb{N}}\subset\Theta$, we write,

$\mathcal{I}^{\prime}(u_{k},v_{k})=\mathcal{I}^{\prime}|_{\Theta}(u_{k},v_{k})-\lambda_{k}\mathcal{K}^{\prime}(u_{k},v_{k}),$

that is,

$\displaystyle\left\langle\mathcal{I}^{\prime}(u_{k},v_{k}),(u_{k},v_{k})\right\rangle=\left\langle\mathcal{I}^{\prime}|_{\Theta}(u_{k},v_{k}),(u_{k},v_{k})\right\rangle-\lambda_{k}\left\langle\mathcal{K}^{\prime}(u_{k},v_{k}),(u_{k},v_{k})\right\rangle.$

We notice that
$\left\langle\mathcal{I}^{\prime}(u_{k},v_{k}),(u_{k},v_{k})\right\rangle=0$
on L.H.S., as $\\{(u_{k},v_{k})\\}_{k\in\mathbb{N}}\subset\Theta$. On R.H.S.,
the sequence $\\{(u_{k},v_{k})\\}_{k\in\mathbb{N}}$ be such that,

$\mathcal{I^{\prime}}|_{\Theta}(u_{k},v_{k})\to 0\,\text{ as }\,k\to\infty.$

This implies,

$\left\langle\mathcal{I^{\prime}}|_{\Theta}(u_{k},v_{k}),(u_{k},v_{k})\right\rangle\to
0\,\text{ as }\,k\to\infty.$

Using Proposition 3.1, we conclude that $\lambda_{k}\to 0$. As a consequence,
we notice that,

$\displaystyle\lim_{k\to\infty}\left\langle\mathcal{I}^{\prime}(u_{k},v_{k}),(\phi_{1},\phi_{2})\right\rangle=0\;\;\text{for
all}\,(\phi_{1},\phi_{2})\in X.$

Hence, we obtain

$\displaystyle\begin{cases}(-\Delta)^{s_{1}}_{p_{1}}u_{k}&=u_{k}|u_{k}|^{\alpha-1}|v_{k}|^{\beta+1}+f_{1}+f_{k}\,\text{
in }\,\Omega,\\\
(-\Delta)^{s_{2}}_{p_{2}}v_{k}&=|u_{k}|^{\alpha+1}v_{k}|v_{k}|^{\beta-1}+f_{2}+g_{k}\,\text{
in }\,\Omega,\\\
&u_{k}=v_{k}=0\;\text{in}\;\mathbb{R}^{N}\setminus\Omega,\end{cases}$

where $\\{f_{k}\\}$ and $\\{g_{k}\\}$ denote two sequences converging strongly
to $0$ in $(X_{0,s_{i},p_{i}}(\Omega))^{*}$ respectively for $i=1,2$. Let us
define $\\{M_{k}\\}_{k\in\mathbb{N}}$ and $\\{N_{k}\\}_{k_{n}}$ be such that
$M_{k}=u_{k}|u_{k}|^{\alpha-1}|v_{k}|^{\beta+1}$ and
$N_{k}=|u_{k}|^{\alpha+1}v_{k}|v_{k}|^{\beta-1}$. We now approach towards the
following proposition regarding $\\{M_{k}\\}$ and $\\{N_{k}\\}$.

######  Proposition 3.2.

The sequences $\\{M_{k}\\}_{k\in\mathbb{N}}$ and
$\\{N_{k}\\}_{k\in\mathbb{N}}$ defined above are bounded in
$X^{*}_{0,s_{1},p_{1}}(\Omega)$ and $X^{*}_{0,s_{2},p_{2}}(\Omega)$
respectively.

###### Proof.

We know that the embedding $X_{0,s_{i},p_{i}}(\Omega)\hookrightarrow
L^{p_{i}^{*}}(\Omega)$ and $L^{(p_{i}^{*})^{\prime}}(\Omega)\hookrightarrow
X^{*}_{0,s_{i},p_{i}}(\Omega)$ are continuous. Our aim is to deduce that
$\\{M_{k}\\}_{k\in\mathbb{N}}$ and $\\{N_{k}\\}_{k\in\mathbb{N}}$ are bounded
in $L^{(p_{i}^{*})^{\prime}}(\Omega)$ respectively for $i=1,2$. To conclude
our result, it is sufficient to consider $\\{M_{k}\\}_{k\in\mathbb{N}}$. Using
the fact that $\\{u_{k}\\}_{k\in\mathbb{N}}$ and
$\\{v_{k}\\}_{k\in\mathbb{N}}$ are bounded respectively in
$X_{0,s_{i},p_{i}}(\Omega)$ for $i=1,2$ (refer Lemma 3.7) and embedding
results, we have the following estimate: for every $\psi\in
L^{p_{1}^{*}}(\Omega)$, there exists a constant $\mathcal{F}^{\prime}>0$ such
that,

