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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 198, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/198\hfil Parabolic inequalities in the half-space]
{Nonexistence of nonnegative solutions for parabolic inequalities 
in the half-space}


\author[E. I. Galakhov, O. A. Salieva, L. A. Uvarova \hfil EJDE-2018/198\hfilneg]
{Evgeny I. Galakhov, Olga A. Salieva, Liudmila A. Uvarova}

\address{Evgeny I. Galakhov \newline
RUDN University, 
ul. Miklukho-Maklaya, 6,
Moscow 117198,  Russia}
\email{galakhov@rambler.ru}

\address{Olga A. Salieva,
MSTU ``Stankin'', 
Vadkovsky per. 1,
Moscow 127055,  Russia}
\email{olga.a.salieva@gmail.com}

\address{Liudmila A. Uvarova
MSTU ``Stankin'', 
Vadkovsky per. 1,
Moscow 127055, Russia}
\email{uvar11@yandex.com}

\thanks{Submitted March 22, 2018. Published December 12, 2018.}
\subjclass[2010]{35K92, 35R45}
\keywords{Quasilinear parabolic inequalities; monotonicity;
\hfill\break\indent  nonexistence of solutions}


\begin{abstract}
 Based on the method of nonlinear capacity, we study the nonexistence of
 nonnegative monotonic solutions for the quasilinear parabolic inequality
 $u_t-\Delta_p u\ge u^q$. Also we study generalizations in the half-space 
 in terms of parameters $p$ and $q$.
\end{abstract}

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

\section{Introduction}

The question about nonexistence of nontrivial nonnegative global solutions to 
nonlinear equation $u_t-Au=g(x)u^q$ and the inequality
$u_t-Au\ge g(x)u^q$, where $A$ is an elliptic operator, 
in different domains is of substantial interest. 
Such inequalities can be understood as nonlinear heat equations with a 
supplementary external source term $f(x,t)=u_t-Au-g(x)u^q \ge 0$. 
The aim of the study is to find the range of values of $q$ such that the 
equation or inequality in question has no-nontrivial nonnegative global solutions,
 i.e.\ the extra heat source leads to blow-up of a local solution.

The results in the whole space $\mathbb{R}^n$ go back to Fujita \cite{Fu}
 who established that solutions to the equation $u_t-\Delta u=u^q$ do not 
exist for $1<q<1+\frac{2}{n}$. Similar nonexistence ranges for much more general 
operators were obtained later in \cite{MP}. As for the half-space, up to our 
knowledge, so far only stationary solutions have been considered. 
The first results in this direction were obtained by  
Berestycki,  Capuzzo Dolcetta and Nirenberg \cite{BCDN} who proved 
nonexistence of solutions to the inequality $-\Delta u\ge u^q$ for $
1<q<\frac{n+1}{n-1}$. The optimality of these results was shown by 
 Birindelli and Mitidieri \cite{BM}. 
Inequalities of the form $Au\ge u^q$ with $A=-\Delta_p$, where $p>1$ and 
$\Delta_p$ is the $p$-Laplace operator defined by 
 $\Delta_p u:=\operatorname{div}(|Du|^{p-2}Du)$, in the half-space with a punched
point or a removed neighborhood of a point on the boundary were studied 
by  Bidaut-V\'eron and Pohozaev \cite{BVP}, and later by 
 V\'eron and A. Porretta \cite{PV}. They obtained results on nonexistence 
of solutions in the domains under study and consequently in the whole half-space 
for $p-1<q<q_{\rm cr}(p,n)$, where $q_{\rm cr}(p,n)=p-1+\frac{p}{\beta_{p,n}}$, 
and $\beta_{p,n}$ is the growth rate of singular solutions near zero, obtained 
explicitly only for $n=2$ ($\beta_{p,2}=\frac{3-p+\sqrt{(p-1)^2+2-p}}{3(p-1)}$).
 One should also note the papers of  Filippucci \cite{F} on critical exponents 
for semilinear inequalities of the form 
$-\operatorname{div}(u^{\alpha}|x|^{\beta}Du)\ge |x|^{\gamma} u^q$ 
in the half-space, of Dancer,  Du and Efendiev \cite{DDE} and of Zou \cite{Z} 
on nonexistence of solutions to the Dirichlet problem
\begin{equation}\label{e3b}
\begin{gathered}
-\Delta_p u = u^q, \quad x\in\mathbb{R}^n_+,\\
u(x)= 0, \quad x\in\partial\mathbb{R}^n_+,
\end{gathered}
\end{equation}
for a nonlinear equation with a $p$-Laplace operator in a half-space, as well
as those of  Farina,  Montoro and  Sciunzi \cite{FMS1}--\cite{FMS3}
on monotonicity of essentially bounded solutions of the same problem,
which implies their nonexistence for a certain range of $q$.
Elliptic problems with singular coefficients near unbounded sets were considered,
in particular, in \cite{GS1,GS2}.

