\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 30, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/30\hfil Nonexistence of solutions]
{Nonexistence of solutions to some inequalities and systems with
singular coefficients on the boundary}

\author[L. Uvarova, O. Salieva, E. Galakhov \hfil EJDE-2017/30\hfilneg]
{Liudmila Uvarova, Olga Salieva, Evgeny Galakhov}

\address{Liudmila Uvarova \newline
 Moscow State Technological University ``Stankin'',
 Russia}
\email{uvar11@yandex.ru}

\address{Olga Salieva \newline
 Moscow State Technological University ``Stankin'',
 Russia}
\email{olga.a.salieva@gmail.com}

\address{Evgeny Galakhov \newline
 Peoples' Friendship University of Russia,
 Moscow, Russia}
\email{galakhov@rambler.ru}

\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted August 6, 2016. Published January 27, 2017.}
\subjclass[2010]{35J62, 35J70, 35J75, 35B44}
\keywords{Elliptic inequalities; p-Laplace; nonexistence;
 singular coefficients}

\begin{abstract}
 We obtain sufficient conditions for the nonexistence of positive solutions
 to some elliptic inequalities and systems containing the p-Laplace operators
 and coefficients possessing singularities on the boundary.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

The problem of sufficient conditions for nonexistence of solutions to systems 
of nonlinear elliptic differential equations and inequalities with singular 
coefficients has been studied by many authors.
For the Laplacian and heat operator with a point singularity inside the domain, 
pioneering results in this direction were obtained by  Brezis and  Cabr\'e \cite{BC}
 by means of comparison principles.
For higher order operators that do not satisfy the comparison principle, 
Pohozaev \cite{P} suggested the nonlinear capacity method.
Later it was developed in joint works with  Mitidieri and other authors 
(see, in particular, the monograph \cite{MPb} and references therein). 
This method allowed one to obtain a number of new sharp sufficient conditions 
of non-solvability of differential inequalities in various functional classes. 
The method is based on deriving asymptotically optimal a priori estimates 
of the solutions by means of algebraic analysis of the integral form of 
the inequality under consideration with a special choice of test functions. 
Applications of this method to different types of elliptic inequalities and 
systems containing degeneracy, point singularities, gradient terms etc. 
can be found, for example, in \cite{FS,FPR,LiLi}.

In the present paper, a modification of the nonlinear capacity method is used 
in order to obtain dimension independent sufficient conditions of non-solvability 
for some quasilinear elliptic inequalities in a bounded domain with coefficients 
having singularities near the boundary. This distinguishes the problem setting 
suggested here from the aforementioned works in this field, where singularities 
appeared at single points or at infinity. In \cite{LiLi}, some results 
concerning the case of boundary singularities are also obtained, but they are 
dimension dependent.

For the proof of nonexistence results by the nonlinear capacity method, 
test functions with different geometrical structure of the support are 
constructed, which takes into account the specific nature of problems under 
consideration. Our first results in this direction were published in \cite{GS1, GS2}.

The rest of the paper consists of two sections.  In \S2, we establish nonexistence 
results for scalar quasilinear elliptic inequalities, and in \S3, for systems 
of such inequalities.

From here on, letter $c$ denotes different positive constants, which may depend 
on the parameters of the problems under consideration.


\section{Scalar inequalities}

Consider the problem
\begin{equation}\label{eq:1}
\begin{gathered}
-\operatorname{div}(|Du|^{p-2}Du) \ge f(x)u^{q}|Du|^{s},  \quad x\in\Omega,\\
u(x)\ge 0,  \quad x\in\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain with a smooth boundary,
 $f(x)\in C(\Omega)$ is a positive function.

Solutions to \eqref{eq:1} will be understood in the weak (distributional) 
sense according to the following definition.

\begin{definition} \label{def2.1} \rm
A nonnegative function $u\in W^{1,p}_{\rm loc}(\Omega)$ will be called a weak 
(distributional) solution of  \eqref{eq:1} if 
$f(x)u^q |Du|^s\in L^1_{\rm loc}(\Omega)$ and for each nonnegative test function $\psi\in C_0^1(\Omega)$ it holds
\begin{equation}\label{eq:1a}
\int_{\Omega} |Du|^{p-2}(Du,D\psi)\,dx \ge \int_{\Omega} f(x)u^q |Du|^s\psi\,dx.
\end{equation}
\end{definition}

\begin{remark} \label{rmk2.1}\rm
 Similarly to \cite{MPb}, it can be shown that if such a solution exists and 
is strictly positive in $\Omega$, then \eqref{eq:1a} still holds for test 
functions of the form $\psi=u^{\gamma}\varphi$ with $\gamma\in\mathbb{R}$ and 
$\varphi\in C_0^1(\Omega)$. If $u$ vanishes somewhere in $\Omega$ and $\gamma<0$, 
one can use test functions $\psi=(u+\delta)^{\gamma}\varphi$ and take 
$\delta\to 0_+$, which yields the same results as in the previous case. 
Therefore we will assume in the sequel that $u>0$ whenever it exists.
\end{remark}


We use the notation $\rho(x)={\rm dist}(x,\partial\Omega)$, and
\[
\Omega_{k\eta}=\{x\in\Omega:\,\rho(x)\ge k\eta\} \quad (\eta>0,\;k=1,2).
\]

\begin{theorem} \label{thm2.1} 
Let $f(x)\ge c\rho^{-\alpha}(x)$ $(x\in\Omega)$ with some constant $c>0$, 
$p>1$, $q>p-1$, $s>0$, and $\alpha>q+1$.
Then problem \eqref{eq:1} has no nontrivial (distinct from a constant a.e.) 
weak solutions.
\end{theorem}


