\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Nonexistence of solutions \hfil EJDE--2002/54} {EJDE--2002/54\hfil Darko \v Zubrini\'c \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 54, pp. 1--8. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Nonexistence of solutions for quasilinear elliptic equations with $p$-growth in the gradient % \thanks{ {\em Mathematics Subject Classifications:} 35J25, 35J60, 45J05. \hfil\break\indent {\em Key words:} Quasilinear elliptic, existence, nonexistence, geometry of domains. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted April 17, 2002. Published June 11, 2002.} } \date{} % \author{Darko \v Zubrini\'c} \maketitle \begin{abstract} We study the nonexistence of weak solutions in $W^{1,p}_{{\rm loc}}(\Omega)$ for a class of quasilinear elliptic boundary-value problems with natural growth in the gradient. Nonsolvability conditions involve general domains with possible singularities of the right-hand side. In particular, we show that if the data on the right-hand side are sufficiently large, or if inner radius of $\Omega$ is large, then there are no weak solutions. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{cor}[theorem]{Corollary}%[section] \newtheorem{prop}[theorem]{Proposition}%[section] \newtheorem{remark}[theorem]{Remark} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 \def\esssup{\mathop{\rm ess\,sup}} \def\essinf{\mathop{\rm ess\,inf}} \def\wo#1#2#3{W^{#1,#2}_0(#3)} \def\osc{\mathop{\rm osc}} \def\ov#1{\overline{#1}} \def\od#1#2{\frac{d#1}{d#2}} \section{Introduction} The aim of this article is to study nonsolvability in the weak sense of the quasilinear elliptic distribution equation \begin{equation}\label{pde} \begin{array}{ll} -\Delta_p u=F(x,u,\nabla u)\quad\mbox{in ${\cal D'}(\Omega)$,} &\\ \phantom{-\Delta_p}u=0\hfill\mbox{on $\partial \Omega$,}& \end{array} \end{equation} in the Sobolev space $W^{1,p}_0(\Omega)$. Note that we do not assume the solutions being essentially bounded. Here $\Omega$ is a domain in $\mathbb{R}^N$, $N\ge1$, $1
1$ is defined by $p'=\frac p{p-1}$.
We define weak solutions as functions $u\in W^{1,p}_0(\Omega)$
which satisfy
equation (\ref{pde}) in the weak sense:
$$
\int_{\Omega}[|\nabla u|^{p-2}\nabla u\cdot\nabla\phi-F(x,u,\nabla
u)\,\phi]\,dx=0,\quad\forall\phi\in C_0^{\infty}(\Omega).
$$
Nonsolvability for (\ref{pde}) has been studied for
$F=\tilde g_0|x|^m+\tilde f_0|\nabla u|^{p}$,
with
$\Omega=B_R(0)$, $\tilde f_0>0$, $\tilde g_0>0$, and
with solutions in the class of radial, decreasing and bounded
functions, see
Pa\v si\'c \cite{kan} and in Korkut, Pa\v si\'c, \v
Zubrini\'c \cite{kpz2}.
The aim of this article is to extend the
nonsolvability results obtained in \cite{kan} and \cite{kpz2} for radial
solutions of quasilinear elliptic problem in a ball to general
domains in $\mathbb{R}^N$. We deal with nonexistence of unbounded solutions as well.
To this end we use a combination of
results obtained in \cite{kpz2}
with the Tolksdorf comparison principle, see \cite{tolks}.
Existence of weak solutions for problems with strong dependence
on the gradient has been studied by Rakotoson
\cite{rakot86},
Boccardo, Murat and Puel \cite{bmp},
Maderna, Pagani and Salsa \cite{mps}, Ferone, Posteraro and Rakotoson \cite{fpr},
%Cho \and\ Che \cite{cc},
Korkut, Pa\v si\'c and \v Zubrini\'c \cite{kpz1},
\cite{kpz2}, Tuomela \cite{Tu}, see also the references therein.
Our main result is stated in
Theorem~\ref{nonex} below. As we have said,
nonsolvability conditions involve geometry of $\Omega$ with respect
to eventual singularities on the right-hand side.
