# On Vizing’s problem for triangle-free graphs

Ross J. Kang Korteweg–de Vries Institute for Mathematics, University of
Amsterdam, Netherlands. Partially supported by a Vidi grant (639.032.614) and
the Gravitation Programme NETWORKS (024.002.003) of the Dutch Research Council
(NWO<EMAIL_ADDRESS>Matthieu Rosenfeld LIRMM, Univ. Montpellier, CNRS,
Montpellier, France<EMAIL_ADDRESS>
###### Abstract

We prove that $\chi(G)\leq\lceil(\Delta+1)/2\rceil+1$ for any triangle-free
graph $G$ of maximum degree $\Delta$ provided $\Delta\geq 524$. This gives
tangible progress towards an old problem of Vizing, in a form cast by Reed. We
use a method of Hurley and Pirot, which in turn relies on a new counting
argument of the second author.

##  1 Introduction

Over half a century ago, Vizing [33] wrote (this is a translation from Russian
[34] adapted to modern symbol usage, with $\omega$, $\Delta$, and $\chi$
denoting clique number, maximum degree, and chromatic number, respectively):

> ‘If $\Delta(G)$ is the maximum degree of a vertex in a graph $G$, it is
> clear that $\chi(G)\leq\Delta(G)+1$. Brooks showed in 1941 that
> $\chi(G)\leq\Delta(G)$ whenever $\Delta(G)\geq 3$ and
> $\omega(G)\leq\Delta(G)$. Further investigations could be conducted, taking
> into account a more exact relation between $\Delta$ and $\omega$. Perhaps
> one should start with estimates of the chromatic number of a graph without
> triangles ($\omega=2$) and with given maximal degree for vertices.’

This problem has its roots in even older questions and results about triangle-
free graphs with arbitrarily high chromatic number [32, 36, 26, 14]. Deep
connections with the off-diagonal Ramsey numbers, especially with the classic
work of Ajtai, Komlós, Szemerédi [1, 2] and Shearer [30], became evident
through the seminal works of Johansson [20] and, more recently, Molloy [22].

######  Theorem 1 (Molloy [22], cf. [20]).

As $\Delta\to\infty$, it holds that $\chi(G)\leq(1+o(1))\Delta/\log\Delta$ for
any triangle-free graph $G$ of maximum degree $\Delta$.

This result has as a simple corollary an upper bound on the off-diagonal
Ramsey number $R(3,k)$ that asymptotically (as $k\to\infty$) matches the
longstanding bound of Shearer [30].

It transpires that the link between the task of finding either large
independent sets or good colourings of triangle-free graphs and facets of the
Lovász local lemma [15], such as the hard-core model (see [29]) or the entropy
compression method [24, 25], is fundamental. It underpins many recent
developments in the area, see e.g. [6, 7, 12, 13, 18, 19, 22].

This work is no exception. To wit, a ‘counting trick’ introduced recently by
second author [28] has shone a new light on the entropy compression method
(see [35]). This ‘trick’ is the main ingredient of an elegant new proof of
Theorem 1 devised by Hurley and Pirot [18, 19]. Here we observe how these
recent developments yield tangible progress towards the old question above, in
a sense that Vizing may have originally intended. Our focus is the following
conjecture111One could rightly argue that Vizing has asked for the value of
$\chi_{\omega=2}(\Delta):=\sup\\{\chi(G):\text{$G$ is a triangle-free graph of
maximum degree $\Delta$}\\}$ for all $\Delta$, but here we take the more
concrete Conjecture 2 as our point of reference, which clearly is sufficient
challenging..

######  Conjecture 2.

It holds that $\chi(G)\leq\lceil(\Delta+1)/2\rceil+1$ for any triangle-free
graph $G$ of maximum degree $\Delta$.

While Theorem 1 is only known to be sharp up to a multiplicative factor
$2+o(1)$ (as $\Delta\to\infty$), the bound in Conjecture 2 is an ‘exact
relation’ in the sense that it is attained by the $4$-regular, $4$-chromatic
Chvátal graph [11] (as well as by, for example, odd cycles of length $5$ or
more, the Petersen graph, and the Clebsch graph). The conjecture is a special
case ($\omega=2$) of Reed’s conjecture [27], which asserts that any graph of
clique number $\omega$ and maximum degree $\Delta$ has chromatic number at
most $\lceil\frac{1}{2}(\omega+\Delta+1)\rceil$. (The most significant general
progress to date on Reed’s conjecture is due to Hurley, de Joannis de Verclos
and the first author [17].) The bound in Conjecture 2 may have earlier
provenance. That is, Kostochka [21], motivated by Vizing’s problem, proved the
very same bound holds under the much stronger condition of girth being at
least $4(\Delta+2)\log\Delta$. Borodin and Kostochka [8] and Catlin [10],
independently, proved an upper bound of $\lceil 3(\Delta+1)/4\rceil$. Brooks’
theorem [9] implies the conjecture for $\Delta\leq 4$. We wish to separately
emphasise the current smallest open case.

######  Conjecture 3 ($\Delta=5$ of Conjecture 2).

Every $5$-regular triangle-free graph has chromatic number at most $4$.

This aligns with one of the few remaining cases of an old and largely refuted
conjecture of Grünbaum [16]; he actually conjectured the negation of this.

Obviously Theorem 1 implies the existence of some $\Delta_{0}$ such that the
bound asserted in Conjecture 2 holds for all $\Delta\geq\Delta_{0}$. But to
the best of our knowledge, no deliberate effort was made to explicitly bound
$\Delta_{0}$. This is likely due to the technicality of earlier methods used
to prove Theorem 1 and its precursors. The situation has changed and the
result we highlight in this note is as follows.

######  Theorem 4.

The bound asserted in Conjecture 2 holds for all $\Delta\geq 524$.

We prove this by adapting the method of Hurley and Pirot to this version of
the triangle-free colouring problem. We note that one straightforwardly
derives an estimate of around $30000$ for $\Delta_{0}$ from Hurley and Pirot’s
main result [19, Thm. 2.10]. Our objective was not to fully optimise the value
$524$ and we anticipate further estimates on the least $\Delta_{0}$ as above.
Rather we have indicated how a satisfying resolution to Vizing’s triangle-free
problem, as interpreted through Conjecture 2 and Reed’s conjecture, is now
within a stone’s throw.

Later in the note we consider some related problems for bipartite or
$C_{4}$-free graphs.

###  1.1 Notation and preliminaries

We write $\deg(v)$ for the degree of $v\in V(G)$.

We will establish Theorem 4 in a more general context than chromatic number,
not only in terms of list colouring, but also in an even more refined context,
that of local colouring (as in [12, 13], for instance). For this, we need some
extra terminology. Our notation is mostly standard; we collect it here for
convenience.

Given a graph $G$, a list-assignment of $G$ is a mapping $L:V(G)\to
2^{{\mathbb{N}}}$. Given a positive integer function on the vertices
$k:V(G)\to\mathbb{N}$, we say that $L$ is a $k$-list-assignment if $|L(v)|\geq
k(v)$ for all $v\in V(G)$. Given a colouring $c:V(G)\to\mathbb{N}$ of $G$, we
say that $c$ is an $L$-colouring if $c(v)\in L(v)$ for all $v\in V(G)$. We
will be preoccupied with proper colourings $c$, that is, with $c(v)\neq
c(v^{\prime})$ for any edge $vv^{\prime}\in E(G)$, and sometimes we will omit
the term proper if there is no possible confusion.

Recall that the list chromatic number $\chi_{\ell}(G)$ of $G$ is the least
integer $k$ such that $G$ admits an $L$-colouring for any $k$-list-assignment
$L$ of $G$.

We also work with partial colourings $c:S\to\mathbb{N}$ for some $S\subsetneq
V(G)$. Given a (partial) proper $L$-colouring $c$ of $G$ and $v\in V(G)$, we
write $L_{c}(v)$ for the set of colours from $L(v)$ that are still available
to $v$, that is, $L_{c}(v)=L(v)\setminus\\{c(u):u\in N(v)\\}$. For any
subgraph $H\subseteq G$, we write $C_{L}(H)$ for the set of proper
$L$-colourings of $H$. In Section 2, it will be essential to have good control
on how the order of $C_{L}$ evolves.

In Section 3, we will make simple use of the following symmetric form of the
Lovász Local Lemma [15, 31].

######  Lemma 5.

Consider a set $\mathcal{E}$ of (bad) events such that for each
$A\in\mathcal{E}$

  1. 1.

$\Pr(A)\leq p<1$, and

  2. 2.

$A$ is mutually independent of a set of all but at most $d$ of the other
events.

If $epd\leq 1$, then with positive probability none of the events in
$\mathcal{E}$ occur.

In the next section, we need the following simple technical result that will
allow us to apply Jensen’s inequality in the proof of our main result.

######  Lemma 6.

For all $c>0$, $a>0$ and $b>1$, the function $z\mapsto
cz(1-\frac{1}{b})^{ab/z}$ is convex over $(0,+\infty)$.

###### Proof.

This is easily verified by differentiating twice. The second derivative is

$z\mapsto
c\frac{\left(1-\frac{1}{b}\right)^{ab/z}\left(ab\ln\left(1-\frac{1}{b}\right)\right)^{2}}{z^{3}},$

which is positive for all $z\in(0,+\infty)$. ∎

##  2 Main bound

With the above terminology, we can fully state and prove our main result.

