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Boundary value problems, $p$-Laplacian, nonlocal problems
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\begin{document}
\title[On Elliptic Problem with Perturbed $p-$Laplace Operator]
{On Elliptic Problem with Perturbed $p-$Laplace Operator}
\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 and prove weak existence theorem
without monotonicity assumption.
\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 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,\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. In should be noted that in
the linear case ($p=2$) the monotonicity property is steady with respect to 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.
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
$$(v,w)=\int_Mv(x)w(x)\,dx.$$
Let $G:H^{1,p}_0(M)\to H^{1,p}(M),\quad p\ge 2$ be a bounded linear
operator. Suppose, there exists such a bounded operator $G^+:L^p(M)\to
L^p(M)$ that the following equality holds:
\begin{equation}
\label{commut}
\partial_i G=G^+\partial_i.
\end{equation}
There is a conjugated operator
$$G^{+*}:L^{p'}(M)\to
L^{p'}(M),\quad \frac{1}{p}+\frac{1}{p'}=1.$$
Similarly, assume that a bounded operator
$G^{+*+}:H^{-1,p'}(M)\to H^{-1,p'}(M)$ satisfy the equation:
\begin{equation}\label{conj}
\partial_i G^{+*}=G^{+*+}\partial_i.\end{equation}
Introduce operator $A_\eps=I+\eps G$, here $I$ is the identity operator.
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 $G^{+*+}f\in L^{p'}(M)$ 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}
\subsection*{Applications}
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 changing 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 this operator
on $H^{-1,p'}(M)$ for any $g\in H^{-1,p}(M)$ we put:
$$(R^{+*+}_q g,h)=(g,R_qh),\quad h\in H^{1,p'}_0(M).$$
Now it is easy to see that formulas (\ref{commut}) and (\ref{conj})
will be fulfilled if as an operator $G$ we take the following linear
combination:
$$G=\sum_{k=1}^la_kR_{q_k}\quad a_j \in \mathbb{R},\quad q_j>0.$$
In the conclusion of this section, note that there are
another sources of proper operators $G$. They are linear combinations of shift operators
of the form $\sum_{k=1}^la_kv(x+\tau_k),\quad v(x)\in L^r(M),\quad
\tau_k\in \mathbb{R}^m.$ Taking into account the arguments above, one sees that the
operator $G$ can also be a linear combination of
dilation/compressions and shifts operators.
\section{Proof of Theorem \protect\ref{main_t}}Consider the operators
$A^\iota_\eps=I+\eps G^\iota,\quad \iota\in\{+,+*,+*+\}.$
Here $I$ is the identity operator of the corresponding space.
Define a liner function $J_\eps: H^{1,p}_0(M)\to \mathbb{R}$ by the formula
$J_\eps(v)=(A_\eps^{+*+}f,v).$
We are going to minimize this function
on the level set of a
function $$F_\eps: H^{1,p}_0(M)\to \mathbb{R},\quad
F_\eps(v)=\int_M\sum_{i=1}^m|\partial_i A_\eps v|^p\,dx.$$
\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
The following Lemma states the coercivity of the operator $\Delta_p A_\eps$
when the parameter $\eps$ is small.
\begin{lem}\label{co_p_Lapl}
There is a 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
$$B_\eps=\{v\in H^{1,p}_0(M)\mid F_\eps(v)=1\}.$$
For any $\eps,\quad |\eps|<\eps_1$ the function $J_\eps\mid_{B_\eps}$ 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 $B_\eps$ is bounded in
$H^{1,p}_0(M)$, but the space $H^{1,p}_0(M)$ is compactly embedded in
$L^p(M)$. Consequently, $B_\eps$ 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)=(A^{+*+}_\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|