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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 216, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/216\hfil Blow-up of solutions]
{Blow-up of solutions to systems of nonlinear inequalities 
 with singularities on unbounded sets}

\author[E. Galakhov, O. Salieva, L. Uvarova \hfil EJDE-2014/216\hfilneg]
{Evgeny Galakhov, Olga Salieva, Liudmila Uvarova}  % in alphabetical order

\address{Evgeny Galakhov \newline
Peoples' Friendship University of Russia,
Miklukho-Maklaya str. 6,
Moscow, 117198, Russia}
\email{galakhov@rambler.ru}

\address{Olga Salieva \newline
Moscow State Technological University ``Stankin'',
Vadkovsky lane 3a,
Moscow, 125994, Russia}
\email{olga.a.salieva@gmail.com}

\address{Liudmila Uvarova \newline
Moscow State Technological University ``Stankin'',
Vadkovsky lane 3a,
Moscow, 125994, Russia}
\email{uvar11@yandex.ru}

\thanks{Submitted September 14, 2014. Published October 14, 2014.}
\subjclass[2000]{35J47, 35J48, 35J60}
\keywords{System of nonlinear differential inequalities;  blow-up;
 solvability}

\begin{abstract}
  We establish conditions for the blow-up of solutions to several 
  systems of nonlinear differential inequalities, with singularities
  on unbounded sets.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

In recent years, many mathematicians study global solvability of nonlinear
partial differential equations and inequalities with singular coefficients.
Here nonlinear terms can depend both on the values of the unknown function
and on its derivatives. This problem is not only interesting in its own right
but also has important mathematical and physical applications.
Thus, Liouville type theorems on nonexistence of positive solutions to nonlinear
equations in the whole space or in the half-space can be used to obtain a
priori estimates of solutions to respective problems in bounded domains
\cite{ACM, CMM}. On the other hand, the class under consideration includes,
 in particular, Hamilton--Jacobi and Korteweg--de Vries equations that play
 an important role in contemporary mathematical physics \cite{P2,P3}.

In \cite{DAM}--\cite{S} and their references, sharp necessary
conditions for existence of global solutions to different classes of second-order
elliptic equations with gradient terms were obtained. The proofs are based
either on ODE techniques for radially symmetric solutions or on the nonlinear
capacity method, which was suggested in \cite{P1} and extensively developed
in \cite{MP}.

In this article we obtain sufficient conditions for nonexistence of solutions
for several classes of systems of inequalities that have singular coefficients
on unbounded sets, such as straight lines and planes, as well as smooth
curves and surfaces in $\mathbb{R}^N$. Here we obtain nonexistence results
in natural functional classes, that is, assuming only minimal local
integrability properties, unlike our previous paper \cite{GS} where lower
bounds for local integrals were required. Some further related results for
scalar inequalities were included into \cite{GS2}.

The main results of the paper are formulated in section 2.
In section 3, a nonexistence theorem is proven for a system of higher-order
elliptic inequalities  with a nonlinearity dependent only on the values of
the unknown function $u$, and in section 4, a similar result is obtained
for a system of inequalities with a nonlinearity containing $|Du|$.
In sections 5 and 6, the respective statements are extended to systems
of second-order elliptic inequalities with a nonlinear principal part.

\section{Main results}

We assume that the set $S$ satisfies a geometrical condition which
is based on an idea from our paper \cite{GS}, and modified according to our
problem setting and functional classes under consideration.
To formulate it, we need some extra notation.

Let $\varepsilon>0$ and $x\in\mathbb{R}^N$. Denote $\rho(x)=\operatorname{dist}(x,S)$,
\[
B_{\varepsilon}(0)=\{x\in\mathbb{R}^N:\,|x|\le\varepsilon\},
\]
and
\[
S_{\varepsilon}=\{x\in\mathbb{R}^N:\,\rho(x)<\varepsilon\}.
\]
For $R>0$, introduce the set
\[
S^R=\overline{S_R\setminus S_{1/R}}\cap \overline{B_R(0)}.
\]
Now we can formulate our assumption on $S$.

\begin{itemize}
\item[(H1)] Suppose that there exists a family of functions
$\xi_R\in C^{2k}_0(\mathbb{R}^N\setminus S;[0,1])$ such that
\begin{equation} \label{e0}
\xi_R(x)=\begin{cases}
0 & (x\in S_{1/(2R)}\cup (\mathbb{R}^N\setminus S_{2R})),\\
1 & (x\in S_{R}\setminus S_{1/R})
\end{cases}
\end{equation}
and there exists a constant $c>0$ such that
\begin{equation} \label{e7}
|D^{\alpha} \xi_R(x)|\le c\rho^{-|\alpha|} \quad (x\in\mathbb{R}^N).
\end{equation}
\end{itemize}

\begin{example} \label{examp1}\rm
 We can consider as the set $S$ a hyperplane
$S=\Pi_n=\{x=(x_1,\dots,x_n)\in\mathbb{R}^N \colon x_n=0\}$.
In that case we can choose $\xi_R(x)=\tilde\xi_R(x_n)$, where
\[
\tilde\xi_R(x_n)=\begin{cases}
0 & (|x_n|\le \frac{1}{2R} \mbox{ or } |x_n|\ge 2R),\\
1 & (\frac{1}{R} \le |x_n| \le R)
\end{cases}
\]
See Figure \ref{fig1}.
\end{example}
\setlength{\unitlength}{.8pt}
\begin{figure}[ht]
\begin{center}
\begin{picture}(400,125)(0,50)
\put(205,145){$\tilde \xi_R$}
\put(200,50){\vector(0,1){100}}
\put(40,100){\line(4,1){80}}
\put(40,97){$\cdot$}
\put(120,120){\line(1,0){40}}
\put(120,97){$\cdot$}
\put(160,120){\line(1,-1){20}}
\put(160,97){$\cdot$}
\put(180,97){$\cdot$}
\put(220,100){\line(1,1){20}}
\put(220,97){$\cdot$}
\put(240,120){\line(1,0){40}}
\put(240,97){$\cdot$}
\put(280,120){\line(4,-1){80}}
\put(280,97){$\cdot$}
\put(360,97){$\cdot$}
\put(0,100){\vector(1,0){400}}
\put(40,85){$-2R$}
\put(105,85){$-R$}
\put(150,85){$-\frac{1}{R}$}
\put(170,85){$-\frac{1}{2R}$}
\put(205,85){$0$}
\put(215,85){$\frac{1}{2R}$}
\put(235,85){$\frac{1}{R}$}
\put(275,85){$R$}
\put(350,85){$2R$}
\put(390,85){$x_n$}
\put(202,120){$1$}
\put(200,120){$\cdot$}
\end{picture}
\end{center}
\caption{The function $\tilde\xi_R(x_n)$}
\label{fig1}
\end{figure}