######  Theorem 7.

Define $k:x\mapsto\lceil(x+1)/2\rceil+1$. Let $G$ be a triangle-free graph of
minimum degree $524$. Then $G$ admits a proper $L$-colouring for any
$(k\circ\deg)$-list-assignment $L$ of $G$.

Note that this statement directly implies Theorem 4 even with the stronger
parameter of list chromatic number $\chi_{\ell}(G)$ of $G$. Similarly, Theorem
7 but without the minimum degree condition would be a much stronger form of
Conjecture 2. Theorem 7 will follow immediately by combining the following two
lemmata.

For any positive integers $\Delta_{0}$, $\ell$, $t$ and a non-decreasing
function $k:\mathbb{N}\rightarrow\mathbb{N}$, we say that
$(\Delta_{0},\ell,t,k)$ has Property $\mathbf{(P)}$ if for all integers
$\delta\geq\Delta_{0}$,

  1. 1.

$0<t<\ell<k(\delta)<\delta$, and

  2. 2.

$\left(k(\delta)-\frac{t\delta}{\ell}\right)\left(1-\frac{1}{t+1}\right)^{\frac{(t+1)(\delta-t\delta/\ell)}{k(\delta)-t\delta/\ell}}\geq\ell\,.$

######  Lemma 8.

Let $\Delta_{0}$, $\ell$ and $t$ be positive integers and
$k:\mathbb{N}\to\mathbb{N}$ be a non-decreasing function such that
$(\Delta_{0},\ell,k,t)$ has Property $\mathbf{(P)}$. Let $G$ be a triangle-
free graph with minimum degree at most $\Delta_{0}$, and let $L$ be a
$(k\circ\deg)$-list-assignment of $G$. Then, for any $v\in V(G)$,

$\displaystyle|C_{L}(G)|/|C_{L}(G-v)|\geq\ell\,.$

For brevity, we allow ourselves to write $k$ instead of $k\circ\deg$.

###### Proof.

The proof is by induction on the order of the vertex set $V(G)$. The case
$V(G)=\\{v\\}$ is trivial. We suppose that our claim holds for all strict
subgraphs of $G$ and our goal is to prove that this holds for $G$. We consider
what happens if we choose a colouring $\mathbf{c}$ uniformly at random from
$C_{L}(G-v)$.

Our induction hypothesis implies that for all $u\subseteq V(G)-v$,

$|C_{L}(G-v)|\geq\ell|C_{L}(G-v-u)|\,.$ (IHP)

Fix a vertex $u\in N(v)$. Let $F=\\{c\in C_{L}(G-v):|L_{\mathbf{c}}(u)|\leq
t\\}$. The restriction of any coloring $c\in F$ to $G-v-u$ is in
$C_{L}(G-v-u)$. Moreover, any coloring from $C_{L}(G-v-u)$ is the restriction
of at most $t$ such colourings, hence, $|F|\leq t|C_{L}(G-v-u)|$. We deduce
that

$\mathbb{P}(|L_{\mathbf{c}}(u)|\leq
t)=\frac{|F|}{|C_{L}(G-v)|}\leq\frac{t|C_{L}(G-v-u)|}{\ell|C_{L}(G-v-u)|}=\frac{t}{\ell}\,.$

Let $\mathbf{t}_{v}=\left|\\{u\in N(v):|L_{\mathbf{c}}(u)|\leq t\\}\right|$.
Then by linearity of expectation

$\mathbb{E}[\mathbf{t}_{v}]\leq\frac{t\deg(v)}{\ell}\,.$ (1)

Let $G_{0}=G-v-N(v)$, and consider the distribution of $\mathbf{c}$
conditioned on $\mathbf{c}|_{G_{0}}=c_{0}$ for some $c_{0}\in C_{L}(G_{0})$.
Note that for all $u\in N(v)$, $L_{\mathbf{c}}(u)=L_{c_{0}}(u)$. Conditioned
on $\mathbf{c}|_{G_{0}}=c_{0}$, the colours of the neighbours of $v$ are
conditionally independent and uniformly chosen from the respective sets
$L_{c_{0}}(u)$.

The problem is reduced to the following elementary statement. We are given
sets $L_{\mathbf{c}}(u)$ of colours, for each $u\in N(v)$, and random
variables $\mathbf{X}_{u}\in L_{\mathbf{c}}(u)$, for each $u\in N(v)$, each
chosen independently and uniformly from the respective set. Writing
$\mathbf{X}=|L(v)-\\{\mathbf{X}_{u}:u\in N(v)\\}|$ (note that it is possible
that $X_{u}\not\in L(v)$), we have that
$\frac{|C_{L}(G)|}{|C_{L}(G-v)|}=\mathbb{E}[|L_{\mathbf{c}}(v)|]\geq\mathbb{E}[\mathbf{X}]$.
The problem is thus reduced to showing that $\mathbb{E}[\mathbf{X}]\geq\ell$.

In order to bound $\mathbb{E}[\mathbf{X}]$, we condition on the values of
$\mathbf{X}_{u}$ for each $u$ such that $|L_{u}|\leq t$, and we write $B$ for
the (random) set of those values. For each $u\in L(v)$, we write the indicator
variable $Y_{u}=[u\in\mathbf{X}]$, hence $\mathbf{X}=\sum\limits_{u\in
L(v)\setminus B}\mathbb{E}[Y_{u}]$. By linearity of expectation, the
conditional expectation of $\mathbf{X}$ given $c_{0}$ is

$\displaystyle\mathbb{E}[\mathbf{X}\mid\mathbf{c}|_{G_{0}}=c_{0}]$
$\displaystyle=\sum_{j\in L(v)\setminus
B}\prod_{\begin{subarray}{c}L_{c_{0}}(u)\ni j\\\ |L_{c_{0}}(u)|>t\\\ u\in
N(v)\end{subarray}}\left(1-\frac{1}{|L_{c_{0}}(u)|}\right)$
$\displaystyle\geq(k(v)-|B|)\left(\prod_{j\in L(v)\setminus
B}\prod_{\begin{subarray}{c}L_{c_{0}}(u)\ni j\\\ |L_{c_{0}}(u)|>t\\\ u\in
N(v)\end{subarray}}\left(1-\frac{1}{|L_{c_{0}}(u)|}\right)\right)^{1/(k(v)-|B|)}$
$\displaystyle\geq(k(v)-|B|)\left(\prod_{\begin{subarray}{c}u\in N(v)\\\
|L_{c_{0}}(u)|>t\end{subarray}}\prod_{j\in L_{c_{0}}(u)\cap L(v)\setminus
B}\left(1-\frac{1}{|L_{c_{0}}(u)|}\right)\right)^{1/(k(v)-|B|)}$
$\displaystyle\geq(k(v)-|B|)\left(\prod_{\begin{subarray}{c}u\in N(v)\\\
|L_{c_{0}}(u)|>t\end{subarray}}\left(1-\frac{1}{|L_{c_{0}}(u)|}\right)^{|L_{c_{0}}(u)|}\right)^{1/(k(v)-|B|)}$
$\displaystyle\geq(k(v)-|B|)\left(1-\frac{1}{t+1}\right)^{\frac{(t+1)(\deg(v)-|B|)}{k(v)-|B|}}$
$\displaystyle=(k(v)-|B|)\left(1-\frac{1}{t+1}\right)^{(t+1)}\left(1-\frac{1}{t+1}\right)^{\frac{(t+1)(\deg(v)-k(v))}{k(v)-|B|}}\,,$

where the first inequality is an application of the AM–GM inequality. By Lemma
6, this last quantity is a convex function of $k(v)-|B|$ (with
$c=(1-1/(t+1))^{t+1}$, $a=\deg(v)-k(v)$ and $b=t+1$) provided $k(v)>|B|$.

Thus, we can use Jensen’s inequality to average over all values of $c_{0}$ as
follows:

$\displaystyle\mathbb{E}[\mathbf{X}]$
$\displaystyle\geq\mathbb{E}\left[(k(v)-|B|)\left(1-\frac{1}{t+1}\right)^{\frac{(t+1)(\deg(v)-|B|)}{k(v)-|B|}}\right]$
$\displaystyle\geq\mathbb{E}\left[k(v)-|B|\right]\left(1-\frac{1}{t+1}\right)^{\frac{(t+1)(\deg(v)-k(v)+\mathbb{E}\left[k(v)-|B|\right])}{\mathbb{E}\left[k(v)-|B|\right]}}.$

By definition $|B|\leq\mathbf{t}_{v}$. Hence, by equation (1), we have

$\mathbb{E}\left[k(v)-|B|\right]\geq k(v)-\frac{t\deg(v)}{\ell}\,.$

This and the previous series of inequalities imply

$\mathbb{E}[\mathbf{X}]\geq\left(k(v)-\frac{t\deg(v)}{\ell}\right)\left(1-\frac{1}{t+1}\right)^{\frac{(t+1)(\deg(v)-t\deg(v)/\ell)}{k(v)-t\deg(v)/\ell}}\,,$

which by our theorem hypothesis implies $\mathbb{E}(\mathbf{X})\geq\ell$ as
desired. ∎

######  Lemma 9.

Let $\Delta_{0}=524$, $\ell=8$, $t=1$ and $k$ be the function such that for
all $\delta\in\mathbb{N}$, $k=\lceil\frac{\delta+1}{2}\rceil+1$. The
quadruplet $(\Delta_{0},\ell,t,k)$ has Property $\mathbf{(P)}$.

