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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
$\{y(t)\hat{v}_\eps+th\mid |t|<t_0\}$ is a curve
 on the manifold $B_\eps.$ This curve passes across the point $\hat{v}_\eps$
 and has the tangent vector $h$ at this point.

To solve equation (\ref{tos}) we use the implicit function theorem.
Observing that $\ph(1,0)=F_\eps(\hat{v}_\eps)=1$ and by formula
\begin{align}
\ph_y(y,t)\mid_{(y,t)=(1,0)}&=F_\eps'(y\hat{v}_\eps+th)\hat{v}_\eps\mid_{(y,t)=(1,0)}\nonumber\\
&=
F_\eps'(\hat{v}_\eps)\hat{v}_\eps=pF_\eps(\hat{v}_\eps)=p\ne 0,\label{nondf}\end{align}
we obtain desired function $y(t)$.

Note that the derivative of the function $y(t)$ vanishes at zero:
\begin{equation}
\label{y_t=0}
y_t(0)=0.\end{equation}

Indeed, since $F_\eps(y(t)\hat{v}_\eps+th)=1$ for $|t|<t_0$, we have
\begin{align}
\frac{d}{dt}\Big|_{t=0}F_\eps(y(t)\hat{v}_\eps+th)&=
F'_\eps(y(0)\hat{v}_\eps)(y_t(0)\hat{v}_\eps+h)\nonumber\\&=
y_t(0)F'_\eps(\hat{v}_\eps)\hat{v}_\eps+F'_\eps(\hat{v}_\eps)h=0.\label{dy=0}\end{align}
Combining  formula (\ref{dy=0}) with (\ref{hker}) and  (\ref{nondf})
we obtain (\ref{y_t=0}).

Since $y(t)\hat{v}_\eps+th\in B_\eps$ for $|t|<t_0$, and by Lemma \ref{min}
 a function $\rho(t)=J_\eps(y(t)\hat{v}_\eps+th)$
  attains
its local minimum at $t=0$ so we have:
$$
\rho_t(0)=J'_\eps(y_t(0)\hat{v}_\eps+h)=0.$$
This formula and formula (\ref{y_t=0}) imply: $J'_\eps h=0$.
\endproof

According to the Lagrange multipliers theorem,
formula (\ref{ker_incl}) implies that there exists such a constant
$\lambda_\eps$ that $\lambda_\eps F'_\eps(\hat{v}_\eps)=J'_\eps$.
In other words by formulas (\ref{derJ}), (\ref{derF}) for any $h\in H^{1,p}_0(M)$
it follows that:
\begin{equation}
\label{gen}
\lambda_\eps 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)=(A^{+*+}_\eps f,h).
\end{equation}
Using formulas (\ref{commut}), (\ref{conj}) write:
\begin{align}
\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)&=\sum_{i=1}^m(|\partial_i A_\eps\hat{v}_\eps|^{p-2}\partial_i A_\eps\hat{v}_\eps,
A^{+}_\eps\partial_i h)\nonumber\\
=\sum_{i=1}^m(A^{+*}_\eps|\partial_i A_\eps\hat{v}_\eps|^{p-2}\partial_i A_\eps\hat{v}_\eps,
\partial_i h)&=-\sum_{i=1}^m(\partial_iA^{+*}_\eps|\partial_i A_\eps\hat{v}_\eps|^{p-2}\partial_i A_\eps\hat{v}_\eps,
 h)\nonumber\\
 &=
-\sum_{i=1}^m(A^{+*+}_\eps\partial_i(|\partial_i A_\eps\hat{v}_\eps|^{p-2}\partial_i A_\eps\hat{v}_\eps),
 h).\nonumber
\end{align}
This formula and formula (\ref{gen}) imply:
\begin{equation}
\label{v}
-\lambda_\eps p\sum_{i=1}^mA^{+*+}_\eps\partial_i(|\partial_i A_\eps\hat{v}_\eps|^{p-2}\partial_i A_\eps\hat{v}_\eps)
=A^{+*+}_\eps f.\end{equation}
It is easy to see that the operator $A^{+*+}_\eps:H^{-1,p}(M)\to H^{-1,p}(M)$ is
invertible, provided $\eps$ is small enough, so that equation (\ref{v}) is
equivalent to the following one:
\begin{equation}
\label{v1}
-\lambda_\eps p\sum_{i=1}^m\partial_i(|\partial_i A_\eps\hat{v}_\eps|^{p-2}\partial_i A_\eps\hat{v}_\eps)
=f.\end{equation}
Take  a constant $\tau_\eps$ so that the equation $-\lambda_\eps p|\tau_\eps|^{p-2}\tau_\eps=1$
is satisfied, and substituting
$\hat{v}_\eps=\tau_\eps u$ in (\ref{v1}) we see that the  function $u$ solves (\ref{main_eq}).

The Theorem is proved.



%\subsection*{Acknowledgement}
%The author would like to thank ...?




\begin{thebibliography}{99}

\bibitem{6} A. B. Antonevich, The index and normal solvability of general
elliptic boundary value problems with a finite group of shifts on the
boundary. Differential Equations, 8 (1972).


\bibitem{10y} M. Guedda and L. Veron, Local and global properties of
solutions of quasilinear elliptic equations. J. Differential Equations 76
(1988) 159-189.

\bibitem{14y}J. L. Lions, Quelques M\'ethodes de R\'esolution des
Probl$\rm{\grave{e}}$mes aux Limites Non-lin\'eaires. Dunod, Paris, 1969.

\bibitem{19y}A. Szulkin, Ljusternik-Schnirelmann theory on $C^1$
manifolds. Ann. Inst. H. Poincar\'e. Anal. Non Lin\'eaire 5 (1988) 119-139.

\bibitem{1}  G.G. Onanov A. L. Skubachevski{\u\i}, Differential equations
with diviating arguments in stationary problems in the mechanics of
deforming media., Soviet Appl. Mech. 15 (1979), 391-397. MR 80e:73055.

\bibitem{15y}M. Otani, Existence and nonexistence of nontrivial solutions
of some nonlinear degenerate elliptic equations. J. Funct. Anal. 76 (1988)
140-159.

\bibitem{16y}M. Otani and T. Teshima, On the first eigenvalue of some
quasilinear elliptic equations. Proc. Japan Acad. Ser. A Math. Sci. 64
(1988) 8-10

\bibitem{7}V. S. Rabinovich, The solvability of differential-difference
equations in $\mathbb{R}^n$ and in a half-space, Soviet Math. Dokl.
19 (1978) 1498-1502 (1979). MR 80h:35125.

\bibitem{ross1} L.E. Rossovski{\u\i}, The coercivity of
functional-differential equations. Math. Notes, 59(1996), 75-82, MR
97b:35184.

\bibitem{ross2}L.E. Rossovski{\u\i},
Boundary Value Problems for Elliptic Functional-Differential Equations
with Dilatations and Compressions of the Arguments.
Proceedings of Moscow Math. Soc. vol.
62 (2001).

\bibitem{8}A. L. Skubachevski{\u\i}, Elliptic functional differential
equations and applications, Operator Theory, Advances and Applications,
Birkh$\mathrm{\ddot{a}}$user, Basel-Boston-Berlin, 1996.

\bibitem{2} M A. Vorontsov N. G. Iroshnikov and R. L. Abernathy,
Diffractive patterns in a nonlinear optical two-dimensional fidback
system with field rotation, Chaos, Solitons and Fractals,  4 (1994)



\end{thebibliography}
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