As an illustration, below we provide a simple consequence
involving inner radius of
domain $\Omega$, that we define by
\begin{equation}
r(\Omega)=\sup\{r>0: \exists
x_1\in\Omega,\,\,B_r(x_1)\subseteq\Omega\}.\nonumber
\end{equation}
Here we also mention a nonsolvability result related to problem
(\ref{pde}),
involving inner radius of $\Omega$, obtained
in
Wang and Gao \cite{wg}, which complements an existence
result of Hachim and Gossez \cite{elgossez}, involving outer radius of domain.
These two papers deal with quasilinear elliptic problems
in which the nonlinearity on the right-hand side
does not depend on the gradient. We also mention a recent paper
of Bidaut-V\'eron and Poho\v zaev \cite{bvp} dealing with
nonexistence results for nonlinear elliptic problems
with nonlinearities $\ge|x|^\sigma u^Q$, where $\sigma\in\mathbb{R}$, $Q>0$.
Here we treat nonlinearities of different type.
From Ferone, Posteraro, Rakotoson \cite[Theorem 3.3]{fpr} it follows, under very general
conditions on $F(x,\eta,\xi)$, that if $|\Omega|$ is sufficiently small
then there exists a weak solution of (\ref{pde}). We obtain a
complementary result, showing that
if $\Omega$ is has sufficiently large inner radius, then (\ref{pde})
has no weak solutions.
Equivalently, if a domain $\Omega$ is fixed, and if the data entering
the right-hand side of (\ref{pde}) are sufficiently large, then
(\ref{pde}) does not possess weak solutions.
For the reader's convenience we
state a special case of our main result formulated in
Theorem~\ref{nonex}.
\begin{cor}[Nonexistence]\label{nonc}
Assume that $\Omega$ is a domain in
$\mathbb{R}^N$ and there exist positive real numbers $\tilde g_0$ and
$\tilde f_0$ such that
\begin{equation}\label{gfFF}
F(x,\eta,\xi)\ge \tilde g_0+\tilde f_0|\xi|^p
\end{equation}
for a.e.\ $x\in\Omega$, and all $\eta\in\mathbb{R}$, $\xi\in\mathbb{R}^N$.
Assume that
$r(\Omega)<\infty$, where $r(\Omega)$ is inner radius of $\Omega$, and
\begin{equation}
\tilde g_0\cdot\tilde f_0^{p-1}\cdot r(\Omega)^p\ge C,
\end{equation}
where $C$ is explicit positive constant in (\ref{Cnon}) with
$m_0=0$.
Then (\ref{pde}) has no nonnegative weak
solutions in the space $W^{1,p}_0(\Omega)$.
\end{cor}
\paragraph{Remark 1.} It is possible to prove another variant of
nonexistence result stated in Corollary~\ref{nonc}
when $r(\Omega)=\infty$. Assume that $F(x,\eta,\xi)\ge \tilde g_1$, where
$\tilde g_1$ is a positive constant. Then it can be proved that
equation (\ref{pde})
has no weak solutions in the space $W^{1,p}_{{\rm loc}}(\Omega)\cap
L^{\infty}(\Omega)$. Note that here we have a weaker assumption on
$F(x,\eta,\xi)$ than in (\ref{gfFF}), but a smaller function space in which we claim to
have nonexistence of weak solutions
than in Corollary~\ref{nonc}. To show this nonexistence result,
assume by contradiction that there exists a solution
$u\in W^{1,p}_{{\rm loc}}(\Omega)\cap L^{\infty}(\Omega)$. It suffices to use
oscillation estimate in Korkut, Pa\v si\'c, \v Zubrini\'c
\cite[Proposition~12]{kpz1}:
\begin{equation}\label{inner}
\osc_{\Omega} u\ge
C\cdot r(\Omega)^{p'}\essinf_{\Omega\times(0,\infty)\times\mathbb{R}^N}
F(x,\eta,\xi)^{p'-1}.