###### Proof.

By definition of $k$, $t$ and $\ell$, we have for all $\delta>0$ that

  * •

$k(\delta)-t\delta/\ell\geq\frac{3\delta}{8}+1$, and

  * •

$\frac{\delta-t\delta/\ell}{k(\delta)-t\delta/\ell}\leq\frac{1-t/\ell}{3/8}=\frac{7}{3}$.

We can now verify that

$\displaystyle\left(k(\delta)-\frac{t\delta}{\ell}\right)\left(1-\frac{1}{t+1}\right)^{\frac{(t+1)(\delta-t\delta/\ell)}{k(\delta)-t\delta/\ell}}\geq\left(\frac{3\delta}{8}+1\right)2^{-14/3}\geq
8.01>\ell\,,$

where we use the fact that $\delta\geq 540$ for the second inequality.

For each of the remaining cases, with $524\leq\delta<540$, one can (for
instance with the help of a computational mathematical engine) verify the
inequality by direct computation. This completes the establishment of Property
$\mathbf{(P)}$. ∎

##  3 The bipartite case

Conjecture 2 is quite trivial in the special case of bipartite $G$. But in
consideration of Theorem 7, the following two conjectures, the latter of which
is stronger than the former, might be interesting to investigate.

######  Conjecture 10.

It holds that $\chi_{\ell}(G)\leq\lceil(\Delta+1)/2\rceil+1$ for any bipartite
graph $G$ of maximum degree $\Delta$.

######  Conjecture 11.

For $k:x\mapsto\lceil(x+1)/2\rceil+1$, any bipartite graph $G$ admits a proper
$L$-colouring for any $(k\circ\deg)$-list-assignment $L$ of $G$.

Note that Conjecture 10 is related to a vexing conjecture of Alon and
Krivelevich [4], which posits a bound logarithmic rather than linear in
$\Delta$. We also point out that the conclusion of Conjecture 11 is known to
be false if we take $k$ to be any sublinear function, by an inductively
defined tree construction [12, Prop. 11]. The settings of Conjectures 10 and
11 could be a testbed for progress on Conjecture 2. For example, the smallest
open case $\Delta=5$ of Conjecture 10 is an attractive analogue of Conjecture
3: is the list chromatic number of a $5$-regular bipartite graph always at
most $4$?

We repeat a standard coupon collection-inspired argument [3] to provide the
following step towards Conjecture 10. The method is distinct from that used
for Theorem 4, but the outcome is a useful benchmark for the triangle-free
case.

######  Theorem 12.

The bound asserted in Conjecture 10 holds for all $\Delta\geq 142$.

For this, we need to prove the next two lemmata.

######  Lemma 13.

Let $G$ be a star graph with $d$ leaves $v_{1},\dots,v_{d}$ and a central
vertex $v$. Let $L$ be a $k$-list-assignment for some positive integer $k$.
Let $\mathbf{c}$ be the random partial $L$-colouring where $\mathbf{c}(v_{i})$
is chosen independently and uniformly from $L(v_{i})$ for each $i\in[d]$. Then

$\displaystyle\mathbb{P}(L_{\mathbf{c}}(v)=\emptyset)\leq\left(1-\left(1-\frac{1}{k}\right)^{d}\right)^{k}\,.$

###### Proof.

Let $C=\mathbf{c}(\\{v_{1},\ldots,v_{d}\\})$. Notice that for each $\alpha\in
L(v)$,

$\displaystyle\mathbb{P}(\alpha\in C)=1-\left(1-\frac{1}{k}\right)^{d}\,,$

Moreover, for $\alpha,\alpha^{\prime}\in L(v)$ the events $\alpha\in C$ and
$\alpha^{\prime}\in C$ are negatively correlated. (See [3, Claim] for the
argument in fullest detail.) Hence

$\mathbb{P}(L_{\mathbf{c}}(v)=\emptyset)=\mathbb{P}(L(v)\subseteq
C)\leq\prod_{\alpha\in C}\mathbb{P}(\alpha\in
C)\leq\left(1-\left(1-\frac{1}{k}\right)^{d}\right)^{k}\,.\qed$

######  Lemma 14.

Under the same assumptions as the previous lemma, with additionally the choice
$k=\lceil(d+1)/2\rceil+1$ and $d=140$ or $d\geq 142$, we have that

$\displaystyle\mathbb{P}(L_{\mathbf{c}}(v)=\emptyset)\leq\frac{1}{ed(d-1)}\,.$

###### Proof.

Note that for $k$ being any linear function of $d$, the bound given in Lemma
13 approaches zero exponentially in $d$ as $d\to\infty$, while the claimed
bound here approaches zero much more slowly, that is, quadratically in $d$.
Thus there exists some $d_{0}$ for which the desired inequality holds for all
$d\geq d_{0}$. In what follows we argue how to explicitly bound $d_{0}$.

First use $1-x\geq e^{x/(x-1)}$ to deduce that

$\displaystyle\left(1-\frac{1}{k}\right)^{d}\geq e^{d/(1-k)}\,.$

Then using $1-x\leq e^{-x}$ we obtain via Lemma 13 that

$\mathbb{P}(L_{\mathbf{c}}(v)=\emptyset)\leq e^{-ke^{d/(1-k)}}\,.$

Thus, considering the function

$f(d)=e^{-ke^{d/(1-k)}}\cdot ed^{2}\,,$

it suffices for the desired inequality if we show that $f(d)\leq 1$.

Let us assume that $d$ is even in which case we may take $k=d/2+2$ and then
for the optimisation treat $f$ as a real function. Computing the derivative,

$f^{\prime}(d)=de^{-1-2/(1-k)-ke^{d/(1-k)}+1}\left(\frac{12d-d^{3}}{8(1-k)^{2}}+2e^{-d/(1-k)}\right).$

The last factor in this expression determines the sign of $f^{\prime}(d)$, and
it is straightforward to check that it is negative when $d\geq 64$. Thus,
since $f(157)\leq 1$, we have shown that the inequality holds for all even
$d\geq 157$.

We argue similarly for the case of odd $d$, except with $k=(d+1)/2+1$.

For each of the finitely many remaining cases of $d$ claimed, one can verify
(for example with assistance from a computational mathematical engine) that

$\left(1-\left(1-\frac{1}{k}\right)^{d}\right)^{k}\cdot ed(d-1)\leq 1$

by a direct computation. By Lemma 13, this completes the proof. ∎

With this last lemma, we can prove the result.

###### Proof of Theorem 12.

By a standard reduction, we may assume without loss of generality that the
bipartite graph $G$ is $\Delta$-regular. We write the bipartition as
$V(G)=A\cup B$. Let $L$ be any $k$-list-assignment of $G$ for
$k=\lceil(\Delta+1)/2\rceil+1$. Let $\mathbf{c}$ be the random partial
$L$-colouring where $\mathbf{c}(w)$ is chosen independently and uniformly from
$L(w)$ for each $w\in B$. For each $v\in A$, let $T_{v}$ be the event that
$L_{\mathbf{c}}(v)=\emptyset$. By Lemma 14,
$e\cdot\mathbb{P}(L_{\mathbf{c}}(v)=\emptyset)\cdot d(d-1)\leq 1$. Since each
event $T_{v}$ is mutually independent of all other events $T_{u}$ apart from
those such that $u$ has a common neighbour in $G$ with $v$ (of which there are
at most $d(d-1)$), it follows from the Lovász Local Lemma that with positive
probability none of the events $T_{v}$ occur. In that eventuality, each vertex
in $A$ has at least one choice of unimpeded colour from its list, which
implies a proper $L$-colouring of $G$. ∎

Neither of Theorems 7 and 12 implies the other. It seems to us that the
argument for the latter works poorly with regard to local colourings.

##  4 Concluding remarks

We have demonstrated how Conjectures 2 and 10 outright are temptingly within
reach. We encourage further research on these problems. We end with yet
another related problem. We distinguish this one because to us it seems
intuitively feasible, yet not directly amenable to either of the techniques in
Sections 2 and 3.

######  Conjecture 15.

It holds that $\chi(G)\leq\lceil(\Delta+1)/2\rceil+1$ for any $C_{4}$-free
graph $G$ of maximum degree $\Delta$.

The result of Borodin and Kostochka [8] and Catlin [10] implies an upper bound
of $\lceil 3(\Delta+1)/4\rceil$ here also (indeed more generally for
$K_{4}$-free $G$). An $O(\Delta/\log\Delta)$ bound was observed in [5, Cor.
2.4], which was recently tightened to a $(1+o(1))\Delta/\log\Delta$ bound in
[13, Thm. 4] (in fact for any forbidden cycle length $\ell=\Delta^{o(1)}$ as
$\Delta\to\infty$). One can derive from [19, Thm. 2.10] that the bound in
Conjecture 15 holds for all $\Delta$ at least around $60000$. We remark that
while the fractional chromatic number version of Conjecture 2 is known (see
[23, Sub. 21.3]), the fractional analogue of Conjecture 15 may already be
nontrivial.

As the Chvátal graph contains $4$-cycles while the Petersen graph does not, it
might well be that $\lceil\Delta/2\rceil+1$ is the ‘correct’ target in
Conjecture 15. Similarly, since $K_{3,3}$ has list chromatic number $3$, the
same expression $\lceil\Delta/2\rceil+1$ might be ‘correct’ in Conjecture 10.
Concretely, is there a $4$-regular $C_{4}$-free graph of chromatic number $4$?
A $4$-regular bipartite graph of list chromatic number $4$?

