# Some recent results on singular $p$-Laplacian systems

Umberto Guarnotta, Roberto Livrea
Dipartimento di Matematica e Informatica, Università di Palermo,
Via Archirafi 34, 90123 Palermo, Italy
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

Salvatore A. Marano
Dipartimento di Matematica e Informatica, Università di Catania,
Viale A. Doria 6, 95125 Catania, Italy
E-mail<EMAIL_ADDRESS>

Corresponding author.

###### Abstract

Some recent existence, multiplicity, and uniqueness results for singular
$p$-Laplacian systems either in bounded domains or in the whole space are
presented, with a special attention to the case of convective reactions. A
extensive bibliography is also provided.

Keywords: quasi-linear elliptic system, gradient dependence, singular term,
entire solution, strong solution.

AMS Subject Classification: 35-02, 35J62, 35J75, 35J92.

##  1 Introduction

This survey paper can be divided into two parts. The first (cf. Section 3)
treats singular quasi-linear Dirichlet systems in bounded domains. So, we
study problems of the type

$\left\\{\begin{aligned} -\Delta_{p}u=f(x,u,v,\nabla u,\nabla v)\;\;&\mbox{in
$\Omega$,}\;\;u>0\;\;&&\mbox{in $\Omega$,}\;\;u\lfloor_{\partial\Omega}=0,\\\
-\Delta_{q}v=g(x,u,v,\nabla u,\nabla v)\;\;&\mbox{in
$\Omega$,}\;\;v>0\;\;&&\mbox{in
$\Omega$,}\;\;v\lfloor_{\partial\Omega}=0,\end{aligned}\right.$ (1.1)

where $1<p,q<\infty$, the symbol $\Delta_{r}$ denotes the $r$-Laplace
operator, namely

$\Delta_{r}u:={\rm div}(|\nabla u|^{r-2}\nabla u),$

$\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $N\geq 3$, having a smooth
boundary $\partial\Omega$, while $f,g\in
C^{0}(\Omega\times(\mathbb{R}^{+})^{2}\times(\mathbb{R}^{N})^{2})$ turn out
singular at zero with respect the solution $(u,v)$ or even its gradient
$(\nabla u,\nabla v)$.

If $p=q=2$ then various special, often non-convective, cases of (1.1) have
been thoroughly investigated; see Section 3.1 below. In particular, the
monograph [33] gives a nice introduction to so-called singular Gierer-
Meinhardt systems. Here, we simply make a short account on some recent
existence, multiplicity, or uniqueness results when $(p,q)\neq(2,2)$, as well
as the relevant technical approaches. Regarding this, let us also point out
Chapter 7 of [57].

The second part (cf. Section 4) carries out a similar analysis for singular
quasi-linear systems in the whole space. Hence, it deals with situations like

$\left\\{\begin{aligned} -\Delta_{p}u=f(x,u,v,\nabla u,\nabla v)\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;u>0\;\;&&\mbox{in $\mathbb{R}^{N}$,}\\\
-\Delta_{q}v=g(x,u,v,\nabla u,\nabla v)\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;v>0\;\;&&\mbox{in
$\mathbb{R}^{N}$.}\end{aligned}\right.$ (1.2)

To the best of our knowledge, no previous book or survey on problem (1.2) is
already available.

Both parts exhibit four sub-sections. The first represents a historical sketch
of the semi-linear setting. The next two address existence, multiplicity, and
uniqueness in the non-convective case. The fourth is devoted to singular
systems with convection.

It may be useful to emphasize that [38] treats non-local singular systems,
which, for the sake of brevity, are not examined here.

We apologize in advance for possibly forgetting significant works, but the
literature on (1.1)–(1.2) is by now quite extensive and our knowledge somewhat
limited.

##  2 Basic notation

Let $X(\Omega)$ be a real-valued function space on a nonempty measurable set
$\Omega\subseteq\mathbb{R}^{N}$. If $u_{1},u_{2}\in X(\Omega)$ and
$u_{1}(x)<u_{2}(x)$ a.e. in $\Omega$ then we simply write $u_{1}<u_{2}$. The
meaning of $u_{1}\leq u_{2}$, etc. is analogous. Put

$X(\Omega)_{+}:=\left\\{u\in X(\Omega):u\geq 0\right\\}.$

The symbol $u\in X_{\rm loc}(\Omega)$ means that $u:\Omega\to\mathbb{R}$ and
$u\lfloor_{K}\in X(K)$ for all nonempty compact subset $K$ of $\Omega$. Given
$1<r<+\infty$, define

$r^{\prime}:=\frac{r}{r-1}\,.$

We denote by $\lambda_{1,r}$ the first eigenvalue of the operator
$-\Delta_{r}$ in $W^{1,r}_{0}(\Omega)$. If $r<N$ then

$r^{*}:=\frac{Nr}{N-r}\,.$

Let us next recall the notion and some relevant properties of the so-called
Beppo Levi spaces $\mathcal{D}^{1,r}_{0}(\mathbb{R}^{N})$, systematically
studied for the first time by Deny and Lions [25]. Set

$\mathcal{D}^{1,r}:=\left\\{z\in L^{1}_{\rm loc}(\mathbb{R}^{N}):|\nabla z|\in
L^{r}(\mathbb{R}^{N})\right\\}$

and write $\mathcal{R}$ for the equivalence relation that identifies two
elements in $\mathcal{D}^{1,r}$ whose difference is a constant. The quotient
set $\mathcal{\dot{D}}^{1,r}$, endowed with the norm

$\|u\|_{1,r}:=\left(\int_{\mathbb{R}^{N}}|\nabla u(x)|^{r}{\rm
d}x\right)^{1/r},$

turns out complete. Indicate with $\mathcal{D}^{1,r}_{0}(\mathbb{R}^{N})$ the
subspace of $\mathcal{\dot{D}}^{1,r}$ defined as the closure of
$C^{\infty}_{0}(\mathbb{R}^{N})$ under $\|\cdot\|_{1,r}$, namely

$\mathcal{D}^{1,r}_{0}(\mathbb{R}^{N}):=\overline{C^{\infty}_{0}(\mathbb{R}^{N})}^{\|\cdot\|_{1,r}}.$

$\mathcal{D}^{1,r}_{0}(\mathbb{R}^{N})$, usually called Beppo Levi space, is
reflexive and continuously embeds in $L^{r^{*}}(\mathbb{R}^{N})$, i.e.,

$\mathcal{D}^{1,r}_{0}(\mathbb{R}^{N})\hookrightarrow
L^{r^{*}}(\mathbb{R}^{N}).$ (2.1)

Consequently, if $u\in\mathcal{D}^{1,r}_{0}(\mathbb{R}^{N})$ then $u$ vanishes
at infinity, meaning that the set
$\\{x\in\mathbb{R}^{N}:|u(x)|\geq\varepsilon\\}$ has finite measure for any
$\varepsilon>0$. In fact, by Chebichev’s inequality and (2.1), one has

$|\\{x\in\mathbb{R}^{N}:\,|u(x)|\geq\varepsilon\\}|\leq\varepsilon^{-r^{*}}\|u\|_{r^{*}}^{r^{*}}\leq(c\varepsilon^{-1}\|u\|_{1,r})^{r^{*}}<+\infty,$

where $c>0$ is the best constant related to (2.1). The monographs [28, 52, 68]
provide an exhaustive introduction on this topic.

Finally,

$a\vee b:=\max\\{a,b\\},\;\;a\wedge
b:=\min\\{a,b\\}\quad\forall\,a,b\in\mathbb{R}.$

If $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ then

$d(x):={\rm dist}(x,\partial\Omega),\quad x\in\overline{\Omega},$

while $\Omega^{\prime}\subset\subset\Omega$ means
$\overline{\Omega^{\prime}}\subseteq\Omega$.

