11institutetext: E. AZROUL 22institutetext: Laboratory of Mathematical
Analysis and Applications, Faculty of sciences Dhar El Mahraz, Sidi Mohamed
Ben Abdellah University,Fes, Morocco, 22email<EMAIL_ADDRESS>33institutetext: A. ABNOUNE 44institutetext: Laboratory of Mathematical
Analysis and Applications, Faculty of sciences Dhar El Mahraz, Sidi Mohamed
Ben Abdellah University,Fes, Morocco, 44email<EMAIL_ADDRESS>55institutetext: M.T.K. ABBASSI 66institutetext: Laboratory of Mathematical
ALgebra and geometry, Faculty of sciences Dhar El Mahraz, Sidi Mohamed Ben
Abdellah University,Fes, Morocco, 66email<EMAIL_ADDRESS>
#  Quasi-linear elliptic equations with data in $L^{1}$ on a compact
Riemannian manifold

E. AZROUL  A. ABNOUNE and M.T.K. ABBASSI

###### Abstract

This work is dedicated to the study of quasi-linear elliptic problems with
$L^{1}$ data, the simple model will be the next equation on $(M,g)$ a compact
Riemannian manifold.

$-\Delta_{p}u=f$

Where $f\in L^{1}(M)$ .Our goal is to develop the functional framework and
tools that are necessary to prove the existence and the uniqueness of the
solution for the previous problem. Notice that our argument can be used to
deal with a more general class of quasi-linear equations.

Introduction

This article is dedicated to the study of quasi-linear elliptic equations with
data in $L^{1}(M)$, the major difficulty encountered when one is interested in
such problems is that the classical theories of existence, either using
variational methods or compactness methods, are not applicable. Hence the need
to use new techniques to prove the existence and uniqueness of solutions for
such problems.

Note that the importance of trying to solve problems with data in $L^{1}$ is
not limited to a purely theoretical framework, but also for applicable
reasons, to be convinced it is recommended to the reader various references as
for example: $[1],[2]$ and $[3]$ where different examples of equations having
an application in physics are presented.

The first significant advance in this direction is due to Stampacchia in
$[3],$ where he considers second-order linear elliptic operators with non-
regular data of the form

$L(u)=f$

where

$L(u)=\sum_{i,j=1}^{N}a_{ij}\frac{\partial^{2}u}{\partial
x_{ij}}+\sum_{i=1}^{N}\frac{\partial u}{\partial x_{i}}+cu$

where $a_{ij},b_{i},c$ are functions with specific hypotheses .

In his famous works, Stampacchia uses the notion of duality to solve these
classes of problems. Existence and uniqueness results have been proved in this
direction thanks to the linear character and the ”regularizing effect” of the
operator. Note that in the case where $L\equiv\Delta$ then the notion of
duality coincides with the notion of solution in the sense of distributions.
In the linear framework and with the notion of duality,we can even consider
data measures.

The extension of Stampacchia’s work to non-linear operators has been done by
several mathematicians. The first works were realized by Boccardo, Mu-rat,
Gallouet and their collaborators. The main difficulties for non-operators
linear consists of two points:

  1. 1.

The sense in which the solution is defined (the meaning of the good solution
and the method of its construction).

  2. 2.

The uniqueness of the ”good” solution.

Note that the second question is legitimate given Serrin’s counter example for
the non-uniqueness of the solution, see $[4]$.

To go beyond the first difficulty we proceed by approximation by returning In
the variational framework, the main step is to demonstrate properties of the
solutions for approximate problems that remain conserved by passing to the
limit. This passage is feasible by imposing natural conditions on the space of
the test functions.

concerning the second difficulty,we demonstrate partial results, especially
for the $\Delta_{p}$ operator we are able to demonstrate the uniqueness of the
solution. It will be noted that the uniqueness of the solution is usually
true.

We organize this work in two sections.In the first section we briefly recall
the functional spaces of Sobolev and Marcinkiewicz, on a compact Riemannian
manifold, which will be very useful in this paper.In Subsection 1.2.3 we
define the notion of the weak solution. Using variational techniques we prove
the existence and the uniqueness of the energy solution for the problem :

$-\Delta_{p}u=f,\quad u\in W_{0}^{1,p}(M)$

where $(M,g)$ a Riemannian manifold.This result is a natural extension of the
Lax-Milgram Theorem to the non-linear case in a Euclidean space. In Section
1.3 we present the proof of the Picone inequality in its general version on a
compact Riemannian manifold and as a consequence we obtain a comparison
principle for quasi-linear problems with a ”concave” term compared to
Laplacian.This result generalizes that of Brezis-Kamin in $[5]$ for the
Laplacian, see $[6]$.The second section is dedicated to define the notion of
entropy on a compact Riemannian manifold, in which we will study our problem.
We begin by defining the functional framework that will be given using the
truncation function, ie we analyze the functional properties of $T_{k}(u)$
instead of $u.$ Note that $T_{k}:\mathbb{R}\rightarrow\mathbb{R}$ defined by :

$T_{k}(s)=\left\\{\begin{array}[]{ccc}{s}&{\text{ if }}&{|s|\leq k}\\\
{k\operatorname{sign}(s)}&{\text{ if }}&{|s|>k}\end{array}\right.$

After giving the definition of solution in the sense of entropy, we prove the
existence and uniqueness of the solution in this context, some properties of
the entropy solution in Marcinkiewicz’s spaces on Riemannian manifolds will be
deduced. At the end of the section some generalizations for non-homogeneous
quasi-linears operators and with second members that may depend on $u$ will be
presented, see $[1]$.

##  1 Preliminaries

###  1.1 some definitions

######  Definition 1 (Equi-integrable functions in $L^{1}$)

Let X be a set of $\mathbb{R}^{N}$. We say that a sequence
$\left\\{f_{n}\right\\}$ of functions of $L^{1}(X)$ is equi-integrable if, for
all $\varepsilon>0$, there exists $\delta>0$ such as meas $(E)<\delta$ with
$E\subset X$ will result for all $n$,

$\int_{E}\left|f_{n}(x)\right|dx\leq\varepsilon.$

We use often the next result of compactness in $L^{1}$.