Further we assume that the set $S$ satisfies assumption (H1).
We formulate our first result for a system of nonlinear elliptic inequalities
\begin{equation}\label{eq:2.7.1}
\begin{gathered}
-\Delta_p u \ge a(x)v^{q_{1}} \quad (x\in\mathbb{R}^N\setminus S),\\
-\Delta_q v \ge b(x)u^{p_{1}} \quad (x\in\mathbb{R}^N\setminus S),\\
u(x),\,v(x)\ge 0 \quad (x\in\mathbb{R}^N\setminus S),
\end{gathered}
\end{equation}
where $p,q,p_1,q_1>1$,  $p-1<p_1$, $q-1<q_1$,
$a,b \in C(\mathbb{R}^N\setminus S)$ are nonnegative functions such that
$a(x) \ge a_0\rho^{-\alpha}|x|^{\beta}$,
$b(x) \ge b_0\rho^{-\gamma}|x|^{\delta}$ for $x \in \mathbb{R}^N\setminus S$,
$a_0,b_0>0$, $\alpha,\beta,\gamma,\delta\in\mathbb{R}$, $\rho(x)=\operatorname{dist}(x,S)$.
Introduce the quantities
\begin{equation}\label{eq:2.7.7a}
\begin{gathered}
\sigma_1=\frac{(|\alpha|-\beta)(q-1)+(|\gamma|-\delta-q)q_1)(p-1)-pp_1q_1}{p_1
q_1-(p-1)(q-1)},\\
\sigma_2=\frac{(|\gamma|-\delta)(p-1)+(|\alpha|-\beta-p)p_1)(q-1)-qp_1q_1}{p_1
q_1-(p-1)(q-1)}.
\end{gathered}
\end{equation}


\begin{theorem} \label{thm1}
 Let $N+\min\{\sigma_1,\sigma_2\}\le 0$. Then system \eqref{eq:2.7.1}
has no nontrivial (nonzero) solution.
\end{theorem}


\begin{example} \label{examp2} \rm
 For a system of inequalities \eqref{eq:2.7.1}
with $p=q=2$:
\begin{equation}\label{eq:2.7.1a}
\begin{gathered}
-\Delta u \ge a(x)v^{q_{1}} \quad (x\in\mathbb{R}^N\setminus S),\\
-\Delta v \ge b(x)u^{p_{1}} \quad  (x\in\mathbb{R}^N\setminus S),\\
u(x),\,v(x)\ge 0 \quad (x\in\mathbb{R}^N\setminus S),
\end{gathered}
\end{equation}
the quantities defined in \eqref{eq:2.7.7a} take the form
\begin{equation}\label{eq:2.7.0xl}
\begin{gathered}
\sigma_1=\frac{(\alpha-\beta+(\gamma-\delta-2)q_1)-2p_1q_1}{p_1
q_1-1},\\
\sigma_2=\frac{(\gamma-\delta+(\alpha-\beta-2)p_1)-2p_1q_1}{p_1
q_1-1}.
\end{gathered}
\end{equation}
\end{example}

Further we consider the system
\begin{equation}\label{eq:1}
\begin{gathered}
-\Delta_p u \ge a(x)|Dv|^{q_{1}} \quad (x\in\mathbb{R}^N\setminus S),\\
-\Delta_q v \ge b(x)|Du|^{p_{1}} \quad (x\in\mathbb{R}^N\setminus S).
\end{gathered}
\end{equation}
For this system  one has the following result.

\begin{theorem} \label{thm2}
Let
\begin{align*}
&\max\{(p-1)(\alpha(q-1)+q_1(\beta+1)), (q-1)(\beta(p-1)+p_1(\alpha+1))\} \\
&\ge N(p_1 q_1-(p-1)(q-1))-p_1 q_1.
\end{align*}
Then system \eqref{eq:1} has no nontrivial (non-constant) solution.
\end{theorem}

We also consider the systems of higher-order differential inequalities
\begin{equation}\label{eq:9}
\begin{gathered}
(-\Delta)^k u \ge a(x)|Dv|^q  \quad (x\in\mathbb{R}^N\setminus S),\\
(-\Delta)^l v \ge b(x)|Du|^p \quad (x\in\mathbb{R}^N\setminus S)
\end{gathered}
\end{equation}
and
\begin{equation}\label{eq:9a}
\begin{gathered}
(-\Delta)^k u \ge a(x)v^q  \quad (x\in\mathbb{R}^N\setminus S),\\
(-\Delta)^l v \ge b(x)u^p  \quad (x\in\mathbb{R}^N\setminus S),\\
u\ge 0, \quad v\ge 0  \quad (x\in\mathbb{R}^N),
\end{gathered}
\end{equation}
where $k,l\in\mathbb{N}$, $p,q>1$.