##  3 Problems in bounded domains

###  3.1 The case $p=2$

As far as we know, the study of singular semi-linear elliptic systems in
bounded domains started with the paper [14], devoted to Gierer-Meinhardt’s
type [37] problem

$\left\\{\begin{aligned} -\Delta u=-u+\frac{u}{v}\;\;&\mbox{in
$\Omega$,}\;\;u>0\;\;&&\mbox{in $\Omega$,}\;\;u\lfloor_{\partial\Omega}=0,\\\
-\Delta v=-\alpha v+\frac{u}{v}\;\;&\mbox{in $\Omega$,}\;\;v>0\;\;&&\mbox{in
$\Omega$,}\;\;v\lfloor_{\partial\Omega}=0,\end{aligned}\right.$ (3.1)

where $\Omega$ denotes a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 1$,
while $\alpha\in\mathbb{R}^{+}$. More general right-hand sides were then
considered in [15, 48, 32]. The work [56] investigates the system

$\left\\{\begin{aligned} -\Delta u=\lambda
u^{q_{1}}-\frac{u^{p_{1}}}{v^{\beta_{1}}}\;\;&\mbox{in
$\Omega$,}\;\;u>0\;\;&&\mbox{in $\Omega$,}\;\;u\lfloor_{\partial\Omega}=0,\\\
-\Delta v=\mu v^{q_{2}}-\frac{u^{p_{2}}}{v^{\beta_{2}}}\;\;&\mbox{in
$\Omega$,}\;\;v>0\;\;&&\mbox{in
$\Omega$,}\;\;v\lfloor_{\partial\Omega}=0.\end{aligned}\right.$ (3.2)

Here, $q_{1},q_{2},\beta_{1},\beta_{2}\in]0,1[$,
$p_{1},p_{2}\in\mathbb{R}^{+}$, and $\lambda,\mu$ are two positive parameters.
Define

$\sigma:=\frac{p_{2}(1-q_{2})}{(1+\beta_{2})(1-q_{1})},\quad\eta:=\frac{\beta_{1}(1-q_{1})}{(1-p_{1})(1-q_{2})}.$

An adequate sub-super-solution methods yields the following

######  Theorem 3.1 ([56], Theorem 1.1).

If $q_{1}<p_{1}$ then there exists $c_{1}>0$ such that (3.2) admits a solution
$(u,v)\in C^{1,\alpha}(\overline{\Omega})^{2}$ for all $\lambda>0$, $\mu\geq
c_{1}\lambda^{\sigma}$.
If $q_{1}\geq p_{1}$ then there exists $c_{2}>0$ such that (3.2) has no
solution once $\mu>0$ and $\lambda<c_{2}\mu^{-\eta}$.

Two years before, in 2008, Hernandez, Mancebo, and Vega [45] established the
existence of classical solutions to the problem

$\left\\{\begin{aligned} \mathcal{L}_{1}u=f(x,u,v)\;\;&\mbox{in
$\Omega$,}\;\;u>0\;\;&&\mbox{in $\Omega$,}\;\;u\lfloor_{\partial\Omega}=0,\\\
\mathcal{L}_{2}v=g(x,u,v)\;\;&\mbox{in $\Omega$,}\;\;v>0\;\;&&\mbox{in
$\Omega$,}\;\;v\lfloor_{\partial\Omega}=0,\end{aligned}\right.$ (3.3)

where $\mathcal{L}_{i}$ denotes a linear, second-order, uniformly elliptic
operator in non-divergence form while
$f,g:\Omega\times(\mathbb{R}^{+})^{2}\to\mathbb{R}$ are smooth enough. They
examined both the cooperative and the non-cooperative situations; see [45,
Theorems 2.1–2.2]. Here, cooperative means that the reaction
$f(x,\cdot,\cdot)$ and $g(x,\cdot,\cdot)$ turn out increasing on
$\mathbb{R}^{+}$ with respect to each variable separately. A uniqueness result
is also obtained provided (3.3) turns out cooperative and concave, namely

$f(x,\tau s,\tau t)>\tau f(x,s,t),\;\;g(x,\tau s,\tau t)>\tau g(x,s,t)$

for all $\tau\in]0,1[$ and $(x,s,t)\in\Omega\times(\mathbb{R}^{+})^{2}$; cf.
[45, Theorem 2.3]. Many special cases are finally discussed. Singular systems
driven by other linear, second-order, elliptic operators in divergence form
were treated in [8].

Let us next point out that the paper [44] contains a uniqueness result (cf.
[44, Theorem 2.2]) where no cooperative structure is supposed.

In 2015, Ghergu [31] thoroughly investigated both existence and uniqueness of
classical solutions to the model system

$\left\\{\begin{aligned} -\Delta u=u^{\alpha_{1}}+v^{\beta_{1}}\;\;\mbox{in
$\Omega$,}\;\;u>0\;\;\mbox{in $\Omega$,}\;\;&&u\lfloor_{\partial\Omega}=0,\\\
-\Delta v=v^{\alpha_{2}}+u^{\beta_{2}}\;\;\mbox{in
$\Omega$,}\;\;v>0\;\;\mbox{in
$\Omega$,}\;\;&&v\lfloor_{\partial\Omega}=0\end{aligned}\right.$ (3.4)

for $\alpha_{i}\vee\beta_{1}<0$, $i=1,2$. The work [55] (see also [30])
contains a similar analysis on the problem

$\left\\{\begin{aligned} -\Delta
u=K_{1}(x)u^{\alpha_{1}}v^{\beta_{1}}\;\;\mbox{in
$\Omega$,}\;\;u>0\;\;\mbox{in $\Omega$,}\;\;&&u\lfloor_{\partial\Omega}=0,\\\
-\Delta v=K_{2}(x)v^{\alpha_{2}}u^{\beta_{2}}\;\;\mbox{in
$\Omega$,}\;\;v>0\;\;\mbox{in
$\Omega$,}\;\;&&v\lfloor_{\partial\Omega}=0,\end{aligned}\right.$ (3.5)

where, as before, $\alpha_{i}\vee\beta_{i}<0$ and $K_{i}\in
C^{\alpha}(\Omega)$, $i=1,2$.

The existence of solutions to singular convective elliptic systems was firstly
studied by Alves and Moussaoui [2] in 2014.

######  Theorem 3.2 ([2], Sections 3–4).

Let $-\alpha_{i},\beta_{i}\in[0,1[$ and $g_{i}\in
C^{0}(\mathbb{R}^{2N},\mathbb{R}^{+})$ be bounded, $i=1,2$. Then the problem

$\left\\{\begin{aligned} -\Delta u=v^{\alpha_{1}}\pm
v^{\beta_{1}}+g_{1}(\nabla u,\nabla v)\;\;&\mbox{in
$\Omega$,}\;\;u>0\;\;&&\mbox{in $\Omega$,}\;\;u\lfloor_{\partial\Omega}=0,\\\
-\Delta v=u^{\alpha_{2}}\pm u^{\beta_{2}}+g_{2}(\nabla u,\nabla
v)\;\;&\mbox{in $\Omega$,}\;\;v>0\;\;&&\mbox{in
$\Omega$,}\;\;v\lfloor_{\partial\Omega}=0\end{aligned}\right.$

admits a solution $(u,v)\in H^{1}_{0}(\Omega)^{2}\cap C^{2}(\Omega)^{2}$.

Finally, the very recent work [18] treats quasi-linear Schrödinger elliptic
problems wiht both singular and convective reactions, while [11] concerns
singular systems having quadratic gradient.

###  3.2 Existence and multiplicity

To shorten notation, given $p,q\in]1,+\infty[$, write

$\begin{split}X^{p,q}(\Omega)&:=W^{1,p}(\Omega)\times W^{1,q}(\Omega),\quad
X^{p,q}_{\rm loc}(\Omega):=W^{1,p}_{\rm loc}(\Omega)\times W^{1,q}_{\rm
loc}(\Omega),\\\ &X^{p,q}_{0}(\Omega):=W^{1,p}_{0}(\Omega)\times
W^{1,q}_{0}(\Omega).\end{split}$

From what we know, the first existence result for singular quasi-linear
problems dates back to 2007 and examines the case

$\left\\{\begin{aligned}
-\Delta_{p}u&=v^{\alpha_{1}}+v^{\beta_{1}}\;\;\mbox{in
$\Omega$,}\;\;&&u>0\;\;\mbox{in $\Omega$,}\;\;u\lfloor_{\partial\Omega}=0,\\\
-\Delta_{q}v&=u^{\alpha_{2}}+u^{\beta_{2}}\;\;\mbox{in
$\Omega$,}\;\;&&v>0\;\;\mbox{in
$\Omega$,}\;\;v\lfloor_{\partial\Omega}=0.\end{aligned}\right.$ (3.6)

Here, $\alpha_{i}<0<\beta_{i}$, while $N\geq 2$.

######  Theorem 3.3 ([1], Theorem 1.1).

If $\frac{2N}{N+1}\leq p,q<N$ and $0<-\alpha_{i},\beta_{i}<\theta_{i}$,
$i=1,2$, with

$\theta_{1}:=\min\left\\{1,p-1,\frac{q^{\prime}}{p^{\prime}}\right\\},\quad\theta_{2}:=\min\left\\{1,q-1,\frac{p^{\prime}}{q^{\prime}}\right\\},$

then (3.6) has a weak solution $(u,v)\in X^{p,q}_{0}(\Omega)$.

Its proof employs a result on nonlinear eigenvalue problems with lack of
bifurcation due to Rabinowitz [65] and a Hardy–Sobolev type inequality [46].