######  Lemma 1 (Lemma of Vitali : compactness in $L^{1}$)

Let $X$ a finite measure set for the Lebesgue measure of $\mathbb{R}^{N}.$ Let
$\left\\{f_{n}\right\\}$ a sequence of functions of $L^{1}(X)$ which converges
everywhere to $f,$ and which is equi-integrable. Then $f\in L^{1}(X)$ and
$\left\\{f_{n}\right\\}$ converges strongly to $f$ in $L^{1}(X)$.

###  1.2 Functional spaces

#### Sobolev spaces $W^{k,p}(M)$

see$[7]$ and $[8]$

Let $(M,g)$ a Riemannian manifold, for an integer $k$ and $u\in
C{{}^{\infty}}(M)$ , $\nabla^{k}u$ represents the $k-th$ of the covariant
derivative of $u$ (with the Convention $\nabla^{0}{u}=u$ ) . and the norm of
$k-th$ of covariant derivative on a local map is given by the formula :

$|\nabla^{k}{u}|=g^{{i_{1}}{j_{1}}}.......g^{{i_{k}}{j_{k}}}({\nabla^{k}{u})}_{{{i_{1}....i_{k}}}}({\nabla^{k}{u})}_{{{j_{1}....j_{k}}}}$

where the Einstein summation convention is adopted.

We also recall the notion of Riemannian measureon manifolds , let
$\\{{U_{i}},{\Phi_{i}}\\}$ be any atlas of M . There exists a partition of
unity $\\{{U_{i}},{\Phi_{i}},{\eta_{i}}\\}$ subordinate to
$\\{{U_{i}},{\Phi_{i}}\\}$ Give a continuous function
$f:M\rightarrow\mathbb{R}$ we define the integrale as follows

$\int_{M}fd\sigma_{g}=\sum_{i}\int_{\Phi(U_{i})}({\eta_{i}}\sqrt{detg}f\circ\Phi_{i})dx.$

where $dx$ is the Lebesgue measure on $\mathbb{R}^{n}.$

Let be $p\geq 1$ a real, and $k$ a positive integer.

$L^{p}{(M)}=\\{u:M\to\mathbb{R}\quad measurable\
/\int_{M}{|u|^{p}d\sigma_{g}<\infty\\}}$

$C_{k}^{p}(M)$ functions space $u\in C^{\infty}$ such as ${|\nabla^{j}{u}|\in
L^{p}{(M)}}$ for $j=0,....,k$

$C_{k}^{p}(M)=\\{u\in C^{\infty}\ /\forall
j=0,....,k\quad\int_{M}{|\nabla^{j}{u}|^{p}d\sigma_{g}<\infty\\}}$

###### Definition 2

The Sobolev space $W^{k,p}(M)$ is the complete space $C^{p}_{k}(M)$ for the
norm

$\|u\|_{W^{k,p}(M)}=\sum_{j=0}^{k}\|\nabla^{j}u\|_{L^{p}(M)}$

$\|u\|_{W^{1,p}(M)}=\|\nabla u\|_{p}+\|u\|_{p}$

###### Definition 3

We must recall the notion of the geodesic distance for every curve :

$\varUpsilon:\ [a,b\ ]\to M$

We define the length of $\varUpsilon$ by :

$l(\varUpsilon)=\int_{a}^{b}\sqrt{g(\varUpsilon(t))(\frac{d\varUpsilon}{dt},\frac{d\varUpsilon}{dt})}dt$

###### Remark 1

For $x,y\in M$ defining a distance $d_{g}$ by :

$d_{g}(x,y)=inf\\{l(\varUpsilon):\varUpsilon:[0,1]\to
M\quad.\quad\varUpsilon(0)=x\quad,\quad\varUpsilon(1)=y\\}$

By the theorem of Hopf-Rinow, we obtain that if $M$ a Riemannian manifold then
compact for all $x,y$ in $M$ can be joined by a minimizing curve $\varUpsilon$
i.e $l(\varUpsilon)=d_{g}(x,y)$

###### Proposition 1

If $p=2$, $W^{k,2}(M)$ is a Hilbert space space for the scalar product

$\left(u,v\right)_{H^{k}}=\sum_{j=0}^{k}\left(\nabla^{j}u,\nabla^{j}v\right)_{L^{2}}$

.

###### Proposition 2

If $p>1$ then $W^{k,p}(M)$ is reflexive.

###### Proposition 3

Any reflex normalized space is a Banach space. Then if $p>1$ then $W^{k,p}(M)$
is Banach.

###### Definition 4

The Sobolev space $W_{0}^{k,p}(M)$ is the closure of $\mathcal{D}(M)$ in
$W^{k,p}(M)$ .

###### Theorem 1.1

If $(M,g)$ is complete, then for all $p\geq 1$ $W^{1,p}_{0}(M)=W^{1,p}(M)$.

Embeddings of Sobolev:See $[7]$.

###### Lemma 2

Let $(M,g)$ a complete Riemannian manifold of dimension $n$. Suppose that
inclusion $W^{1,1}(M)\subset L^{\frac{n}{(n-1)}}(M)$ is valid. Then, for a
whole real number $1\leq q<p$ and an integer $0\leq m<k$ which verify
$\frac{1}{p}=\frac{1}{q}-\frac{(k-m)}{n}$, $W^{k,q}(M)\subset W^{m,p}(M)$.

###### Remark 2

Note that the proof of the Lemma LABEL:lem1 shows that if $A\in\mathbb{R}$ is
such that $\forall~{}~{}u\in W^{1,1}(M)$,

$\left(\int_{M}|u|^{n/(n-1)}d\sigma_{g}\right)^{(n-1)/n}\leq
A\int_{M}\left(|\nabla u|+|u|\right)d\sigma_{g})$

So, for all $1\leq q<n$ and all $u\in W^{1,q}(M)$,

$\left(\int_{M}|u|^{p}d\sigma_{g}\right)^{1/p}\leq\frac{Ap(n-1)}{n}\left\\{\left(\int_{M}|\nabla
u|^{q}d\sigma_{g}\right)^{1/q}+\left(\int_{M}|u|^{q}d\sigma_{g}\right)^{1/q}\right\\}$

Where $1/p=1/q-1/n.$

###### Theorem 1.2

Let $(M,g)$ a compact Riemannian manifold of dimension $n$. For a real number
$1\leq q<p$ and an integer $0\leq m<k$ which verify
$\frac{1}{p}=\frac{1}{q}-\frac{(k-m)}{n}$, $W^{k,q}(M)\subset W^{m,p}(M)$.