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# Positive solutions for infinite semipositone$\diagup$positone quasilinear
elliptic systems with singular and superlinear terms

Brahim Khodja
Mathematic Department, Badji-Mokhtar Annaba University, 23000 Annaba Algeria
e-mail address<EMAIL_ADDRESS>

Abdelkrim Moussaoui
Biology Department, A. Mira Bejaia University,
Targa Ouzemour 06000 Bejaia, Algeria
e-mail<EMAIL_ADDRESS>
###### Abstract

We establish existence and regularity of positive solutions for a class of
quasilinear elliptic systems with singular and superlinear terms. The approach
is based on sub-supersolution methods for systems of quasilinear singular
equations and the Schauder’s fixed point Theorem.

2000 Mathematics Subject Classification: 35J75; 35J48; 35J92

Keywords: Singular systems; Sub-supersolutions; Schauder’s fixed point
theorem; Regularity

##  1 Introduction and main result

Let $\Omega\subset\mathbb{R}^{N}$ $\left(N\geq 2\right)$ is a bounded domain
with $C^{1,\alpha}$-boundary $\partial\Omega$, $\alpha\in(0,1)$, and let
$1<p,q\leq N$. We deal with the following quasilinear singular elliptic
problem

$\left\\{\begin{array}[]{ll}-\Delta_{p}u=\lambda
u^{\alpha_{1}}+v^{\beta_{1}}&\text{in }\Omega,\\\
-\Delta_{q}v=u^{\alpha_{2}}+\lambda v^{\beta_{2}}&\text{in }\Omega,\\\
u,v>0&\text{in }\Omega,\\\ u,v=0&\text{on }\partial\Omega,\end{array}\right.$
(1.1)

where $\lambda$ is a real parameter. Here $\Delta_{p}$ and $\Delta_{q}$ denote
the $p$-Laplacian and $q$-Laplacian differential operators defined by
$\Delta_{p}u=div(\left|\nabla u\right|^{p-2}\nabla u)$ and
$\Delta_{q}v=div(\left|\nabla v\right|^{q-2}\nabla v)$, respectively. We
consider system (1.1) in a singular case assuming that

$\begin{array}[]{c}-1<\alpha_{1},\beta_{2}<0.\end{array}$ (1.2)

We explicitly observe that under assumption (1.2) and depending on the sign of
a real number $\lambda$, it holds

$\lim_{s\rightarrow 0^{+}}(\lambda
s^{\alpha_{1}}+s^{\beta_{1}})=\lim_{s\rightarrow 0^{+}}(s^{\alpha_{2}}+\lambda
s^{\beta_{2}})=\left\\{\begin{array}[]{cc}+\infty&\text{if }\lambda>0\\\
-\infty&\text{if }\lambda<0.\end{array}\right.$

Therefore, system (1.1) can be referred to as an infinite positone problem if
$\lambda>0$ and as an infinite semipositone problem if $\lambda<0$.

The principle fact in this work is that the singularity in problem (1.1) comes
out through nonlinearities which are $(p-1)$-superlinear and
$(q-1)$-superlinear near $+\infty$. Namely, we assume that

$\alpha_{2}>q-1\text{ \ and \ }\beta_{1}>p-1.$ (1.3)

In this context, system (1.1) has a cooperative structure, that is, for $u$
(resp. $v$) fixed the right term in the first (resp. second) equation of (1.1)
is increasing in $v$ (resp. $u$). Further, according to (1.3) we have

$\lim_{s\rightarrow+\infty}(\lambda
s^{\alpha_{1}}+s^{\beta_{1}})/s^{p-1}=\lim_{s\rightarrow+\infty}(s^{\alpha_{2}}+\lambda
s^{\beta_{2}})/s^{q-1}=+\infty.$

This type of problem is rare in the literature. According to our knowledge,
only a positone-type singular system with superlinear terms was examined in
[21]. There the authors considered problem (1.1) depending on two positive
parameters in the whole space $\mathbb{R}^{N}$. The existence of a positive
entire solution is shown provided the parameters are sufficiently small.

The sublinear condition $\alpha_{2}<q-1$ and $\beta_{1}<p-1$ for singular
systems of type (1.1) have been thoroughly investigated. For a complete
overview on the study of the infinite positone problem (1.1) we refer to [1,
2, 15, 17], while for the study of the infinite semipositone problem (1.1), we
cite [5, 13, 14]. We also mention [6, 7] focusing on the semilinear case of
(1.1), that is, when $p=q=2$.

Another important class of singular problems considered in the literature is
the following

$\left\\{\begin{array}[]{ll}-\Delta_{p}u=u^{\alpha_{1}}v^{\beta_{1}}&\text{in
}\Omega,\\\ -\Delta_{q}v=u^{\alpha_{2}}v^{\beta_{2}}&\text{in }\Omega,\\\
u,v>0&\text{in }\Omega,\\\ u,v=0&\text{on }\partial\Omega.\end{array}\right.$
(1.4)

Relevant contributions regarding the cooperative case of system (1.4), that is
$\alpha_{2},\beta_{1}>0$, can be found in [8, 9, 18]. With regard to the
complementary situation $\alpha_{2},\beta_{1}<0$ which is the so called
competitive structure for system (1.4), we quote the papers [9, 17, 19]. The
semilinear case in (1.1) (i.e. $p=q=2$) was examined in [6, 12, 20] where the
linearity of the principal part is essentially used. It is worth pointing out
that in the aforementioned works, singular problem (1.4) was examined only
under the sublinear condition $\max\\{\alpha_{1},\beta_{1}\\}<p-1$ and
$\max\\{\alpha_{2},\beta_{2}\\}<q-1$. The assumptions imposed therein,
especially in [19], are not satisfied for our system (1.1) under hypothesis
(1.3).

Our main concern is the question of existence of a (positive) smooth solution
for a singular system a class of elliptic systems where the nonlinearities
besides a singular terms have superlinear terms. The main result is formulated
in the next theorem.

###### Theorem 1.1

Assume (1.2) and (1.3) hold. Then system (1.1) has a (positive) solution
$\left(u,v\right)$ in $C_{0}^{1,\gamma}(\overline{\Omega})\times
C_{0}^{1,\gamma}(\overline{\Omega})$ for some $\gamma\in(0,1)$.

The proof of Theorem 1.1 is done in section 3. The main technical difficulty
consists in the presence of singular terms in system (1.1) that can occur
under hypothesis (1.2). This difficulty is heightened by the superlinear
character of (1.1) that arise from (1.3). Our approach is chiefly based on
Theorem 2.1 proved in Section 2 via Schauder’s fixed point theorem (see [22])
and adequate truncations. This is a version of the sub-supersolution method
for quasilinear singular elliptic systems with cooperative structure. We
mention that in Theorem 2.1 no sign condition is required on the right-hand
side nonlinearities and so it can be used for large classes of quasilinear
singular problems. A significant feature of our result lies in the obtaining
of the sub- and supersolution. Due to the superlinear character of the
nonlinearities in (1.1), the latter cannot be constructed easily. At this
point, the choice of suitable functions with an adjustment of adequate
constants is crucial. Here we emphasize that the obtained sub- and
supersolution are quite different from functions considered in the
aforementioned papers, especially those constructed in [19].

This article is organized as follows. In section 2 we state and prove a
general theorem about sub and supersolution method for singular systems.
Section 3 contains the proof of Theorem 1.1.

##  2 Sub and supersolution theorem

Given $1<p<+\infty$, the spaces $L^{p}(\Omega)$ and $W_{0}^{1,p}(\Omega)$ are
endowed with the usual norms $\|u\|_{p}=(\int_{\Omega}|u|^{p}\ dx)^{1/p}$ and
$\|u\|_{1,p}=(\int_{\Omega}|\nabla u|^{p}\ dx)^{1/p}$, respectively. In the
sequel, corresponding to $1<p<+\infty$, we denote $p^{\prime}=\frac{p-1}{p}$.
We will also use the spaces $C(\overline{\Omega})$ and

$C_{0}^{1,\gamma}(\overline{\Omega})=\\{u\in
C^{1,\gamma}(\overline{\Omega}):u=0\ \mbox{on $\partial\Omega$}\\}$

with $\gamma\in(0,1)$. We denote by $\lambda_{1,p}$ and $\lambda_{1,q}$ the
first eigenvalue of $-\Delta_{p}$ on $W_{0}^{1,p}(\Omega)$ and of
$-\Delta_{q}$ on $W_{0}^{1,q}(\Omega)$, respectively. Let $\phi_{1,p}$ be the
normalized positive eigenfunction of $-\Delta_{p}$ corresponding to
$\lambda_{1,p}$, that is

$-\Delta_{p}\phi_{1,p}=\lambda_{1,p}\phi_{1,p}^{p-1}\ \text{in }\Omega,\ \
\phi_{1,p}=0\ \text{on }\partial\Omega,\text{ \ }\|\phi_{1}\|_{p}=1$

Similarly, let $\phi_{1,q}$ be the normalized positive eigenfunction of
$-\Delta_{q}$ corresponding to $\lambda_{1,q}$, that is

$-\Delta_{q}\phi_{1,q}=\lambda_{1,q}\phi_{1,q}^{q-1}\ \text{in }\Omega,\text{
\ }\phi_{1,q}=0\ \text{on }\partial\Omega,\text{ \ }\|\phi_{2}\|_{q}=1.$

For later use we set

$R=\max\left\\{\underset{\overline{\Omega}}{\max}\phi_{1,p},\underset{\overline{\Omega}}{\max}\phi_{1,q}\right\\}.$
(2.5)

We denote by $d(x)$ the distance from a point $x\in\overline{\Omega}$ to the
boundary $\partial\Omega$, where $\overline{\Omega}=\Omega\cup\partial\Omega$
is the closure of $\Omega\subset\mathbb{R}^{N}$. It is known that we can find
a constant $l>0$ such that

$\phi_{1,p}(x),\phi_{1,q}(x)\geq ld(x)\text{ for all }x\in\Omega,$ (2.6)

where $d(x):=dist(x,\partial\Omega)$ (see, e.g., [10]).