Now, consider the general problem

$\left\\{\begin{aligned} -\Delta_{p}u&=f(x,u,v)\;\;\mbox{in
$\Omega$,}\;\;&&u>0\;\;\mbox{in $\Omega$,}\;\;u\lfloor_{\partial\Omega}=0,\\\
-\Delta_{q}v&=g(x,u,v)\;\;\mbox{in $\Omega$,}\,\;&&v>0\;\;\mbox{in
$\Omega$,}\;\;v\lfloor_{\partial\Omega}=0,\\\ \end{aligned}\right.$ (3.7)

where $f,g:\Omega\times(\mathbb{R}^{+})^{2}\to\mathbb{R}$ fulfill
Carathéodory’s conditions. In 2012, El Manouni, Perera, and Shivaji
investigated (3.7) under the assumptions below.

  * $({\rm a}_{1})$

The functions $t\mapsto f(x,s,t)$ and $s\mapsto g(x,s,t)$ are increasing in
$\mathbb{R}^{+}$, namely system (3.7) is cooperative.

  * $({\rm a}_{2})$

For every $0<s_{0}\leq s_{1}$ and $0<t_{0}\leq t_{1}$ one has

$\sup_{\Omega\times[s_{0},s_{1}]\times]0,t_{1}]}f(x,s,t)<+\infty,\;\;\sup_{\Omega\times]0,s_{1}]\times[t_{0},t_{1}]}g(x,s,t)<+\infty,$

as well as

$\sup_{\Omega\times[s_{0},s_{1}]\times[t_{0},t_{1}]}\max\left\\{|f(x,s,t)|,|g(x,s,t)|\right\\}<+\infty.$

They seek solutions $(u,v)\in X^{p,q}_{\rm loc}(\Omega)$ that satisfy the
differential equations in the sense of distributions, i.e.,

$\begin{split}\int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi\,{\rm
d}x=\int_{\Omega}f(\cdot,u,v)\varphi\,{\rm d}x\quad\forall\,\varphi\in
C^{\infty}_{0}(\Omega),\\\ \int_{\Omega}|\nabla v|^{q-2}\nabla
v\cdot\nabla\psi\,{\rm d}x=\int_{\Omega}g(\cdot,u,v)\psi\,{\rm
d}x\quad\forall\,\psi\in C^{\infty}_{0}(\Omega),\end{split}$

and, moreover, $u,v\in C^{0}(\overline{\Omega})$.

A pair $(\underline{u},\underline{v})\in X^{p,q}(\Omega)$ is called a sub-
solution of problem (3.7) provided $f(\cdot,\underline{u},\underline{v})\in
L^{p^{\prime}}(\Omega)$, $g(\cdot,\underline{u},\underline{v})\in
L^{q^{\prime}}(\Omega)$, and

$-\Delta_{p}\underline{u}\leq f(\cdot,\underline{u},\underline{v})\;\;\mbox{in
$\Omega$,}\;\;-\Delta_{p}\underline{v}\leq
g(\cdot,\underline{u},\underline{v})\;\;\mbox{in
$\Omega$,}\;\;\underline{u}\vee\underline{v}\leq 0\;\>\mbox{on
$\partial\Omega$.}$

A super-solution $(\overline{u},\overline{v})$ is defined similarly, by
reversing all the above inequalities.

Let $\\{\varepsilon_{n}\\}\subseteq\mathbb{R}^{+}$ satisfy $\varepsilon_{n}\to
0$. Set, for every $n\in\mathbb{N}$,
$(x,s,t)\in\Omega\times(\mathbb{R}^{+})^{2}$,

$f_{n}(x,s,t):=f(x,s\vee\varepsilon_{n},t\vee\varepsilon_{n}),\;\;g_{j}(x,s,t):=g(x,s\vee\varepsilon_{n},t\vee\varepsilon_{n}),$

and consider the sequence of regularized systems

$\left\\{\begin{aligned} -\Delta_{p}u&=f_{n}(x,u,v)\;\;&&\mbox{in
$\Omega$,}\;\;u\lfloor_{\Omega}=0,\\\ -\Delta_{q}v&=g_{n}(x,u,v)\;\;&&\mbox{in
$\Omega$,}\;\;v\lfloor_{\Omega}=0.\end{aligned}\right.$ (3.8)

######  Theorem 3.4 ([27], Theorem 3.1).

Suppose $({\rm a}_{1})$–$({\rm a}_{2})$ hold. If, for each $n\in\mathbb{N}$,
there exist a sub-solution $(\underline{u}_{n},\underline{v}_{n})$ and a
super-solution $(\overline{u}_{n},\overline{v}_{n})$ to (3.8) such that
$(\underline{u}_{n},\underline{v}_{n})\leq(\overline{u}_{n},\overline{v}_{n})$,

$\inf_{n\in\mathbb{N}}\operatorname*{ess\,inf}_{\Omega^{\prime}}\,(\underline{u}_{n}\wedge\underline{v}_{n})>0$

whenever $\Omega^{\prime}\subset\subset\Omega$, and

$\sup_{n\in\mathbb{N}}\operatorname*{ess\,sup}_{\Omega}\,(\overline{u}_{n}\vee\overline{v}_{n})<+\infty,$

then (3.7) admits a distributional solution $(u,v)\in C^{1,\alpha}_{\rm
loc}(\Omega)^{2}\cap C^{0}(\overline{\Omega})^{2}$.

Sufficient conditions for the existence of sub-super-solution pairs to (3.8)
are given in [27, Sections 4–5]. As an example, via Theorem 3.4 one can show
that the model problem (cf. (3.4))

$\left\\{\begin{aligned} -\Delta_{p}u=u^{\alpha_{1}}+\mu
v^{\beta_{1}}\;\;\mbox{in $\Omega$,}\;\;u>0\;\;\mbox{in
$\Omega$,}\;\;&&u\lfloor_{\partial\Omega}=0,\\\
-\Delta_{q}v=v^{\alpha_{2}}+\mu u^{\beta_{2}}\;\;\mbox{in
$\Omega$,}\;\;v>0\;\;\mbox{in
$\Omega$,}\;\;&&v\lfloor_{\partial\Omega}=0,\end{aligned}\right.$

where $\alpha_{1}\vee\alpha_{2}<0$ and $\beta_{1}\wedge\beta_{2}\geq 0$,
possesses a solution provided $\mu\geq 0$ is small enough. Moreover, singular
semi-positone systems, which means both $\lim_{s\to 0^{+}}f(x,s,t)=-\infty$
uniformly in $(x,t)$ and $\lim_{t\to 0^{+}}g(x,s,t)=-\infty$ uniformly with
respect to $(x,s)$, are investigated. Let us also mention the papers [51],
[47], and [16]. In particular, [16] deals with a non-cooperative system.

One year later, Giacomoni, Hernandez, and Sauvy obtained the general results
below, where $f,g\in C^{1}(\Omega\times(\mathbb{R}^{+})^{2})$, through a
different notion of sub-super-solution.

We say that $(\underline{u},\underline{v}),(\overline{u},\overline{v})\in
X^{p.q}_{\rm loc}(\Omega)\cap C^{0}(\overline{\Omega})^{2}$ are a sub-super-
solution pair to (3.7) when $\underline{u}$, $\underline{v}$, $\overline{u}$,
$\overline{v}$ are locally uniformly positive,
$(\underline{u},\underline{v})\leq(\overline{u},\overline{v})$, and for any
$(u,v)\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$ one
has

$-\Delta_{p}\underline{u}\leq
f(\cdot,\underline{u},v),\;-\Delta_{p}\underline{v}\leq
g(\cdot,u,\underline{v}),\;-\Delta_{p}\overline{u}\geq
f(\cdot,\overline{u},v),\;-\Delta_{p}\overline{v}\geq g(\cdot,u,\overline{v})$

in $\Omega$. The set ${\cal
C}:=[\underline{v},\overline{v}]\times[\underline{u},\overline{u}]$ is usually
called trapping region; cf. [10].

######  Theorem 3.5 ([35], Theorem 2.1).

Let $(\underline{u},\underline{v}),(\overline{u},\overline{v})\in
X^{p,q}_{0}(\Omega)$ be a sub-super-solution pair to (3.7) fulfilling:

  * $({\rm a}_{3})$

$\overline{u}\leq c_{1}d^{\gamma_{1}}$ and $\overline{v}\leq
c_{2}d^{\gamma_{2}}$, with appropriate $c_{1},c_{2},\gamma_{1},\gamma_{2}>0$.