###### Theorem 1.3

(Rellich-Kondrakov’s Theorem): Let $(M,g)$ a compact Riemannian manifold of
$n$ dimension , $j\geq 0$ and $m\geq 1$ two integers, $q\geq 1$ and $p$ two
real numbers that verify $1\leq p<nq/(n-mq)$, the inclusion
$W^{j+m,q}(M)\subset W^{j,p}(M)$ is compact

###### Corollary 1

Let $(M,g)$ a compact Riemannian manifold of $n$ dimension. For everything
$1\leq q<n$ and $p\geq 1$ such as $\frac{1}{p}>\frac{1}{q}-\frac{1}{n}$, the
inclusion $W^{1,q}(M)\subset L^{p}(M)$ is compact.

###### Lemma 3

(Inequality of Poincare): Let $D$ a regular domain is bounded in a Riemannian
manifold $M$ and $1\leq p<\infty$. Then there is a constant $A$ such as:

$\left(\int_{D}|u-u_{D}|^{p}d\sigma_{g}\right)^{\frac{1}{p}}\leq
A\left(\int_{D}|\nabla u|^{p}d\sigma_{g}\right)^{\frac{1}{p}},$

for everything $u\in W^{1,p}_{loc}(M)$, where
$u_{D}=\frac{1}{vol(D)}\int_{D}ud\sigma_{g}$ is the mean value of $u$ on $D$

By combining this lemma with the Holder inequality, we obtain:

###### Corollary 2

There exists a constant $c=c_{D}$ such that

$\displaystyle\int_{D}|u-u_{D}|d\sigma_{g}\leq c_{D}\left(\int_{M}|\nabla
u|^{p}d\sigma_{g}\right)^{\frac{1}{p}}~{}~{}~{}~{}~{}~{}\forall~{}u\in
W^{1,p}_{loc}(M)$

#### Marcinkiewicz’s spaces.

###### Definition 5

$[7]$ Let $f:M\rightarrow\mathbb{R}$ be a measurable function, its
distribution function

$\phi_{f}(k)=\operatorname{meas}\bigg{\\{}x\in M:|f(x)|>k\bigg{\\}}\quad k>0,$

###### Definition 6

Let $0<q<\infty$ and $(M,g)$ Riemannian manifold, the space Marcinkiewicz
$\mathcal{M}^{q}(M)$ is the set of functions measurable
$f:M\rightarrow\mathbb{R}$ such as

$\phi_{f}(k)\leq Ck^{-q},~{}~{}C<\infty,$

Marcinkiewicz’s space $\mathcal{M}^{q}(M)$ is defined the norm

$\|f\|_{\mathcal{M}^{q}(M)}=\inf\Bigg{\\{}C:\phi_{f}(k)\leq
Ck^{-q},\quad\text{for all}\quad k>0\Bigg{\\}}.$

is a Banach space.

Note that if $f\in L^{q}(M),$ we have

$\int_{\\{|f|>k\\}}d\sigma_{g}\leq\int_{M}\left|\frac{f}{k}\right|^{q}d\sigma_{g}\leq
k^{-q}\int_{M}|f|^{q}d\sigma_{g},$

so

$\phi_{f}(k)\leq k^{-q}\|f\|_{q}^{q}$

and as a conclusion will have $L^{q}(M)\subset\mathcal{M}^{q}(M).$

For analyze the properties of the spaces $\mathcal{M}^{q}(M),$ needs some the
next lemma

###### Lemma 4

$Iff\in L^{q}(M)$ so

$\int_{M}|f(x)|^{q}dx=q\int_{0}^{+\infty}t^{q-1}\phi_{f}(t)dt.$

Proof.

We start with the case where $q=1$. Let

$H(t)=\left\\{\begin{array}[]{lll}{1}&{\text{ if }}&{t>0,}\\\ {0}&{\text{ if
}}&{t<0,}\end{array}\right.$

so

$H(|f(x)|-k)=\left\\{\begin{array}[]{ll}{1}&{\text{ if ~{}~{}}|f(x)|>k,}\\\
{0}&{\text{ if ~{}~{}}|f(x)|<k,}\end{array}\right.$

so we have

$\displaystyle\int_{0}^{+\infty}\phi_{f}(k)dk_{g}$
$\displaystyle=\int_{0}^{+\infty}\left[\int_{M}H(|f(x)|-k)d\sigma_{g}\right]dk_{g}$
$\displaystyle=\int_{M}\left[\int_{0}^{+\infty}H(|f(x)|-k)dk_{g}\right]d\sigma_{g},\quad\text{
( Fubini) }$
$\displaystyle=\int_{M}\left[\int_{\\{|f(x)|>k\\}}1dk_{g}\right]d\sigma_{g}=\int_{M}\left[\int_{0}^{|f(x)|}1dk_{g}\right]d\sigma_{g},$
$\displaystyle=\int_{M}|f(x)|d\sigma_{g},$

so
$\displaystyle\int_{0}^{+\infty}\phi_{f}(k)dk_{g}=\displaystyle\int_{M}|f(x)|d\sigma_{g}$
and the result is demonstrated.

We now consider the general case $q>1.$ Ask $g(x)=|f(x)|^{q}$ so, $g\in
L^{1}(M)$ and

$\phi_{g}(k)=\operatorname{meas}\bigg{\\{}|g|>k\bigg{\\}}=\operatorname{mes}\bigg{\\{}|f|^{q}>k\bigg{\\}}=\operatorname{meas}\bigg{\\{}|f|>k^{\frac{1}{q}}\bigg{\\}},$

i.e. $\quad\phi_{g}(k)=\phi_{f}\left(k^{\frac{1}{q}}\right),$

so

$\int_{M}|g(x)|d\sigma_{g}=\int_{0}^{+\infty}\phi_{f}\left(k^{\frac{1}{q}}\right)dk_{g},$

Ask $t=k^{\frac{1}{q}}$ so $k=t^{q}$ and $dk=qt^{q-1},$ so that

$\int_{M}|f(x)|^{q}d\sigma_{g}=q\int_{0}^{+\infty}t^{q-1}\phi_{f}(t)dt.\blacksquare$

###### Corollary 3

If $q\in]1,\infty[,$ so

$L^{q}(M)\subset\mathcal{M}^{q}(M)\subset
L^{q-\varepsilon}(M)\quad\forall\varepsilon>0,$

and for all $q,\hat{q}\in[1,\infty[$ we have

$\mathcal{M}^{q}(M)\subset\mathcal{M}^{\hat{q}}(M)\quad\text{if}\quad
q\geq\hat{q}.$

Proof.