Let us introduce the problem

$\left\\{\begin{array}[]{ll}-\Delta_{p}u=f(x,u,v)&\text{in }\Omega,\\\
-\Delta_{q}v=g(x,u,v)&\text{in }\Omega,\\\ u,v>0&\text{in }\Omega,\\\
u,v=0&\text{on }\partial\Omega,\end{array}\right.$ (2.7)

where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ $\left(N\geq 2\right)$
with smooth boundary, $1<p,q<\infty$ and
$f,g:\Omega\times(0,+\infty)\times(0,+\infty)\rightarrow\mathbb{R}$ are
continuous functions which can exhibit singularities when the variables $u$
and $v$ approach zero. We consider system (2.7) with cooperative structure
assuming that for $u$ (resp. $v$) fixed the nonlinearity $f$ (resp. $g$) is
increasing in $v$ (resp. $u$). This makes the sub-supersolution techniques
applicable for (2.7). For systems without cooperative structure, i.e.
competitive systems, additional assumptions are required (see [9]).

We recall that a sub-supersolution for (2.7) is any pair
$(\underline{u},\underline{v})$,
$(\overline{u},\overline{v})\in(W_{0}^{1,p}(\Omega)\cap
L^{\infty}(\Omega))\times(W_{0}^{1,q}(\Omega)\cap L^{\infty}(\Omega))$ for
which there hold
$(\overline{u},\overline{v})\geq(\underline{u},\underline{v})$ in $\Omega$,

$\begin{array}[]{l}\int_{\Omega}\left|\nabla\underline{u}\right|^{p-2}\nabla\underline{u}\nabla\varphi\
dx-\int_{\Omega}f(x,\underline{u},\omega_{2})\varphi\ dx\\\
+\int_{\Omega}\left|\nabla\underline{v}\right|^{q-2}\nabla\underline{v}\nabla\psi\
dx-\int_{\Omega}g(x,\omega_{1},\underline{v})\psi\ dx\leq 0,\end{array}$
$\begin{array}[]{l}\int_{\Omega}\left|\nabla\overline{u}\right|^{p-2}\nabla\overline{u}\nabla\varphi\
dx-\int_{\Omega}f(x,\overline{u},\omega_{2})\varphi\ dx\\\
+\int_{\Omega}\left|\nabla\overline{v}\right|^{q-2}\nabla\overline{v}\nabla\psi-\int_{\Omega}g(x,\omega_{1},\overline{v})\psi\
dx\geq 0,\end{array}$

for all $\left(\varphi,\psi\right)\in W_{0}^{1,p}\left(\Omega\right)\times
W_{0}^{1,q}\left(\Omega\right)$ with $\varphi,\psi\geq 0$ a.e. in $\Omega$ and
for all $\left(\omega_{1},\omega_{2}\right)\in
W_{0}^{1,p}\left(\Omega\right)\times W_{0}^{1,q}\left(\Omega\right)$
satisfying $\underline{u}\leq\omega_{1}\leq\overline{u}$ and
$\underline{v}\leq\omega_{2}\leq\overline{v}$ a.e. in $\Omega$ (see [4, p.
269]). The main goal in this section is to prove Theorem 2.1 below, which is a
key point in the proof of Theorem 1.1.

###### Theorem 2.1

Let $\left(\underline{u},\underline{v}\right),$
$\left(\overline{u},\overline{v}\right)\in C^{1}(\overline{\Omega})\times
C^{1}(\overline{\Omega})$ be a sub and supersolution pairs of (2.7) and
suppose there exist constants $k_{1},k_{2}>0$ and $-1<\alpha,\beta<0$ such
that

$\begin{array}[]{c}\left|f(x,u,v)\right|\leq k_{1}d(x)^{\alpha}\text{ and
}\left|g(x,u,v)\right|\leq k_{2}d(x)^{\beta}\text{ in
}\Omega\times[\underline{u},\overline{u}]\times[\underline{v},\overline{v}].\end{array}$
(2.8)

Then system (2.7) has a positive solution $(u,v)$ in
$C_{0}^{1,\gamma}(\overline{\Omega})\times
C_{0}^{1,\gamma}(\overline{\Omega})$ for certain $\gamma\in(0,1).$

Proof.

For each $(z_{1},z_{2})\in C(\overline{\Omega})\times C(\overline{\Omega}),$
let $(u,v)\in W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega)$ be the unique
solution of the problem

$\left\\{\begin{array}[]{ll}-\Delta_{p}u=\widetilde{f}(x,z_{1},z_{2})&\text{in
}\Omega,\\\ -\Delta_{q}v=\widetilde{g}(x,z_{1},z_{2})&\text{in }\Omega,\\\
u,v>0&\text{in }\Omega,\\\ u,v=0&\text{on }\partial\Omega,\end{array}\right.$
(2.9)

where

$\begin{array}[]{c}\widetilde{f}(x,z_{1},z_{2})=f(x,\widetilde{z}_{1},\widetilde{z}_{2})\text{
and
}\widetilde{g}(x,z_{1},z_{2})=g(x,\widetilde{z}_{1},\widetilde{z}_{2})\end{array}$
(2.10)

with

$\widetilde{z}_{1}=\min(\max(z_{1},\underline{u}),\overline{u})\text{ and
}\widetilde{z}_{2}=\min(\max(z_{2},\underline{v}),\overline{v}).$ (2.11)

On account of (2.11) it follows that
$\underline{u}\leq\widetilde{z}_{1}\leq\overline{u}$ and
$\underline{v}\leq\widetilde{z}_{2}\leq\overline{v}$. Then, bearing in mind
(2.8) we have

$\left|\widetilde{f}(x,z_{1},z_{2})\right|\leq k_{1}d(x)^{\alpha}\text{ and
}\left|\widetilde{g}(x,z_{1},z_{2})\right|\leq k_{2}d(x)^{\beta}\text{ for
a.e. }x\in\Omega.$ (2.12)

We point out that the estimates (2.12) enable us to deduce that

$\widetilde{f}(x,z_{1},z_{2})\in W^{-1,p^{\prime}}(\Omega)\text{ and
}\widetilde{g}(x,z_{1},z_{2})\in W^{-1,q^{\prime}}(\Omega).$

This is a consequence of Hardy-Sobolev inequality (see, e.g., [1, Lemma 2.3]).
Then the unique solvability of $(u,v)$ in (2.9) is readily derived from Minty-
Browder Theorem (see, e.g., [3]).

Let us introduce the operator

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

We note from (2.9) that the fixed point of $\mathcal{T}$ coincide with the
weak solution of (2.7). Consequently, to achieve the desired conclusion it
suffices to prove that $\mathcal{T}$ has a fixed point. To this end we apply
Schauder’s fixed point theorem. Using (2.12) there exists $\gamma\in(0,1)$
such that $(u,v)\in C_{0}^{1,\gamma}(\overline{\Omega})\times
C_{0}^{1,\gamma}(\overline{\Omega})$ and
$\left\|u\right\|_{C_{0}^{1,\gamma}(\overline{\Omega})},\left\|u\right\|_{C_{0}^{1,\gamma}(\overline{\Omega})}\leq
C,$ where $C>0$ is independent of $u$ and $v$ (see [11, Lemma 3.1]). Then the
compactness of the embedding $C_{0}^{1,\gamma}(\overline{\Omega})\subset
C(\overline{\Omega})$ implies that $\mathcal{T(}C(\overline{\Omega})\times
C(\overline{\Omega}))$ is a relatively compact subset of
$C(\overline{\Omega})\times C(\overline{\Omega})$.

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

$\begin{array}[]{c}\int_{\Omega}\left|\nabla u_{n}\right|^{p-2}\nabla
u_{n}\nabla\varphi=\int_{\Omega}\widetilde{f}(x,z_{1,n},z_{2,n})\varphi\
dx\end{array}$ (2.13)

and

$\begin{array}[]{c}\int_{\Omega}\left|\nabla v_{n}\right|^{q-2}\nabla
v_{n}\nabla\psi=\int_{\Omega}\widetilde{g}(x,z_{1,n},z_{2,n})\psi\
dx\end{array}$ (2.14)

for all $\left(\varphi,\psi\right)\in W_{0}^{1,p}(\Omega)\times
W_{0}^{1,q}(\Omega)$. Inserting $(\varphi,\psi)=(u_{n},v_{n})$ in (2.13) and
(2.14), using (2.8) we get

$\left\|u_{n}\right\|_{1,p}=\int_{\Omega}\widetilde{f}(x,z_{1,n},z_{2,n})u_{n}\
dx\leq\int_{\Omega}d^{\alpha}u_{n}dx$ (2.15)

and

$\left\|v_{n}\right\|_{1,q}=\int_{\Omega}\widetilde{g}(x,z_{1,n},z_{2,n})v_{n}\
dx\leq\int_{\Omega}d^{\beta}v_{n}dx.$ (2.16)

