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Boundary value problems, $p$-Laplacian, nonlocal problems
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\begin{document}
\title[Solvability of Perturbed $p-$Laplace Equation]
{Solvability of Perturbed $p-$Laplace Equation}
\author[Oleg Zubelevich]{Oleg Zubelevich\\ \\\tt
Department of Differential Equations and Mathematical Physics\\
Peoples Friendship University of Russia\\
Ordzhonikidze st., 3, 117198, Moscow, Russia\\
E-mail: ozubel@yandex.ru}
\email{ozubel@yandex.ru}
\curraddr{2-nd Krestovskii Pereulok 12-179, 129110, Moscow, Russia}
%\address{Department (\# 803) of Differential Equations
%$Moscow State Aviation Institute
%Volokolamskoe Shosse 4, 125993, Moscow, Russia}
%\date{}
\thanks{Partially supported by grants RFBR 05-01-01119.}
\subjclass[2000]{35J60}
\keywords{Boundary value problems, $p$-Laplacian, nonlocal problems.}
\begin{abstract}We consider elliptic boundary value problem
with perturbed $p-$Laplace operator, $p>1$ and prove weak existence
theorem.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}{Definition}[section]
\section{Introduction}
The $p-$Laplacian appears in the study of flow through porus media
($p=3/2$), nonlinear elasticity ($p\ge 2$) and glaciology ($p\in
(1,4/3]$). We refer to \cite{14y} for more background
material.
For $p-$Laplace equation with right hand side homogeneous in $u$,
existence and nonexistence results were obtained by many authors, see for example
\cite{10y}, \cite{15y}, \cite{14y}, \cite{19y}. Variational methods were
employed in \cite{16y} and others when trying to find positive solutions.
There are applications
which provide elliptic problems with nonlocal linear operators situated inside the
$p-$Laplacian:
\begin{equation}\label{eq0}\Delta_p A u=f(x)\in H^{-1,p'}(M),\quad u\mid_{\partial
M}=0,\quad \frac{1}{p}+\frac{1}{p'}=1,\end{equation}
here $A$ is a certain bounded linear operator.
If operator $A$ is a small perturbation of the identity then, as it is not
hard to show, the problem remains coercive, as for monotonicity, this property is
much delicate to check, and possibly it is destroyed even under
the small perturbation. It should be noted that in
the linear case ($p=2$) the monotonicity property is steady under the
small perturbation.
On the other hand, using standard variational technique
we have to consider weak convergent sequences but the operator $A$ has not necessarily
to be continuous with respect to weak convergence.
Our version of variational method
allows to prove the
existence theorem avoiding of using the monotonicity or weak convergence.
Linear case of equation (\ref{eq0}) with
nonlocal operator of the argument's dilation/compression
$$
Au=\sum_{i=-k}^ka_i u(q^ix),\quad q>1,\quad a_j\in
\mathbb{R}$$
was studied in \cite{ross1}, \cite{ross2}. In these articles the condition
of closeness to identity for operator $A$ is not assumed. The case of
coefficients $a_k$ dependent on $x$ was considered in
\cite{ross_3} in the linear setup.
Boundary value problems for elliptic functional-differential equations
have been studied in the articles \cite{6}, \cite{7} and others.
Boundary value problems for elliptic
equations with shifts in the space variables were
considered in \cite{1}, \cite{2}.
A theory of boundary value problems for elliptic differential-difference
equation in a bounded domain was constructed in \cite{8}.
\section{Main theorem}
Let $M$ be a bounded domain in $\mathbb{R}^m$ with smooth boundary
$\partial M$ and $x=(x_1,\ldots,x_m)$ be coordinates in $\mathbb{R}^m$.
Denote by $\partial_i$ the partial derivative
in the variable $x_i$.
For any $v\in L^r(M),\quad r\ge 1$ and $w\in L^{r'}(M),\quad 1/r+1/r'=1$ we put
$$(v,w)=\int_Mv(x)w(x)\,dx.$$
Denote by $\mathcal{W}$ a space of bounded linear operators
$G:H^{1,p}_0(M)\to H^{1,p}(M),\quad p>1$ with the following properties.