  * $({\rm a}_{4})$

There exist $c_{3},c_{4}>0$, $\delta_{1},\delta_{2}\in\mathbb{R}$ such that

$|f(\cdot,u,v)|\leq c_{3}d^{\delta_{1}},\,\;|g(\cdot,u,v)|\leq
c_{4}d^{\delta_{2}}\,\;\forall\,(u,v)\in{\cal C}.$

  * $({\rm a}_{5})$

For suitable $c_{5},c_{6},\sigma_{1},\sigma_{2}>0$ one has

$\left|\frac{\partial f}{\partial s}(\cdot,u,v)\right|\leq
c_{5}d^{\delta_{1}-\sigma_{1}},\;\left|\frac{\partial g}{\partial
t}(\cdot,u,v)\right|\leq c_{6}d^{\delta_{2}-\sigma_{2}},\;(u,v)\in{\cal C}.$

Assume further that

$\delta_{1}>-2+\frac{1}{p}+(\sigma_{1}-\gamma_{1})^{+},\;\;\delta_{2}>-2+\frac{1}{q}+(\sigma_{2}-\gamma_{2})^{+}.$

Then (3.7) admits a weak solution $(u,v)\in{\cal C}$.

######  Theorem 3.6 ([35], Theorem 2.3).

Suppose $(\underline{u},\underline{v}),(\overline{u},\overline{v})$ is a sub-
super-solution pair to (3.7) complying with $({\rm a}_{3})$. If

$\frac{\partial f}{\partial t}(x,s,t)>0,\;\;\frac{\partial g}{\partial
s}(x,s,t)>0\quad\forall\,(x,s,t)\in\Omega\times(\mathbb{R}^{+})^{2}$ (3.9)

and there exist $c_{7},c_{8}>0$, $\eta_{1},\eta_{2}\in\mathbb{R}$ such that

$\left|\frac{\partial f}{\partial s}(\cdot,u,v)\right|\leq
c_{7}d^{\eta_{1}},\;\;\left|\frac{\partial g}{\partial
t}(\cdot,u,v)\right|\leq c_{8}d^{\eta_{2}}\;\;\forall\,(u,v)\in{\cal C},$

then (3.7) possesses a distributional solution $(u,v)\in{\cal C}$.

Proofs are based on a very nice, non-trivial use of Schauder’s fixed point
theorem. Applications to the model problem

$\left\\{\begin{aligned}
-\Delta_{p}u=K_{1}(x)u^{\alpha_{1}}\,v^{\beta_{1}}\;\;&\mbox{in
$\Omega$,}\;\;u>0\;\;&&\mbox{in $\Omega$,}\;\;u\lfloor_{\partial\Omega}=0,\\\
-\Delta_{q}v=K_{2}(x)v^{\alpha_{2}}\,u^{\beta_{2}}\;\;&\mbox{in
$\Omega$,}\;\;v>0\;\;&&\mbox{in
$\Omega$,}\;\;v\lfloor_{\partial\Omega}=0,\end{aligned}\right.$ (3.10)

where $K_{1},K_{2}$ satisfy appropriate conditions while

$\alpha_{1}<p-1,\;\;\alpha_{2}<q-1,\;\;(p-1-\alpha_{1})(q-1-\alpha_{2})>|\beta_{1}\beta_{2}|>0,$
(3.11)

are given. Evidently, (3.10) becomes (3.5) for $p=q=2$. See also [21], where
$p=q<N$, $K_{1}\in L^{\infty}(\Omega)_{+}$, $K_{2}\in L^{\gamma}(\Omega)_{+}$
for some $\gamma>\frac{N}{p}$,

$0<\beta_{1}<1\wedge(p-1),\;\;-1<\alpha_{1}<p-\beta_{1}-1,\;\;\alpha_{2}:=\beta_{1}-1,\;\;\beta_{2}:=\alpha_{1}+1\,.$

A special case of (3.10) was treated in [58] (cf. in addition [20]), namely

$\left\\{\begin{aligned}
-\Delta_{p}u=u^{\alpha_{1}}\,v^{\beta_{1}}\;\;&\mbox{in
$\Omega$,}\;\;u>0\;\;&&\mbox{in $\Omega$,}\;\;u\lfloor_{\partial\Omega}=0,\\\
-\Delta_{q}v=v^{\alpha_{2}}\,u^{\beta_{2}}\;\;&\mbox{in
$\Omega$,}\;\;v>0\;\;&&\mbox{in
$\Omega$,}\;\;v\lfloor_{\partial\Omega}=0.\end{aligned}\right.$ (3.12)

######  Theorem 3.7 ([58], Theorem 1.1).

Let $1<p,q\leq N$ and let $\alpha_{i},\beta_{i}$ satisfy

$\begin{split}-2+\frac{1}{p}\leq\alpha_{1}<0,\;\;-2+\frac{1}{q}\leq\alpha_{2}<0,\\\
0<\beta_{1}<\frac{q}{p}(p-1-\alpha_{1}),\;\;0<\beta_{2}<\frac{p}{q}(q-1-\alpha_{2}).\end{split}$
(3.13)

Then (3.12) has a weak solution $(u,v)\in X^{p,q}_{0}(\Omega)\cap
L^{\infty}(\Omega)^{2}$.

######  Remark 3.8.

It should be noted that (3.13) forces (3.11). Moreover, both (3.9) and (3.13)
basically entail $({\rm a}_{1})$.

System (3.12), with a competitive interaction between the two components $u$
and $v$ was thoroughly studied by Giacomoni, Schindler, and Takac [36]. Recall
that (3.7) is said to be competitive if

  * $({\rm a}^{\prime}_{1})$

The functions $t\mapsto f(x,s,t)$ and $s\mapsto g(x,s,t)$ are decreasing on
$\mathbb{R}^{+}$.

In the situation above this means $\alpha_{i}\vee\beta_{i}<0$. Under the sub-
homogeneity condition

$(p-1-\alpha_{1})(q-1-\alpha_{2})>\beta_{1}\beta_{2}$ (3.14)

and suitable upper bounds regarding $\alpha_{i},\beta_{i}$, they proved that
(3.12) admits a solution $(u,v)\in X^{p,q}_{0}(\Omega)\cap
C^{0,\alpha}(\overline{\Omega})^{2}$. When $p=q$, see also [69].

Existence results for problem (3.7) where the competitive structure $({\rm
a}^{\prime}_{1})$ is allowed can be found in [59, 60].

The recent work [4] examines singular $(p(x),q(x))$-Laplacian problems with
singularity coming through logarithmic reactions that involve variable
exponents growth conditions; cf. also [3, 63].

As far as we know, till today, much less attention has been paid to
multiplicity of solutions. Actually, we can only mention the papers [67, 12,
26, 5, 50]. The first deals with singular $p(x)$-Laplacian systems while the
second is devoted to quasi-linear problems driven by
$(\Phi_{1},\Phi_{2})$-Laplace operators. Theorem 1 in [26] considers the case
when $f,g$ do not depend on $x$, are positive, entail a cooperative structure,
and, roughly speaking, cross the first eigenvalue at infinity, namely

$\lim_{s\to+\infty}\frac{f(s,t)}{s^{p-1}}=J_{1}>\lambda_{1,p},\quad\lim_{t\to+\infty}\frac{g(s,t)}{t^{q-1}}=J_{2}>\lambda_{1,q}.$

Two smooth solutions are obtained combining sub-super-solution methods with
the Leray-Schauder topological degree. A different approach is adopted in [5].
The differential operators, which include the $r$-Laplacian as a special case,
are neither homogeneous nor linear, while

$f(x,s,t):=K_{1}(x)s^{\alpha_{1}}+\frac{\partial h}{\partial s}(x,s,t),\quad
g(x,s,t):=K_{2}(x)t^{\alpha_{2}}+\frac{\partial h}{\partial t}(x,s,t),$

where $K_{i}\in L^{\infty}(\Omega)$, $\alpha_{i}<0$, and $h\in
C^{1}(\overline{\Omega}\times\mathbb{R}^{2})$ satisfy appropriate conditions
that permit the use of variational methods. Theorem 1.7 of [5] gives two
positive solutions provided $\max_{i=1,2}\|K_{i}\|_{\infty}$ is sufficiently
small. Finally, [50] addresses singular $p(x)$-Laplacian systems with
nonlinear boundary conditions.

###  3.3 Uniqueness

Unless the semi-linear case, uniqueness of solutions looks a difficult matter,
even for problem (3.12). In fact, as pointed out by Giacomoni, Hernandez, and
Moussaoui [34], equations with quasi-linear elliptic operators exhibit
additional troubles for obtaining the validity of the strong comparison
principle, which requires the $C^{1}$-regularity of solutions. If it cannot be
obtained (as in the strongly singular setting $\alpha_{i}+\beta_{i}<-1$) then
one can still try to use a suitable variant of the well known Krasnoleskii’s
argument [49].
Let us first examine the cooperative case $\beta_{1}\wedge\beta_{2}>0$.
Proposition 3.1 of [34] basically yields

######  Theorem 3.9 ([34], Theorem 3.2).

If $-1<\alpha_{i}+\beta_{i}<0<\beta_{i}$, $i=1,2$, then (3.12) admits a unique
weak solution $(u,v)\in X^{p,q}_{0}(\Omega)\cap
C^{1,\alpha}(\overline{\Omega})^{2}$. Moreover, $u,v\geq c_{1}d$ for some
$c_{1}>0$.