$\displaystyle\Big{|}\int_{\Omega}M_{k}\psi\,dx\Big{|}$
$\displaystyle\leq\int_{\Omega}|M_{k}||\psi|\,dx\leq\int_{\Omega}|u_{k}|^{\alpha}|v_{k}|^{\beta+1}|\psi|\,dx$
(3.23)
$\displaystyle\leq|u_{k}|_{L^{p_{1}^{*}}(\Omega)}^{\alpha}|v_{k}|_{L^{p_{2}^{*}}(\Omega)}^{\beta+1}|\psi|_{L^{p_{1}^{*}}(\Omega)}\leq\mathcal{F}^{\prime}|\psi|_{L^{p_{1}^{*}}(\Omega)}.$

We note that (3.23) is valid due to Hölder’s inequality as,

$\frac{\alpha+1}{p_{1}^{*}}+\frac{\beta+1}{p_{2}^{*}}=1.$

Using (3.23), for every $\psi\in
L^{(p_{1}^{*})^{\prime}}(\Omega)\setminus\\{0\\}$, we have,

$\frac{\left|\int_{\Omega}M_{k}\psi\,dx\right|}{\|\psi\|_{L^{p_{1}^{*}}}}\leq\mathcal{F}^{\prime}.$

This implies,

$\sup_{\psi\in
L^{p_{1}^{*}}(\Omega)\setminus\\{0\\}}\frac{\Big{|}\int_{\Omega}M_{k}\psi\,dx\Big{|}}{|\psi|_{L^{p_{1}^{*}}(\Omega)}}=|M_{k}|_{L^{(p_{1}^{*})^{\prime}}(\Omega)}\leq\mathcal{F}^{\prime}.$

We conclude that $\\{M_{k}\\}_{k\in\mathbb{N}}$ is bounded in
$L^{(p_{i}^{*})^{\prime}}(\Omega)$ and thereupon, is bounded in
$X^{*}_{0,s_{1},p_{1}}(\Omega)$. This completes the proof.

∎

######  Proposition 3.3.

The sequences $\\{M_{k}\\}_{k\in\mathbb{N}}$ and
$\\{N_{k}\\}_{k\in\mathbb{N}}$ are bounded in $L^{1}(\Omega)$. Moreover, we
have,

$\lim_{k\to\infty}\int_{\Omega}M_{k}\phi_{1}\,dx=\int_{\Omega}M^{*}\phi_{1}\,dx,\,\forall\,\phi_{1}\in
L^{\infty}(\Omega),$

and

$\lim_{k\to\infty}\int_{\Omega}N_{k}\phi_{2}\,dx=\int_{\Omega}N^{*}\phi_{2}\,dx,\,\forall\,\phi_{2}\in
L^{\infty}(\Omega),$

where $M^{*}=u^{*}|u^{*}|^{\alpha-1}|v^{*}|^{\beta+1}$,
$N^{*}=|u^{*}|^{\alpha+1}v^{*}|v^{*}|^{\beta-1}$ and $(u^{*},v^{*})$ is as
defined in Lemma 3.7.

###### Proof.

Using Hölder inequality, we can see that $\\{M_{k}\\}_{k\in\mathbb{N}}$ and
$\\{N_{k}\\}_{k\in\mathbb{N}}$ are bounded in $L^{1}(\Omega)$. Using Lemma
3.7, we obtain the existence of $u^{*}$ and $v^{*}$ which furthermore
guarantees that some subsequences denoted again by
$\\{M_{k}\\}_{k\in\mathbb{N}}$ and $\\{N_{k}\\}_{k\in\mathbb{N}}$ converge
a.e. in $\Omega$ to $M^{*}$ and $N^{*}$ respectively. In a similar fashion, we
can show that $M^{*}$ and $N^{*}$ belong to $L^{1}(\Omega)$. Let