For any operator
$G\in \mathcal{W}$
there exists a bounded operator $G^+:L^p(M)\to
L^p(M)$ such that:
\begin{equation}
\label{commut}
\partial_i G=G^+\partial_i.
\end{equation}
Let
$$G^{+*}:L^{p'}(M)\to
L^{p'}(M),\quad \frac{1}{p}+\frac{1}{p'}=1$$ be conjugated operator,
then there is a bounded operator
$G^{+*+}:H^{-1,p'}(M)\to H^{-1,p'}(M)$ satisfying the equation:
\begin{equation}\label{conj}
\partial_i G^{+*}=G^{+*+}\partial_i.\end{equation}
Denote by $\mathcal{L}(H^{1,p}_0(M))$ the space of bounded linear
operators of $H^{1,p}_0(M)$ to itself.
Let $\eps\in \mathbb{R}$ be a real parameter, then
introduce an operator $A_\eps:H^{1,p}_0(M)\to H^{1,p}(M)$ by the formula:
\begin{equation}
\label{mainsum}
A_\eps=\mathrm{id}_{H^{1,p}_0(M)}+\eps\sum_{j=1}^lG_jT_j,\quad G_j\in \mathcal{W},
\quad T_j\in \mathcal{L}(H^{1,p}_0(M)).\end{equation}
Consider the $p$-Laplace operator:
$$\Delta_p v=\sum_{i=1}^{m}\partial_i(|\partial_i v|^{p-2}\partial_i v).$$
The main object of our study is
the following elliptic problem:
\begin{equation}\label{main_eq}
\Delta_p A_\eps u=f(x)\in L^{p'}(M),\quad
u\mid_{\partial M}=0.\end{equation}
\begin{theo}
\label{main_t}If $T_j^*G^{+*+}_j f\in L^{p'}(M),\quad j=1,\ldots,l$ then
there exists such a positive constant $\tilde{\eps}$ that for any
$\eps,\quad |\eps|<\tilde{\eps}$
problem (\ref{main_eq}) has a weak solution $u\in H^{1,p}_0(M)$ i.e.
for any $h\in H^{1,p}_0(M)$ one
has
$$-\sum_{i=1}^m(|\partial_i A_\eps u|^{p-2}\partial_i A_\eps u,\partial_i h)=(f,h).$$
\end{theo}
\section{Applications: Dilation/Compression Operators}
In this section we regard the
domain $M$ as star-shaped with respect to the origin.
Define an operator $\sigma:L^r(M)\to L^r(\mathbb{R}^m),\quad r\ge 1$ as follows:
$$\sigma v= \begin{cases}
v(x) & \text{if}\quad x\in M, \\
0 & \text{if}\quad x\in \mathbb{R}^m\backslash \overline{M}.
\end{cases}$$
Use this operator to construct
$R_qv=(\sigma v)(qx),\quad q>0.$ This
operator dilates/compresses the graph of the function $v(x)$ in $q$ times,
here $q$ is a constant.
One can show that the operator $R_q$ is bounded as an operator of the space
$H^{1,r}_0(M),\quad r\ge 1$
to $H^{1,r}(M)$ and as an operator of $L^r(M)$ to itself.
The operator $R_q$ commutes with the partial derivatives in the following
fashion:
$$
\partial_i R_q=R^+_q\partial_i,\quad R_q^+=qR_q.$$
The operator $R^*_q$ can also be written in the explicit form:
letting $R^*_q=q^{-m}R_{q^{-1}}$ and by the change of variable
$x\mapsto q^{-1}x$ in the integral we make sure that $(R_qv,w)=(v,R^*_qw)$.
So, one has: $R^{+*}_q=q^{1-m}R_{q^{-1}}$ and
$R^{+*+}_q=q^{-m}R_{q^{-1}}=R_q^*$.