For system \eqref{eq:9}, there holds the following theorem.

\begin{theorem} \label{thm3} Let
\begin{align*}
\max\Big\{&|\alpha|+\beta+(|\gamma|+\delta+2l-1+(2k-1)p)q,\\
&|\gamma|+\delta+(|\alpha|+\beta+2k-1+(2l-1)q)p\Big\} 
\ge N(pq-1).
\end{align*}
Then system \eqref{eq:9} has no nontrivial (non-constant) solution.
\end{theorem}

For system \eqref{eq:9a}, one has

\begin{theorem} \label{thm4}
 Let
\[
\max\Big\{|\alpha|+\beta+(|\gamma|+\delta+2l+2kp)q,
|\gamma|+\delta+(|\alpha|+\beta+2k+2lq)p\Big\} \ge N(pq-1).
\]
Then system \eqref{eq:9a} has no nontrivial (nonzero) solution.
\end{theorem}

\section{Proof of Theorem \ref{thm1}}

Suppose that there exists $(u,v)$ -- a nontrivial solution of system
\eqref{eq:2.7.1}. Let $\varphi_R\in
C^{\infty}_{0}(\overline{\mathbb{R}^N};\mathbb{R}_+)$
be a family of test functions to be specified below.

Multiplying the first inequality \eqref{eq:2.7.1} by
$u_{\varepsilon}^{\lambda}\varphi_R$ and the second one by
$v_{\varepsilon}^{\lambda}\varphi_R$, where $u_{\varepsilon}=u+\varepsilon$,
$v_{\varepsilon}=v+\varepsilon$, $\varepsilon>0$ and $\max\{1-p,1-q\}<\lambda<0$, we obtain
\begin{gather}\label{eq:2.7.4x}
\int a(x)v^{q_{1}}u_{\varepsilon}^{\lambda}\varphi_R\,d x \le c\lambda\int
|Du|^{p}u_{\varepsilon}^{\lambda-1}\varphi_R\,d x + \int
|Du|^{p-1}|D\varphi_R| u_{\varepsilon}^{\lambda}\,d x,\\
 \label{eq:2.7.5x}
\int b(x)u^{p_{1}}v_{\varepsilon}^{\lambda}\varphi_R \,d x \le c\lambda\int
|Dv|^{q}v_{\varepsilon}^{\lambda-1}\varphi_R \,d x\ + \int
|Dv|^{q-1}|D\varphi_R|v_{\varepsilon}^{\lambda}\,d x.
\end{gather}
Application of Young's inequality to the first terms on the right-hand
sides of the obtained relations results in
\begin{gather}\label{eq:2.7.4}
\int a(x)v^{q_{1}}u_{\varepsilon}^{\lambda}\varphi_R\,d x +
\frac{c|\lambda|}{2}\int |Du|^{p}u_{\varepsilon}^{\lambda-1}\varphi_R\,d x
\le c_{\lambda}\int \frac{|D\varphi_R|^{p}}{\varphi_R^{p-1}}
u_{\varepsilon}^{\lambda+p-1}\,d x, \\
\label{eq:2.7.5}
\int b(x)u^{p_{1}}v_{\varepsilon}^{\lambda}\varphi_R \,d x +
\frac{c|\lambda|}{2}\int  |Dv|^{q}v_{\varepsilon}^{\lambda-1}\varphi_R
\,d x\le d_{\lambda}\int
\frac{|D\varphi_R|^{q}}{\varphi_R^{q-1}}v_{\varepsilon}^{\lambda+q-1}\,d x,
\end{gather}
where the constants $c_{\lambda}$ and $d_{\lambda}$ depend only on $p,q$,
and $\lambda$. Further, multiplying each differential inequality
\eqref{eq:2.7.1} by $\varphi_R$ and integrating by parts, we arrive at
\begin{gather}\label{eq:2.7.6}
\int a(x)v^{q_{1}}\varphi_R \,d x \le \Big( \int
|Du|^{p}u^{\lambda-1}_{\varepsilon}\varphi_R \,d x\Big)^{\frac{p-1}{p}}
\Big(\int\frac{|D\varphi_R|^{p}}{\varphi_R^{p-1}}u^{(1-\lambda)(p-1)}_{\varepsilon}
\,d x \Big)^{1/p}\,, \\
\label{eq:2.7.7}
\int b(x)u^{p_{1}}\varphi_R\,d x \le \Big( \int
|Dv|^{q}v^{\lambda-1}_{\varepsilon}\varphi_R\,d x\Big)^{\frac{q-1}{q}}
\Big(\int\frac{|D\varphi_R|^{q}}{\varphi_R^{q-1}}v^{(1-\lambda)(q-1)}_{\varepsilon}\,d
x \Big)^{1/q}.
\end{gather}
Combining \eqref{eq:2.7.4}--\eqref{eq:2.7.7} and taking $\varepsilon \to 0$,
we obtain a priori estimates
\begin{gather}\label{eq:2.7.8}
\int a(x)v^{q_{1}}\varphi_R\,d x \le D_{\lambda}
\Big(\int \frac{|D\varphi_R|^{p}}{\varphi_R^{p-1}}u^{\lambda+p-1}\,d x
\Big)^{\frac{p-1}{p}} \Big(
\int\frac{|D\varphi_R|^{p}}{\varphi_R^{p-1}}u^{(1-\lambda)(p-1)}\,d
x \Big)^{1/p}\,, \\
\label{eq:2.7.9}
\int b(x)u^{p_{1}}\varphi_R \,d x \le E_{\lambda}
\Big(\int\frac{|D\varphi_R|^{q}}{\varphi_R^{q-1}}v^{\lambda+q-1}\,d x
\Big)^{\frac{q-1}{q}}\Big(\int\frac{|D\varphi_R|^{q}}
{\varphi_R^{q-1}}v^{(1-\lambda)(q-1)}\,d x \Big)^{1/q}\,,
\end{gather}
where $D_{\lambda}$, $E_{\lambda}>0$ depend only on $p,\,q$, and
$\lambda$.

Applying the H\"older inequality with exponent $r$ to the first integral on
the right-hand side of \eqref{eq:2.7.8}, we obtain
\begin{equation}\label{eq:2.7.10}
\begin{aligned}
&\Big(\int \frac{|D\varphi_R|^{p}}{\varphi_R^{p-1}}u^{\lambda+p-1} \,d x
\Big)^{\frac{p-1}{p}} \\
&\le  \Big(\int b(x)u^{(\lambda+p-1)r}\varphi_R \,d x \Big)^{\frac{p-1}{pr}}
\Big(\int b^{-\frac{r'}{r}}(x)
\frac{|D\varphi_R|^{pr'}}{\varphi_R^{pr'-1}}\,d x\Big)^{\frac{p-1}{pr'}},
\end{aligned}
\end{equation}
where $\frac{1}{r}+\frac{1}{r'}=1$.