$|N_{k}-N^{*}|_{L^{1}(\Omega)}<(\mathcal{F}^{\prime})^{R_{\alpha\beta}}$

where $R_{\alpha\beta}>0$ is such that

$\frac{1}{R_{\alpha\beta}}=\frac{\alpha}{p_{1}^{*}}+\frac{\beta+1}{p_{2}^{*}}.$

We will now apply Egoroff’s Theorem [5]. For fix $\varepsilon>0$, there exists
a measurable subset $\mathcal{A}$ of $\Omega$ such that

$\displaystyle\big{|}\Omega\setminus\mathcal{A}\big{|}^{\frac{1}{p_{1}^{*}}}<\frac{\varepsilon}{2\mathcal{F}^{\prime}}.$

and the sequence $\\{N_{k}\\}_{k\in\mathbb{N}}$ converges uniformly to $N^{*}$
in $\mathcal{A}$. As a consequence, we obtain,

$\displaystyle\int_{\Omega}|N_{k}-N^{*}|\,dx=\int_{\Omega\setminus\mathcal{A}}|N_{k}-N^{*}|\,dx+\int_{\mathcal{A}}|N_{k}-N^{*}|\,dx.$

Using uniform convergence of $N_{k}$ to $N^{*}$ in $\mathcal{A}$, we note that
there exists $k_{\varepsilon}>0$ such that for $k\geq k_{\varepsilon}$, we
have,

$\displaystyle|N_{k}(x)-N^{*}(x)|\leq\frac{\varepsilon}{2|\Omega|},\;\;\forall
x\in\mathcal{A}.$

So, for all $k\geq k_{\varepsilon}$, we have,

$\displaystyle\int_{\mathcal{A}}|N_{k}(x)-N^{*}(x)|\,dx\leq\int_{\mathcal{A}}\frac{\varepsilon}{2|\Omega|}\,dx\leq\frac{\varepsilon}{2}.$
(3.24)

On another note, we observe that

$\displaystyle\int_{\Omega\setminus\mathcal{A}}|N_{k}-N^{*}|\,dx$
$\displaystyle=\int_{\Omega}|N_{k}-N^{*}|\chi_{\Omega\setminus\mathcal{A}}\,dx$
$\displaystyle\leq\Big{(}\int_{\Omega}|N_{k}-N^{*}|\,dx\Big{)}^{\frac{1}{R_{\alpha\beta}}}\big{|}\Omega\setminus\mathcal{A}\big{|}^{\frac{1}{p_{1}^{*}}},$

Using the above $L^{1}(\Omega)$ bound of $\\{N_{k}-N^{*}\\}_{k\in\mathbb{N}}$
and
$\big{|}\Omega\setminus\mathcal{A}\big{|}^{\frac{1}{p_{1}^{*}}}<\frac{\varepsilon}{2\mathcal{F}^{\prime}}$,
we get

$\int_{\Omega\setminus\mathcal{A}}|N_{k}-N^{*}|\,dx\leq\frac{\varepsilon}{2}.$
(3.25)

Combining (3.24) and (3.25) we see that,

$\int_{\Omega}|N_{k}-N^{*}|\,dx\leq\varepsilon.$

Thus, $\\{N_{k}\\}_{k\in\mathbb{N}}$ converges to $N^{*}$ in $L^{1}(\Omega)$.
Consequently, without loss of generality, we get
$\\{N_{k}\\}_{k\in\mathbb{N}}$ converges weakly to $N^{*}$ in $L^{1}(\Omega)$
and similar arguments will lead to $\\{M_{k}\\}_{k\in\mathbb{N}}$ converges
weakly to $M^{*}$ in $L^{1}(\Omega)$. This finishes the proof. ∎

We now provide a direct consequence of Proposition 3.3. Firstly, observing
that $\\{M_{k}\\}_{k\in\mathbb{N}}$ is bounded in $L^{1}(\Omega)$, we obtain
the strong convergence of $[u_{k}]_{s_{1},p_{1}}^{p_{1}}$ to
$[u^{*}]^{p_{1}}_{s_{1},p_{1}}$ in $L^{t_{1}}(\Omega)$ for every $t_{1}<p_{1}$
and $[v_{k}]_{s_{2},p_{2}}^{p_{2}}$ to $[v^{*}]^{p_{2}}_{s_{2},p_{2}}$ in
$L^{t_{2}}(\Omega)$ for every $t_{2}<p_{2}$. As a follow-up, we note that (see
Lemma 3.3, [9]),