Assuming that $f\in\mathcal{M}^{q}(M)$ and $\varepsilon>0,$ we have

$\displaystyle\int_{M}|f(x)|^{q-\varepsilon}d\sigma_{g}$
$\displaystyle=\int_{\\{|f|\leq
1\\}}|f(x)|^{q-\varepsilon}d\sigma_{g}+\int_{\\{|f|>1\\}}|f(x)|^{q-\varepsilon}d\sigma_{g},$
$\displaystyle\leq c_{1}+\int_{\\{|f|\leq
1\\}}|f(x)|^{q-\varepsilon}d\sigma_{g},$ $\displaystyle\leq
c_{1}+\int_{M}|f(x)|^{q-\varepsilon}1_{\\{|f|>1\\}}d\sigma_{g},$
$\displaystyle\leq c_{1}+\int_{M}|g(x)|^{q-\varepsilon}d\sigma_{g},$

where $g(x)=|f(x)|1_{\\{|f|>1\\}},$ so

$\phi_{g}(t)=\operatorname{meas}\\{|g|>t\\}\leq\operatorname{meas}\left\\{|f(x)|1_{\\{|f|>1\\}}>t\right\\}$

implies that

$\displaystyle\int_{M}|f(x)|^{q-\varepsilon}d\sigma_{g}$ $\displaystyle\leq
c_{1}+\int_{M}|g(x)|^{q-\varepsilon}d\sigma_{g},$ $\displaystyle\leq
c_{1}+(q-\varepsilon)\int_{0}^{+\infty}t^{q-\varepsilon-1}\phi_{g}(t)dt,$
$\displaystyle\leq
c_{1}+(q-\varepsilon)\int_{0}^{1}t^{q-\varepsilon-1}\phi_{g}(t)dt+(q-\varepsilon)\int_{1}^{+\infty}t^{q-\varepsilon-1}\phi_{g}(t)dt,$
$\displaystyle\leq
c_{1}+c_{2}(q-\varepsilon)\int_{0}^{1}t^{q-\varepsilon-1}dt+c_{3}(q-\varepsilon)\int_{1}^{+\infty}t^{q-\varepsilon-1}t^{-q}dt,$
$\displaystyle\leq
c_{1}+c_{2}(q-\varepsilon)+c_{3}(q-\varepsilon)\int_{1}^{+\infty}t^{-\varepsilon-1}dt,$
$\displaystyle\leq
c_{4}+(q-\varepsilon)\left[-\frac{t^{-\varepsilon}}{\varepsilon}\right]_{1}^{+\infty},$
$\displaystyle\leq C<\infty.$

so $\quad f\in L^{q-\varepsilon}(M),$ and then it results that
$\mathcal{M}^{q}(M)\subset L^{q-\varepsilon}(M).$ As
$L^{q}(M)\subset\mathcal{M}^{q}(M)\subset
L^{q-\varepsilon}(M)\subset\mathcal{M}^{q-\varepsilon}(M)\quad$ for all
$q\in]1,\infty[$ and for all $\varepsilon>0,$ so we deduct for all
$q,\hat{q}\in]1,\infty[$ wa have

$\mathcal{M}^{q}(M)\subset\mathcal{M}^{\hat{q}}(M)\quad si\quad
q\geq\hat{q}.~{}~{}\blacksquare$

#### Elliptic problems and the concept of the weak solution.

Let $p>1$ and $(M,g)$ a compact Riemannian manifold , for $u\in
W_{0}^{1,p}(M),$ we can consider the continuous linear form $-\Delta_{p}u$
over $W_{0}^{1,p}(M)$ defined by

$\left\langle-\Delta_{p}u,v\right\rangle\equiv\int_{M}|\nabla u|^{p-2}\nabla
u\nabla vd\sigma_{g}.$

It’s clear that
$-\Delta_{p}u\in\left(W_{0}^{1,p}(M)\right)^{\prime}=W_{0}^{-1,p^{\prime}}(M)$
and
$\left\|-\Delta_{p}u\right\|_{W_{0}^{-1,p^{\prime}}(M)}=\|u\|_{W_{0}^{1,p}(M)}$.

As a consequence, we have the next definition

###### Definition 7

let $f\in W_{0}^{-1,p^{\prime}}(m),$ we say that u is a weak solution of the
problem

$\left\\{\begin{array}[]{ccc}{-\Delta_{p}u=f}&{\text{in }}&{M,}\\\
{u(x)=0}&{\text{ on }}&{\partial M,}\end{array}\right.$

in $W_{0}^{1,p}(M)$ if and only if

$\int_{M}|\nabla u|^{p-2}\nabla u\nabla\varphi
d\sigma_{g}=\left\langle-\Delta_{p}u,\varphi\right\rangle\quad\forall\varphi\in
W_{0}^{1,p}(M).$

The next inequalities will be systematically used in this work.

###### Lemma 5

Let $\xi_{1},\xi_{2}\in M,$ we have

  1. 1)

If $p\leq 2,$

$\left|\xi_{1}+\xi_{2}\right|^{p}-\left|\xi_{1}\right|^{p}-p\left|\xi_{1}\right|^{p-2}\left\langle\xi_{1},\xi_{2}\right\rangle\leq
C(p)\left|\xi_{2}\right|^{p},$

$\left|\xi_{2}\right|^{p}-\left|\xi_{1}\right|^{p}-p\left|\xi_{1}\right|^{p-2}\left\langle\xi_{1},\xi_{2}-\xi_{1}\right\rangle\geq
C(p)\frac{\left|\xi_{2}-\xi_{1}\right|^{2}}{\left(\left|\xi_{2}\right|+\left|\xi_{1}\right|\right)^{2-p}}.$

  2. 2)

If $p>2,$

$\left|\xi_{1}+\xi_{2}\right|^{p}-\left|\xi_{1}\right|^{p}-p\left|\xi_{1}\right|^{p-2}\left\langle\xi_{1},\xi_{2}\right\rangle\leq\frac{p(p-1)}{2}\left(\left|\xi_{1}\right|+\left|\xi_{2}\right|\right)^{p-2}\left|\xi_{2}\right|^{2},$
$\left|\xi_{2}\right|^{p}-\left|\xi_{1}\right|^{p}-p\left|\xi_{1}\right|^{p-2}\left\langle\xi_{1},\xi_{2}-\xi_{1}\right\rangle\geq\frac{C(p)}{2^{p}-1}\left|\xi_{2}-\xi_{1}\right|^{p}.$

For the demonstration, see $[9]$ and $[10]$.