To define operator $R^{+*+}_q$ on $H^{-1,p'}(M)$
note that any element $g\in H^{-1,p'}(M)$ presents as follows
\cite{adams}:
$$g=g_0+\sum_{i=1}^m\partial_ig_i,\quad g_j\in L^{p'}(M),\quad
j=0,\ldots,m.$$
Now we put:
$$(R^{+*+}_q g,h)=(g_0,R_qh)-\sum_{i=1}^m(g_i,\partial_iR_qh),\quad h\in H^{1,p}_0(M).$$
It is easy to see that formulas (\ref{commut}) and (\ref{conj})
are fulfilled and $R_q\in \mathcal{W}.$
Show that an operator
\begin{equation}
\label{gen_sum_c_d}
Q_\eps=\mathrm{id}_{H^{1,p}_0(M)}+
\eps\sum_{j=1}^{l}a_j(x)R_{q_j},\quad a_j\in C^1(\overline{M})
\end{equation}
can be presented in the form (\ref{mainsum}).
Consider the cases $q_j\ge 1$ and $q_j<1$ separately.
Introduce an operator of multiplication by $a$ by the
formula: $U_av=av(x)$. If $a(x)\in C^1(\overline{M})$ then
$U_{a(x)}\in\mathcal{L}(H^{1,p}_0(M))$.
Assume $q_j\ge 1$. Then obviously we have:
$a_{j}(x)R_{q_j}=R_{q_j}U_{a_j(q_j^{-1}x)}$.
If $q_j<1$ take a function $\widetilde{a}_j(x)\in C^1(\overline{M})$
such that $\widetilde{a}_j(x)=a_j(q_j^{-1}x)$ for $x\in q_j\overline{M}$,
then we have $a_{j}(x)R_{q_j}=R_{q_j}U_{\widetilde{a}_j(x)}$.
\section{Proof of Theorem \protect\ref{main_t}}
Define operator $B_\eps:H^{-1,p'}(M)\to H^{-1,p'}(M)$ by the formula:
$$B_\eps=\mathrm{id}_{H^{-1,p'}(M)}+\eps\sum_{j=1}^lT_j^*G^{+*+}_j.$$
Consider a linear function
$J_\eps(v)=(B_\eps f,v).$
We are going to minimize this function
on the level set of a
function $$F_\eps: H^{1,p}_0(M)\to \mathbb{R},\quad
F_\eps(v)=\int_M\sum_{i=1}^m|\partial_i A_\eps v|^p\,dx.$$
Such a variational scheme is not quite standard: problems with
homogeneous right hand side are usually treated in the opposite way:
it is the function $F_\eps(v)$ whose conditional extremum is looked for.
Though our case differs
from that one, the proposed technique provides much more direct and simple
way to obtain existence results even for the
homogeneous problems (in the subcritical case
of course !).
Since
this technique gives possibility to avoid from dealing with
weak convergent sequences one can apply it to a problem which contains
nonlocal linear operators in the right hand
side.
\begin{lem}
\label{pprel}
Take a positive constant $\eps_0$ and a constant $\gamma \ge 1$.