In this article  we consider the nonexistence of nonnegative solutions for 
the parabolic inequality $u_t-\Delta_p u\ge ax_n^{\gamma}u^q$ in the half-space.
Based  on the method of nonlinear capacity \cite{MP,P},
we obtain sufficient conditions for nonexistence of solutions. 
Similar results for elliptic inequalities and systems can be found in \cite{GS3}.

The rest of this article consists of three sections. 
\S2 has our main results, 
\S3 contains a proof in the semilinear case, and 
\S4 the quasilinear case.

\section{Formulation of main results}


Denote $\mathbb{R}^n_+=\{x=(x_1,\dots,x_n)\in\mathbb{R}^n:\,x_n>0\}$.
Let $p>1$, $q>p-1$, $a>0$, $\gamma\in\mathbb{R}$, and let 
$u_0\in C(\mathbb{R}^n_+)$ be a nonnegative function.  Consider the problem
\begin{equation} \label{eq:5.4.1xx}
\begin{gathered}
u_t-\Delta_p u\ge ax_n^{\gamma}u^q, \quad (x,t)\in\mathbb{R}^n_+\times\mathbb{R}_+,\\
u(x,0)=u_0(x), \quad x\in\mathbb{R}^n_+,\\
u(x,t)\ge 0, \quad (x,t)\in\mathbb{R}^n_+\times\mathbb{R}_+.
\end{gathered}
\end{equation}
We understand its weak solutions in the following sense.



\begin{definition} \label{def2.1} \rm
A weak solution of problem \eqref{eq:5.4.1xx} is a nonnegative function 
$u\in C^{2,1}(\mathbb{R}^n_+\times\mathbb{R}_+)$, which satisfies the integral inequality
\[
\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} (|Du|^{p-2}(Du,D\varphi)-u\varphi_t)\,dx\,dt
\ge \int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} ax_n^{\gamma} u^q \varphi\,dx\,dt
+\int_{\mathbb{R}^n_+} u_0 \varphi\,dx
\]
for any nonnegative $\varphi\in C^{\infty}(\mathbb{R}^n_+\times\mathbb{R}_+)$ such that
$\varphi(x,t)\equiv 0$ for $(x,t)\in\partial \mathbb{R}^n_+\times\mathbb{R}_+$ (that is, for $x_n=0$).
\end{definition}


Weak solutions of the problems considered below are defined in a similar way.
In the case $p=2$, we obtain the following result.

\begin{theorem} \label{thm2.1} 
Let $a>0$,$\gamma>-2$, and $1<q\le 1+\frac{\gamma+2}{n+1}$.
Then \eqref{eq:5.4.1xx} with $p=2$:
\begin{equation} \label{eq:5.3.1xx}
\begin{gathered}
u_t-\Delta u\ge ax_n^{\gamma}u^q, \quad (x,t)\in\mathbb{R}^n_+\times\mathbb{R}_+,\\
u(x,0)=u_0(x), \quad  x\in\mathbb{R}^n_+,\\
u(x,t)\ge 0, \quad (x,t)\in\mathbb{R}^n_+\times\mathbb{R}_+
\end{gathered}
\end{equation}
has no nonnegative nontrivial weak solutions $u$.
\end{theorem}

For other values of $p\ne 2$, we obtain a nonexistence result in a smaller 
functional class of solutions (with an additional property of monotonicity).