 For other definitions of a solution, the nonexistence condition can be different. 
In particular, for the so-called very weak solution in the semilinear case $p=2$, 
it becomes $\alpha>2$ (see, e.g., the survey \cite{DH}).


\begin{proof}[Proof of Theorem \ref{thm2.1}]
 Assume that there exists a nontrivial weak solution $u$ of inequality \eqref{eq:1}.
 Introduce a family of functions $\varphi_{\eta}\in C^1_{0}(\Omega;[0,1])$ of the 
form $\varphi_{\eta}(x)=\xi^{\lambda}_{\eta}(x)$ with
\begin{gather}\label{eq:4a}
\xi_{\eta}(x)=\begin{cases}
1, & x\in\Omega_{2\eta},\\
0, & x\not\in\Omega_{\eta},
\end{cases} \\
\label{eq:5a}
|D\xi_{\eta}(x)| \le c\eta^{-1} \quad (x\in\Omega)
\end{gather}
and $\lambda>0$ sufficiently large. Then, using a test function
 $\psi=u^{\gamma}\varphi_{\eta}$ with $1-p<\gamma<0$ in \eqref{eq:1a}, we obtain
\begin{align*}
&\int_{\Omega} f(x)u^{q+\gamma}|Du|^{s}\varphi_{\eta}\,dx \\
&\le\int_{\Omega}(|Du|^{p-2}Du, D(u^{\gamma}\varphi_{\eta}))\,dx\\
&=\gamma \int_{\Omega}u^{\gamma-1}|Du|^{p} \varphi_{\eta} \,dx
+ \int_{\Omega} u^{\gamma}|Du|^{p-2}(Du, D\varphi_{\eta})\,dx \\
&\le \gamma \int_{\Omega}u^{\gamma-1}|Du|^{p} \varphi_{\eta} \,dx
+ \int_{\Omega} u^{\gamma}|Du|^{p-1} |D\varphi_{\eta}|\,dx,
\end{align*}
whence 
\[
 \int_{\Omega} f(x)u^{q+\gamma}|Du|^{s}\varphi_{\eta}\,dx 
+ |\gamma| \int_{\Omega}u^{\gamma-1}|Du|^{p} \varphi_{\eta} \,dx 
\le \int_{\Omega} u^{\gamma}|Du|^{p-1} |D\varphi_{\eta}|\,dx.
\]
Representing the integrand on the right-hand side of  this inequality in the form
\[
 2^{-y/s} u^{\frac{(q+\gamma)y}{s}}|Du|^{y}f^{y/s}\varphi_{\eta}^{y/s} 
 2^{y/s} u^{\frac{\gamma s-(q+\gamma)y}{s}} |Du|^{p-1-y}|D\varphi_{\eta}|
 f^{-y/s}\varphi_{\eta}^{-y/s},
\]
where $y$ will be chosen below, and applying the parametric Young inequality 
with the exponent $s/y$, we obtain
\begin{align*}
 &\frac{1}{2}\int_{\Omega} f(x)u^{q+\gamma}|Du|^{s}\varphi_{\eta}\,dx 
+ |\gamma| \int_{\Omega}u^{\gamma-1}|Du|^{p} \varphi_{\eta} \,dx \\
&\le c\int_{\Omega} u^{\frac{\gamma s-(q+\gamma)y}{s-y}} 
|Du|^{\frac{(p-1-y)s}{s-y}}|D\varphi_{\eta}|^{\frac{s}{s-y}} 
f^{-\frac{y}{s-y}}\varphi_{\eta}^{-\frac{y}{s-y}}\,dx.
\end{align*}
Apply the Young inequality with the exponent $z$,
\begin{equation}\label{eq:5b}
\begin{aligned}
&c\int_{\Omega} u^{\frac{\gamma s-(q+\gamma)y}{s-y}} 
|Du|^{\frac{(p-1-y)s}{s-y}}|D\varphi_{\eta}|^{\frac{s}{s-y}}
 f^{-\frac{y}{s-y}}\varphi_{\eta}^{-\frac{y}{s-y}}\,dx  \\
&\le \frac{|\gamma|}{2}\int_{\Omega} u^{\frac{(\gamma s-(q+\gamma)y)z}{s-y}}
 |Du|^{\frac{(p-1-y)sz}{s-y}}\varphi_{\eta} \,dx \\
&\quad + c\int_{\Omega} |D\varphi_{\eta}|^{\frac{sz'}{s-y}} 
f^{-\frac{yz'}{s-y}}\varphi_{\eta}^{1-\frac{sz'}{s-y}}\,dx,
\end{aligned}
\end{equation}
where $\frac{1}{z}+\frac{1}{z'}=1$.

We choose $y$ and $z$ so that
\begin{gather*}
 (p-1-y)sz=p(s-y),\\
 \frac{\gamma s-(q+\gamma)y}{s-y} z=\gamma-1,
\end{gather*}
i.e.,
\begin{gather*}
 y=y_{\gamma}=\frac{s(p+\gamma-1)}{p(q+\gamma)-s(\gamma-1)}, \\
 z=z_{\gamma}=\frac{p[p(q+\gamma)-s(\gamma-1)-(p+\gamma-1)]}{(p-1)(p(q+\gamma)
-s(\gamma-1))-s(p+\gamma-1)}.
\end{gather*}
Note that for $\gamma=0$, by our assumptions $q>p-1>0$ and $s>0$, we have
\begin{gather*}
\frac{s}{y_0}=\frac{pq+s}{p-1}>\frac{pq+s}{q}=p+\frac{s}{q}>p>1,\\
z_0=\frac{p(q-1)+s+1}{(p-1)q}=1+\frac{q-(p-1)+s}{p(q-1)}>1.
\end{gather*}
Hence by continuity, for $|\gamma|$ sufficiently small, one has 
$\frac{s}{y_{\gamma}}>1$ and $z_{\gamma}>1$, as required for applying the 
Young inequality.