$\frac{|u_{k}(x)-u_{k}(y)||^{p_{1}-2}(u_{k}(x)-u_{k}(y))}{|x-y|^{\frac{N+p_{1}s_{1}}{p^{\prime}_{1}}}}\rightharpoonup\frac{|u^{*}(x)-u^{*}(y)||^{p_{1}-2}(u^{*}(x)-u^{*}(y))}{|x-y|^{\frac{N+p_{1}s_{1}}{p^{\prime}_{1}}}}$
(3.26)

weakly in $L^{p^{\prime}_{1}}(\mathbb{R}^{2N})$ and

$\frac{|v_{k}(x)-v_{k}(y)||^{p_{2}-2}(v_{k}(x)-v_{k}(y))}{|x-y|^{\frac{N+p_{2}s_{2}}{p^{\prime}_{2}}}}\rightharpoonup\frac{|v^{*}(x)-v^{*}(y)||^{p_{2}-2}(v^{*}(x)-v^{*}(y))}{|x-y|^{\frac{N+p_{2}s_{2}}{p^{\prime}_{2}}}}$
(3.27)

weakly in $L^{p^{\prime}_{2}}(\mathbb{R}^{2N})$. We now take care of an
important corollary stated below.

######  Corollary 3.1.

There holds,

$\lim_{k\to\infty}\int_{\Omega}M_{k}\phi_{1}\,dx=\int_{\Omega}M^{*}\phi_{1}\,dx\,\text{
for all }\,\phi_{1}\in X_{0,s_{1},p_{1}}(\Omega)$

and

$\lim_{k\to\infty}\int_{\Omega}N_{k}\phi_{2}\,dx=\int_{\Omega}N^{*}\phi_{2}\,dx\,\text{
for all }\,\phi_{2}\in X_{0,s_{2},p_{2}}(\Omega).$

###### Proof.

We notice that the proof holds true by Proposition 3.3 if $\phi_{1}$ and
$\phi_{2}$ belong to $C_{c}^{\infty}(\Omega)$. We now suppose that $\phi_{1}$
belongs to $X_{0,s_{1},p_{1}}(\Omega)$. Our objective is to show that

$\displaystyle\int_{\Omega}M_{k}\phi_{1}\,dx\to\int_{\Omega}M^{*}\phi_{1}\,dx.$
(3.28)

We choose $\varepsilon>0$ sufficiently small. Since $C_{c}^{\infty}(\Omega)$
is dense in $X_{0,s_{1},p_{1}}(\Omega)$, there exists $\Phi_{\varepsilon}\in
C_{c}^{\infty}(\Omega)$ such that
$[\phi_{1}-\Phi_{\varepsilon}]_{s_{1},p_{1}}\leq\varepsilon$. Indeed, we have,

$\int_{\Omega}M_{k}\phi_{1}\,dx-\int_{\Omega}M^{*}\phi_{1}\,dx=\int_{\Omega}(M_{k}-M^{*})(\phi_{1}-\Phi_{\varepsilon})\,dx+\int_{\Omega}(M_{k}-M^{*})\Phi_{\varepsilon}\,dx,$

which yields,

$\Bigg{|}\int_{\Omega}M_{k}\phi_{1}\,dx-\int_{\Omega}M^{*}\phi_{1}\,dx\Bigg{|}\leq\int_{\Omega}|M_{k}-M^{*}||\phi_{1}-\Phi_{\varepsilon}|\,dx+{\Big{|}\int_{\Omega}(M_{k}-M^{*})\Phi_{\varepsilon}\,dx\Big{|}.}$
(3.29)

Now, using Hölder’s inequality, we have,

$\int_{\Omega}|M_{k}-M^{*}||\phi_{1}-\Phi_{\varepsilon}|\,dx\leq\Big{(}\int_{\Omega}|M_{k}-M^{*}|^{R_{\alpha,\beta}}\Big{)}^{\frac{1}{R_{\alpha,\beta}}}\Big{(}\int_{\Omega}|\phi_{1}-\Phi_{\varepsilon}|^{p_{1}^{*}}\,dx\Big{)}^{\frac{1}{p_{1}^{*}}}.$

Applying the continuous embedding $X_{0,s_{1},p_{1}}(\Omega)\hookrightarrow
L^{p_{1}^{*}}(\Omega)$, we get,
$|\phi_{1}-\Phi_{\varepsilon}|_{L^{p_{1}^{*}}(\Omega)}\leq\frac{\varepsilon}{2}.$
Using Proposition 3.3 for $M_{k}$, for $\varepsilon>0$ sufficiently small and
$k$ large enough, we deduce that