Choosing the exponent $r$ so that $(\lambda+p-1)r=p_{1}$,
from \eqref{eq:2.7.8} and \eqref{eq:2.7.10}
we have
\begin{equation}\label{eq:2.7.9a}
\begin{aligned}
\int a(x)v^{q_{1}}\varphi_R \,d x
&\le D_{\lambda}\Big(\int b(x)u^{p_{1}} \varphi_R\,d x
\Big)^{\frac{p-1}{pr}}
\Big(\int b^{-\frac{r'}{r}}(x)
\frac{|D\varphi_R|^{pr'}} {\varphi_R^{pr'-1}} \,d x
\Big)^{\frac{p-1}{pr'}} \\
&\quad\times \Big(\int\frac{|D\varphi_R|^{p}}{\varphi_R^{p-1}}
u^{(1-\lambda)(p-1)}\,d x \Big)^{1/p}.
\end{aligned}
\end{equation}
Applying the H\"older inequality with exponent $y>1$ to the last
integral on the right-hand side of  \eqref{eq:2.7.9a}, we obtain
\begin{equation}\label{eq:2.7.10a}
\begin{aligned}
\int \frac{|D\varphi_R|^{p}}{\varphi_R^{p-1}}
u^{(1-\lambda)(p-1)}\,d x
&\le \Big(\int b(x)u^{(1-\lambda)(p-1)y}\varphi_R\,d x\Big)^{1/y}\\
&\quad\times \Big(\int b^{-\frac{y'}{y}}(x) \frac{|D\varphi_R|^{py'}}{\varphi_R^{py'-1}}\,d
x\Big)^{1/y'},
\end{aligned}
\end{equation}
where $\frac{1}{y}+\frac{1}{y'}=1$.

Choosing $y$ in \eqref{eq:2.7.10a} so that
$(1-\lambda)(p-1)y=p_{1}$ and taking into account \eqref{eq:2.7.9a},
we reach the estimate
\begin{align*}
\int a(x)v^{q_{1}}\varphi_R\,d x
&\le D_{\lambda}\Big(\int
b(x)u^{p_{1}} \varphi_R \,d x \Big)^{\frac{p-1}{pr}}
\Big(\int b^{-\frac{r'}{r}}(x)
\frac{|D\varphi_R|^{pr'}}{\varphi_R^{pr'-1}}\,d x\Big)^{\frac{p-1}{pr'}}  \\
&\quad \times \Big(\int b(x)u^{p_{1}}\varphi_R\,d x\Big)^{\frac{1}{py}}
\Big(\int b^{-\frac{y'}{y}}(x)\frac{|D\varphi_R|^{py'}}
{\varphi_R^{py'-1}}\,d x \Big)^{\frac{1}{py'}}\,;
\end{align*}
i.e.,
\begin{equation}\label{eq:2.7.12}
\begin{aligned}
&\int a(x)v^{q_1} \varphi_R \,d x \\
&\le D_{\lambda} \Big(\int
b(x)u^{p_1}\varphi_R \,d x\Big)^{\frac{p-1}{pr}+\frac{1}{py}}
 \Big( \int b^{-\frac{r'}{r}}(x) \frac{|
D\varphi_R |^{pr'}} {\varphi_R^{pr'-1}} \,d x
\Big)^{\frac{p-1}{pr'}} \\
&\quad \times \Big( \int b^{-\frac{y'}{y}}(x)
\frac{|D\varphi_R|^{py'}} {\varphi_R^{py'-1}} \,d x
\Big)^{\frac{1}{py'}},
\end{aligned}
\end{equation}
where the exponents $r$ and $y$ are chosen so that
\begin{equation}\label{eq:2.7.13}
\begin{gathered}
\frac{1}{y} + \frac{1}{y'} = 1, \quad (1-\lambda)(p-1)y = p_1,\\
\frac{1}{r} + \frac{1}{r'} = 1, \quad (\lambda+p-1)r = p_1.
\end{gathered}
\end{equation}
Note that such choice of $r$ and $y$ is possible due to our hypotheses
on $p$ and $p_1$ provided that $\lambda < 0$ is small enough in absolute value.
Similarly, choosing $s$ and $z$ such that
\begin{equation}\label{eq:2.7.14}
\begin{gathered}
\frac{1}{z} + \frac{1}{z'} = 1, \quad (1-\lambda)(q-1)z = q_1, \\
\frac{1}{s} + \frac{1}{s'} = 1, \quad (\lambda+q-1)s = q_1,
\end{gathered}
\end{equation}
and estimating the right-hand side of \eqref{eq:2.7.9} by