Since $-1<\alpha,\beta<0$, by virtue of the Hardy-Sobolev inequality (see,
e.g., [1]), the last integrals in (2.15) and (2.16) are finite which in turn
imply that $\\{u_{n}\\}$ and $\\{v_{n}\\}$ are bounded in
$W_{0}^{1,p}(\Omega)$ and $W_{0}^{1,q}(\Omega),$ respectively. So, passing to
relabeled subsequences, we can write the weak convergence in
$W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}\left(\Omega\right)$

$\begin{array}[]{c}(u_{n},v_{n})\rightharpoonup\left(u,v\right)\end{array}$
(2.17)

for some $\left(u,v\right)\in W_{0}^{1,p}(\Omega)\times
W_{0}^{1,q}\left(\Omega\right)$. Setting $\varphi=u_{n}-u$ in (2.13) and
$\psi=v_{n}-v$ in (2.14), we find that

$\begin{array}[]{l}\int_{\Omega}\left|\nabla u_{n}\right|^{p-2}\nabla
u_{n}\nabla(u_{n}-u)=\int_{\Omega}\widetilde{f}(x,z_{1,n},z_{2,n})(u_{n}-u)\
dx\end{array}$

and

$\begin{array}[]{l}\int_{\Omega}\left|\nabla v_{n}\right|^{p-2}\nabla
v_{n}\nabla(v_{n}-v)=\int_{\Omega}\widetilde{g}(x,z_{1,n},z_{2,n})(v_{n}-v)\
dx.\end{array}$

Lebesgue’s dominated convergence theorem ensures

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

The $S_{+}$-property of $-\Delta_{p}$ on $W_{0}^{1,p}\left(\Omega\right)$ and
of $-\Delta_{q}$ on $W_{0}^{1,q}\left(\Omega\right)$ (see, e.g. [16,
Proposition 3.5]), along with (2.17), implies

$\begin{array}[]{c}u_{n}\rightarrow u\text{ in }W_{0}^{1,p}(\Omega)\text{ and
}v_{n}\rightarrow v\text{ in }W_{0}^{1,q}(\Omega).\end{array}$

Then, through (2.13), (2.14) and the invariance of $C(\overline{\Omega})\times
C(\overline{\Omega})$ by $\mathcal{T}$, we infer that
$\left(u,v\right)=\mathcal{T(}z_{1},z_{2})$. On the other hand, from (2.13)
and (2.14) we know that the sequence $\\{\left(u_{n},v_{n}\right)\\}$ is
bounded in $C_{0}^{1,\gamma}(\overline{\Omega})\times
C_{0}^{1,\gamma}(\overline{\Omega})$ for certain $\gamma\in(0,1)$. Since the
embedding $C_{0}^{1,\gamma}(\overline{\Omega})\subset C(\overline{\Omega})$ is
compact, along a relabeled subsequence there holds
$(u_{n},v_{n})\rightarrow(u,v)$ in $C(\overline{\Omega})\times
C(\overline{\Omega})$. We conclude that $\mathcal{T}$ is continuous.

We are thus in a position to apply Schauder’s fixed point theorem to the map
$\mathcal{T}$, which establishes the existence of $(u,v)\in
C(\overline{\Omega})\times C(\overline{\Omega})$ satisfying
$(u,v)=\mathcal{T}(u,v).$

Let us justify that

$\underline{u}\leq u\leq\overline{u}\text{ and }\underline{v}\leq
v\leq\overline{v}\text{ in }\Omega.$

Put $\zeta=(\underline{u}-u)^{+}$ and suppose $\zeta\neq 0$. Then, bearing in
mind that system (2.7) is cooperative, from (2.11), (2.9) and (2.10), we infer
that

$\begin{array}[]{c}\int_{\\{u<\underline{u}\\}}|\nabla u|^{p-2}\nabla
u\nabla\zeta\ dx=\int_{\Omega}|\nabla u|^{p-2}\nabla u\nabla\zeta\
dx=\int_{\\{u<\underline{u}\\}}\widetilde{f}(x,u,v)\zeta\ dx\\\ \\\
=\int_{\\{u<\underline{u}\\}}f(x,\widetilde{u},\widetilde{v})\zeta\
dx=\int_{\\{u<\underline{u}\\}}f(x,\underline{u},\widetilde{v})\zeta\
dx\geq\int_{\\{u<\underline{u}\\}}|\nabla\underline{u}|^{p-2}\nabla\underline{u}\nabla\zeta\
dx.\end{array}$

This implies that

$\begin{array}[]{c}\int_{\\{u<\underline{u}\\}}(|\nabla u|^{p-2}\nabla
u-|\nabla\underline{u}|^{p-2}\nabla\underline{u})\nabla\zeta\ dx\leq
0,\end{array}$

a contradiction. Hence $u\geq\underline{u}$ in $\Omega$. A quite similar
argument provides that $v\geq\underline{v}$ in $\Omega$. In the same way, we
prove that $u\leq\overline{u}$ and $v\leq\overline{v}$ in $\Omega$.

Finally, thanks to [11, Lemma 3.1] one has $(u,v)\in
C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega})$
for some $\gamma\in(0,1)$. This completes the proof.

##  3 Proof of the main result

This section is devoted to the proof of Theorem 1.1. It relies on sub-
supersolution techniques shown by Theorem 2.1.

Let $y_{1}$ and $y_{2}$ be the unique solutions of the problems

$\left\\{\begin{array}[]{ll}-\Delta_{p}y_{1}=y_{1}^{\alpha_{1}}&\text{ in
}\Omega\\\ y_{1}>0&\text{ in }\Omega\\\ y_{1}=0&\text{ on
}\partial\Omega\end{array}\right.\text{ \ and \
}\left\\{\begin{array}[]{ll}-\Delta_{q}y_{2}=y_{2}^{\beta_{2}}&\text{ in
}\Omega\\\ y_{2}>0&\text{ in }\Omega\\\ y_{2}=0&\text{ on
}\partial\Omega\end{array}\right.$ (3.18)

respectively. They verify the estimates

$c_{1}\phi_{1,p}(x)\leq y_{1}(x)\leq c_{2}\phi_{1,p}(x)\text{ \ and \
}c_{3}\phi_{1,q}(x)\leq y_{2}(x)\leq c_{4}\phi_{1,q}(x),$ (3.19)

with constants $c_{2}\geq c_{1}>0$ and $c_{4}\geq c_{3}>0$ (see [10]). For
$\delta>0$ sufficiently small we denote

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

and $\mu=\mu(\delta)>0$ a constant such that

$\begin{array}[]{c}\phi_{1,p}\left(x\right),\phi_{1,q}\left(x\right)\geq\mu\text{
in }\Omega\backslash\Omega_{\delta}.\end{array}$ (3.20)

Let $\underline{u}$ and $\underline{v}$ satisfy

$-\Delta_{p}\underline{u}(x)=C\left\\{\begin{array}[]{ll}y_{1}^{\alpha_{1}}(x)&\text{if
\ }x\in\Omega\backslash\overline{\Omega}_{\delta}\\\
-y_{1}^{\alpha_{1}}(x)&\text{if \
}x\in\Omega_{\delta}\end{array}\right.,\text{ }\underline{u}=0\text{ \ on
}\partial\Omega$ (3.21)

and

$-\Delta_{q}\underline{v}(x)=C\left\\{\begin{array}[]{ll}y_{2}^{\beta_{2}}(x)&\text{if
\ }x\in\Omega\backslash\overline{\Omega}_{\delta}\\\
-y_{2}^{\beta_{2}}(x)&\text{if \ }x\in\Omega_{\delta}\end{array}\right.,\text{
}\underline{v}=0\text{ \ on }\partial\Omega\text{,}$ (3.22)

with a constant $C>1$ to be chosen later on. The Hardy-Sobolev inequality (see
e.g. [1]) guarantees that the right hand side of (3.21) and (3.22) are in
$W^{-1,p^{\prime}}(\Omega)$ and $W^{-1,q^{\prime}}(\Omega)$, respectively.
This allows to apply the Minty-Browder theorem (see [3, Theorem V.15]) to
deduce the existence of unique solutions $\underline{u}$ and $\underline{v}$
for problems (3.21) and (3.22), respectively. Moreover, (3.18), (3.19),
(3.21), (3.22) and the monotonicity of the operators $-\Delta_{p}$ and
$-\Delta_{q}$ together with [11, Corollary 3.1] imply that

$\begin{array}[]{c}\frac{c_{1}}{2}C^{\frac{1}{p-1}}\phi_{1,p}(x)\leq\underline{u}(x)\leq
c_{2}C^{\frac{1}{p-1}}\phi_{1,p}(x)\text{ and
}\frac{c_{3}}{2}C^{\frac{1}{q-1}}\phi_{1,q}(x)\leq\underline{v}(x)\leq
c_{4}C^{\frac{1}{q-1}}\phi_{1,q}(x)\text{ in }\Omega.\end{array}$ (3.23)

For $\lambda\geq 0,$ the positivity of
$\underline{u},\underline{v},y_{1},y_{2}$ and $C$ enable us to have

$\begin{array}[]{c}-Cy_{1}^{\alpha_{1}}-\lambda\underline{u}^{\alpha_{1}}\leq
0\leq\underline{v}^{\beta_{1}}\text{ in }\Omega_{\delta}\end{array}$ (3.24)

and

$\begin{array}[]{c}-Cy_{2}^{\beta_{2}}-\lambda\underline{v}^{\beta_{2}}\leq
0\leq\underline{u}^{\alpha_{2}}\text{ in }\Omega_{\delta}.\end{array}$ (3.25)