\begin{theorem} \label{thm2.2} 
 Let $a>0$, $\gamma>-p$, $q\ge\max(1,p-1)$, $\gamma(p-2)>p(1-q)$, and
\[
[(n+1)(q-1)-\gamma](q-p+1)-p(q-1)-\gamma(p-2)<0.
\]
Then \eqref{eq:5.4.1xx} has no nonnegative nontrivial weak solutions $u$ 
such that $u(x',\cdot,t)$ is monotonic in $x_n$ for each $x'\in\mathbb{R}^{n-1}$
and $t>0$.
\end{theorem}

\begin{corollary} \label{coro2.1} 
Let $a>0$ and $\max(1,p-1)\le q\le p-1+\frac{p}{n+1}$.
Then the problem
\begin{equation} \label{eq:5.4.1xxb}
\begin{gathered}
u_t-\Delta_p u\ge au^q, \quad (x,t)\in\mathbb{R}^n_+\times\mathbb{R}_+,\\
u(x,0)=u_0(x), \quad x\in\mathbb{R}^n_+,\\
u(x,t)\ge 0, \quad (x,t)\in\mathbb{R}^n_+\times\mathbb{R}_+
\end{gathered}
\end{equation}
(that is, \eqref{eq:5.4.1xx} with $\gamma=0$)
 has no nonnegative nontrivial weak solutions $u$ such that $u(x',\cdot,t)$
is monotonic in $x_n$ for each $x'\in\mathbb{R}^{n-1}$ and $t>0$.
\end{corollary}

Evidently, the above corollary  follows from Theorem \ref{thm2.2} in the 
case $\gamma=0$.

\begin{remark} \label{rmk2.3} \rm
 Nonexistence results can be obtained in the same class of monotonic solutions 
for the problem
\begin{equation} \label{eq:5.4.1yy}
\begin{gathered}
u_t+\Delta_p u\ge ax_n^{\gamma}u^q, \quad (x,t)\in\mathbb{R}^n_+\times\mathbb{R}_+,\\
u(x,0)=u_0(x), \quad x\in\mathbb{R}^n_+,\\
u(x,t)\ge 0, \quad (x,t)\in\mathbb{R}^n_+\times\mathbb{R}_+,
\end{gathered}
\end{equation}
where the operator $\Delta_p$ has the opposite sign (see \cite{GS3}).
Although the result in \cite{GS3} is formulated for monotonically
nondecreasing solutions, its proof is valid for non-increasing ones as well.
\end{remark}