Suppose now

$\alpha_{1}+\beta_{i}<-1,\;i=1,2,\;\;0<\beta_{1}<p-1,\;\;0<\beta_{2}<q-1,$
(3.15)

and denote by $\gamma,\theta\in]0,1[$ the unique solution to the system

$\left\\{\begin{aligned}
(\gamma-1)(p-1)-1&=\alpha_{1}\gamma+\beta_{1}\theta\\\
(\theta-1)(q-1)-1&=\beta_{2}\gamma+\alpha_{2}\theta.\end{aligned}\right.$
(3.16)

######  Theorem 3.10 ([34], Theorem 3.3).

Let (3.15)–(3.16) be satisfied. Assume also that

$\alpha_{1}\gamma+\beta_{1}\theta>\frac{1}{p}-2,\;\;\beta_{2}\gamma+\alpha_{2}\theta>\frac{1}{q}-2.$
(3.17)

Then (3.12) possesses a unique weak solution $(u,v)\in X^{p,q}_{0}(\Omega)$
fulfilling

$c_{2}(\varphi_{1,p}^{\gamma},\varphi_{1,q}^{\theta})\leq(u,v)\leq
c_{3}(\varphi_{1,p}^{\gamma},\varphi_{1,q}^{\theta})$

with appropriate $c_{2},c_{3}>0$.

We next examine the competitive case $\beta_{i}<0$. Via the sub-homogeneity
condition (3.14), Theorem 2.2 of [36] considers various possible choices of
exponents. Here, for the sake of brevity, we will present only one.

######  Theorem 3.11 ([36], Theorem 2.2).

Suppose $\alpha_{i}\vee\beta_{i}<0$. Under (3.14), (3.16), and (3.17), the
same conclusion of Theorem 3.10 is true.

###  3.4 Systems with convection terms

In 2017, Motreanu, Moussaoui, and Zhang [61] treated the general problem

$\left\\{\begin{aligned} -\Delta_{p}u=f(u,v,\nabla u,\nabla v)\;\;&\mbox{in
$\Omega$,}\;\;u>0\;\;&&\mbox{in $\Omega$,}\;\;u\lfloor_{\partial\Omega}=0,\\\
-\Delta_{q}v=g(u,v,\nabla u,\nabla v)\;\;&\mbox{in
$\Omega$,}\;\;v>0\;\;&&\mbox{in
$\Omega$,}\;\;v\lfloor_{\partial\Omega}=0,\end{aligned}\right.$ (3.18)

where
$f,g:(\mathbb{R}^{+})^{2}\times(\mathbb{R}^{N}\setminus\\{0\\})^{2}\to\mathbb{R}^{+}$
are continuous functions satisfying the growth conditions below.

  * $({\rm a}_{6})$

There exist $\mu_{i},\hat{\mu}_{i}>0$ and $\alpha_{i},\beta_{i}\in\mathbb{R}$,
such that

$\displaystyle\mu_{1}(s+|\xi_{1}|)^{\alpha_{1}}(t+|\xi_{2}|)^{\beta_{1}}\leq
f(s,t,\xi_{1},\xi_{2})\leq\hat{\mu}_{1}(s+|\xi_{1}|)^{\alpha_{1}}(t+|\xi_{2}|)^{\beta_{1}}$
$\displaystyle\mu_{2}(s+|\xi_{1}|)^{\beta_{2}}(t+|\xi_{2}|)^{\alpha_{2}}\leq
g(s,t,\xi_{1},\xi_{2})\leq\hat{\mu}_{2}(s+|\xi_{1}|)^{\beta_{2}}(t+|\xi_{2}|)^{\alpha_{2}}$

for all
$(s,t,\xi_{1},\xi_{2})\in(\mathbb{R}^{+})^{2}\times(\mathbb{R}^{N}\setminus\\{0\\})^{2}$.

  * $({\rm a}_{7})$

One has

$\beta_{1}\vee\beta_{2}<0,\quad\alpha_{1}\alpha_{2}>0,\quad|\alpha_{1}|<p-1+\beta_{1},\quad|\alpha_{2}|<q-1+\beta_{2}.$

Since $\beta_{1}\vee\beta_{2}<0$, system (3.18) has a competitive structure.
Moreover, the right-hand sides may exhibit singularities in both the solution
and its gradient.

Combining comparison arguments with a priori estimates yields the next

######  Theorem 3.12 ([61], Theorem 1.1).

If $({\rm a}_{6})$–$({\rm a}_{7})$ hold then (3.18) admits a solution
$(u,v)\in C^{1,\alpha}(\overline{\Omega})^{2}$.

Three years later, also the cooperative case was examined; see [9]. Now,
$1<p,q<N$ while
$f,g:(\mathbb{R}^{+})^{2}\times\mathbb{R}^{2N}\to\mathbb{R}^{+}$ are
continuous and fulfill

  * $({\rm a}_{8})$

There exist
$\mu_{i},\hat{\mu}_{i},\beta_{i},\gamma_{i},\theta_{i}\in\mathbb{R}^{+}$,
$\alpha_{i}\in]-1,0[$, $i=1,2$, such that

$\begin{split}\mu_{1}s^{\alpha_{1}}t^{\beta_{1}}\leq
f(s,t,\xi_{1},\xi_{2})\leq\hat{\mu}_{1}s^{\alpha_{1}}t^{\beta_{1}}+|\xi_{1}|^{\gamma_{1}}+|\xi_{2}|^{\theta_{1}},\\\
\mu_{2}s^{\beta_{2}}t^{\alpha_{2}}\leq
g(s,t,\xi_{1},\xi_{2})\leq\hat{\mu}_{2}s^{\beta_{2}}t^{\alpha_{2}}+|\xi_{1}|^{\gamma_{2}}+|\xi_{2}|^{\theta_{2}}\end{split}$

for all $(s,t,\xi_{1},\xi_{2})\in(\mathbb{R}^{+})^{2}\times\mathbb{R}^{2N}$.

  * $({\rm a}_{9})$

One has $\alpha_{i}+\beta_{i}\geq 0$, as well as

$\max\\{-\alpha_{1}+\beta_{1},\gamma_{1},\theta_{1}\\}<p-1,\;\;\max\\{-\alpha_{2}+\beta_{2},\gamma_{2},\theta_{2}\\}<q-1.$

######  Theorem 3.13 ([9], Theorem 1).

Under $({\rm a}_{8})$–$({\rm a}_{9})$, problem (3.18) possesses a solution
$(u,v)\in C^{1}_{0}(\overline{\Omega})^{2}$. Moreover, $c_{1}d\leq u,v\leq
c_{2}d$ for suitable $c_{1},c_{2}>0$.

A further interesting contribution in contained in [22]. The very recent paper
[40] (see also [41]) establishes the existence of infinitely many solutions to
the Neumann problem

$\left\\{\begin{aligned} -\Delta_{p}u=f(x,u,v,\nabla u,\nabla v)\;\;&\mbox{in
$\Omega$,}\;\;u>0\;\;&&\mbox{in $\Omega$,}\;\;\frac{\partial
u}{\partial\nu}=0\;\;\mbox{on $\partial\Omega$,}\\\
-\Delta_{q}v=g(x,u,v,\nabla u,\nabla v)\;\;&\mbox{in
$\Omega$,}\;\;v>0\;\;&&\mbox{in $\Omega$,}\;\;\frac{\partial
v}{\partial\nu}=0\;\;\mbox{on $\partial\Omega$,}\end{aligned}\right.$ (3.19)

where $1<p,q<+\infty$,
$f,g:\Omega\times(\mathbb{R}^{+})^{2}\times\mathbb{R}^{2N}\to\mathbb{R}$
satisfy Caratheodory’s conditions while $\nu$ denotes the outer unit normal to
$\partial\Omega$.
The sub-linear case is first investigated using the next assumptions.

  * $({\rm a}_{10})$

There exist $\alpha_{i}<0<\beta_{i}$, $\gamma_{1},\delta_{1}\in[0,p-1[$,
$\gamma_{2},\delta_{2}\in[0,q-1[$, and $a_{i},b_{i},c_{i}\in
L^{\infty}(\Omega)$ such that

$\displaystyle|f(x,s,t,\xi_{1},\xi_{2})|\leq
a_{1}(x)s^{\alpha_{1}}t^{\beta_{1}}+b_{1}(x)(|\xi_{1}|^{\gamma_{1}}+|\xi_{2}|^{\delta_{1}})+c_{1}(x),$
$\displaystyle|g(x,s,t,\xi_{1},\xi_{2})|\leq
a_{2}(x)s^{\beta_{2}}t^{\alpha_{2}}+b_{2}(x)(|\xi_{1}|^{\gamma_{2}}+|\xi_{2}|^{\delta_{2}})+c_{2}(x)$

for all
$(x,s,t,\xi_{1},\xi_{2})\in\Omega\times(\mathbb{R}^{+})^{2}\times\mathbb{R}^{2N}$.