To demonstrate the existence of a weak solution for the previous problem, one
often uses variational techniques and arguments of minimization of the convex
functional ones. More precisely, we have the next result.

###### Theorem 1.4

$[11]$ Let $V$ be a reflexive Banach space, $K\subset V$ is a closed non-empty
convex, and $J:K\rightarrow\mathbb{R}\cup\\{+\infty\\}$ a semi-continue
inferiorly coercive function weakly on $K.$

So $\inf_{u\in K}J(u)<\infty\quad$and $\quad\exists u_{0}\in K,\quad
J\left(u_{0}\right)=\min_{u\in K}J(u).$

Moreover, if J is strictly convex, $u_{0}$ is unique. If $J$ is differentiable
in the sense of Gateaux and $K$ then open $J^{\prime}\left(u_{0}\right)=0.$

###### Theorem 1.5

Let$(M,g)$ a Riemannian manifold and $p\in]1,\infty[.$ We suppose that $f\in
L^{q}(M)$ with $q\geq\overline{q}=\dfrac{Np}{N(p-1)+p},$ so there exists a
unique solution $u\in W_{0}^{1,p}(M)$ of the problem

$\left\\{\begin{aligned} -\Delta_{p}u=f&\quad\text{ in }\quad&M,\\\
u=0&\quad\text{if }&\partial M.\end{aligned}\right.$

###  1.3 Inequality of Picone for the $p-$ Laplacian and application.

We begin by formulating the inequality of Picone punctual for the case of $p-$
Laplacian.

###### Theorem 1.6

Let $v>0,u\geq 0$ two positive class $C^{1}$ functions, we pose

$L(u,v)=|\nabla u|^{p}+(p-1)\frac{u^{p}}{v^{p}}|\nabla
v|^{p}-p\frac{u^{p-1}}{v^{p-1}}|\nabla v|^{p-2}\nabla v\nabla u.$
$R(u,v)=|\nabla u|^{p}-\nabla\left(\frac{u^{p}}{v^{p-1}}\right)|\nabla
v|^{p-2}\nabla v.$

so $L(u,v)=R(u,v),L(u,v)\geq 0$ and $L(u,v)=0,$ almost everywhere. in $M$ of
$u=kv$ in each Connected component of $M.$

The proof of Theorem 6 is simple, it is based on the development of term
$\nabla\left(\dfrac{u^{p}}{v^{p-1}}\right)|\nabla v|^{p-2}\nabla v.$ To apply
the Picone inequality to nonlinear elliptic equations we need to prove an
extension of Theorem $6$ in $W_{0}^{1,p}(M),$ more precisely we have the next
lemma

###### Lemma 6

Let $v\in W^{1,p}(M)$ such as $v\geq\delta>0$ in $M.$ so for all $u\in
C_{0}^{\infty}(M),$ $u\geq 0$

$\int_{M}|\nabla
u|^{p}\geq\int_{M}\left(\frac{|u|^{p}}{v^{p-1}}\right)\left(-\Delta_{p}v\right).$

Proof.

As $v\in W^{1,p}(M)$ and $v\geq\delta>0$ in $M,$ then it exists a sequence
$\left\\{v_{n}\right\\}$ regular functions such as

$\left\\{\begin{array}[]{c}{v_{n}\rightarrow v\operatorname{\quad}{in}\quad
W^{1,p}(M),v_{n}\in C^{1}(M)}\\\ {v_{n}\rightarrow v\quad a.e,\quad\text{ and
}v_{n}>\dfrac{\delta}{2}\text{\quad}{in}\quad M.}\end{array}\right.$

As a consequence of the continuity of the operator $-\Delta_{p}$
$\bigg{(}\operatorname{of}\quad W^{1,p}(M)$ in $W^{-1,p^{\prime}}(M),$
$p^{\prime}=\dfrac{p}{p-1}\bigg{)}$ we get that
$-\Delta_{p}v_{n}\rightarrow-\Delta_{p}v$ in
$W^{-1,~{}p^{\prime}}(M),p^{\prime}=\dfrac{p}{p-1}\cdot(\operatorname{see}[24]).$
En using the identity of Picone at $v_{n},$ it results

$|\nabla u|^{p}\geq\nabla\left(\frac{u^{p}}{v_{n}^{p-1}}\right)\left|\nabla
v_{n}\right|^{p-2}\nabla v_{n}.$

as

$\displaystyle\int_{M}-\Delta_{p}v_{n}\frac{u^{p}}{v_{n}^{p-1}}$
$\displaystyle=\int_{M}\left|\nabla v_{n}\right|^{p-2}\left\langle\nabla
v_{n},\nabla\left(\frac{u^{p}}{v_{n}^{p-1}}\right)\right\rangle$
$\displaystyle=p\int_{M}\frac{u^{p-1}}{v_{n}^{p-1}}\left|\nabla
v_{n}\right|^{p-2}\left\langle\nabla v_{n},\nabla
u\right\rangle-(p-1)\int_{M}\frac{u^{p}}{v_{n}^{p}}\left|\nabla
v_{n}\right|^{p}.$

Using the hypothesis on the Dominated convergence theorem we conclude

$\int_{M}|\nabla
u|^{p}\geq\int_{M}\left(\frac{-\Delta_{p}v}{v^{p-1}}\right)u^{p},\quad u\in
C_{0}^{\infty}(M),u\geq 0.$