$\displaystyle\Big{|}\int_{\Omega}(M_{k}-M^{*})\Phi_{\varepsilon}\,dx\Big{|}\leq\frac{\varepsilon}{2}$

Combining altogether we obtain using (3.29),

$\displaystyle\Big{|}\int_{\Omega}M_{k}\phi_{1}\,dx-\int_{\Omega}M^{*}\phi_{1}\,dx\Big{|}\leq\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon,$

thereby proving (3.28). In a similar fashion, we acquire

$\displaystyle\int_{\Omega}N_{k}\phi_{2}\,dx\to\int_{\Omega}N^{*}\phi_{2}\,dx,$

for all $\phi_{2}\in X_{0,s_{2},q_{2}}(\Omega)$. This wraps-up the proof. ∎

The subsequent proposition is also thought-provoking for the readers and
announced below:

######  Proposition 3.4.

The pair $(u^{*},v^{*})$ obtained in Lemma 3.7 is a solution of problem
$(\mathcal{P}_{f_{1},f_{2}})$.

###### Proof.

Let $\phi_{i}\in X_{0,s_{i},p_{i}}(\Omega)$ for $i=1,2$. We define
$\mathcal{I}^{\prime}|_{u}$ and $\mathcal{I}^{\prime}|_{v}$ for $(u,v)\in X$
by

$\displaystyle\left\langle\mathcal{I}^{\prime}|_{u}(u,v),\phi_{1}\right\rangle=\left\langle\mathcal{I}^{\prime}(u,v),(\phi_{1},0)\right\rangle$

and

$\displaystyle\left\langle\mathcal{I}^{\prime}|_{v}(u,v),\phi_{2}\right\rangle=\left\langle\mathcal{I}^{\prime}(u,v),(0,\phi_{2})\right\rangle.$

Putting $u=u_{k}$ and $v=v_{k}$ we get,

$\displaystyle\left\langle\mathcal{I}^{\prime}|_{u}(u_{k},v_{k}),\phi_{1}\right\rangle=\left\langle(-\Delta)^{s_{1}}_{p_{1}}u_{k},\phi_{1}\right\rangle-\int_{\Omega}u_{k}|u_{k}|^{\alpha-1}|v_{k}|^{\beta+1}\phi_{1}\,dx-\left\langle
f_{1},\phi_{1}\right\rangle-\left\langle f_{k},\phi_{1}\right\rangle$ (3.30)

and

$\displaystyle\left\langle\mathcal{I}^{\prime}|_{v}(u_{k},v_{k}),\phi_{2}\right\rangle=\left\langle(-\Delta)^{s_{2}}_{p_{2}}v_{k},\phi_{2}\right\rangle-\int_{\Omega}|v_{k}|^{\alpha+1}v_{k}|v_{k}|^{\beta+1}\phi_{2}\,dx-\left\langle
f_{2},\phi_{2}\right\rangle-\left\langle g_{k},\phi_{2}\right\rangle.$ (3.31)

Note that

$\frac{\phi_{1}(x)-\phi_{1}(y)}{|x-y|^{\frac{N+s_{1}p_{1}}{p_{1}}}}\in
L^{p_{1}}(\mathbb{R}^{N})\;\;\text{and}\;\;\frac{\phi_{2}(x)-\phi_{2}(y)}{|x-y|^{\frac{N+s_{2}p_{2}}{p_{2}}}}\in
L^{p_{2}}(\mathbb{R}^{N})$

which gives

$\lim_{k\to\infty}\left\langle(-\Delta)^{s_{1}}_{p_{1}}u_{k},\phi_{1}\right\rangle=\left\langle(-\Delta)^{s_{1}}_{p_{1}}u^{*},\phi_{1}\right\rangle\;\;\text{and}\;\;\lim_{k\to\infty}\left\langle(-\Delta)^{s_{2}}_{p_{2}}v_{k},\phi_{2}\right\rangle=\left\langle(-\Delta)^{s_{2}}_{p_{2}}v^{*},\phi_{2}\right\rangle.$

On taking the limit and using (3.26) and (3.27) we obtain,

$\displaystyle\left\langle\mathcal{I}^{\prime}|_{u}(u_{k},v_{k}),\phi_{1}\right\rangle=\left\langle(-\Delta)^{s_{1}}_{p_{1}}u^{*},\phi_{1}\right\rangle-\int_{\Omega}u^{*}|u^{*}|^{\alpha-1}|v^{*}|^{\beta+1}\phi_{1}\,dx-\left\langle
f_{1},\phi_{1}\right\rangle$

and

$\displaystyle\left\langle\mathcal{I}^{\prime}|_{v}(u_{k},v_{k}),\phi_{2}\right\rangle=\left\langle(-\Delta)^{s_{2}}_{p_{2}}v^{*},\phi_{2}\right\rangle-\int_{\Omega}|v^{*}|^{\alpha+1}|v^{*}|^{\beta+1}v^{*}\phi_{2}\,dx-\left\langle
f_{2},\phi_{2}\right\rangle.$