  * $({\rm a}_{11})$

There are
$\\{h_{n}\\},\\{\hat{h}_{n}\\},\\{k_{n}\\},\\{\hat{k}_{n}\\},\\{C_{n}\\}\subseteq\mathbb{R}^{+}$,
with $C_{n}\to+\infty$, satisfying $h_{n}<k_{n}<h_{n+1}$,
$\hat{h}_{n}<\hat{k}_{n}<\hat{h}_{n+1}$,

$\begin{split}f(x,k_{n},t,\xi_{1},\xi_{2})&\leq 0\leq
f(x,h_{n},t,\xi_{1},\xi_{2}),\\\ g(x,s,\hat{k}_{n},\xi_{1},\xi_{2})&\leq 0\leq
g(x,s,\hat{h}_{n},\xi_{1},\xi_{2})\end{split}$ (3.20)

for all
$(x,s,t,\xi_{1},\xi_{2})\in\Omega\times[h_{n},k_{n}]\times[\hat{h}_{n},\hat{k}_{n}]\times
B_{\mathbb{R}^{N}}(C_{n})^{2}$, $n\in\mathbb{N}$, and

$\lim_{n\to\infty}\frac{h_{n}^{\alpha_{1}}\hat{k}_{n}^{\beta_{1}}}{C_{n}^{p-1}}=0,\quad\lim_{n\to\infty}\frac{\hat{h}_{n}^{\alpha_{2}}k_{n}^{\beta_{2}}}{C_{n}^{q-1}}=0.$
(3.21)

######  Theorem 3.14 ([40], Theorem 4.2).

If $({\rm a}_{10})$–$({\rm a}_{11})$ hold then (3.19) has a sequence of
solutions $\\{(u_{n},v_{n})\\}\subseteq C^{1}(\overline{\Omega})^{2}$ such
that $(u_{n},v_{n})<(u_{n+1},v_{n+1})$ for every $n\in\mathbb{N}$. Moreover,
$\displaystyle{\lim_{n\to\infty}}u_{n}=\displaystyle{\lim_{n\to\infty}}v_{n}=+\infty$
uniformly in $\overline{\Omega}$ once $h_{n},\hat{h}_{n}\to+\infty$.

As regards the super-linear case, denote by $({\rm a}^{\prime}_{10})$
condition $({\rm a}_{10})$ written for $\gamma_{1},\delta_{1}>p-1$ and
$\gamma_{2},\delta_{2}>q-1$. Similarly,

  * $({\rm a}^{\prime}_{11})$

There exist
$\\{h_{n}\\},\\{\hat{h}_{n}\\},\\{k_{n}\\},\\{\hat{k}_{n}\\},\\{C_{n}\\}\subseteq\mathbb{R}^{+}$,
with $C_{n}\to 0$, satisfying $k_{n+1}<h_{n}<k_{n}$,
$\hat{k}_{n+1}<\hat{h}_{n}<\hat{k}_{n}$ for all $n\in\mathbb{N}$ and
(3.20)–(3.21).

######  Theorem 3.15 ([40], Theorem 4.3).

Under $({\rm a^{\prime}}_{10})$–$({\rm a^{\prime}}_{11})$, problem (3.19)
possesses a sequence of solutions $\\{(u_{n},v_{n})\\}\subseteq
C^{1}(\overline{\Omega})^{2}$ such that $(u_{n+1},v_{n+1})<(u_{n},v_{n})$ for
every $n\in\mathbb{N}$. Moreover,
$\displaystyle{\lim_{n\to\infty}}u_{n}=\displaystyle{\lim_{n\to\infty}}v_{n}=0$
uniformly in $\overline{\Omega}$ once $k_{n},\hat{k}_{n}\to 0$.

An easy example of nonlinearities, with both singular and convective terms,
that fulfill (3.20)–(3.21) is the following.

######  Example 3.16.

Set, for every
$(x,s,t,\xi_{1},\xi_{2})\in\Omega\times(\mathbb{R}^{+})^{2}\times\mathbb{R}^{2N}$,

$\begin{split}f(x,s,t,\xi_{1},\xi_{2})=\sin\frac{1}{s}\,\left(s^{\alpha_{1}}t^{\beta_{1}}-|\xi_{1}|^{\gamma_{1}}-|\xi_{2}|^{\delta_{1}}\right),\\\
g(x,s,t,\xi_{1},\xi_{2})=\cos\frac{1}{t}\,\left(s^{\beta_{2}}t^{\alpha_{2}}-|\xi_{1}|^{\gamma_{2}}-|\xi_{2}|^{\delta_{2}}\right),\end{split}$

where

$\gamma_{1}\wedge\delta_{1}>\alpha_{1}+\beta_{1}>p-1,\quad\gamma_{2}\wedge\delta_{2}>\alpha_{2}+\beta_{2}>q-1.$

To check (3.20)–(3.21) one can pick $C_{n}=\frac{1}{n}$,

$h_{n}=\frac{1}{\pi/2+2\pi n},\;k_{n}=\frac{1}{-\pi/2+2\pi
n},\;\hat{h}_{n}=\frac{1}{2\pi+2\pi n},\;\hat{k}_{n}=\frac{1}{\pi+2\pi n}.$

##  4 Problems on the whole space

###  4.1 The case $p=2$

In 2009, Moussaoui, Khodja, and Tas [62] studied the following singular,,
semi-linear elliptic, Gierer-Meinhardt’s type system:

$\left\\{\begin{aligned} -\Delta
u+\alpha_{1}(x)u=a_{1}(x)\frac{1}{v^{q}}\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;u>0\;\;&&\mbox{in $\mathbb{R}^{N}$,}\\\ -\Delta
v+\alpha_{2}(x)v=a_{2}(x)\frac{u^{r}}{v^{s}}\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;v>0\;\;&&\mbox{in $\mathbb{R}^{N}$,}\\\ u(x)\to
0,\;v(x)\to 0\;\;&\mbox{as $|x|\to\infty$,}\end{aligned}\right.$ (4.1)

where, roughly speaking, $\alpha_{i},a_{i}\in L^{\infty}_{\rm
loc}(\mathbb{R}^{N})_{+}$, $a_{i}$ satisfies suitable integrability
conditions, $q,r,s>0$, and $r\leq s+1<2$. A solution
$(u,v)\in\mathcal{D}^{1,2}_{0}(\mathbb{R}^{N})^{2}$ is obtained via Schauder’s
fixed point theorem. Previous papers on the same subject are [23, 24], whilst,
excepting [6, 7], we were not able to find more recent contributions.

Let us next point out that, even for the semi-linear case, the question
whether a singular elliptic problem on the whole space admits multiple
positive solutions is an open question.

Finally, as regards uniqueness, we mention [6, Theorem 1.1], which deals with
the system

$\left\\{\begin{aligned} -\Delta
u+\alpha(x)u^{2}=a(x)\frac{v^{1-p}}{u^{p}}\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;u>0\;\;&&\mbox{in $\mathbb{R}^{N}$,}\\\ -\Delta
v+\alpha(x)v^{2}=a(x)\frac{u^{1-p}}{v^{p}}\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;v>0\;\;&&\mbox{in
$\mathbb{R}^{N}$,}\end{aligned}\right.$ (4.2)

where $N\geq 3$, $p\in]\frac{1}{2},1[$, $\alpha\in L^{1}(\mathbb{R}^{N})\cap
L^{\frac{q}{q-2}}(\mathbb{R}^{N})$ for some $q\in]2,2^{*}[$, $\alpha>0$, and
$0\leq a\leq\alpha$.

Existence and uniqueness of solutions to singular convective elliptic problems
was firstly studied by Benrhouma [7] in 2017, who considered the system

$\left\\{\begin{aligned} -\Delta u+\alpha_{1}\frac{|\nabla
u|^{2}}{u}=\frac{p}{p+q}a(x)u^{p-1}v^{q}+b_{1}(x)\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;u>0\;\;&&\mbox{in $\mathbb{R}^{N}$,}\\\ -\Delta
v+\alpha_{2}\frac{|\nabla
v|^{2}}{v}=\frac{q}{p+q}a(x)u^{p}v^{q-1}+b_{2}(x)\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;v>0\;\;&&\mbox{in
$\mathbb{R}^{N}$,}\end{aligned}\right.$

with $p\wedge q>1$, $p+q<2^{*}-1$, $a$ fulfilling appropriate integrability
conditions, $\alpha_{i}>\frac{N+2}{4}$, and $b_{i}\in
L^{2^{*}}(\mathbb{R}^{N})_{+}\cap L^{\infty}(\mathbb{R}^{N})$, $i=1,2$.