the H\"older inequality, we obtain
\begin{equation}\label{eq:2.7.15}
\begin{aligned}
&\int b(x)u^{p_1} \varphi_R \,d x \\
&\le E_\lambda \Big(\int a(x)v^{q_1} \varphi_R \,d x \Big)^{\frac{q-1}{qs} +
\frac{1}{qz}}
\Big(\int a^{-\frac{s'}{s}}(x) \frac{|D\varphi_R |^{qs'}}{\varphi_R^{qs'-1}}
\,d x\Big)^{\frac{q-1}{qs'}} \\
&\quad\times \Big(\int a^{-\frac{z'}{z}}(x)
\frac{|D\varphi_R|^{qz'}}{\varphi_R^{qz'-1}}
\,d x \Big)^{\frac{1}{qz'}}.
\end{aligned}
\end{equation}
Combining \eqref{eq:2.7.12} and \eqref{eq:2.7.15}, we finally arrive at
\begin{equation}\label{eq:2.7.16}
\begin{aligned}
&\Big(\int a(x)v^{q_1} \varphi_R \,d x \Big)^{1 - mn} \\
&\le D_{\lambda} E_{\lambda}^N \Big(\int a^{-\frac{s'}{s}}(x)
\frac{|D\varphi_R|^{qs'}}{\varphi_R^{qs'-1}}\,d x
\Big)^{\frac{n(q-1)}{qs'}} 
\Big(\int a^{-\frac{z'}{z}}(x)
\frac{|D\varphi_R|^{qz'}}{\varphi_R^{qz'-1}}\,d x \Big)^{\frac{n}{qz'}} \\
&\quad\times \Big(\int b^{-\frac{r'}{r}}(x)
\frac{|D\varphi_R|^{pr'}}{\varphi_R^{pr'-1}}\,d x \Big)^{\frac{p-1}{pr'}}
\Big(\int b^{-\frac{y'}{y}}(x)
\frac{|D\varphi_R|^{py'}}{\varphi_R^{py'-1}}\,d x \Big)^{\frac{1}{py'}}
\end{aligned}
\end{equation}
and
\begin{equation}\label{eq:2.7.17}
\begin{aligned}
&\Big(\int b(x)u^{p_1} \varphi_R \,d x \Big)^{1-mn} \\
&\le E_{\lambda} D_{\lambda}^m \Big( \int b^{-\frac{r'}{r}}(x)
\frac{|D\varphi_R |^{pr'}}{\varphi_R^{pr'-1}}\,d
x\Big)^{\frac{m(p-1)}{pr'}} 
\Big(\int b^{-\frac{y'}{y}}(x)
\frac{|D\varphi_R|^{py'}}{ \varphi_R^{py'-1}} \,d
x\Big)^{\frac{m}{py'}} \\
&\quad\times \Big(\int a^{-\frac{s'}{s}}(x)
\frac{|D\varphi_R|^{qs'}}{\varphi_R^{qs'-1}}\,d x
\Big)^{\frac{q-1}{qs'}}
\Big(\int a^{-\frac{z'}{z}}(x) \frac{
|D\varphi_R|^{qz'}}{\varphi_R^{qz'-1}}\,d x
\Big)^{\frac{1}{qz'}},
\end{aligned}
\end{equation}
\begin{equation}\label{eq:2.7.18}
n:=\frac{p-1}{pr}+\frac{1}{py}, \quad
m:=\frac{q-1}{qs}+\frac{1}{qz}.
\end{equation}
Simple calculations taking into account \eqref{eq:2.7.13} and
\eqref{eq:2.7.14} give explicit values of $m$ and $n$, namely,
\begin{equation}\label{eq:2.7.19}
m=\frac{q-1}{q_1}, \quad n=\frac{p-1}{p_1}.
\end{equation}
Our assumptions imply that the exponent on the left-hand side of
\eqref{eq:2.7.16}, \eqref{eq:2.7.17} is such that
\[
1-mn=\frac{p_1 q_1-(p-1)(q-1)}{p_1 q_1} > 0.
\]
Thus from \eqref{eq:2.7.17} and our assumptions on $a$ and $b$ we have
\begin{equation}\label{eq:2.7.6a}
\begin{gathered}
\int_{(S^R\setminus S^{1/R})\cap B_R(0)} a(x)v^{q_1} \,d x \le
CR^{N+\sigma_1}, \\
\int_{(S^R\setminus S^{1/R})\cap B_R(0)} b(x)u^{p_1} \,d x
\le CR^{N+\sigma_2}.
\end{gathered}
\end{equation}
Taking $R\to\infty$ in \eqref{eq:2.7.6a}, under condition 
$N+\min\{\sigma_1,\sigma_2\}\le 0$ we come to a contradiction,
 which completes the proof of Theorem \ref{thm1}.