For such $y$ and $z$, and $\varphi_{\eta}$ with properties 
\eqref{eq:4a}, \eqref{eq:5a} and sufficiently large $\lambda>0$, \eqref{eq:5b} 
implies
\begin{equation}\label{eq:5c}
\begin{aligned}
&\frac{1}{2}\int_{\Omega} f(x)u^{q+\gamma}|Du|^{s}\varphi_{\eta}\,dx 
+ \frac{|\gamma|}{2} \int_{\Omega}u^{\gamma-1}|Du|^{p} \varphi_{\eta} \,dx \\
&\le c\eta^{\frac{\alpha(p+\gamma-1)-p(q+\gamma)+s\gamma+q-p+1}{q+s-p+1}}.
\end{aligned}
\end{equation}
Taking $\eta\to 0_+$, for sufficiently small $\gamma<0$ we obtain a contradiction
to the assumed non-triviality of $u$, which proves the theorem.
\end{proof}

Similar arguments yield an analogous result for the problem with variable exponents
\begin{equation}\label{eq:1x}
\begin{gathered}
-\operatorname{div}(|Du|^{p(x)-2}Du) \ge f(x)u^{q(x)}|Du|^{s(x)}, 
 \quad x\in\Omega,\\
u(x)\ge 0,  \quad x\in\Omega,
\end{gathered}
\end{equation}
where $p(x), q(x), s(x), f(x)\in C(\Omega)$ are appropriate positive functions. 
This problem will be considered in detail in future article.

\section{Systems of inequalities}

In this section we consider the system of inequalities
\begin{equation}\label{eq:3.7.1}
\begin{gathered}
-\operatorname{div}(|Du|^{p-2}Du) \ge f(x)v^{q_1}|Dv|^{q_2},  \quad x\in\Omega,\\
-\operatorname{div}(|Dv|^{q-2}Dv)  \ge g(x)u^{p_1}|Du|^{p_2},  \quad x\in\Omega,\\
u,\,v\ge 0,  \quad x\in\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain with a smooth boundary.

We assume that $p,q>1$, and $f,g \in C(\Omega)$ are positive functions 
such that $f(x) \ge a_0\rho^{-\alpha}(x)$, $g(x) \ge b_0\rho^{-\beta}(x)$ 
for $x \in \Omega$, where $a_0,b_0>0$.

The solutions of  \eqref{eq:3.7.1} will be understood in the weak (distributional) 
sense according to the following definition.

\begin{definition} \label{def3.1}\rm
 A pair of nonnegative functions 
$(u,v)\in W^{1,p}_{\rm loc}(\Omega)\cap W^{1,q}_{\rm loc}(\Omega)$
 are called a weak (distributional) solution of  \eqref{eq:3.7.1} 
if $f(x)v^{q_1}|Dv|^{q_2}\in L^1_{\rm loc}(\Omega)$, 
$g(x)u^{p_1}|Dv|^{p_2}\in L^1_{\rm loc}(\Omega)$, 
and for any nonnegative test functions $\psi_1,\psi_2\in C_0^1(\Omega)$ it holds
\begin{equation}\label{eq:3.7.1a}
\begin{gathered}
 \int_{\Omega} |Du|^{p-2}(Du,D\psi_1)\,dx 
 \ge \int_{\Omega} f(x)v^{q_1} |Dv|^{q_2}\psi_1\,dx,\\
 \int_{\Omega} |Dv|^{q-2}(Dv,D\psi_2)\,dx 
 \ge \int_{\Omega} g(x)u^{p_1} |Du|^{p_2}\psi_2\,dx.
\end{gathered}
\end{equation}
\end{definition}


Similarly to Remark \ref{rmk2.1}, we can assume that $u>0$ and $v>0$ whenever they exist, 
and use test functions of the form $\psi_1=u^{\gamma}\varphi$ and 
$\psi_2=v^{\gamma}\varphi$ with $\varphi\in C_0^1(\Omega)$.

\begin{theorem} \label{thm3.1} 
Let $p_1+p_2>p-1$, $q_1+q_2>q-1$ and either
\begin{equation}\label{eq:3.7.19a}
(\beta-1-p_1)(q_1+q_2)+(\alpha-1-q_1)(q-1)>0
\end{equation}
or
\begin{equation}\label{eq:3.7.19b}
(\alpha-1-q_1)(p_1+p_2)+(\beta-1-p_1)(p-1)>0.
\end{equation}
Then problem \eqref{eq:3.7.1} has no nontrivial solutions.
\end{theorem}

\begin{proof} 
Let $(u,v)$ be a nontrivial solution of system \eqref{eq:3.7.1}, and 
$\varphi_{\eta}\in C^{\infty}_{0}(\Omega;[0,1])$ be functions of the same 
form as in the proof of Theorem \ref{thm2.1}, which satisfy \eqref{eq:4a} and \eqref{eq:5a}.