$\blacksquare$

In a more general context, we have the next result

###### Theorem 1.7

if $u\in W_{0}^{1,p}(M)$, $u\geq 0$, $v\in W_{0}^{1,p}(M)$, $-\Delta_{p}v\geq
0$ is a measure of Radon bounded, $\left.v\right|_{\partial M}=0,v\gneq 0,$ so

$\int_{M}|\nabla
u|^{p}\geq\int_{M}\left(\frac{u^{p}}{v^{p-1}}\right)\left(-\Delta_{p}v\right).$

Proof.  According to the principle of Maximum strong we have $v>0$ in $M.$
(See $[13])$. We pose $v_{m}(x)=v(x)+\dfrac{1}{m},m\in\mathbb{N}.$ So
$\Delta_{p}v_{m}=\Delta_{p}v$ and $\left\\{v_{m}\right\\}$ converges in
$W^{1,p}(M)$ and a.e to $v.$ Therefore, using Lemma $1.4,$ on gets the result
for all $\phi\in C_{0}^{\infty}(M),\phi\geq 0.$ Now in the general case, by
density we deduce the existence of $u_{n}\rightarrow u$ in $W_{0}^{1,p}(M)$,
$u_{n}\in C_{0}^{\infty}(M)$ et $u_{n}\geq 0$, so

$\int_{M}\left|\nabla
u_{n}\right|^{p}\geq\int_{M}\left(\frac{-\Delta_{p}v_{n}}{v_{n}^{p-1}}\right)u_{n}^{p}=\int_{M}\left(\frac{-\Delta_{p}v}{v_{n}^{p-1}}\right)u_{n}^{p}.$

By the hypothesis imposed on $u$ and according to the Lemma of Fatou we obtain
the result. $\blacksquare$

#### Comparison principle.

As application of lemma $6$, we demonstrate the next comparison result .

###### Lemma 7

Let $f$ be a continuous positive function such that
$\dfrac{f(u)}{u^{p-1}}\downarrow$ with $1<p.$ We suppose that $u,v\in
W_{0}^{1,p}(M)\cap C^{1}(M)$ are such that

$\left\\{\begin{array}[]{ll}{-\Delta_{p}u\geq f(u),}&{u>0\text{ in }M}\\\
{-\Delta_{p}v\leq f(v),}&{v>0\text{ in }M}\end{array}\right.$

so $u\geq v$ in $M$

Proof.  previous inequality implies that

$\frac{-\Delta_{p}u}{u^{p-1}}+\frac{\Delta_{p}v}{v^{p-1}}\geq\frac{f(u)}{u^{p-1}}-\frac{f(v)}{v^{p-1}}.$

Multiply by $w=\left(v^{p}-u^{p}\right)^{+},$ we get that

$\displaystyle\int_{M}\left(\frac{-\Delta_{p}u}{u^{p-1}}+\frac{\Delta_{p}v}{v^{p-1}}\right)\left(v^{p}-u^{p}\right)^{+}$
$\displaystyle\geq\int_{M}\left(\frac{f(u)}{u^{p-1}}-\frac{f(v)}{v^{p-1}}\right)\left(v^{p}-u^{p}\right)^{+}$
$\displaystyle=\int_{[v>-1]}\left(\frac{f(u)}{u^{p-1}}-\frac{f(v)}{v^{p-1}}\right)\left(v^{p}-u^{p}\right)^{+}$

By the assumption on $f,$ we conclude that the term on the right in the
previous equality is positive. On the other hand as
$w=\left(v^{p}-u^{p}\right)^{+},$ so $\nabla w=p\left(v^{p-1}\nabla
v-u^{p-1}\nabla u\right)\chi_{[v\geq u]}$, so

$\displaystyle\int_{M}\left(\frac{-\Delta_{p}u}{u^{p-1}}+\frac{\Delta_{p}v}{v^{p-1}}\right)w=\int_{M}|\nabla
u|^{p-2}\left\langle\nabla
u,\nabla\left(\frac{w}{u^{p-1}}\right)\right\rangle-\int_{M}|\nabla
v|^{p-2}\left\langle\nabla
v,\nabla\left(\frac{w}{v^{p-1}}\right)\right\rangle$ $\displaystyle=$
$\displaystyle\int_{M}|\nabla u|^{p-2}\left\langle\nabla u,\frac{u^{p-1}\nabla
w-(p-1)u^{p-2}w\nabla u}{u^{2(p-1)}}\right\rangle$ $\displaystyle-$
$\displaystyle\int_{M}|\nabla v|^{p-2}\left\langle\nabla v,\frac{v^{p-1}\nabla
w-(p-1)v^{p-2}w\nabla v}{v^{2(p-1)}}\right\rangle$ $\displaystyle=$
$\displaystyle\int_{M\cap[v>u]}\left[\nabla\left.u\right|^{p-2}\langle\nabla
u,\nabla v\rangle-(p-1)\frac{v^{p}}{u^{p}}|\nabla u|^{p}-|\nabla
u|^{p}\right]$ $\displaystyle+$
$\displaystyle\int_{M\cap[v>u]}\left[p\frac{u^{p-1}}{v^{p-1}}|\nabla
v|^{p-2}\langle\nabla v,\nabla u\rangle-(p-1)\frac{u^{p}}{v^{p}}|\nabla
v|^{p}-|\nabla v|^{p}\right]$ $\displaystyle=$
$\displaystyle\int_{M\cap[v>u]}K_{1}(x)d\sigma_{g}+\int_{M\cap[v>u]}K_{2}(x)d\sigma_{g}$

and as $u>0$ and $v>0$ in $m$, using the Picone inequality, $K_{1}\leq 0$ and
$K_{2}\leq 0$. So

$\int_{m}\left(\frac{-\Delta_{p}u}{u^{p-1}}+\frac{\Delta_{p}v}{v^{p-1}}\right)w\leq
0$

and Consequently,

$\int_{m\cap[v\geq
u]}\left(\frac{f(u)}{u^{p-1}}-\frac{f(v)}{v^{p-1}}\right)\left(v^{p}-u^{p}\right)\leq
0.$

But on the set $[v>u]$, $\dfrac{f(u)}{u^{p-1}}-\dfrac{f(v)}{v^{p-1}}\geq 0,$
so $|[v>u]|=0,$ and we deduce that $v\leq u.$

Easily demonstrates the extension using Lemma $7$ $\blacksquare$

######  Lemma 8 (Comparison principle)

Let $u,v\in W_{0}^{1,p}(M)\cap C^{1}(M)$ such as

$\left\\{\begin{array}[]{cc}{-\Delta_{p}u\geq h(x)f(u),}&{u>0\text{ in }M}\\\
{-\Delta_{p}v\leq h(x)f(v),}&{v>0\text{ in }M}\end{array}\right.$

where $h$ is a positive function such that $h\neq 0.$ So $u\geq v$ in $M.$

###### Remark 3

The result of Lemma $8$ is valid if $h(x)=|x|^{-p}$.