\section{Proof of Theorem \ref{thm2}}

Multiplying inequalities \eqref{eq:1} by the test function 
$\varphi_R\in C_0^1(\mathbb{R}^N;[0,1])$ and
integrating by parts, we obtain
\begin{gather*}
\int_{\mathbb{R}^N} a(x)|Dv|^{q_1}\varphi_R(x)\,dx 
\le \int_{\mathbb{R}^N} (|Du|^{p-2}Du,D\varphi_R)\,dx,  \\
\int_{\mathbb{R}^N} b(x)|Du|^{p_1}\varphi_R(x)\,dx 
\le \int_{\mathbb{R}^N} (|Dv|^{q-2}Dv,D\varphi_R)\,dx,
\end{gather*}
which because of relations
\[
(|Du|^{p-2}Du,D\varphi_R) \le |Du|^{p-1}|D\varphi_R|, \quad 
(|Dv|^{q-2}Dv,D\varphi_R) \le |Dv|^{q-1}|D\varphi_R|
\]
and the H\"older inequality, results in
\begin{gather}\label{eq:3}
\begin{aligned}
&\int_{\mathbb{R}^N} a(x)|Dv|^{q_1}\varphi_R(x)\,dx \\
&\le \Big(\int_{\mathbb{R}^N} b(x)|Du|^{p_1}\varphi_R(x)\,dx\Big)^{\frac{p-1}{p_1}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{p-1}{p_1-p+1}}(x)
|D\varphi_R|^{\frac{p_1}{p_1-p+1}}\varphi_R^{1-\frac{p_1}{p_1-p+1}}(x)\,dx
\Big)^{\frac{p_1-p+1}{p_1}},
\end{aligned} \\
\label{eq:4}
\begin{aligned}
& \int_{\mathbb{R}^N} b(x)|Du|^{p_1}\varphi_R(x)\,dx   \\
&\le \Big(\int_{\mathbb{R}^N} a(x)|Dv|^{q_1}\varphi_R(x) \,dx\Big)^{\frac{q-1}{q_1}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} a^{-\frac{q-1}{q_1-q+1}}(x)
|D\varphi_R|^{\frac{q_1}{q_1-q+1}}\varphi_R^{1-\frac{q_1}{q_1-q+1}}(x)\,dx
\Big)^{\frac{q_1-q+1}{q_1}}.
\end{aligned}
\end{gather}
Substituting \eqref{eq:3} into \eqref{eq:4} and vice versa, we obtain
\begin{align*}
&\Big(\int_{\mathbb{R}^N} a(x)|Dv|^{q_1}\varphi_R(x)\,dx\Big)^{1-\frac{(p-1)(q-1)}{p_1 q_1}}\\
&\le \Big(\int_{\mathbb{R}^N} a^{-\frac{q-1}{q_1-q+1}}(x)|D\varphi_R|^{\frac{q_1}{q_1-q+1}}
 \varphi_R^{1-\frac{q_1}{q_1-q+1}}(x)\,dx\Big)^{\frac{(p-1)(q_1-q+1)}{p_1 q_1}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{p-1}{p_1-p+1}}(x)|D\varphi_R|
^{\frac{p_1}{p_1-p+1}}\varphi_R^{1-\frac{p_1}{p_1-p+1}}(x)\,dx
 \Big)^{\frac{p_1-p+1}{p_1}},
\end{align*}
\begin{align*}
&\Big(\int_{\mathbb{R}^N} b(x)|Du|^{p_1}\varphi_R(x)\,dx\Big)^{1-\frac{(p-1)(q-1)}{p_1 q_1}} \\
&\le \Big(\int_{\mathbb{R}^N} a^{-\frac{q-1}{q_1-q+1}}(x)|D\varphi_R|^{\frac{q_1}{q_1-q+1}}
 \varphi_R^{1-\frac{q_1}{q_1-q+1}}(x)\,dx\Big)^{\frac{q_1-q+1}{q_1}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{p-1}{p_1-p+1}}(x)|D\varphi_R|
^{\frac{p_1}{p_1-p+1}}\varphi_R^{1-\frac{p_1}{p_1-p+1}}(x)\,dx
 \Big)^{\frac{(q-1)(p_1-p+1)}{p_1 q_1}};
\end{align*}
i. e.,
\begin{gather}\label{eq:5}
\begin{aligned}
&\int_{\mathbb{R}^N} a(x)|Dv|^{q_1}\varphi_R(x)\,dx  \\
&\le \Big(\int_{\mathbb{R}^N} a^{-\frac{q-1}{q_1-q+1}}(x)|D\varphi_R|^{\frac{q_1}{q_1-q+1}}
 \varphi_R^{1-\frac{q_1}{q_1-q+1}}(x)\,dx\Big)
 ^{\frac{(p-1)(q_1-q+1)}{p_1 q_1 - (p-1)(q-1)}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{p-1}{p_1-p+1}}(x)|D\varphi_R|
^{\frac{p_1}{p_1-p+1}}\varphi_R^{1-\frac{p_1}{p_1-p+1}}(x)\,dx
\Big)^{\frac{q_1 (p_1-p+1)}{p_1 q_1 - (p-1)(q-1)}},
\end{aligned}\\
\label{eq:6}
\begin{aligned}
&\int_{\mathbb{R}^N} b(x)|Du|^{p_1}\varphi_R(x)\,dx  \\
&\le \Big(\int_{\mathbb{R}^N} a^{-\frac{q-1}{q_1-q+1}}(x)
 |D\varphi_R|^{\frac{q_1}{q_1-q+1}}\varphi_R^{1-\frac{q_1}{q_1-q+1}}(x)\,dx
\Big)^{\frac{p_1 (q_1-q+1)}{p_1 q_1 - (p-1)(q-1)}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{p-1}{p_1-p+1}}(x)
 |D\varphi_R|^{\frac{p_1}{p_1-p+1}}\varphi_R^{1-\frac{p_1}{p_1-p+1}}(x)\,dx
\Big)^{\frac{(q-1)(p_1-p+1)}{p_1 q_1 - (p-1)(q-1)}}.
\end{aligned}
\end{gather}

Choosing test function $\varphi_R\in C_0^1(\mathbb{R}^N;[0,1])$ so that
\[
\varphi_R(x)=\begin{cases}
1 & (|x|\le R),\\
0 & (|x|\ge 2R),
\end{cases}
\]
and
\begin{equation}\label{eq:2}
|D\varphi_R(x)|\le cR^{-1} \quad (x\in\mathbb{R}^N),
\end{equation}
we obtain
\begin{gather*}
\int_{B_R(0)} a(x)|Dv|^{q_1}\,dx 
\le cR^{N-\frac{(p-1)((|\alpha|+\beta)(q-1)+q_1(|\gamma|+\delta+1))
+p_1 q_1}{p_1 q_1-(p-1)(q-1)}},
\\
\int_{B_R(0)} b(x)|Du|^{p_1}\,dx \le cR^{N-\frac{(q-1)((|\gamma|+\delta)(p-1)
+p_1(|\alpha|+\beta+1))+p_1 q_1}{p_1 q_1-(p-1)(q-1)}}.
\end{gather*}
Taking $R\to\infty$, we complete the proof similarly to Theorem \ref{thm1}.