\section{Proof of Theorem \ref{thm2.1}}

We use  the method of nonlinear capacity \cite{MP,P}. 
We choose a family of nonnegative test functions 
$\xi_{R,T}^{\lambda}\in C^1_0(\mathbb{R}^n)$ such that $\lambda>0$
(to be specified below), $R$ and $T$ are some positive parameters, 
and $\xi_{R,T}(x)=\prod_{k=1}^{N-1} \chi_R(x_k)\cdot\chi_R(x_n-3R)\cdot \chi_T(t)$ 
with
\begin{equation} \label{eq:5.3.4}
\chi_R(s)=\begin{cases}
1 &\text{if } s\le R,\\
0 &\text{if } s\ge 2R,
\end{cases}
\end{equation}
where
\begin{equation} \label{eq:5.3.5}
|D\chi_R(s)|\le cR^{-1}, \quad s\in\mathbb{R}_+.
\end{equation}
Multiply both sides of \eqref{eq:5.3.1xx} by $\xi_{R,T}^{\lambda} x_n$ and
integrate by parts. After elementary transformations we obtain
\begin{equation} \label{eq:5.3.6}
\begin{aligned}
&a\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q}\xi_{R,T}^{\lambda} x_n^{\gamma+1}\,dx\,dt  \\
&\le \int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u \cdot |\Delta(\xi^{\lambda}_{R,T} x_n)|\,dx\,dt
 + \int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u \cdot
 \big|\frac{\partial \xi_{R,T}^{\lambda}}{\partial t}\big| x_n\,dx\,dt.
\end{aligned}
\end{equation}
Application of the parametric Young inequality to both integrals on the
right-hand side of \eqref{eq:5.3.6} yields
\begin{equation} \label{eq:5.3.8}
\begin{aligned}
\frac{a}{2}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q}
 \xi^{\lambda}_{R,T} x_n^{\gamma+1}\,dx\,dt 
&\le c\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} |D\xi_{R,T}|^{\frac{2q}{q+1}}\xi_{R,T}
 ^{\lambda-\frac{2q}{q+1}} x_n^{\frac{q-\gamma-1}{q-1}}\,dx\,dt \\
&\quad + c\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} |\chi'_T(t)|^{\frac{q}{q-1}}
\chi_T^{\lambda-\frac{q}{q-1}}x_n^{-\frac{\gamma+1}{q-1}}\,dx\,dt\\
& :=I_1(R,T)+I_2(R,T).
\end{aligned}
\end{equation}
For $\lambda>\frac{2q}{q-1}$, the integral $I_1(R,T)$ can be estimated as
\begin{equation} \label{eq:5.3.9}
I_1(R,T)\le �R^{n-\frac{q+\gamma+1}{q-1}}T
\end{equation}
and $I_2(R,T)$ as
\begin{equation} \label{eq:5.3.9a}
I_2(R,T)\le �R^{n-\frac{\gamma+1}{q-1}}T^{1-\frac{q}{q-1}}.
\end{equation}
From \eqref{eq:5.3.8}--\eqref{eq:5.3.9a} we obtain
\begin{equation} \label{eq:5.3.11}
\frac{a}{2}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q}\xi_{R,T}^{\lambda} x_n^{\gamma+1}\,dx\,dt
\le c(R^{n-\frac{q+\gamma+1}{q-1}}T+R^{n-\frac{\gamma+1}{q-1}}T^{1-\frac{q}{q-1}}).
\end{equation}
Choosing $T=R^{\theta}$ with $\theta>0$ such that both terms are of the same
order and taking $R\to\infty$, we obtain
\[
\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q} x_n^{\gamma+1}\,dx\,dt = 0,
\]
which contradicts the assumption of non-triviality of the solution.
This completes the proof of Theorem \ref{thm2.1}.