Using a test function $\psi_1=u^{\gamma}\varphi_{\eta}$ in  
 \eqref{eq:3.7.1a}, and $\psi_2=v^{\gamma}\varphi_{\eta}$ in \eqref{eq:3.7.1a},
 where $\gamma$ is a number such that $p_1+p_2-p+1<\gamma<0$, 
$q_1+q_2-q+1<\gamma<0$, we obtain
\begin{gather}\label{eq:3.7.4x}
\int fv^{q_1}|Dv|^{q_2}u^{\gamma}\varphi_{\eta}\,dx 
\le \gamma\int u^{\gamma-1}|Du|^{p}\varphi_{\eta}\,dx 
 + \int u^{\gamma}|Du|^{p-1}|D\varphi_{\eta}| \,dx, \\
\label{eq:3.7.5x}
\int gu^{p_1}|Du|^{p_2}v^{\gamma} \varphi_{\eta} \,dx 
\le \gamma\int v^{\gamma-1}|Dv|^{q}\varphi_{\eta} \,dx
 + \int v^{\gamma}|Dv|^{q-1}|D\varphi_{\eta}|\,dx.
\end{gather}
We  use the representations
\begin{gather}\label{eq:3.7.16}
u^{\gamma}|Du|^{p-1}  
= u^{a_1} |Du|^{b_1} \varphi_{\eta}^{\frac{1}{c_1}} u^{\gamma-a_1} |Du|^{p-1-b_1} 
\varphi_{\eta}^{-\frac{1}{c_1}}, \\
\label{eq:3.7.17}
v^{\gamma}|Dv|^{q-1}  = v^{a_2} |Du|^{b_2} \varphi_{\eta}^{\frac{1}{c_2}}
 v^{\gamma-a_2} |Dv|^{q-1-b_2} \varphi_{\eta}^{-\frac{1}{c_2}},
\end{gather}
to apply to the right-hand sides of \eqref{eq:3.7.4x} and \eqref{eq:3.7.5x} 
the parametric Young inequality with exponents denoted by $c_1$ and $c_2$, 
respectively. We choose the parameters so that
\begin{equation}\label{eq:3.7.12}
\begin{gathered}
a_1 c_1 = \gamma-1,\quad 
b_1 c_1 = p,\\
 \frac{\gamma-a_1}{p-1-b_1}=\frac{p_1}{p_2},
\end{gathered}
\end{equation}
and
\begin{equation}\label{eq:3.7.13}
\begin{gathered}
a_2 c_2 = \gamma-1,\quad 
b_2 c_2 = q,\\
 \frac{\gamma-a_2}{q-1-b_2}=\frac{q_1}{q_2}.
\end{gathered}
\end{equation}
The purpose of this choice consists in preparation to the consequent 
application of the H\"older inequality, in order to obtain, under a 
suitable choice of the parameters, $\int bu^{p_1}|Du|^{p_2} \varphi_{\eta} \,dx$ 
and $ \int av^{q_1}|Dv|^{q_2} \varphi_{\eta} \,d x$.