\section{Proof of Theorem \ref{thm3}}

Multiplying inequalities \eqref{eq:9} by the test function
 $\varphi_R\in C_0^{2k-1}(\mathbb{R}^N;[0,1])$
and integrating by parts, we obtain
\begin{gather*}
\int_{\mathbb{R}^N} a(x)|Dv|^q \varphi_R(x)\,dx 
\le \int_{\mathbb{R}^N} (Du,D((-\Delta)^{k-1}\varphi_R))\,dx,  \\
\int_{\mathbb{R}^N} b(x)|Du|^p \varphi_R(x)\,dx 
\le \int_{\mathbb{R}^N} (Dv,D((-\Delta)^{l-1}\varphi_R))\,dx,
\end{gather*}
which by relations
\begin{gather*}
(Du,D((-\Delta)^{k-1}\varphi_R))\le |Du|\cdot|D((-\Delta)^{k-1}\varphi_R)|,\\
(Dv,D((-\Delta)^{l-1}\varphi_R))\le |Dv|\cdot|D((-\Delta)^{l-1}\varphi_R)|,
\end{gather*}
and the H\"older inequality,  results in
\begin{gather}\label{eq:10}
\begin{aligned}
&\int_{\mathbb{R}^N} a(x)|Dv|^q\varphi_R(x)\,dx  \\
&\le \Big(\int_{\mathbb{R}^N} b(x)|Du|^p\varphi_R(x)\,dx\Big)^{1/p} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{1}{p-1}}(x)
|D((-\Delta)^{k-1}\varphi_R)|^{\frac{p}{p-1}}\varphi_R^{-\frac{1}{p-1}}
(x)\,dx\Big)^{\frac{p-1}{p}},
\end{aligned} \\
\label{eq:11}
\begin{aligned}
&\int_{\mathbb{R}^N} b(x)|Du|^p\varphi_R(x)\,dx  \\
&\le \Big(\int_{\mathbb{R}^N} a(x)|Dv|^q\varphi_R(x) \,dx\Big)^{1/q} \\
&\quad\times \Big(\int_{\mathbb{R}^N} a^{-\frac{1}{q-1}}(x)
|D((-\Delta)^{l-1}\varphi_R)|^{\frac{q}{q-1}}\varphi_R^{-\frac{1}{q-1}}(x)\,dx
\Big)^{\frac{q-1}{q}}.
\end{aligned}
\end{gather}
Substituting \eqref{eq:10} into \eqref{eq:11} and vice versa, we obtain
\begin{align*}
&\Big(\int_{\mathbb{R}^N} a(x)|Dv|^q\varphi_R(x)\,dx\Big)^{1-\frac{1}{pq}}  \\
&\le \Big(\int_{\mathbb{R}^N} a^{-\frac{1}{q-1}}(x)|D((-\Delta)^{l-1}
 \varphi_R)|^{\frac{q}{q-1}}\varphi_R^{-\frac{1}{q-1}}(x)\,dx\Big)
^{\frac{q-1}{pq}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{1}{p-1}}(x)|D((-\Delta)^{k-1}
\varphi_R)|^{\frac{p}{p-1}}\varphi_R^{-\frac{1}{p-1}}(x)\,dx\Big)^{\frac{p-1}{p}},
\end{align*}
\begin{align*}
&\Big(\int_{\mathbb{R}^N} b(x)|Du|^p\varphi_R(x)\,dx\Big)^{1-\frac{1}{pq}}  \\
&\le \Big(\int_{\mathbb{R}^N} a^{-\frac{1}{q-1}}(x)|D((-\Delta)^{l-1}\varphi_R)
 |^{\frac{q}{q-1}}\varphi_R^{-\frac{1}{q-1}}(x)\,dx\Big)^{\frac{q-1}{q}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{1}{p-1}}(x)|D((-\Delta)^{k-1}
\varphi_R)|^{\frac{p}{p-1}}\varphi_R^{-\frac{1}{p-1}}(x)\,dx\Big)
^{\frac{p-1}{pq}};
\end{align*}
i.e.,
\begin{gather}\label{eq:12}
\begin{aligned}
& \int_{\mathbb{R}^N} a(x)|Dv|^q\varphi_R(x)\,dx  \\
&\le \Big(\int_{\mathbb{R}^N} a^{-\frac{1}{q-1}}(x)|D((-\Delta)^{l-1}
 \varphi_R)|^{\frac{q}{q-1}}\varphi_R^{-\frac{1}{q-1}}(x)\,dx\Big)
 ^{\frac{q-1}{pq-1}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{1}{p-1}}(x)|D((-\Delta)^{k-1}
\varphi_R)|^{\frac{p}{p-1}}\varphi_R^{-\frac{1}{p-1}}(x)\,dx\Big)
^{\frac{(p-1)q}{pq-1}},
\end{aligned}\\
\label{eq:13}
\begin{aligned}
&\int_{\mathbb{R}^N} b(x)|Du|^p\varphi_R(x)\,dx  \\
&\le \Big(\int_{\mathbb{R}^N} a^{-\frac{1}{q-1}}(x)|D((-\Delta)^{l-1}\varphi_R)
|^{\frac{q}{q-1}}\varphi_R^{-\frac{1}{q-1}}(x)\,dx\Big)^{\frac{p(q-1)}{pq-1}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{1}{p-1}}(x)|D((-\Delta)^{k-1}\varphi_R)
|^{\frac{p}{p-1}}\varphi_R^{-\frac{1}{p-1}}(x)\,dx\Big)^{\frac{p-1}{pq-1}}.
\end{aligned}
\end{gather}

Denote $K=\max\{k,l\}$. Choosing the test function
$\varphi_R\in C_0^{2K-1}(\mathbb{R}^N;[0,1])$ so that
\[
\varphi_R(x)=\begin{cases}
1 & (|x|\le R),\\
0 & (|x|\ge 2R),
\end{cases}
\]
and
\begin{equation}\label{eq:15}
|D^{\mu}\varphi_R(x)|\le cR^{-1} \quad (x\in\mathbb{R}^N;0\le|\mu|\le 2K-1),
\end{equation}
we have
\begin{gather*}
\int_{B_R(0)} a(x)|Dv|^q\,dx \le cR^{N-\frac{|\alpha|
+\beta+(|\gamma|+\delta+2l-1+(2k-1)p)q}{pq-1}}, \\
\int_{B_R(0)} b(x)|Du|^p\,dx \le cR^{N-\frac{|\gamma|
+\delta+(|\alpha|+\beta+2k-1+(2l-1)q)p}{pq-1}}.
\end{gather*}
Taking $R\to\infty$, we complete the proof similarly to Theorems \ref{thm1} and 
\ref{thm2}.