\section{Proof of Theorem \ref{thm2.2}}

Now, using the same family of test functions $\xi_{R,T}$ as in the previous proof, 
we multiply both parts of \eqref{eq:5.4.1xxb} by $u^{\alpha}\xi_{R,T}^{\lambda} x_n$, 
where $\alpha<0$ will be specified below, and integrate by parts. 
After elementary transformations we obtain
\begin{equation} \label{eq:5.4.6}
\begin{aligned}
&a\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q+\alpha}\xi_{R,T}^{\lambda} x_n^{\gamma+1}\,dx\,dt
+ |\alpha|\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha-1}|Du|^p\xi_{R,T}^{\lambda} x_n\,dx\,dt
\\
&\le \int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha} |Du|^{p-1} |D\xi^{\lambda}_{R,T}| x_n\,dx\,dt
 + \int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u_t u^{\alpha}\xi_{R,T}^{\lambda} x_n\,dx\,dt\\
&\quad +\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha} |Du|^{p-2}
 \frac{\partial  u}{\partial  x_n}\xi^{\lambda}_{R,T}\,dx\,dt.
\end{aligned}
\end{equation}
Application of the parametric Young inequality to the first integral on the 
right-hand side of \eqref{eq:5.4.6} yields
\begin{equation} \label{eq:5.4.7}
\begin{aligned}
&a\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q+\alpha}\xi^{\lambda}_{R,T} x_n^{\gamma+1}\,dx\,dt
 + \frac{|\alpha|}{2}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha-1}|Du|^p\xi^{\lambda}_{R,T}
 x_n\,dx\,dt  \\
&\le c\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha+p-1} |D\xi^{\lambda}_{R,T}|^p
  \xi_{R,T}^{\lambda(1-p)} x_n\,dx\,dt \\
&\quad +\frac{1}{\alpha+1} \int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha+1}
 (\xi_{R,T}^{\lambda})_t x_n\,dx\,dt \\
&\quad +\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha} |Du|^{p-2}
 \frac{\partial  u}{\partial  x_n}\xi^{\lambda}_{R,T}\,dx\,dt.
\end{aligned}
\end{equation}
Applying the parametric Young inequality to the first two integrals on 
the right-hand side of \eqref{eq:5.4.7} once more, we obtain
\begin{equation} \label{eq:5.4.8}
\begin{aligned}
&\frac{a}{2}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q+\alpha}\xi^{\lambda}_{R,T} x_n^{\gamma+1}
 \,dx\,dt
 + \frac{|\alpha|}{2}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha-1}|Du|^p\xi^{\lambda}_{R,T}
 x_n\,dx\,dt  \\
&\le c\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} |D\xi_{R,T}|^{\frac{p(q+\alpha)}{q-p+1}}
 \xi_{R,T}^{\lambda-\frac{p(q+\alpha)}{q-p+1}} x_n^{\frac{q+\alpha-(\alpha+p-1)
 (\gamma+1)}{q-p+1}}\,dx\,dt \\
&\quad + c\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} |\chi'_T(t)|^{\frac{q+\alpha}{q-1}}
\chi_T^{\lambda-\frac{q+\alpha}{q-1}}x_n^{\frac{q+\alpha-(\alpha+1)
 (\gamma+1)}{q-1}}\,dx\,dt  \\
&\quad + \int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha} |Du|^{p-2}
 \frac{\partial  u}{\partial  x_n}\xi_{R,T}^{\lambda}\,dx\,dt \\
&:=I_1(R,T)+I_2(R,T)+I_3(R,T).
\end{aligned}
\end{equation}
For $\lambda>\frac{pq}{q-p+1}$ and
\begin{equation} \label{eq:5.4.8a}
\alpha>\frac{n(q-p+1)-(q+\gamma-1)(p-1)}{p+\gamma}
\end{equation}
the integral $I_1(R,T)$ and $I_2(R,T)$ can be estimated as
\begin{gather} \label{eq:5.4.9}
I_1(R,T)\le R^{n-\frac{(p-1)(q+\alpha)+(\alpha+p-1)(\gamma+1)}{q-p+1}}T,\\
 \label{eq:5.4.9a}
I_2(R,T)\le R^{n+\frac{q+\alpha-(\alpha+1)(\gamma+1)}{q-1}}
 T^{1-\frac{q+\alpha}{q-1}}.
\end{gather}
If $\frac{\partial  u}{\partial  x_n}\ge 0$, then $I_3(R,T)<0$. Estimate the integral