Solving the systems of equations \eqref{eq:3.7.12} and \eqref{eq:3.7.13}, 
we obtain
\begin{equation}\label{eq:3.7.14}
\begin{gathered}
 a_1 = \frac{(\gamma-1)((p-1)p_1-\gamma p_2)}{pp_1+p_2(1-\gamma)},\\
 b_1 = \frac{p((p-1)p_1-\gamma p_2)}{pp_1+p_2(1-\gamma)},\\
 c_1 = \frac{pp_1+p_2(1-\gamma)}{(p-1)p_1-\gamma p_2},
\end{gathered}
\end{equation}
and
\begin{equation}\label{eq:3.7.15}
\begin{gathered}
 a_2 = \frac{(\gamma-1)((q-1)q_1-\gamma q_2)}{qq_1+q_2(1-\gamma)},\\
 b_2 = \frac{q((q-1)q_1-\gamma q_2)}{qq_1+q_2(1-\gamma)},\\
 c_2 = \frac{qq_1+q_2(1-\gamma)}{(q-1)q_1-\gamma q_2}.
\end{gathered}
\end{equation}
Substituting \eqref{eq:3.7.14} and \eqref{eq:3.7.15} in \eqref{eq:3.7.16} 
and \eqref{eq:3.7.17}, we have the representations
\begin{gather*}
\begin{aligned}
u^{\gamma}|Du|^{p-1} 
&  = u^{\frac{(\gamma-1)((p-1)p_1-\gamma p_2)}{pp_1+p_2(1-\gamma)}} 
 |Du|^{\frac{p((p-1)p_1-\gamma p_2)}{pp_1+p_2(1-\gamma)}} 
 \varphi_{\eta}^{\frac{(p-1)p_1-\gamma p_2}{pp_1+p_2(1-\gamma)}}  \\
& \quad \times u^{\frac{p_1(p+\gamma-1)}{pp_1+p_2(1-\gamma)}} 
 |Du|^{\frac{p_2(p+\gamma-1)}{pp_1+p_2(1-\gamma)}}
  \varphi_{\eta}^{-\frac{(p-1)p_1-\gamma p_2}{pp_1+p_2(1-\gamma)}},
\end{aligned} 
\\
\begin{aligned}
v^{\gamma} |Dv|^{q-1} 
& =  v^{\frac{(\gamma-1)((q-1)q_1-\gamma q_2)}{qq_1+q_2(1-\gamma)}} 
|Dv|^{\frac{q((q-1)q_1-\gamma q_2)}{qq_1+q_2(1-\gamma)}} 
 \varphi_{\eta}^{\frac{(q-1)q_1-\gamma q_2}{qq_1+q_2(1-\gamma)}}  \\
& \quad \times v^{\frac{q_1(q+\gamma-1)}{qq_1+q_2(1-\gamma)}} 
|Dv|^{\frac{q_2(q+\gamma-1)}{qq_1+q_2(1-\gamma)}} \varphi_{\eta}^{-\frac{(q-1)q_1
 -\gamma q_2}{qq_1+q_2(1-\gamma)}}.
\end{aligned}
\end{gather*}
Note that for $\gamma=0$ we have
\[
c_1=\frac{qq_1+q_2}{(q-1)q_1}>\frac{(q-1)q_1+q_2}{(q-1)q_1}
=1+\frac{q_2}{(q-1)q_1}>1
\]
and similarly $c_2>1$. Hence the same inequalities $c_1>1$ and $c_2>1$ hold by 
continuity for $|\gamma|$ sufficiently small. Thus, applying to the right-hand 
sides of \eqref{eq:3.7.4x} and \eqref{eq:3.7.5x} the parametric Young 
inequality with the exponents $c_1$ and $c_2$ from \eqref{eq:3.7.14} 
and \eqref{eq:3.7.15} respectively, we arrive at
\begin{gather*}
\begin{aligned}
&\int fv^{q_1}|Dv|^{q_2}u^{\gamma} \varphi_{\eta}\,dx 
 +\frac{|\gamma|}{2}\int u^{\gamma-1}|Du|^{p}\varphi_{\eta}\,dx \\
&\le c_{\gamma}\int u^{\frac{p_1(p+\gamma-1)}{p_1+p_2}}
 |Du|^{\frac{p_2(p+\gamma-1)}{p_1+p_2}}
\frac{|D\varphi_{\eta}|^{\frac{pp_1+p_2(1-\gamma)}{p_1+p_2}}}
 {\varphi_{\eta}^{\frac{pp_1+p_2(1-\gamma)}{p_1+p_2}-1}}\,dx,
\end{aligned}\\
\begin{aligned}
&\int gu^{p_1}|Du|^{p_2}v^{\gamma}\varphi_{\eta} \,dx 
+\frac{|\gamma|}{2}\int v^{\gamma-1}|Dv|^{q}\varphi_{\eta}\,dx\\
&\le d_{\gamma}\int v^{\frac{q_1(q+\gamma-1)}{q_1+q_2}}|Dv|
 ^{\frac{q_2(q+\gamma-1)}{q_1+q_2}}
\frac{|D\varphi_{\eta}|^{\frac{qq_1+q_2(1-\gamma)}{q_1+q_2}}}
 {\varphi_{\eta}^{\frac{qq_1+q_2(1-\gamma)}{q_1+q_2}-1}}\,dx,
\end{aligned}
\end{gather*}
where the constants $c_{\gamma}$ and $d_{\gamma}$ depend only on 
$p,q,p_1,q_1,p_2,q_2$ and $\gamma$. 
Applying the H\"older inequality 
with the exponents
\begin{gather*}
d_1=\frac{p_1+p_2}{p+\gamma-1},\quad d'_1=\frac{p_1+p_2}{p_1+p_2-p-\gamma+1},\\
d_2=\frac{q_1+q_2}{q+\gamma-1},\quad d'_2=\frac{q_1+q_2}{q_1+q_2-q-\gamma+1}
\end{gather*}
respectively (note that under our assumptions for $\gamma=0$
\[
d_1=\frac{p_1+p_2}{p-1}>1,\quad d_2=\frac{q_1+q_2}{q-1}>1
\]
and hence by continuity $d_1>1$ and $d_2>1$ for any $|\gamma|$ sufficiently small), 
we obtain
\begin{gather}\label{eq:3.7.4}
\begin{aligned}
&\int f v^{q_1}|Dv|^{q_2}u^{\gamma}\varphi_{\eta}\,dx +
\frac{|\gamma|}{2}\int u^{\gamma-1}|Du|^{p}\varphi_{\eta}\,dx\\
&\le c_{\gamma}\Big(\int g u^{p_1}|Du|^{p_2}\varphi_{\eta}\,dx
 \Big)^{\frac{p+\gamma-1}{p_1+p_2}} \\
&\quad\times \Big(\int g^{-\frac{p+\gamma-1}{p_1+p_2-p-\gamma+1}} 
\frac{|D\varphi_{\eta}|^{\frac{pp_1+p_2(1-\gamma)}{p_1+p_2-p-\gamma+1}}}{\varphi_{\eta}^{\frac{pp_1+p_2(1-\gamma)}
{p_1+p_2-p-\gamma+1}-1}}\,dx\Big)^{\frac{p_1+p_2-p-\gamma+1}{p_1+p_2}},
\end{aligned}\\
\label{eq:3.7.5}
\begin{aligned}
&\int g u^{p_1}|Du|^{p_2}v^{\gamma} \varphi_{\eta}\,dx +
\frac{|\gamma|}{2}\int v^{\gamma-1}|Dv|^{q}\varphi_{\eta}\,dx \\
&\le d_{\gamma}\Big(\int f v^{q_1}|Dv|^{q_2}\varphi_{\eta}\,dx
 \Big)^{\frac{q+\gamma-1}{q_1+q_2}} \\
&\quad\times \Big(\int f^{-\frac{q+\gamma-1}{q_1+q_2-q-\gamma+1}} 
\frac{|D\varphi_{\eta}|^{\frac{qq_1+q_2(1-\gamma)}{q_1+q_2-
q-\gamma+1}}}{\varphi_{\eta}^{\frac{qq_1+q_2(1-\gamma)}{q_1+q_2-q-\gamma+1}-1}}
\,dx\Big)^{\frac{q_1+q_2-q-\gamma+1}{q_1+q_2}}.
\end{aligned}
\end{gather}
Further, using test functions $\psi_1=\psi_2=\varphi_{\eta}$ in \eqref{eq:3.7.1a},
 we obtain
\begin{gather}\label{eq:3.7.20}
\int av^{q_1}|Dv|^{q_2}\varphi_{\eta} \,dx 
\le  \int |Du|^{p-1}|D\varphi_{\eta}| \,dx\,, \\
\label{eq:3.7.21}
\int bu^{p_1}|Du|^{p_2}\varphi_{\eta}\,dx \le \int |Dv|^{q-1}|D\varphi_{\eta}| \,dx\,.
\end{gather}
We use the representation
\begin{gather}\label{eq:3.7.18}
|Du|^{p-1}  = u^{a_3} |Du|^{b_3} \varphi_{\eta}^{\frac{1}{c_3}} u^{-a_3} 
|Du|^{p-1-b_3} (g\varphi_{\eta})^{\frac{1}{d_3}} g^{-\frac{1}{d_3}} 
\varphi_{\eta}^{-\frac{1}{c_3}-\frac{1}{d_3}}, \\
\label{eq:3.7.19}
|Dv|^{q-1}  = v^{a_4} |Dv|^{b_4} \varphi_{\eta}^{\frac{1}{c_4}} v^{-a_4} 
|Dv|^{q-1-b_4} (a\varphi_{\eta})^{\frac{1}{d_4}} f^{-\frac{1}{d_4}} 
\varphi_{\eta}^{-\frac{1}{c_4}-\frac{1}{d_4}},
\end{gather}
for applying to the right-hand sides of \eqref{eq:3.7.20} and \eqref{eq:3.7.21} 
the triple Young inequality, with the exponents  $c_3$, $d_3$, $e_3$ and $c_4$, 
$d_4$, $e_4$ respectively. Here we choose the parameters so that
\begin{equation}\label{eq:3.7.22}
\begin{gathered}
a_3 c_3 = \gamma-1,\quad 
b_3 c_3 = p,\quad
a_3 d_3 = -p_1,\\
(p-1-b_3) d_3 = p_2,\quad
\frac{1}{c_3}+\frac{1}{d_3}+\frac{1}{e_3}=1,
\end{gathered}
\end{equation}
and
\begin{equation}\label{eq:3.7.23}
\begin{gathered}
a_4 c_4 = \gamma-1,\quad
b_4 c_4 = q,\quad
a_4 d_4 = -q_1,\\
(q-1-b_4) d_4 = q_2,\quad 
 {\frac{1}{c_4}+\frac{1}{d_4}+\frac{1}{e_4}=1.}
\end{gathered}
\end{equation}
Solving the systems of equations \eqref{eq:3.7.22} and \eqref{eq:3.7.23}, we obtain
\begin{equation}\label{eq:3.7.24}
\begin{gathered}
 a_3 = \frac{(\gamma-1)p_1(p-1)}{pp_1+p_2(1-\gamma)},\\
 b_3 = \frac{p p_1(p-1)}{pp_1+p_2(1-\gamma)},\\
 c_3 = \frac{pp_1+p_2(1-\gamma)}{p_1(p-1)},\\
 d_3 = \frac{pp_1+p_2(1-\gamma)}{(p-1)(1-\gamma)},\\
 e_3 = \frac{pp_1+p_2(1-\gamma)}{p_1+(p_2-p+1)(1-\gamma)},
\end{gathered}
\end{equation}
and
\begin{equation}\label{eq:3.7.25}
\begin{gathered}
a_4 = \frac{(\gamma-1)q_1(q-1)}{qq_1+q_2(1-\gamma)},\\
b_4 = \frac{qq_1(q-1)}{qq_1+q_2(1-\gamma)},\\
c_4 = \frac{qq_1+q_2(1-\gamma)}{q_1(q-1)},\\
d_4 = \frac{qq_1+q_2(1-\gamma)}{(q-1)(1-\gamma)},\\
e_4 = \frac{qq_1+q_2(1-\gamma)}{q_1+(q_2-q+1)(1-\gamma)}.
\end{gathered}
\end{equation}
Note that for $\gamma=0$ one has
\begin{gather*}
 c_3 = \frac{pp_1+p_2}{p_1(p-1)}=\frac{p_1(p-1)+p_1+p_2}{p_1(p-1)}
 =1+\frac{p_1+p_2}{p_1(p-1)}>1,\\
 d_3 = \frac{pp_1+p_2}{p-1}=\frac{p_1(p-1)+p_1+p_2}{p-1}
 =p_1+\frac{p_1+p_2}{p-1}>p_1>1,\\
 e_3 = \frac{pp_1+p_2}{p_1+p_2-p+1}>\frac{p_1+p_2}{p_1+p_2-p+1}>1,
\end{gather*}
and similar estimates for $c_4,d_4,e_4$. 
Then it follows by continuity that for $|\gamma|$ sufficiently small 
all these exponents also exceed 1, similarly to the previous arguments.