For $\lambda<0$, (3.19), (3.23) and (1.2) imply

$\begin{array}[]{c}-Cy_{1}^{\alpha_{1}}-\lambda\underline{u}^{\alpha_{1}}\leq(-Cc_{2}^{\alpha_{1}}-\lambda(\frac{c_{1}}{2}C^{\frac{1}{p-1}})^{\alpha_{1}})\phi_{1,p}^{\alpha_{1}}\leq
0\leq\underline{v}^{\beta_{1}}\text{ in }\Omega_{\delta}\end{array}$ (3.26)

and

$\begin{array}[]{c}-Cy_{2}^{\beta_{2}}-\lambda\underline{v}^{\beta_{2}}\leq\left(-C^{q-1}c_{4}^{\beta_{2}}-\lambda(\frac{c_{3}}{2}C^{\frac{1}{q-1}})^{\beta_{2}}\right)\phi_{1,q}^{\beta_{2}}\leq
0\leq\underline{u}^{\alpha_{2}}\text{ in }\Omega_{\delta},\end{array}$ (3.27)

provided that $C$ is sufficiently large. Now we deal with the corresponding
estimates on $\Omega\backslash\overline{\Omega}_{\delta}.$ If $\lambda\geq 0$
we get from (3.19), (3.23), (3.20), (2.6) and (1.2) that

$\begin{array}[]{l}(Cy_{1}^{\alpha_{1}}-\lambda\underline{u}^{\alpha_{1}})\underline{v}^{-\beta_{1}}\leq
Cy_{1}^{\alpha_{1}}\underline{v}^{-\beta_{1}}\leq
C^{1-\frac{\beta_{1}}{q-1}}c_{1}^{\alpha_{1}}(\frac{c_{3}}{2}l)^{-\beta_{1}}\phi_{1,p}^{\alpha_{1}-\beta_{1}}\\\
\leq
C^{1-\frac{\beta_{1}}{q-1}}c_{1}^{\alpha_{1}}(\frac{c_{3}}{2}l)^{-\beta_{1}}\mu^{\alpha_{1}-\beta_{1}}\leq
1\ \ \text{in }\Omega\backslash\overline{\Omega}_{\delta}\end{array}$ (3.28)

and

$\begin{array}[]{l}(Cy_{2}^{\beta_{2}}-\lambda\underline{v}^{\beta_{2}})\underline{u}^{-\alpha_{2}}\leq
Cy_{2}^{\beta_{2}}\underline{u}^{-\alpha_{2}}\leq
C^{1-\frac{\alpha_{2}}{p-1}}c_{3}^{\beta_{2}}(\frac{c1}{2}l)^{-\alpha_{2}}\phi_{1,q}^{\beta_{2}-\alpha_{2}}\\\
\leq
C^{1-\frac{\alpha_{2}}{p-1}}c_{3}^{\beta_{2}}(\frac{c1}{2}l)^{-\alpha_{2}}\mu^{\beta_{2}-\alpha_{2}}\leq
1\ \ \text{in }\Omega\backslash\overline{\Omega}_{\delta},\end{array}$ (3.29)

provided that $C$ is sufficiently large. For $\lambda<0,$ (3.19), (3.23),
(3.20), (2.6) and (1.2) imply

$\begin{array}[]{l}(Cy_{1}^{\alpha_{1}}-\lambda\underline{u}^{\alpha_{1}})\underline{v}^{-\beta_{1}}\leq\left(Cc_{1}^{\alpha_{1}}-\lambda
C^{\frac{\alpha_{1}}{p-1}}(\frac{c_{1}}{2})^{\alpha_{1}}\right)C^{-\frac{\beta_{1}}{q-1}}(\frac{c_{3}}{2}l)^{-\beta_{1}}\phi_{1,p}^{\alpha_{1}-\beta_{1}}\\\
=C^{1-\frac{\beta_{1}}{q-1}}c_{1}^{\alpha_{1}}\left(1-\lambda
C^{\frac{\alpha_{1}}{p-1}-1}2^{-\alpha_{1}}\right)(\frac{c_{3}}{2}l)^{-\beta_{1}}\phi_{1,p}^{\alpha_{1}-\beta_{1}}\\\
\leq C^{1-\frac{\beta_{1}}{q-1}}c_{1}^{\alpha_{1}}\left(1-\lambda
2^{-\alpha_{1}}\right)(\frac{c_{3}}{2}l)^{-\beta_{1}}\mu^{\alpha_{1}-\beta_{1}}\leq
1\ \text{in\ }\Omega\backslash\overline{\Omega}_{\delta}\end{array}$

and

$\begin{array}[]{l}(Cy_{2}^{\beta_{2}}-\lambda\underline{v}^{\beta_{2}})\underline{u}^{-\alpha_{2}}\leq\left(Cc_{3}^{\beta_{2}}-\lambda
C^{\frac{\beta_{2}}{q-1}}(\frac{c_{3}}{2})^{\beta_{2}}\right)C^{-\frac{\alpha_{2}}{p-1}}(\frac{c_{1}}{2}l)^{-\alpha_{2}}\phi_{1,q}^{\beta_{2}-\alpha_{2}}\\\
=C^{1-\frac{\alpha_{2}}{p-1}}c_{3}^{\beta_{2}}\left(1-\lambda
C^{\frac{\beta_{2}}{q-1}-1}2^{-\beta_{2}}\right)(\frac{c_{1}}{2}l)^{-\alpha_{2}}\phi_{1,q}^{\beta_{2}-\alpha_{2}}\\\
\leq C^{1-\frac{\alpha_{2}}{p-1}}c_{3}^{\beta_{2}}\left(1-\lambda
2^{-\beta_{2}}\right)(\frac{c_{1}}{2}l)^{-\alpha_{2}}\mu^{\beta_{2}-\alpha_{2}}\leq
1\ \text{in\ }\Omega\backslash\overline{\Omega}_{\delta},\end{array}$

provided that $C$ is sufficiently large. This is equivalent to

$\begin{array}[]{c}Cy_{1}^{\alpha_{1}}\leq\lambda\underline{u}^{\alpha_{1}}+\underline{v}^{\beta_{1}}\
\ \text{in \ }\Omega\backslash\overline{\Omega}_{\delta}\end{array}$ (3.30)

and

$\begin{array}[]{c}Cy_{2}^{\beta_{2}}\leq\underline{u}^{\alpha_{2}}+\lambda\underline{v}^{\beta_{2}}\
\ \text{in \ }\Omega\backslash\overline{\Omega}_{\delta}\text{,}\end{array}$
(3.31)

for all $\lambda\in\mathbb{R}.$

Due to the definition of $\underline{u}$ and $\underline{v}$ (see (3.21) and
(3.22)) we actually have

$\begin{array}[]{c}\int_{\Omega}\left|\nabla\underline{u}\right|^{p-2}\nabla\underline{u}\nabla\varphi\text{
}dx=\int_{\Omega\backslash\overline{\Omega}_{\delta}}Cy_{1}^{\alpha_{1}}\varphi\text{
}dx-\int_{\Omega_{\delta}}Cy_{1}^{\alpha_{1}}\varphi\text{ }dx\end{array}$
(3.32)

and

$\begin{array}[]{c}\int_{\Omega}\left|\nabla\underline{v}\right|^{q-2}\nabla\underline{v}\nabla\psi\text{
}dx=\int_{\Omega\backslash\overline{\Omega}_{\delta}}Cy_{2}^{\beta_{2}}\psi\text{
}dx-\int_{\Omega_{\delta}}Cy_{2}^{\beta_{2}}\psi\text{ }dx,\end{array}$ (3.33)

where $\left(\varphi,\psi\right)\in W_{0}^{1,p}\left(\Omega\right)\times
W_{0}^{1,q}\left(\Omega\right)$ with $\varphi,\psi\geq 0$. Then combining
(3.24)-(3.27), (3.30), (3.31) with (3.32), (3.33), it is readily seen that

$\begin{array}[]{c}\int_{\Omega}\left|\nabla\underline{u}\right|^{p-2}\nabla\underline{u}\nabla\varphi\text{
}dx\leq\int_{\Omega}(\lambda\underline{u}^{\alpha_{1}}+\underline{v}^{\beta_{1}})\varphi\text{
}dx\end{array}$

and

$\begin{array}[]{c}\int_{\Omega}\left|\nabla\underline{v}\right|^{q-2}\nabla\underline{v}\nabla\psi\text{
}dx\leq\int_{\Omega}(\underline{u}^{\alpha_{2}}+\lambda\underline{v}^{\beta_{2}})\psi\text{
}dx,\end{array}$

for all $\left(\varphi,\psi\right)\in W_{0}^{1,p}\left(\Omega\right)\times
W_{0}^{1,q}\left(\Omega\right)$ with $\varphi,\psi\geq 0$, showing that
$\left(\underline{u},\underline{v}\right)$ is a subsolution of problem (1.1).