\end{equation}
where $C$ is an explicit positive constant depending only on $p$ and $N$,
and $\osc_\Omega u=\esssup_\Omega u-\essinf_\Omega u$. Since $r(\Omega)=\infty$,
we obtain that $\osc_\Omega
u=\infty$, which contradicts $u\in L^\infty(\Omega)$.
\section{Nonexistence of weak solutions in $W^{1,p}_{{\rm loc}}(\Omega)$}
The main result of this paper is stated in Theorem~\ref{nonex} below.
It complements the existence result stated in Ferone, Posteraro
and Rakotoson \cite[Theorem
3.3]{fpr}.
It also extends
\cite[Theorem~8(c)]{kpz2}, where nonexistence result has been
obtained for $\Omega=B_R(0)$, $F=\tilde g_0|x|^m+\tilde
f_0|\xi|^p$, and in the class of decreasing, radial functions
$u\in\wo1p\Omega\cap L^\infty(\Omega)$. Here we state
nonexistence result for (\ref{pde}) where $\Omega$ can be arbitrary domain
in $\mathbb{R}^N$ (even unbounded), allowing more general nonlinearities
than in \cite{kpz2}, still with strong
dependence in the gradient.
\begin{theorem}[Nonexistence]\label{nonex} Let $\Omega$ be a domain
in $\mathbb{R}^N$ and assume that $m_0>\max\,\{-p,-N\}$. Let there exist
$x_0\in\Omega$ and $R>0$ such that
$B_R(x_0)\subset\Omega$, and
\begin{equation}\label{F>}
F(x,\eta,\xi)\ge\tilde g_0|x-x_0|^{m_0}+\tilde f_0|\xi|^p,
\end{equation}
for a.e.\ $x\in B_R(x_0)$, and all $\eta\ge0$, $\xi\in\mathbb{R}^N$.
Assume that $\tilde g_0$, $\tilde f_0$
are positive real numbers such that
\begin{equation}\label{non}
\tilde g_0\cdot\tilde f_0^{p-1}\cdot R^{m_0+p}> C,
\end{equation}
where
\begin{equation}\label{Cnon}
C=
\left\{
\begin{array}{ll}
\displaystyle
[(m_0+p)(p')^p]^{p-1}(m_0+N) & \mbox{ for $p>N$,}\\[3pt]
\displaystyle
[(m_0+N)(p')^p]^{p-1}(m_0+N) & \mbox{ for $p\le N$.}
\end{array}
\right.
\end{equation}
Then quasilinear elliptic distribution equation $-\Delta_p u=F(x,u,\nabla
u)$ has no weak solutions
$u\in W^{1,p}_{{\rm loc}}(\Omega)$ such that $u\ge0$ on $\partial B_R(x_0)$.
\end{theorem}
Here the condition $u\ge0$ on $\partial B_R(x_0)$ means by definition that
$u^-|_{B_R(x_0)}\in\wo 1p{B_R(x_0)}$, where $u^-=\max\{-u,0\}$.
The proof of Theorem~\ref{nonex} is based on iterative procedure recently introduced by Pa\v
si\'c in \cite{kan}. Following Korkut, Pa\v si\'c
and \v Zubrini\'c \cite{kpz2}
we introduce a sequence of
functions $\omega_n:(0,T]\to\mathbb{R}$, $T=|B|$, $B=B_R(x_1)$, by $\omega_n=z_0+z_1+\dots+z_n$, where
functions $z_k(t)$ are defined inductively by
\begin{equation}\label{zz}
z_{k+1}(t)=f_0\int_0^t\frac{z_k(s)^\delta}{s^\varepsilon}ds,\quad z_0(t)=g_0
t^\gamma,
\end{equation}
with the constants defined by
\begin{equation}\label{cde}
\gamma=1+\frac {m_0}N,\quad\delta=p',\quad \varepsilon=p'(1-\frac1N),
\end{equation}
and $g_0$, $f_0$ are positive constants:
\begin{equation}\label{veza}
g_0=\frac{\tilde g_0}{C_N^{\frac{m_0+p}N}N^{p-1}(m_0+N)},\quad
f_0=\tilde f_0.