###  4.2 Existence and multiplicity

Henceforth, given $p,q\in]1,+\infty[$, we will write

$X^{p,q}(\mathbb{R}^{N}):=\mathcal{D}^{1,p}_{0}(\mathbb{R}^{N})\times\mathcal{D}^{1,q}_{0}(\mathbb{R}^{N}).$

To the best of our knowledge, until 2019, singular elliptic systems in the
whole space were investigated only for $p:=q:=2$, essentially exploiting the
linearity of involved differential operators. The paper [54] considers the
problem

$\left\\{\begin{aligned} -\Delta_{p}u=a_{1}(x)f(u,v)\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;u>0\;\;&&\mbox{in $\mathbb{R}^{N}$,}\\\
-\Delta_{q}v=a_{2}(x)g(u,v)\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;v>0\;\;&&\mbox{in
$\mathbb{R}^{N}$,}\end{aligned}\right.$ (4.3)

where $N\geq 3$ while $1<p,q<N$. Nonlinearities
$f,g:(\mathbb{R}^{+})^{2}\to\mathbb{R}^{+}$ are continuous and fulfill the
condition

  * $({\rm a}_{12})$

There exist $\mu_{i},\hat{\mu}_{i}>0$, $i=1,2$, such that

$\mu_{1}s^{\alpha_{1}}\leq
f(s,t)\leq\hat{\mu}_{1}s^{\alpha_{1}}(1+t^{\beta_{1}}),\;\;\mu_{2}t^{\alpha_{2}}\leq
g(s,t)\leq\hat{\mu}_{2}(1+s^{\beta_{2}})t^{\alpha_{2}}$

for all $s,t\in\mathbb{R}^{+}$, with $-1<\alpha_{i}<0<\beta_{i}$,

$\alpha_{1}+\beta_{2}<p-1,\;\;\alpha_{2}+\beta_{1}<q-1,$ (4.4)

as well as

$\beta_{1}<\frac{q^{*}}{p^{*}}\min\\{p-1,p^{*}-p\\},\;\;\beta_{2}<\frac{p^{*}}{q^{*}}\min\\{q-1,q^{*}-q\\}.$

Coefficients $a_{i}:\mathbb{R}^{N}\to\mathbb{R}^{+}$ satisfy the assumption

  * $({\rm a}_{13})$

$a_{i}\in L^{1}(\mathbb{R}^{N})\cap L^{\zeta_{i}}(\mathbb{R}^{N})$, where

$\frac{1}{\zeta_{1}}\leq
1-\frac{p}{p^{*}}-\frac{\beta_{1}}{q^{*}}\,,\;\;\frac{1}{\zeta_{2}}\leq
1-\frac{q}{q^{*}}-\frac{\beta_{2}}{p^{*}}\,.$

A pair $(u,v)\in X^{p,q}(\mathbb{R}^{N})$ is called a (weak) solution to (4.3)
provided $u,v>0$ and

$\displaystyle\int_{\mathbb{R}^{N}}|\nabla u|^{p-2}\nabla u\nabla\varphi\,{\rm
d}x$ $\displaystyle=\int_{\mathbb{R}^{N}}a_{1}f(u,v)\varphi\,{\rm
d}x\;\;\forall\,\varphi\in\mathcal{D}^{1,p}(\mathbb{R}^{N}),$
$\displaystyle\int_{\mathbb{R}^{N}}|\nabla v|^{q-2}\nabla v\nabla\psi\,dx$
$\displaystyle=\int_{\mathbb{R}^{N}}a_{2}g(u,v)\psi\,dx\;\;\forall\,\psi\in\mathcal{D}^{1,q}(\mathbb{R}^{N}).$

Variational methods do not work, at least in a direct way, because the Euler
functional associated with (4.3) is not well defined. A similar comment holds
for sub-super-solution techniques, that are usually employed in the case of
bounded domains. So, one is naturally led to apply fixed point results. An a
priori estimate in $L^{\infty}(\mathbb{R}^{N})\times
L^{\infty}(\mathbb{R}^{N})$ for solutions of (4.3) is first established by a
Moser’s type iteration procedure and an adequate truncation, which, due to
singular terms, require a specific treatment. Problem (4.3) is next perturbed
by introducing a parameter $\varepsilon>0$. This produces the family of
regularized systems

$\left\\{\begin{aligned} -\Delta_{p}u=a_{1}(x)f(u+\varepsilon,v)\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;u>0\;\;&&\mbox{in $\mathbb{R}^{N}$,}\\\
-\Delta_{q}v=a_{2}(x)g(u,v+\varepsilon)\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;v>0\;\;&&\mbox{in
$\mathbb{R}^{N}$,}\end{aligned}\right.$ (4.5)

whose study yields useful information on the original problem. In fact, the
previous $L^{\infty}$-boundedness still holds for solutions to (4.5),
regardless of $\varepsilon$. Thus, via Schauder’s fixed point theorem, one
gets a solution $(u_{\varepsilon},v_{\varepsilon})$ lying inside a rectangle
given by positive lower bounds, where $\varepsilon$ does not appear, and
positive upper bounds, that may instead depend on $\varepsilon$. Finally,
letting $\varepsilon\to 0^{+}$ and using the $({\rm S})_{+}$-property of the
negative $r$-Laplacian in $\mathcal{D}^{1,r}(\mathbb{R}^{N})$ (see [54,
Proposition 2.2]) yields a weak solution to (4.3).

######  Theorem 4.1 ([54], Theorem 5.1).

Let $({\rm a}_{12})$ and $({\rm a}_{13})$ be satisfied. Then (4.3) has a weak
solution $(u,v)\in X^{p,q}(\mathbb{R}^{N})$, which is essentially bounded.

Very recently, the parametric system

$\left\\{\begin{aligned} -\Delta_{p}u=a_{1}(x)f_{1}(u)+\lambda
b_{1}(x)g_{1}(u)h_{1}(v)\;\;&\mbox{in $\mathbb{R}^{N}$,}\;\;u>0\;\;&&\mbox{in
$\mathbb{R}^{N}$,}\\\ -\Delta_{p}v=a_{2}(x)f_{2}(v)+\mu
b_{2}(x)g_{2}(v)h_{2}(u)\;\;&\mbox{in $\mathbb{R}^{N}$,}\;\;v>0\;\;&&\mbox{in
$\mathbb{R}^{N}$,}\\\ u(x)\to 0,\;v(x)\to 0\;\;\mbox{as
$|x|\to\infty$}\end{aligned}\right.$ (4.6)

where $N\geq 3$, $1<p<N$, $a_{i},b_{i}\in C^{0}(\mathbb{R}^{N})$,
$f_{i},g_{i},h_{i}\in C^{0}(\mathbb{R}^{+},\mathbb{R}^{+})$, $f_{i}$ is
singular at zero, and $\lambda,\mu>0$, was thoroughly investigated in [66].
Under suitable hypotheses, it is shown that there exists an open set
$\Theta\subseteq(\mathbb{R}^{+})^{2}$, whose part of its boundary contained in
$(\mathbb{R}^{+})^{2}$, say $\Gamma$, turns our a continuous monotone curve,
such that (4.6) admits a $C^{1}$-solution if $(\lambda,\mu)\in\Theta$ and has
no solution when
$(\lambda,\mu)\in(\mathbb{R}^{+})^{2}\setminus(\Theta\cup\Gamma)$.

###  4.3 Uniqueness

As far as we know, uniqueness of solutions to singular quasi-linear elliptic
systems in the whole space is still an open problem. Taking inspiration from
[13], a first result has been obtained by Gambera and Guarnotta [29].

###  4.4 Systems with convection terms

The very recent paper [43] treats the problem

$\left\\{\begin{aligned} -\Delta_{p}u=f(x,u,v,\nabla u,\nabla v)\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;u>0\;\;&&\mbox{in $\mathbb{R}^{N}$,}\\\
-\Delta_{q}v=g(x,u,v,\nabla u,\nabla v)\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;v>0\;\;&&\mbox{in
$\mathbb{R}^{N}$,}\end{aligned}\right.$ (4.7)

where $N\geq 3$, $p,q\in]2-\frac{1}{N},N[$, while
$f,g:\mathbb{R}^{N}\times(\mathbb{R}^{+})^{2}\times\mathbb{R}^{2N}\to\mathbb{R}^{+}$
are Carathéodory’s functions satisfying assumptions $({\rm a}_{14})$–$({\rm
a}_{15})$ below. Since $f,g$ depend on the gradient of solutions and equations
are set in the whole space, neither variational methods can be exploited nor
compactness for Sobolev embedding holds. The research started in [54], where
convective terms did not appear, is continued here, along the works [9, 40,
42], which address analogous questions, but concerning a bounded domain.