Next, we construct a supersolution part for problem (1.1). To this end, let
$\widetilde{\Omega}$ be a bounded domain in $\mathbb{R}^{N}$ with
$C^{1,\alpha}$ boundary $\partial\widetilde{\Omega}$, $\alpha\in(0,1)$, such
that $\overline{\Omega}\subset\widetilde{\Omega}.$ We denote by
$\widetilde{\lambda}_{1,p}$ and $\widetilde{\lambda}_{1,q}$ the first
eigenvalue of $-\Delta_{p}$ on $W_{0}^{1,p}(\widetilde{\Omega})$ and of
$-\Delta_{q}$ on $W_{0}^{1,q}(\widetilde{\Omega})$, respectively. Let
$\widetilde{\phi}_{1,p}$ be the normalized positive eigenfunction of
$-\Delta_{p}$ corresponding to $\widetilde{\lambda}_{1,p}$, that is

$-\Delta_{p}\widetilde{\phi}_{1,p}=\widetilde{\lambda}_{1,p}\widetilde{\phi}_{1,p}^{p-1}\
\text{in }\widetilde{\Omega},\ \ \widetilde{\phi}_{1,p}=0\text{ on
}\partial\widetilde{\Omega}.$

Similarly, let $\widetilde{\phi}_{1,q}$ be the normalized positive
eigenfunction of $-\Delta_{q}$ corresponding to $\widetilde{\lambda}_{1,q}$,
that is

$-\Delta_{q}\widetilde{\phi}_{1,q}=\widetilde{\lambda}_{1,q}\widetilde{\phi}_{1,q}^{q-1}\
\text{in }\Omega,\text{ \ }\widetilde{\phi}_{1,q}=0\ \text{on
}\partial\widetilde{\Omega}.$

By the definition of $\widetilde{\Omega}$ and the strong maximum principle,
there exists a constant $\rho>0$ sufficiently small such that

$\widetilde{\phi}_{1,p}\left(x\right),\widetilde{\phi}_{1,q}\left(x\right)>\rho\text{
in }\overline{\Omega}.$ (3.34)

Without lost of generality we assume that

$R=\max\left\\{\underset{\overline{\Omega}}{\max}\widetilde{\phi}_{1,p},\underset{\overline{\Omega}}{\max}\widetilde{\phi}_{1,q}\right\\}.$
(3.35)

Let $\xi_{1},\xi_{2}\in C^{1}\left(\overline{\widetilde{\Omega}}\right)$ be
the solutions of the homogeneous Dirichlet problems:

$\left\\{\begin{array}[]{l}-\Delta_{p}\xi_{1}=C^{\delta(p-1)}\xi_{1}^{\theta_{1}}\text{
in }\widetilde{\Omega}\\\ \xi_{1}=0\text{ on
}\widetilde{\Omega}\end{array}\right.,\text{ \ \
}\left\\{\begin{array}[]{l}-\Delta_{q}\xi_{2}=C^{\delta(q-1)}\xi_{2}^{\theta_{2}}\text{
in }\widetilde{\Omega}\\\ \xi_{2}=0\text{ on
}\widetilde{\Omega},\end{array}\right.$ (3.36)

with constants $\delta,$ $\theta_{1}$ and $\theta_{2}$ satisfying

$\begin{array}[]{c}\theta_{1}\in(\alpha_{1},0),\text{ \
}\theta_{2}\in(\beta_{2},0)\text{ \ and \
}\delta<\min\\{\frac{1}{\theta_{1}},\frac{1}{\theta_{2}}\\}<0.\end{array}$
(3.37)

Functions $\xi_{1}$and $\xi_{2}$ verifying

$\begin{array}[]{c}C^{\delta}c_{0}\widetilde{\phi}_{1,p}\leq\xi_{1}\leq
C^{\delta}c\widetilde{\phi}_{1,p}\text{ \ and \
}C^{\delta}c_{0}^{\prime}\widetilde{\phi}_{1,q}\leq\xi_{2}\leq
C^{\delta}c^{\prime}\widetilde{\phi}_{1,q},\end{array}$ (3.38)

for some positive constants $c_{0},c_{0}^{\prime},c$ and $c^{\prime}$ (see
[10]). Set

$\begin{array}[]{c}(\overline{u},\overline{v})=C^{-\delta}(\xi_{1},\xi_{2}).\end{array}$
(3.39)

Then we have
$\left(\overline{u},\overline{v}\right)\geq\left(\underline{u},\underline{v}\right)$
in $\overline{\Omega}$. Indeed, on the one hand, through (3.21), (3.22) and
(3.36), one has

$-\Delta_{p}\overline{u}\geq-\Delta_{p}\underline{u}\text{ \ and
}-\Delta_{p}\overline{v}\geq-\Delta_{p}\underline{v}\text{ \ in
}\Omega_{\delta}.$

On the other hand, on the basis of (3.21), (3.22), (3.36), (3.38), (3.19),
(3.20), (3.35), (3.37) and for $C$ large enough, we achieve

$\begin{array}[]{l}-\Delta_{p}\overline{u}=C^{-\delta(p-1)}C^{\delta(p-1)}\xi_{1}^{\theta_{1}}=\xi_{1}^{\theta_{1}}\geq
C^{\delta\theta_{1}}(c\widetilde{\phi}_{1,p})^{\theta_{1}}\\\ \geq
C^{\delta\theta_{1}}(cR)^{\theta_{1}}\geq C(c_{1}\mu)^{\alpha_{1}}\geq
C(c_{1}\phi_{1,p})^{\alpha_{1}}=Cy_{1}^{\alpha_{1}}=-\Delta_{p}\underline{u}\text{
\ in }\Omega\backslash\overline{\Omega}_{\delta}.\end{array}$

and

$\begin{array}[]{l}-\Delta_{q}\overline{v}=C^{-\delta(q-1)}C^{\delta(q-1)}\xi_{2}^{\theta_{2}}=\xi_{2}^{\theta_{2}}\geq
C^{\delta\theta_{2}}(c^{\prime}\phi_{1,q})^{\theta_{2}}\\\ \geq
C^{\delta\theta_{2}}(c^{\prime}R)^{\theta_{2}}\geq C(c_{3}\mu)^{\beta_{2}}\geq
C(c_{3}\phi_{1,q})^{\beta_{2}}\geq
Cy_{2}^{\beta_{2}}=-\Delta_{q}\underline{v}\text{ in
}\Omega\backslash\overline{\Omega}_{\delta}.\end{array}$

Then the monotonicity of the operators $-\Delta_{p}$ and $-\Delta_{q}$ leads
to the conclusion.

Now, taking into account (3.37), (3.38), (3.35), (3.34) and (1.2), for all
$\lambda\in\mathbb{R}$, we derive that in $\overline{\Omega}$ one has

$\begin{array}[]{l}\xi_{1}^{\theta_{1}}\geq
C^{\delta\theta_{1}}(c\phi_{1,p})^{\theta_{1}}\geq
C^{\delta\theta_{1}}(cR)^{\theta_{1}}\geq\lambda(c_{0}\rho)^{\alpha_{1}}+(c^{\prime}R)^{\beta_{1}}\\\
\lambda(c_{0}\widetilde{\phi}_{1,p})^{\alpha_{1}}+(c^{\prime}\widetilde{\phi}_{1,q})^{\beta_{1}}\geq\lambda(C^{-\delta}\xi_{1})^{\alpha_{1}}+(C^{-\delta}\xi_{2})^{\beta_{1}}=\lambda\overline{u}^{\alpha_{1}}+\overline{v}^{\beta_{1}}\text{
\ in }\overline{\Omega}\end{array}$ (3.40)

and

$\begin{array}[]{l}\xi_{2}^{\theta_{2}}\geq
C^{\delta\theta_{2}}(c^{\prime}\phi_{1,q})^{\theta_{2}}\geq
C^{\delta\theta_{2}}(c^{\prime}R)^{\theta_{2}}\geq\lambda(cR)^{\alpha_{2}}+(c_{0}^{\prime}\rho)^{\beta_{1}}\\\
\lambda(c\widetilde{\phi}_{1,p})^{\alpha_{1}}+(c_{0}^{\prime}\widetilde{\phi}_{1,q})^{\beta_{1}}\geq\lambda(C^{-\delta}\xi_{1})^{\alpha_{1}}+(C^{-\delta}\xi_{2})^{\beta_{1}}=\lambda\overline{u}^{\alpha_{1}}+\overline{v}^{\beta_{1}}\text{
\ in }\overline{\Omega}\end{array}$ (3.41)

provided that $C$ is sufficiently large. Consequently, it turns out from
(3.36), (3.40) and (3.41) that

$\begin{array}[]{l}\int_{\Omega}\left|\nabla\overline{u}\right|^{p-2}\nabla\overline{u}\nabla\varphi\text{
}dx=\int_{\Omega}\xi_{1}^{\theta_{1}}\varphi\geq\int_{\Omega}(\lambda\overline{u}^{\alpha_{1}}+\overline{v}^{\beta_{1}})\varphi\text{
}dx\end{array}$

and

$\begin{array}[]{l}\int_{\Omega}\left|\nabla\overline{v}\right|^{q-2}\nabla\overline{v}\nabla\psi\text{
}dx=\int_{\Omega}\xi_{2}^{\theta_{2}}\psi\geq\int_{\Omega}\left(\overline{u}^{\alpha_{2}}+\lambda\overline{v}^{\beta_{2}}\right)\psi\text{
}dx,\end{array}$

for all $\left(\varphi,\psi\right)\in W_{0}^{1,p}\left(\Omega\right)\times
W_{0}^{1,q}\left(\Omega\right).$ This proves that the pair
$(\overline{u},\overline{v})$ is a supersolution for problem (1.1).

Finally, owing to Theorem 2.1 problem (1.1) has a positive solution $(u,v)\in
C_{0}^{1,\gamma}(\overline{\Omega})\times
C_{0}^{1,\gamma}(\overline{\Omega})$, for certain $\gamma\in(0,1)$, within
$\left[\underline{u},\overline{u}\right]\times\left[\underline{v},\overline{v}\right]$.
This completes the proof.

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