Moreover, using (3.6) and (3.7) we obtain

$\displaystyle\left\langle(-\Delta)^{s_{1}}_{p_{1}}u^{*},\phi_{1}\right\rangle-\int_{\Omega}u^{*}|u^{*}|^{\alpha-1}|v^{*}|^{\beta+1}\phi_{1}\,dx-\left\langle
f_{1},\phi_{1}\right\rangle=0$

for every $\phi_{1}$ in $X_{0,s_{1},p_{1}}(\Omega)$ and

$\displaystyle\left\langle(-\Delta)^{s_{2}}_{p_{2}}v^{*},\phi_{2}\right\rangle-\int_{\Omega}|v^{*}|^{\alpha+1}|v^{*}|^{\beta+1}v^{*}\phi_{2}\,dx-\left\langle
f_{2},\phi_{2}\right\rangle=0$

for every $\phi_{2}$ in $X_{0,s_{2},p_{2}}(\Omega)$ As a result, we deduce
that,

$\displaystyle\begin{cases}(-\Delta)^{s_{1}}_{p_{1}}u^{*}&=u^{*}|u^{*}|^{\alpha-1}|v^{*}|^{\beta+1}+f_{1}\,\text{
in }\,\Omega,\\\
(-\Delta)^{s_{2}}_{p_{2}}v^{*}&=u^{*}|u^{*}|^{\alpha+1}|v^{*}|^{\beta-1}+f_{2}\,\text{
in }\,\Omega.\end{cases}$

This implies, $(u^{*},v^{*})$ is a weak solution of
$(\mathcal{P}_{f_{1},f_{2}})$. Contrarily, this solution $(u^{*},v^{*})$
satisfies the following:

  * (i)

$\left\langle\mathcal{I}^{\prime}(u^{*},v^{*}),(u^{*},v^{*})\right\rangle=0$,

  * (ii)

$\mathcal{I}(u^{*},v^{*})=m<0$.

By (i), we note that $(u^{*},v^{*})$ belongs to $\Theta$. As $(u^{*},v^{*})$
is the solution of $(\mathcal{P}_{f_{1},f_{2}})$, we obtain (i) by taking into
account $(\phi_{1},\phi_{2})=(u^{*},v^{*})$.

To see the proof of (ii), we note that (i) implies
$m\leq\mathcal{I}(u^{*},v^{*})$, since
$m:=\inf_{(u,v)\in\Theta}\mathcal{I}(u,v)$. Otherwise, due to the weak semi-
continuity of $\mathcal{I}|_{\Theta}$ and
$\mathcal{I}(u_{k},v_{k})<m+\frac{1}{k}$ yields,

$\mathcal{I}(u^{*},v^{*})\leq\liminf_{k\to+\infty}\mathcal{I}(u_{k},v_{k})\leq
m.$

This in turn yields us,

$\displaystyle
m=\lim_{k\to\infty}\mathcal{I}(u_{k},v_{k})=\mathcal{I}(u^{*},v^{*}).$

Using Lemma 3.4, we get $\mathcal{I}(u^{*},v^{*})<0$. This finishes our proof.
∎

######  Remark 3.1.

Before wrapping-up the article, as a future direction it would be inspiring to
study analogous set of problems by considering even more general kernel, again
both fractional and nonlinear, by taking into account rough coefficients.
Thus, by replacing the fractional $(p_{1},p_{2})$ Laplacians in
${(\mathcal{P}_{f_{1},f_{2}})}$ with the nonlinear integro-differential
operator given by

$\mathcal{L}_{K}u(x)=P.V.\int_{\mathbb{R}^{N}}|u(y)-u(x)|^{p-2}(u(y)-u(x))K(x,y)dy,\,x\in\mathbb{R}^{N}$

where the symmetric function $K$ is a Gagliardo-type kernel (namely, an
$(s,p)$-kernel) with measurable coefficients. Owing to this aspiration, we
cite ([7],[8]) for interested readers which treats with the general nonlinear
fractional operators above.

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