\section{Proof of Theorem \ref{thm4}}

Multiplying inequalities \eqref{eq:9a} by the test function
 $\varphi_R\in C_0^{2K}(\mathbb{R}^N;[0,1])$, where $K=\max\{k,l\}$,
and integrating by parts, we obtain
\begin{gather*}
\int_{\mathbb{R}^N} a(x)v^q \varphi_R(x)\,dx 
 \le \int_{\mathbb{R}^N} u \cdot (-\Delta)^k \varphi_R \,dx,  \\
\int_{\mathbb{R}^N} b(x)u^p \varphi_R(x)\,dx 
 \le \int_{\mathbb{R}^N} v \cdot (-\Delta)^l \varphi_R \,dx.
\end{gather*}
Taking into account that
\begin{gather*}
u \cdot (-\Delta)^k \varphi_R \le u\cdot|(-\Delta)^k\varphi_R)|, \\
v \cdot (-\Delta)^l \varphi_R \le v\cdot|(-\Delta)^l \varphi_R)|
\end{gather*}
and  using the H\"older inequality, we arrive at
\begin{gather}\label{eq:10a}
\begin{aligned}
\int_{\mathbb{R}^N} a(x)v^q \varphi_R(x)\,dx  
&\le \Big(\int_{\mathbb{R}^N} b(x)u^p\varphi_R(x)\,dx\Big)^{1/p}  \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{1}{p-1}}(x)|(-\Delta)^k \varphi_R|
^{\frac{p}{p-1}}\varphi_R^{-\frac{1}{p-1}}(x)\,dx\Big)^{\frac{p-1}{p}},
\end{aligned} \\
\label{eq:11a}
\begin{aligned}
\int_{\mathbb{R}^N} b(x)u^p\varphi_R(x)\,dx 
&\le \Big(\int_{\mathbb{R}^N} a(x)v^q\varphi_R(x) \,dx\Big)^{1/q} \\
&\quad \times \Big(\int_{\mathbb{R}^N} a^{-\frac{1}{q-1}}(x)|(-\Delta)^l 
\varphi_R|^{\frac{q}{q-1}}\varphi_R^{-\frac{1}{q-1}}(x)\,dx\Big)^{\frac{q-1}{q}}.
\end{aligned}
\end{gather}
Substituting \eqref{eq:10a} into \eqref{eq:11a} and vice versa, we obtain
\begin{align*}
\Big(\int_{\mathbb{R}^N} a(x)v^q\varphi_R(x)\,dx\Big)^{1-\frac{1}{pq}} 
&\le \Big(\int_{\mathbb{R}^N} a^{-\frac{1}{q-1}}(x)|(-\Delta)^l\varphi_R|
^{\frac{q}{q-1}}\varphi_R^{-\frac{1}{q-1}}(x)\,dx\Big)^{\frac{q-1}{pq}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{1}{p-1}}(x)|(-\Delta)^k 
\varphi_R|^{\frac{p}{p-1}}\varphi_R^{-\frac{1}{p-1}}(x)\,dx\Big)^{\frac{p-1}{p}},
\end{align*}
\begin{align*}
\Big(\int_{\mathbb{R}^N} b(x)u^p\varphi_R(x)\,dx\Big)^{1-\frac{1}{pq}} 
&\le \Big(\int_{\mathbb{R}^N} a^{-\frac{1}{q-1}}(x)|(-\Delta)^l\varphi_R|
^{\frac{q}{q-1}}\varphi_R^{-\frac{1}{q-1}}(x)\,dx\Big)^{\frac{q-1}{q}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{1}{p-1}}(x)|(-\Delta)^k\varphi_R
|^{\frac{p}{p-1}}\varphi_R^{-\frac{1}{p-1}}(x)\,dx\Big)^{\frac{p-1}{pq}};
\end{align*}
i.e.,
\begin{gather}\label{eq:12a}
\begin{aligned}
&\int_{\mathbb{R}^N} a(x)v^q\varphi_R(x)\,dx  \\
&\le \Big(\int_{\mathbb{R}^N} a^{-\frac{1}{q-1}}(x)|(-\Delta)^l 
\varphi_R|^{\frac{q}{q-1}}\varphi_R^{-\frac{1}{q-1}}(x)\,dx\Big)^{\frac{q-1}{pq-1}}\\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{1}{p-1}}(x)|(-\Delta)^k \varphi_R|
^{\frac{p}{p-1}}\varphi_R^{-\frac{1}{p-1}}(x)\,dx\Big)^{\frac{(p-1)q}{pq-1}},
\end{aligned} \\
\label{eq:13a}
\begin{aligned}
&\int_{\mathbb{R}^N} b(x)u^p\varphi_R(x)\,dx  \\
&\le \Big(\int_{\mathbb{R}^N} a^{-\frac{1}{q-1}}(x)|(-\Delta)^l\varphi_R|^{\frac{q}{q-1}}
\varphi_R^{-\frac{1}{q-1}}(x)\,dx\Big)^{\frac{p(q-1)}{pq-1}} \\
&\quad \times \Big(\int_{\mathbb{R}^N} b^{-\frac{1}{p-1}}(x)|(-\Delta)^k \varphi_R|
^{\frac{p}{p-1}}\varphi_R^{-\frac{1}{p-1}}(x)\,dx\Big)^{\frac{p-1}{pq-1}}.
\end{aligned}
\end{gather}
Choosing the test function $\varphi_R\in C_0^{2K}(\mathbb{R}^N;[0,1])$ so that
\[
\varphi_R(x)=\begin{cases}
1 & (|x|\le R),\\
0 & (|x|\ge 2R)
\end{cases}
\]
and estimate \eqref{eq:15} holds, we obtain
\begin{gather*}
\int_{B_R(0)} a(x)|Dv|^q\,dx 
\le cR^{N-\frac{|\alpha|+\beta+(|\gamma|+\delta+2l+2kp)q}{pq-1}}, \\
\int_{B_R(0)} b(x)|Du|^p\,dx 
\le cR^{N-\frac{|\gamma|+\delta+(|\alpha|+\beta+2k+2lq)p}{pq-1}}.
\end{gather*}
Taking $R\to\infty$, we complete the proof similarly to Theorems 
\ref{thm1}, \ref{thm2} and \ref{thm3}.

\subsection*{Acknowledgements}
This research was supported by grants of Russian Foundation of Basic
Research 13-01-12460-ofi-m and No. 14-01-00736 and by a President grant
for government support of the leading scientific schools of the Russian
Federation No. 4479.2014.1.

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\end{document}