As a direct application of Lemma $8,$ we obtain the next uniqueness result

###### Theorem 1.8

The problem

$\left\\{\begin{array}[]{c}{-\Delta_{p}u=\lambda h(x)u^{q}\text{ in }M,\quad
0<q<p-1}\\\ {u>0\text{ in }M}\\\ {\left.u\right|_{\partial
M}=0}\end{array}\right.$

where $h$ is in the conditions of the preceding theorem, admits a unique
solution.

###### Remark 4

In general, we have the same result of uniqueness if we replace $u^{q}$ with a
function of Carateodory $f(x,u)$ such that $\dfrac{f(x,u)}{u^{p-1}}$ is
decreasing uniformly in $x\in M.$ To demonstrate existence we need to impose
more conditions on $f$.

##  2 Theory of existence and uniqueness of solutions for nonlinear elliptic
problems with data in $L^{1}$

###  2.1 Introduction

Consider the problem of form

$\left\\{{\begin{array}[]{*{20}{c}}{-{\Delta_{p}}u}&=&f&{{\rm{in}}}&{M,}\\\
u&=&0&{{\rm{on}}}&{\partial M.}\end{array}}\right.$ (2.1)

where $1<p<\infty,f$ is a measurable function such that $f\in L^{1}(M)$.

There are three difficulties associated with the study of the equation (2.1) .

  1. 1-

Find the direction for which the previous equation is well defined.

  2. 2-

The construction of a solution in the direction obtained.

  3. 3-

Uniqueness of the solution found.

Note that the most general meaning that can be used is the direction of
distribution, ie, u checks

$\int_{M}|\nabla u|^{p-2}\nabla u\nabla\phi d\sigma_{g}=\int_{M}f\phi
d\sigma_{g}\quad\forall\phi\in C_{0}^{\infty}(M)$

except that the problem in this context is who we do not have a construction
argument (the test function space being too ”small”), and the second problem
is the uniqueness of the solution ( the operator is nonlinear). Note that for
the case $p=2,$ the distributional framework is a natural framework for
studying equations with a second member in $L^{1},$ because $\Delta u=0$ in
the distributions sense implies that $u$ is harmonic in the classical sense.

To solve the nonlinear problem we need to introduce a new space
$\tau_{loc}^{1,1}(M)$ in which we can make sense of the gradient of $u,$ which
in general is not locally integrable. So the idea is to work with the
truncations $T_{k}(u)$ of the $u$ solution and expand the space of the test
functions to bounded functions with a gradient in a suitable Lebesgue space.

The arguments we will introduce will be applicable to a class of equations
general form.

$-{\Delta_{p}}u=F(x,u)\quad in\quad{D^{{}^{\prime}}(M)}\quad\quad(1,2)$

Or $F$ is a carathéodory functions, continuous and decreasing in $u$ for $x$
fixed, and measurable in x for $u$ fixed. moreover, $F(x,0)\in L^{1}(M)$ and
$F(x,c)\in L_{loc}^{1}(M)$ if $c\neq 0$ , and if

$G_{c}(x)=\sup_{|u|\leq c}|F(x,u)|,$

so $G_{c}\in L_{loc}^{1}(M)$ for all $c>0$

###  2.2 Functional Framework

Before discussing the concept of the entropy solution, we will present the
functional framework in which the solution is well defined. We start with the
introduction of the truncation operator. For a constant $k>0,$ we define the
function $T_{k}:\mathbb{R}\rightarrow\mathbb{R}$ by

$T_{k}(s)=\left\\{\begin{array}[]{ccc}{s}&{\text{ if }}&{|s|\leq k,}\\\
{k\operatorname{sign}(s)}&{\text{ if }}&{|s|>k.}\end{array}\right.$

So for a measurable function $u$ defined in $M,T_{k}u$ is defined by
$\left(T_{k}u\right)(x)=T_{k}(u(x)).$

we will use in its subsection Functional spaces :

  1. i)

$\tau_{loc}^{1,1}(M)$ is the set of measurable functions
$u:M\rightarrow\mathbb{R}$ such as for all $k>0$ the truncation function
$T_{k}(u)$ in $W_{loc}^{1,1}(M).$

  2. ii)

for $p\in]1,\infty[,\tau_{loc}^{1,p}(M)$ is the subset of
$\tau_{loc}^{1,1}(M)$ composed by functions $u$ such as
$\left|\nabla\left(T_{k}(u)\right)\right|\in L_{loc}^{p}(M)$ for all $k>0$.

  3. iii)

Of even, $\tau^{1,p}(M)$ is the subset of $\tau_{loc}^{1,1}(M)$ composed of
functions $u,$ such as, of more $\left|\nabla T_{k}(u)\right|\in L^{p}(M)$ for
all $k>0$.

  4. iv)

Finally, $\tau_{0}^{1,p}(M)$ is the subset of $\tau^{1,p}(M),$ composed of
functions that can be approximated by class functions $C^{1}$ a compact
support in $M$ in the next sense : a function $u\in\tau^{1,p}(M)$ in
$\tau_{0}^{1,p}(M),$ if fopr all $k>0,$ it exists a Sequence
$\left(\phi_{n}\right)\subset C_{0}^{\infty}(M)$ such as

$\phi_{n}\rightarrow T_{k}(u)\quad\text{ in }\quad L_{loc}^{1}(M)$
$\nabla\phi_{n}\rightarrow\nabla T_{k}(u)\quad\text{in }\quad L^{p}(M)$

This space will play an important role in this work.