Substituting \eqref{eq:3.7.24} and \eqref{eq:3.7.25} in \eqref{eq:3.7.18} and 
\eqref{eq:3.7.19}, we have the representations
\begin{gather*}
\begin{aligned}
|Du|^{p-1} 
& =  u^{\frac{(\gamma-1)p_1(p-1)}{pp_1+p_2(1-\gamma)}}|Du|^{\frac{p p_1(p-1)}
 {pp_1+p_2(1-\gamma)}}
\varphi_{\eta}^{\frac{p_1(p-1)}{pp_1+p_2(1-\gamma)}} \\
& \quad \times u^{\frac{p_1(p-1)(1-\gamma)}{pp_1+p_2(1-\gamma)}} 
 |Du|^{\frac{p_2(p-1)(1-\gamma)}{pp_1+p_2(1-\gamma)}} 
 (b\varphi_{\eta})^{\frac{(p-1)(1-\gamma)}{pp_1+p_2(1-\gamma)}} \\
&\quad \times g^{-\frac{(p-1)(1-\gamma)}{pp_1+p_2(1-\gamma)}}
 \varphi_{\eta}^{\frac{(\gamma-p_1-1)(p-1)}{pp_1+p_2(1-\gamma)}},
\end{aligned}\\
\begin{aligned}
|Dv|^{q-1} 
&  =  v^{\frac{(\gamma-1)q_1(q-1)}{qq_1+q_2(1-\gamma)}}|Dv|^{\frac{qq_1(q-1)}
 {qq_1+q_2(1-\gamma)}}
\varphi_{\eta}^{\frac{q_1(q-1)}{qq_1+q_2(1-\gamma)}}  \\
& \quad \times v^{\frac{q_1(q-1)(1-\gamma)}{qq_1+q_2(1-\gamma)}} 
 |Dv|^{\frac{q_2(q-1)(1-\gamma)}{qq_1+q_2(1-\gamma)}} (b\varphi_{\eta}
 )^{\frac{(q-1)(1-\gamma)}{qq_1+q_2(1-\gamma)}} \\
& \quad \times g^{-\frac{(q-1)(1-\gamma)}{qq_1+q_2(1-\gamma)}}
 \varphi_{\eta}^{\frac{(\gamma-q_1-1)(q-1)}{qq_1+q_2(1-\gamma)}}\,.
\end{aligned}
\end{gather*}
Applying to the right-hand sides of \eqref{eq:3.7.20} and \eqref{eq:3.7.21} 
the triple Young inequality with the exponents 
$c_3$, $d_3$, $e_3$, $c_4$, $d_4$, $e_4$ from \eqref{eq:3.7.24}, \eqref{eq:3.7.25} 
respectively, we arrive at
\begin{equation}\label{eq:3.7.6}
\begin{aligned}
&\int fv^{q_1}|Dv|^{q_2}\varphi_{\eta} \,dx \\
&\le  \Big(\int
u^{\gamma-1}|Du|^{p}\varphi_{\eta} \,dx\Big)^{\frac{p_1(p-1)}{pp_1+p_2(1-\gamma)}} \\
&\quad \times \Big(\int gu^{p_1}|Du|^{p_2}\varphi_{\eta} \,dx
 \Big)^{\frac{(p-1)(1-\gamma)}{pp_1+p_2(1-\gamma)}}  \\
&\quad \Big(\int g^{-\frac{(p-1)(1-\gamma)}{p_1+(p_2-p+1)(1-\gamma)}}
 \frac{|D\varphi_{\eta}|^{\frac{pp_1+p_2(1-\gamma)}{p_1+(p_2-p+1)(1-\gamma)}}}
{\varphi_{\eta}^{\frac{pp_1+p_2(1-\gamma)}{p_1+(p_2-p+1)(1-\gamma)}-1}}\,dx 
\Big)^{\frac{p_1+(p_2-p+1)(1-\gamma)}{pp_1+p_2(1-\gamma)}}\,,
\end{aligned}
\end{equation}
\begin{equation}\label{eq:3.7.7}
\begin{aligned}
&\int gu^{p_1}|Du|^{p_2} \varphi_{\eta} \,dx \\
&\le  \Big(\int v^{\gamma-1}|Dv|^{q}\varphi_{\eta} \,dx
 \Big)^{\frac{q_1(q-1)}{qq_1+q_2(1-\gamma)}} \\
&\quad \times \Big(\int fv^{q_1}|Dv|^{q_2}\varphi_{\eta} \,dx
 \Big)^{\frac{(q-1)(1-\gamma)}{qq_1+q_2(1-\gamma)}}  \\
&\quad \times \Big(\int g^{-\frac{(q-1)(1-\gamma)}{q_1+(q_2-q+1)(1-\gamma)}} 
 \frac{|D\varphi_{\eta}|^{\frac{qq_1+q_2(1-\gamma)}{q_1+(q_2-q+1)(1-\gamma)}}}
{\varphi_{\eta}^{\frac{qq_1+q_2(1-\gamma)}{q_1+(q_2-q+1)(1-\gamma)}-1}}\,dx 
 \Big)^{\frac{q_1+(q_2-q+1)(1-\gamma)}{qq_1+q_2(1-\gamma)}}\,.
\end{aligned}
\end{equation}