A pair $(u,v)\in X^{p,q}(\mathbb{R}^{N})$ such that $u,v>0$ is called:

  * 1)

_distributional solution_ to (4.7) if for every $(\varphi,\psi)\in
C^{\infty}_{0}(\mathbb{R}^{N})^{2}$ one has

$\begin{split}\int_{\mathbb{R}^{N}}|\nabla u|^{p-2}\nabla u\nabla\varphi\,{\rm
d}x&=\int_{\mathbb{R}^{N}}f(\cdot,u,v,\nabla u,\nabla v)\varphi\,{\rm d}x,\\\
\int_{\mathbb{R}^{N}}|\nabla v|^{q-2}\nabla v\nabla\psi\,{\rm
d}x&=\int_{\mathbb{R}^{N}}g(\cdot,u,v,\nabla u,\nabla v)\psi\,{\rm
d}x;\end{split}$ (4.8)

  * 2)

_(weak) solution_ of (4.7) when (4.8) holds for all $(\varphi,\psi)\in
X^{p,q}(\mathbb{R}^{N})$;

  * 3)

_‘strong’ solution_ to (4.7) if $|\nabla u|^{p-2}\nabla u,|\nabla
v|^{q-2}\nabla v\in W^{1,2}_{\rm loc}(\mathbb{R}^{N})$ and the differential
equations are satisfied a.e. in $\mathbb{R}^{N}$.

Obviously, both 2) and 3) force 1), whilst reverse implications turn out
generally false; see also [43, Remark 4.5]. Moreover, as observed at p. 48 of
[68], problems in unbounded domains may admit strong solutions that are not
weak or vice-versa.

Roughly speaking, the technical approach proceeds as follows. The auxiliary
problem

$\left\\{\begin{aligned} -\Delta_{p}u=f(x,u+\varepsilon,v,\nabla u,\nabla
v)\;\;&\mbox{in $\mathbb{R}^{N}$,}\;\;u>0\;\;&&\mbox{in $\mathbb{R}^{N}$,}\\\
-\Delta_{q}v=g(x,u,v+\varepsilon,\nabla u,\nabla v)\;\;&\mbox{in
$\mathbb{R}^{N}$,}\;\;v>0\;\;&&\mbox{in
$\mathbb{R}^{N}$,}\end{aligned}\right.$ (4.9)

$\varepsilon>0$, obtained by shifting appropriate variables of reactions,
which avoids singularities, is first solved. To do this, nonlinear regularity
theory, a priori estimates, Moser’s iteration, trapping region, and fixed
point arguments are employed. Unfortunately, bounds from above alone do not
allow to get a solution of (4.7): treating singular terms additionally
requires some estimates from below. Theorem 3.1 in [19] ensures that solutions
to (4.9) turn out locally greater than a positive constant regardless of
$\varepsilon$. Thus, under the hypotheses below, one can construct a sequence
$\\{(u_{\varepsilon},v_{\varepsilon})\\}\subseteq X^{p,q}(\mathbb{R}^{N})$
such that $(u_{\varepsilon},v_{\varepsilon})$ solves (4.9) for all
$\varepsilon>0$ and whose weak limit as $\varepsilon\to 0^{+}$ is a
distributional solution to (4.7). Next, a localization-regularization
reasoning shows that

$(u,v)\;\mbox{distributional solution}\implies(u,v)\;\mbox{weak solution.}$

Through a recent differentiability result [17, Theorem 2.1] one then has

$(u,v)\;\mbox{distributional solution}\implies(u,v)\;\mbox{strong solution.}$

The assumptions below will be posited.

  * $({\rm a}_{14})$

There exist $\alpha_{i}\in]-1,0]$,
$\beta_{i},\gamma_{i},\delta_{i}\in\mathbb{R}^{+}_{0}$, as well as
$\mu_{i},\hat{\mu}_{i}>0$ such that

$\displaystyle\mu_{1}a_{1}(x)s^{\alpha_{1}}t^{\beta_{1}}\leq
f(x,s,t,\xi_{1},\xi_{2})\leq\hat{\mu}_{1}a_{1}(x)\left(s^{\alpha_{1}}t^{\beta_{1}}+|\xi_{1}|^{\gamma_{1}}+|\xi_{2}|^{\delta_{1}}\right)$
$\displaystyle\mu_{2}a_{2}(x)s^{\beta_{2}}t^{\alpha_{2}}\leq
g(x,s,t,\xi_{1},\xi_{2})\leq\hat{\mu}_{2}a_{2}(x)\left(s^{\beta_{2}}t^{\alpha_{2}}+|\xi_{1}|^{\gamma_{2}}+|\xi_{2}|^{\delta_{2}}\right)$

in $\mathbb{R}^{N}\times(\mathbb{R}^{+})^{2}\times\mathbb{R}^{2N}$. Moreover,

$\beta_{1}\vee\delta_{i}<q-1,\;\;\;\beta_{2}\vee\gamma_{i}<p-1,\;\;i=1,2,$

$a_{1}\in L^{s_{p}}_{\rm loc}(\mathbb{R}^{N})$, with $s_{p}>p^{\prime}N$,
$a_{2}\in L^{s_{q}}_{\rm loc}(\mathbb{R}^{N})$, with $s_{q}>q^{\prime}N$, and
$\displaystyle{\operatorname*{ess\,inf}_{B_{\rho}}a_{i}>0}$ for all $\rho>0$.

  * $({\rm a}_{15})$

There exist $\zeta_{1},\zeta_{2}\in]N,+\infty]$ such that $a_{i}\in
L^{1}(\mathbb{R}^{N})\cap L^{\zeta_{i}}(\mathbb{R}^{N})$, where

$\frac{1}{\zeta_{1}}<1-\frac{p}{p^{*}}-\theta_{1},\quad\frac{1}{\zeta_{2}}<1-\frac{q}{q^{*}}-\theta_{2},$

with

$\theta_{1}:=\max\left\\{\frac{\beta_{1}}{q^{*}},\frac{\gamma_{1}}{p},\frac{\delta_{1}}{q}\right\\}<1-\frac{p}{p^{*}},\quad\theta_{2}:=\max\left\\{\frac{\beta_{2}}{p^{*}},\frac{\gamma_{2}}{p},\frac{\delta_{2}}{q}\right\\}<1-\frac{q}{q^{*}}.$

Further,

$\displaystyle(\beta_{1}\vee\delta_{1})(\beta_{2}\vee\gamma_{2})<(p-1-\gamma_{1})(q-1-\delta_{2}),$
$\displaystyle\frac{1}{s_{p}}+\left(\frac{\gamma_{1}}{p}\vee\frac{\delta_{1}}{q}\right)\leq\frac{1}{2},\qquad\frac{1}{s_{q}}+\left(\frac{\gamma_{2}}{p}\vee\frac{\delta_{2}}{q}\right)\leq\frac{1}{2}.$

######  Example 4.2.

Condition $({\rm a}_{15})$ is fulfilled once $a_{1},a_{2}\in
L^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$ and

$\max\left\\{\frac{\beta_{1}}{q^{*}},\frac{\gamma_{1}}{p},\frac{\delta_{1}}{q}\right\\}<1-\frac{p}{p^{*}},\quad\max\left\\{\frac{\beta_{2}}{p^{*}},\frac{\gamma_{2}}{p},\frac{\delta_{2}}{q}\right\\}<1-\frac{q}{q^{*}}.$

In fact, it suffices to choose $\zeta_{1}:=\zeta_{2}:=+\infty$.

######  Theorem 4.3 ([43], Theorem 1.3).

Under hypotheses $({\rm a}_{14})$–$({\rm a}_{15})$, problem (4.7) admits a
weak and strong solution $(u,v)\in X^{p,q}(\mathbb{R}^{N})$.

######  Remark 4.4.

If we merely seek weak solutions to (4.7) then the request $p,q\in]1,N[$ and a
weaker integrability property of $a_{i}$ suffice; cf. [39, Section 4.2.2].

### Acknowledgement

U. Guarnotta and S.A. Marano were supported by the research project PRA
2020–2022, Linea 3 and Linea 2, resp., ‘MO.S.A.I.C.’ of the University of
Catania.
U. Guarnotta, R. Livrea, and S.A. Marano were supported by the research
project PRIN 2017 ‘Nonlinear Differential Problems via Variational,
Topological and Set-valued Methods’ (Grant No. 2017AYM8XW) of MIUR.

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