$I_3(R,T)$ in the case $\frac{\partial  u}{\partial  x_n}\le 0$. In case $p<2$, using
the H\"older inequality and integrating by parts, we have
\begin{align*}
I_3(R,T)
&=-\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha} |Du|^{p-2}
 \frac{\partial  u}{\partial  x_n}\xi_{R,T}^{\lambda}\,dx\,dt\\
& \le \int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha}
 \Big(-\frac{\partial  u}{\partial  x_n}\Big)^{p-1} \xi^{\lambda}_{R,T}\,dx\,dt  \\
&\le c\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+}
 \Big(-\frac{\partial  u^{1+\frac{\alpha}{p-1}}}{\partial  x_n}\Big)^{p-1}
 \xi^{\lambda}_{R,T}\,dx\,dt \\
&\le c\Big(-\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} \frac{\partial  u^{1+\frac{\alpha}{p-1}}}
 {\partial  x_n}\xi^{\frac{\lambda}{p-1}}_R\,dx\,dt\Big)^{p-1}R^{n(2-p)} \\
&=c\Big(\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{1+\frac{\alpha}{p-1}}
 \frac{\partial  \xi^{\frac{\lambda}{p-1}}_R}{\partial  x_n}\,dx\,dt\Big)^{p-1} R^{n(2-p)} \\
&\le c\Big(\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{1+\frac{\alpha}{p-1}}
 \big|\frac{\partial  \xi^{\frac{\lambda}{p-1}}_R}{\partial  x_n}\big|\,dx\,dt\Big)^{p-1}
 R^{n(2-p)}\\
&\le c\Big(\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q+\alpha}\xi^{\lambda}_{R,T}
  x_n^{\gamma+1}\,dx\,dt\Big)^{\frac{\alpha+p-1}{q+\alpha}}\cdot R^{n(2-p)}  \\
&\quad \times\Big(\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+}
 \big|\frac{\partial  \xi^{\lambda}_{R,T}}{\partial  x_n}
 \big|^{\frac{(q+\alpha)(p-1)}{(q+\alpha-1)(p-1)-\alpha}} \\
&\quad\times \big(\xi_{R,T}^{\lambda(q-p+1)-(q+\alpha)}x_n^{-(\gamma+1)(\alpha+p-1)}
 \big)^{\frac{1}{(q+\alpha-1)(p-1)-\alpha}}\,dx\,dt\Big)
 ^{\frac{(q+\alpha-1)(p-1)-\alpha}{q+\alpha}} \\
&\le \frac{a}{4}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+}
 u^{q+\alpha}\xi^{\lambda}_{R,T} x_n^{\gamma+1}\,dx\,dt \\
&\quad + cR^{\frac{n[(2-p)(q+\alpha)+(q+\alpha-1)(p-1)-\alpha]
 -(q+\alpha+\gamma+1)(p-1)-(\gamma+1)\alpha}{q-p+1}} \\
&=\frac{a}{4}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q+\alpha}\xi^{\lambda}_{R,T}
 x_n^{\gamma+1}\,dx\,dt + cR^{n-\frac{(q+\alpha)(p-1)
 +(\gamma+1)(\alpha+p-1)}{q-p+1}}\,,
\end{align*}
i.e.
\begin{equation} \label{eq:5.4.10}
I_3(R,T)
\le \frac{a}{4}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q+\alpha}\xi^{\lambda}_{R,T}
x_n^{\gamma+1}\,dx\,dt
+ cR^{n-\frac{(q+\alpha)(p-1)+(\gamma+1)(\alpha+p-1)}{q-p+1}}T\,.
\end{equation}
From \eqref{eq:5.4.8}--\eqref{eq:5.4.10} we obtain
\begin{equation} \label{eq:5.4.11}
\begin{aligned}
&\frac{a}{2}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q+\alpha}\xi_{R,T}^{\lambda} x_n^{\gamma+1}
 \,dx\,dt
 + \frac{|\alpha|}{2}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha-1}|Du|^p\xi_{R,T}^{\lambda}
 x_n\,dx\,dt \\
&\le c\big(R^{n-\frac{(q+\alpha)(p-1)+(\gamma+1)(\alpha+p-1)}{q-p+1}}T
 +R^{n-\frac{(\gamma+1)(\alpha+1)}{q-1}}T^{1-\frac{q+\alpha}{q-1}}\big).
\end{aligned}
\end{equation}
Choosing $T=R^{\theta}$ with $\theta>0$ such that both terms are of the same
order and taking $R\to\infty$,  for $\alpha$ satisfying \eqref{eq:5.4.8a} we obtain
\[
\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q+\alpha} x_n^{\gamma+1}\,dx\,dt = 0,
\]
which contradicts the assumption of non-triviality of the solution.
This proves the theorem in the case $p<2$.