Using \eqref{eq:3.7.4} and \eqref{eq:3.7.5}, from the previous estimates we derive
\begin{equation}\label{eq:3.7.8}
\begin{aligned}
&\int fv^{q_1}|Dv|^{q_2}\varphi_{\eta}\,dx \\
&\le D_{\gamma} \Big(\int gu^{p_1}|Du|^{p_2}\varphi_{\eta} \,dx 
 \Big)^{\frac{p_1(p-1)(p+\gamma-1)+(p_1+p_2)(p-1)(1-\gamma)}{(pp_1+p_2(1-\gamma))
 (p_1+p_2)}}  \\
&\quad \times \Big(\int \frac{g^{-\frac{p+\gamma-1}{p_1+p_2-p-\gamma+1}} 
 |D\varphi_{\eta}|^{\frac{pp_1+p_2(1-\gamma)}{p_1+p_2-p-\gamma+1}}}
 {\varphi_{\eta}^{\frac{pp_1+p_2(1-\gamma)}{p_1+p_2-p-\gamma+1}-1}}
\,dx\Big)^{\frac{p_1(p-1)(p_1+p_2-p-\gamma+1)}{(pp_1+p_2(1-\gamma))
 (p_1+p_2)}} \\
&\quad \times \Big(\int g^{-\frac{(p-1)(1-\gamma)}{p_1+(p_2-p+1)(1-\gamma)}} 
 \frac{|D\varphi_{\eta}|^{\frac{pp_1+p_2(1-\gamma)}{p_1+(p_2-p+1)(1-\gamma)}}}
{\varphi_{\eta}^{\frac{pp_1+p_2(1-\gamma)}{p_1+(p_2-p+1)(1-\gamma)}-1}}\,dx 
 \Big)^{\frac{p_1+(p_2-p+1)(1-\gamma)}{pp_1+p_2(1-\gamma)}}\,,
\end{aligned}
\end{equation}
\begin{equation}\label{eq:3.7.9}
\begin{aligned}
&\int gu^{p_1}|Du|^{p_2}\varphi_{\eta}\,dx \\
&\le E_{\gamma}\Big(\int fv^{q_1}|Dv|^{q_2}\varphi_{\eta} \,dx 
 \Big)^{\frac{q_1(q-1)(q+\gamma-1)+(q_1+q_2)(q-1)(1-\gamma)}{(qq_1+q_2(1-\gamma))
 (q_1+q_2)}}  \\
&\quad \times \Big(\int \frac{g^{-\frac{q+\gamma-1}{q_1+q_2-q-\gamma+1}} 
 |D\varphi_{\eta}|^{\frac{qq_1+q_2(1-\gamma)}{q_1+q_2-q-\gamma+1}}}
 {\varphi_{\eta}^{\frac{qq_1+q_2(1-\gamma)}{q_1+q_2-q-\gamma+1}-1}}
 \,dx\Big)^{\frac{q_1(q-1)(q_1+q_2-q-\gamma+1)}{(qq_1+q_2(1-\gamma))(q_1+q_2)}} \\
&\quad \times \Big(\int g^{-\frac{(q-1)(1-\gamma)}{q_1+(q_2-q+1)(1-\gamma)}} 
 \frac{|D\varphi_{\eta}|^{\frac{qq_1+q_2(1-\gamma)}{q_1+(q_2-q+1)(1-\gamma)}}}
 {\varphi_{\eta}^{\frac{qq_1+q_2(1-\gamma)}{q_1+(q_2-q+1)(1-\gamma)}-1}}\,dx
  \Big)^{\frac{q_1+(q_2-q+1)(1-\gamma)}{qq_1+q_2(1-\gamma)}}\,,
\end{aligned}
\end{equation}
where $D_{\gamma}$ and $E_{\gamma}>0$ depend only on $p,q,p_1,q_1,p_2,q_2$ and 
$\gamma$.