\end{equation}
It can be shown that (see \cite[Proposition~1]{kpz2}):
\begin{equation}\label{zm}
z_m (t)=\frac{g_0^{\delta^{m}}f_0^{\sum^{m-1}_{k=0}\delta^{k}}
t^{(1-\varepsilon)\sum^{m-1}_{k=0}\delta^{k}+\gamma\delta^m}}
{\prod^{m}_{k=1}[(1-\varepsilon)\sum^{k-1}_{j=0}\delta^{j}+\gamma\delta^k]^{\delta^{m-k}}}.
\end{equation}
It has been proved in \cite[Proposition 2]{kpz2} that
if
\begin{equation}\label{dce}
\delta>\frac{\varepsilon-1}\gamma+1,\quad \delta>1,\quad \gamma>0,\quad \varepsilon\in\mathbb{R},
\end{equation}
then condition
\begin{equation}\label{hypn}
g_0^{\delta-1}f_0> C_1:=\left\{
\begin{array}{ll}
\displaystyle\frac{[\gamma(\delta-1)-\varepsilon+1]\delta^{\delta'}}
{(\delta-1)T^{\gamma(\delta-1)-\varepsilon+1}}& \mbox{for $\varepsilon<1$,}\\
\displaystyle\frac{\gamma\,\delta^{\delta'}}
{T^{\gamma(\delta-1)-\varepsilon+1}}& \mbox{for $\varepsilon\ge1$.}
\end{array}
\right.
\end{equation}
implies that
$\omega_n(t)\to\infty$ as
$n\to\infty$ for all
$t\in [t^*,T]$, where
\begin{equation}\label{t*}
t^*:=\left\{
\begin{array}{ll}
\displaystyle\big(\frac{[\gamma(\delta-1)-\varepsilon+1]\delta^{\delta'}}
{(\delta-1)f_0g_0^{\delta-1}}\big)^{1/[\gamma(\delta-1)-\varepsilon+1]}& \mbox{for
$\varepsilon<1$,}\\[3pt]
\displaystyle\big(\frac{\gamma\,\delta^{\delta'}}
{f_0g_0^{\delta-1}}\big)^{1/[\gamma(\delta-1)-\varepsilon+1]}& \mbox{for
$\varepsilon\ge1$.}
\end{array}
\right.
\end{equation}
Condition (\ref{non}) is equivalent with (\ref{hypn}), which in
turn is equivalent with $t^*< T$.
We obtain a more precise result in the following lemma.
\begin{lemma}
Assume that conditions (\ref{dce}) and (\ref{hypn}) are fulfilled.
Then we have
\begin{equation}\label{on>}
\omega_n(t)\ge
\frac{d^{n+1}-1}{d-1}\cdot g_0t^\gamma,
\end{equation}
for all $t\in[t^*,T]$,
where $d=\delta^{\delta'-1}>1$.
\end{lemma}
\paragraph{Proof.} It suffices to prove that
\begin{equation}
\frac{z_{n+1}(t)}{z_n(t)}\ge\delta^{\delta'-1},
\end{equation}
for all $t\in[t^*,T]$,
since then
\begin{equation}
\omega_n(t)=z_0(t)+\dots+z_n(t)\ge g_0t^\gamma\sum_{k=0}^n d^k=
g_0t^\gamma\frac{d^{n+1}-1}{d-1},
\end{equation}
Using (\ref{zm}) we obtain that
\begin{equation}\label{q}
\frac{z_{n+1}(t)}{z_n(t)}=\frac{B(t)^{\delta^n}}{\left(\prod_{k=1}^n
D_k^{\delta^{n-k}}\right)^{\delta-1}D_{n+1}},
\end{equation}
where
\begin{equation}
B(t)=g_0^{\delta-1}f_0t^{\gamma(\delta-1)-\varepsilon+1},\quad
D_k=(1-\varepsilon)\sum_{j=0}^{k-1}\delta^j+\gamma\delta^k.