In the case $p>2$, estimates \eqref{eq:5.4.8} and \eqref{eq:5.4.9} are still valid, 
and for the integral $I_3(R,T)$ in the case $\frac{\partial  u}{\partial  x_n}\le 0$
we have
\begin{align*}
I_3(R,T)
&=\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha} |Du|^{p-2}
 \frac{\partial  u}{\partial  x_n}\xi_{R,T}^{\lambda}\,dx\,dt  \\
&= -\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha}|Du|^{p-2}
 \Big(-\frac{\partial  u}{\partial  x_n}\Big)^{\frac{p-2}{p-1}}
 \Big(+\frac{\partial  u}{\partial  x_n}\Big)^{\frac{1}{p-1}} \xi^{\lambda}_{R,T}\,dx\,dt \\
&\le \int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha}|Du|^{p-2+\frac{p-2}{p-1}}
 \Big(-\frac{\partial  u}{\partial  x_n}\Big)^{\frac{1}{p-1}} \xi^{\lambda}_{R,T}\,dx\,dt
\end{align*}
and by the Young inequality, similarly to the previous argument,
\begin{align*}
I_3(R,T) 
&\le \frac{|\alpha|}{4}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha-1}|Du|^p\xi_{R,T}^{\lambda}
  x_n\,dx\,dt  \\
&\quad + c\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha+p-2}
\frac{\partial  u}{\partial  x_n} x_n^{2-p}\xi^{\lambda}_{R,T}\,dx\,dt  \\
&\le \frac{|\alpha|}{4}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha-1}|Du|^p\xi_{R,T}^{\lambda}
  x_n\,dx\,dt \\
&\quad + cR^{2-p}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha+p-1}
\big|\frac{\partial  \xi_{R,T}}{\partial  x_n}\big|\xi^{\lambda-1}_R\,dx\,dt  \\
&\le \frac{|\alpha|}{4}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha-1}|Du|^p
 \xi_{R,T}^{\lambda} x_n\,dx\,dt 
 + \frac{a}{4} \int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q+\alpha}\xi_{R,T}^{\lambda}
 x_n^{\gamma+1}\,dx\,dt  \\
&\quad +cR^{\frac{(2-p)(q+\alpha)}{q-p+1}}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+}
 x_n^{-\frac{(\gamma+1)(\alpha+p-1)}{q-p+1}}
 \big|\frac{\partial  \xi_{R,T}}{\partial  x_n}\big|^{\frac{q+\alpha}{q-p+1}}
 \xi_{R,T}^{\lambda-\frac{q+\alpha}{q-p+1}}\,dx\,dt  \\
&\le \frac{a}{4} \int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q+\alpha}\xi_{R,T}^{\lambda}
  x_n^{\gamma+1}\,dx\,dt + cR^{n-\frac{(p-1)(q+\alpha)+(\alpha+p-1)
 (\gamma+1)}{q-p+1}}T,
\end{align*}
i. e.
\begin{equation} \label{eq:5.4.20}
\begin{aligned}
I_3(R,T)
&\le \frac{|\alpha|}{4}\int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{\alpha-1}|Du|^p\xi_{R,T}^{\lambda}
x_n\,dx\,dt \\
&\quad + \frac{a}{4} \int_{\mathbb{R}_+}\int_{\mathbb{R}^n_+} u^{q+\alpha}\xi_{R,T}^{\lambda}
x_n^{\gamma+1}\,dx\,dt + cR^{n-\frac{(p-1)(q+\alpha)+(\alpha+p-1)(\gamma+1)}{q-p+1}}T,
\end{aligned}
\end{equation}
which together with \eqref{eq:5.4.8} and \eqref{eq:5.4.9} yields
\eqref{eq:5.4.11} again.
The proof can be completed similarly to the previous case.