Then for any $\eps,\quad |\eps|<\eps_0$ and for any $a,b\in \mathbb{R}$
one has
$$|a+\eps b|^\gamma=|a|^\gamma+\eps \psi(a,b,\eps),$$
here the function $\psi$ satisfies the inequality:
$$|\psi(a,b,\eps)|\le c_1(|a|^\gamma+|b|^\gamma).$$
Positive constant $c_1$ depends only on $\eps_0$ and $\gamma$.\end{lem}
\proof
If $a=0$ or $\eps=0$ then the assertion is trivial, otherwise letting $\xi=b/a$
we have:
$$\frac{|\psi(a,b,\eps)|}{|a|^\gamma+|b|^\gamma}=
\frac{||a+\eps b|^\gamma-|a|^\gamma|}{|\eps|(|a|^\gamma+|b|^\gamma)}=
\frac{|\xi|}{1+|\xi|^\gamma}\cdot\frac{||1+\eps\xi|^\gamma-1|}{|\eps\xi|}\le
c_1.$$
\endproof
\begin{lem}\label{co_p_Lapl}
There are positive constants $\eps_1,c_2$ such that for any
$\eps,\quad |\eps|<\eps_1$ and for any $v\in H^{1,p}_0(M)$ one has
$$F_\eps(v)\ge c_2\|v\|^p_{H^{1,p}_0(M)}.$$
\end{lem}
\proof
According to Lemma \ref{pprel} write
\begin{align}
F_\eps(v)&=\int_M\sum_{i=1}^m|\partial_iv+\eps\partial_iGv|^p\,dx=
\int_M\sum_{i=1}^m|\partial_iv|^p+\eps\psi(\partial_iv,\partial_iGv,\eps)\,dx\nonumber\\
&=
\|v\|^p_{H^{1,p}_0(M)}+\eps\int_M\sum_{i=1}^m\psi(\partial_iv,\partial_iGv,\eps)\,dx.\label{l1}
\end{align}
Now the proof follows from the estimate of the last term in (\ref{l1}):
$$\Big|\int_M\sum_{i=1}^m\psi(\partial_iv,\partial_iGv,\eps)\,dx\Big|\le
c_1\int_M\sum_{i=1}^m|\partial_iv|^p+|\partial_iGv|^p\,dx\le
c_3\|v\|^p_{H^{1,p}_0(M)},$$ here $c_3$ is a positive constant independent
on $v$ and $\eps$.\endproof
\begin{lem}
\label{min}Consider a set
$$S_\eps=\{v\in H^{1,p}_0(M)\mid F_\eps(v)=1\}.$$
For any $\eps,\quad |\eps|<\eps_1$ the function $J_\eps\mid_{S_\eps}$ attains its
minimum, say at $\hat{v}_\eps$.
\end{lem}
\proof By the conditions of the Theorem, the function $J_\eps$ can be extended continuously to the
space $L^{p}(M)$, we do not introduce another notation for this extension.
As it follows from Lemma \ref{co_p_Lapl} the set $S_\eps$ is bounded in
$H^{1,p}_0(M)$, but the space $H^{1,p}_0(M)$ is compactly embedded in
$L^p(M)$. Consequently, $S_\eps$ is a compact subset of $L^p(M)$. Thus
the function $J_\eps$ attains its minimum $\hat{v}_\eps$.\endproof
Consider weak derivatives:
\begin{align}
J'_\eps h&=\frac{d}{ds}\Big|_{s=0}J_\eps(\hat{v}_\eps+sh)=(B_\eps f,h)=
J_\eps(h),\quad h\in H^{1,p}_0(M),\label{derJ}\\
F'_\eps(\hat{v}_\eps)h&=\frac{d}{ds}\Big|_{s=0}F_\eps(\hat{v}_\eps+sh)=
p\sum_{i=1}^m(|\partial_i A_\eps\hat{v}_\eps|^{p-2}\partial_i A_\eps\hat{v}_\eps,\partial_i
A_\eps h).\label{derF}
\end{align}
There is standard relation between these derivatives.
This relation is
described as follows.
\begin{lem}
\label{kernels}
For any $\eps,\quad |\eps|<\eps_1$
the following inclusion holds true:
\begin{equation}
\label{ker_incl}
\ker F'_\eps(\hat{v}_\eps)\subseteq \ker
J'_\eps.\end{equation}
\end{lem}
\proof
Fix $\eps,\quad |\eps|<\eps_1$ and let
\begin{equation}\label{hker}
h\in \ker F'_\eps(\hat{v}_\eps).\end{equation}
Define a function of two real arguments by the formula:
$$\ph(y,t)=F_\eps(y\hat{v}_\eps+th).$$
We want to show that for some positive $t_0$ there exists such a function
$y(t)\in C^1(-t_0,t_0),\quad y(0)=1$ that satisfies an equation
\begin{equation}
\label{tos}
\ph(y(t),t)=1.\end{equation} If it would be true, then the set
$\{y(t)\hat{v}_\eps+th\mid |t|