\end{equation}
Let us consider the case $\varepsilon\ge1$ (it is equivalent with $p\le
N$). In this case we have $D_k\le\gamma\delta^k$, which enables us to estimate the
denominator on the right-hand side of (\ref{q}):
\begin{eqnarray}
\Big(\prod_{k=1}^n
D_k^{\delta^{n-k}}\Big)^{\delta-1}D_{n+1}
&\le&\Big(\prod_{k=1}^n
(\gamma\delta^k)^{\delta^{n-k}}\Big)^{\delta-1}\gamma\delta^{n+1}\nonumber\\
&=&\gamma^{\delta^n}\delta^{\frac{(2\delta-1)(\delta^n-1)}{\delta-1}-\delta^n+2}.\nonumber
\end{eqnarray}
Here we have used the identity
$\sum_{k=1}^nk\cdot\delta^{n-k}=\frac{(2\delta-1)(\delta^n-1)}{(\delta-1)^2}-
\frac{\delta^n+n-1}{\delta-1}$. Therefore
$$
\frac{z_{n+1}(t)}{z_n(t)}\ge\delta^{\delta'-1}\left(\frac
{B(t)}{\gamma\cdot\delta^{\delta'}}\right)^{\delta^n}\ge\delta^{\delta'-1},
$$
since $B(t)\ge\gamma\cdot\delta^{\delta'}$ is equivalent with $t\ge t^*$.
It is easy to see that (\ref{q}) holds also with modified $B(t)$
and $D_k$:
\begin{equation}
B(t)=g_0^{\delta-1}f_0t^{\gamma(\delta-1)-\varepsilon+1}(\delta-1),\quad
D_k=(\gamma(\delta-1)-\varepsilon+1)\delta^k+\varepsilon-1.
\end{equation}
Therefore if we assume that $\varepsilon<1$ (that is, $p>N$)
we obtain $D_k\le(\gamma(\delta-1)-\varepsilon+1)\delta^k$, and we can proceed in
the same way as above, by noting that
$B(t)\ge(\gamma(\delta-1)-\varepsilon+1)\delta^{\delta'}$ is equivalent with $t\ge t^*$ also in
this case.
\hfill$\Box$
\begin{lemma} We have
\begin{equation}\label{ode<}
\od{\omega_n}t\le g_0\gamma
t^{\gamma-1}+f_0\frac{\omega_n(t)^\delta}{t^\varepsilon}
\end{equation}
for all $n$ and for all
$t\in(0,T)$.
\end{lemma}
\paragraph{Proof.} Using (\ref{zz}) we obtain
\begin{eqnarray}
\od{\omega_n}t
%&=&\od{}t(z_0(t)+z_1(t)+\dots +z_n(t))\nonumber\\
&=&g_0\gamma
t^{\gamma-1}+f_0\frac{z_0(t)^\delta}{t^\varepsilon}+\dots+f_0\frac{z_{n-1}(t)^\delta}{t^\varepsilon}\nonumber\\
&\le& g_0\gamma
t^{\gamma-1}+f_0\frac{z_0(t)^\delta+\dots+z_{n-1}(t)^\delta+z_n(t)^\delta}
{t^\varepsilon}\nonumber\\
&\le& g_0\gamma
t^{\gamma-1}+f_0\frac{(z_0(t)+\dots+z_{n-1}(t)+z_n(t))^\delta}
{t^\varepsilon},\nonumber
\end{eqnarray}
where in the last inequality we have used $\delta>1$.
\hfill$\Box$
\begin{lemma}\label{unab} Let $m_0>\max\,\{-p,-N\}$ and let us
define, $B=B_R(x_0)$,
\begin{equation}\label{un}
u_n(x)=\int_{C_N|x-x_0|^N}^{|B|}\frac{\omega_n(s)^{p'-1}}{s^{p'(1-\frac1N)}}ds.
\end{equation}
(a) We have $u_n\in\wo1pB\cap C^2(B\setminus\{x_0\})\cap C(\ov B)$, and
\begin{equation}\label{pdeC_1$,
($C_1$ is defined in (\ref{hypn})), which in turn
is equivalent with $t^*