\subsection*{Acknowledgments}
This work was supported by the Ministry of Education and Science of Russia, 
as a state order in the sphere of scientific activities 
(order No. 1.7706.2017/8.9). 
This publication was prepared with  support from RUDN University Program 5-100.

\begin{thebibliography}{00}

\bibitem{AC} C. Azizieh, P. Cl\'ement;
A priori estimates and continuation methods for positive solutions of 
p-Laplace equations, {\it J. Diff. Eqns}, \textbf{179} (2002), 213--245.

\bibitem{BCDN} H. Berestycki, I. Capuzzo Dolcetta, L. Nirenberg;
 Superlinear indefinite elliptic problems and nonlinear Liouville theorems,
 {\it Topol. Methods Nonlinear Anal.}, \textbf{4} (1994), 59--78.

\bibitem{BM} I. Birindelli, E. Mitidieri;
 Liouville theorems for elliptic inequalities and applications,
 {\it Proc. Royal Soc. Edinburgh}, \textbf{128A} (1998), 1217--1247.

\bibitem{BVP} M. F. Bidaut-V\'eron, S. I. Pohozaev;
Nonexistence results and estimates for some nonlinear elliptic problems,
 {\it Journ. Anal. Math.}, \textbf{84} (2001), 1--49.

\bibitem{DDE} E. N. Dancer, Y. Du, M. Efendiev;
Quasilinear elliptic equations on half- and quarter-spaces, 
{\it Adv. Nonlinear Stud.}, \textbf{13} (2013), 115--136.

\bibitem{FMS1} A. Farina, L. Montoro, B. Sciunzi;
Monotonicity and one-dimensional symmetry for solutions of $-\Delta_p u = f(u)$ 
in half-spaces, {\it Calc. Var. and PDE}, \textbf{43} (2012), 123--145.

\bibitem{FMS2} A. Farina, L. Montoro, B. Sciunzi;
 Monotonicity of solutions of quasilinear degenerate elliptic equation in 
half-spaces, {\it Math. Annalen}, \textbf{357} (2013), 855--893.

\bibitem{FMSR} A. Farina, L. Montoro, G. Riey, B. Sciunzi;
Monotonicity of solutions to quasilinear problems with a first-order term 
in half-spaces, {\it Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire,} 
\textbf{32(1)} (2015), 1--22.

\bibitem{FMS3} A. Farina, L. Montoro, and B. Sciunzi, Monotonicity in half-spaces of positive solutions to $-\Delta_p u = f(u)$ in the case $p>2$, https://arxiv.org/pdf/1509.03897.pdf.

\bibitem{F} R. Filippucci;
 A Liouville result on a half space, Recent Trends in Nonlinear Partial 
Differential Equations: Workshop in Honor of Patrizia Pucci�s 60th Birthday, 
May 28 - June 1, 2012, University of Perugia, Perugia, Italy.
James B. Serrin, Enzo L. Mitidieri and Vicentiu D. Radulescu (editors),
 American Mathematical Society, Providence (RI), 2013.

\bibitem{Fu} H. Fujita;
 On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, 
{\it Proc. Sympos. Pure Math.}, \textbf{18} (1968), 138--161.

\bibitem{GS1} E. Galakhov, O. Salieva;
 On blow-up of solutions to differential inequalities with singularities on 
unbounded sets, {\it Journ. Math. Anal. Appl.}, \textbf{408} (2013), 102--113.

\bibitem{GS2} E. Galakhov, O. Salieva;
Blow-up for nonlinear inequalities with singularities on unbounded sets. 
Current Trends in Analysis and its Applications: Proceedings of the IXth ISAAC 
Congress, Birkhauser, Basel, 2015. P. 299-305.

\bibitem{GS3} E. Galakhov, O. Salieva;
On nonexistence of nonnegative monotonic solutions for some quasilinear elliptic 
inequalities in a half-space, {\it Proc. Steklov Inst. Math.}, \textbf{293} (2016), 
140--150.

\bibitem{GS4} E. Galakhov, O. Salieva;
Nonexistence of solutions of the Dirichlet problem for some quasilinear elliptic 
equations in a half-space, {\it Diff. Eq.}, \textbf{52} (2016), 749--760.

\bibitem{MP} E. Mitidieri, S. Pohozaev;
A priori estimates and nonexistence of solutions of nonlinear partial differential 
equations and inequalities, {\it Proc. Steklov Inst. Math.}, 
\textbf{234} (2001), 3--383.

\bibitem{P} S. Pohozaev;
 Essentially nonlinear capacities generated by differential operators, 
{\it Dokl. Math.}, \textbf{357} (1997), 592--594.

\bibitem{PV} A. Porretta, L. V\'eron;
Separable solutions of quasilinear Lane-Emden equations, 
{\it J. Europ. Math. Soc.}, \textbf{15} (2013), 755--774.

\bibitem{S} O. Salieva;
 On monotonicity of solutions of Dirichlet problem for some quasilinear 
elliptic equations in half-spaces, {\it Eurasian Math. J.}, \textbf{6} (2015), 
59--76.

\bibitem{Z} H. Zou;
 A priori estimates and existence for quasi-linear elliptic equations, 
{\it Calc. Var.}, \textbf{33} (2008), 417--437.

\end{thebibliography}

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