Then by \eqref{eq:4a} and \eqref{eq:5a} we have
\begin{equation}\label{eq:3.7.10}
\int fv^{q_1}|Dv|^{q_2}\varphi_{\eta}\,dx \le c\left(\int gu^{p_1}|Du|^{p_2}\varphi_{\eta} \,dx \right)^{\mu_1} \eta^{\nu_1},
\end{equation}
\begin{equation}\label{eq:3.7.11}
\int gu^{p_1}|Du|^{p_2}\varphi_{\eta}\,dx 
\le c\Big(\int fv^{q_1}|Dv|^{q_2}\varphi_{\eta} \,dx \Big)^{\mu_2} \eta^{\nu_2},
\end{equation}
where after simplifying the obtained expressions one gets
\begin{gather*}
 \mu_1=\frac{p-1}{p_1+p_2}, \quad   \nu_1=\frac{(\beta-1-p_1)(p-1)}{p_1+p_2},\\
 \mu_2=\frac{q-1}{q_1+q_2}, \quad   \nu_2=\frac{(\alpha-1-q_1)(q-1)}{q_1+q_2}.
\end{gather*}
Substituting \eqref{eq:3.7.10} in \eqref{eq:3.7.11} and vice versa, we obtain
\begin{gather*}
\int fv^{q_1}|Dv|^{q_2}\varphi_{\eta}\,dx 
\le c\eta^{\frac{[(\beta-1-p_1)(q_1+q_2)+(\alpha-1-q_1)(q-1)](p-1)}
{(p_1+p_2)(q_1+q_2)-(p-1)(q-1)}}, \\
\int gu^{p_1}|Du|^{p_2}\varphi_{\eta}\,dx 
\le c\eta^{\frac{[(\alpha-1-q_1)(p_1+p_2)+(\beta-1-p_1)(p-1)](q-1)}
{(p_1+p_2)(q_1+q_2)-(p-1)(q-1)}}.
\end{gather*}
Passing to the limit as $\eta\to 0_+$, due to \eqref{eq:3.7.19a} 
and \eqref{eq:3.7.19b} we obtain a contradiction, which completes the proof.
\end{proof}


Similar necessary conditions for existence of solutions can be formulated 
for higher order equations and systems \cite{D}, \cite{HMV}, as well as 
for systems of quasilinear elliptic inequalities with variable exponents. 
We leave the latter subject for a future article.

\subsection*{Acknowledgements} 
The second and third author were partially supported by Russian Foundation 
for Basic Research (grant No. 14-01-00736).


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\end{document}