We have the next lemma giving some properties of the preceding spaces

###### Lemma 9

for all $p\in[1,\infty[,$ wa have

  1. 1)

$W_{loc}^{1,p}(M)\subset\tau_{loc}^{1,p}(M)\quad$ et $\quad
W_{0}^{1,p}(M)\subset\tau_{0}^{1,p}(M)$,

  2. 2)

$\tau_{loc}^{1,p}(M)\cap L_{loc}^{\infty}(M)=W_{loc}^{1,p}(M)\cap
L_{loc}^{\infty}(M)$,

  3. 3)

$\nabla T_{k}(u)=\nabla u1_{\\{|u|<k\\}},$

where ${1}_{A}$ denotes the characteristic function of a measurable set $A$.

Proof.

  1. 1)

we have

$\displaystyle u\in W_{loc}^{1,p}(M)$ $\displaystyle\Rightarrow u\in
W_{loc}^{1,1}(M)\text{ et }\nabla u\in L_{loc}^{p}(M),$
$\displaystyle\Rightarrow T_{k}(u)\in W_{loc}^{1,1}(M)\quad\text{ and }\nabla
T_{k}(u)\in L_{loc}^{p}(M)\quad\forall k>0$ $\displaystyle\Rightarrow
u\in\tau_{loc}^{1,p}(M)$

so $W_{loc}^{1,p}(M)\subset\tau_{loc}^{1,p}(M).$ For the second point, we have

$u\in W_{0}^{1,p}(M)\Rightarrow u\in W^{1,p}(M)\quad$ and
$\quad\exists\left\\{\phi_{n}\right\\}\subset C_{0}^{\infty}(M)\quad$ such as

$\left\\{{\begin{array}[]{*{20}{c}}{{\phi_{n}}\,\to
u}&{{\rm{}}}\hfil&{{\rm{in}}}&{{L^{p}}(M)}\\\ {\nabla{\phi_{n}}}&{\to\nabla
u}&{{\rm{in}}}&{{L^{p}}(M)}\end{array}}\right.$

$\Rightarrow
u\in{\tau^{1,p}}(M){\rm{and}}\exists\left\\{{{\phi_{n}}}\right\\}\subset
C_{0}^{\infty}M)\quad such\,as$

$\Rightarrow\left\\{{\begin{array}[]{*{20}{c}}{{\phi_{n}}\to{T_{k}}(u)}&{{\rm{in}}}&{L_{loc}^{1}(M)}\\\
{\nabla{\phi_{n}}\to\nabla{T_{k}}(u)}&{{\rm{in}}}&{{L^{p}}(M)}\end{array}}\right.\quad\forall
k>0,$ $\Rightarrow u\in\tau_{0}^{1,p}(M),$

so $W_{0}^{1,p}(M)\subset\tau_{0}^{1,p}(M).$

  2. 2)

as

$\displaystyle u\in W_{loc}^{1,p}(M)\cap L_{loc}^{\infty}(M)$
$\displaystyle\Rightarrow u\in W_{l,oc}^{1,p}(M)\quad\text{ and }\quad u\in
L_{loc}^{\infty}(M)$ $\displaystyle\Rightarrow
u\in\tau_{loc}^{1,p}(M)\quad\text{ and }\quad u\in L_{loc}^{\infty}(M)$
$\displaystyle\Rightarrow u\in\tau_{loc}^{1,p}(M)\cap L_{loc}^{\infty}(M)$

so $\quad W_{loc}^{1,p}(M)\cap
L_{loc}^{\infty}(M)\subset\tau_{loc}^{1,p}(M)\cap L_{loc}^{\infty}(\Omega).$

We also have

$\displaystyle u\in\tau_{loc}^{1,p}(M)\cap L_{loc}^{\infty}(M)$
$\displaystyle\Rightarrow\quad u\in\tau_{loc}^{1,p}(M)\quad\text{ and }\quad
u\in L_{loc}^{\infty}(M)$ $\displaystyle\Rightarrow\quad T_{k}(u)\in
W_{loc}^{1,1}(M)\quad\text{ and }\quad\nabla T_{k}(u)\in
L_{loc}^{p}(M)\quad\text{ and }\quad u\in L_{loc}^{\infty}(M)$
$\displaystyle\Rightarrow\quad u\in L_{loc}^{p}(M)\quad\text{and }\quad\nabla
u\in L_{loc}^{p}(M)\quad\text{ and }\quad u\in L_{loc}^{\infty}(M)$
$\displaystyle\Rightarrow\quad u\in W_{loc}^{1,p}(M)\quad\text{ and }\quad
u\in L_{loc}^{\infty}(M)$ $\displaystyle\Rightarrow\quad u\in
W_{loc}^{1,p}(M)\cap L_{loc}^{\infty}(M)$

so $\quad\tau_{loc}^{1,p}(M)\cap L_{loc}^{\infty}(M)\subset
W_{loc}^{1,p}(M)\cap L_{loc}^{\infty}(M)$.

So $\quad\tau_{\text{loc}}^{1,p}(M)\cap
L_{loc}^{\infty}(M)=W_{loc}^{1,p}(M)\cap L_{loc}^{\infty}(M)$.

  3. 3)

We have

$T_{k}(u)=\left\\{\begin{array}[]{ccc}{u}&{\text{if }}&{|u|\leq k}\\\
{k\frac{u}{|u|}}&{\text{if }}&{|u|>k}\end{array}\right.$

implies

$\nabla{T_{k}}(u)=\left\\{\begin{array}[]{l}\begin{array}[]{*{20}{c}}{\nabla
u}&\text{ if}&{\left|u\right|\leq k}\end{array}\\\
\begin{array}[]{*{20}{c}}{\nabla u}&\text{
if}&{\left|u\right|>k}\end{array}\end{array}\right.$

so $\quad\nabla T_{k}(u)=\nabla u1_{\\{|u|<k\\}}$. $\blacksquare$

Note that if $u\in\tau_{loc}^{1,1}(M),$ so $\nabla u$ is not defined even in
the sense of distributions, yet we have the next lemma that gives meaning to
$\nabla u.$

###### Lemma 10

$[1]$ Let $u\in\tau_{loc}^{1,1}(M),$ it exists a function
$v:M\rightarrow\mathbb{R}^{N}$ unique measurable such as

$\nabla T_{k}(u)=v1_{\\{|u|<k\\}}\quad a.e.$

in others, $u\in W_{loc}^{1,1}(M)$ if and only if $v\in L_{loc}^{1}(M),$ so
$v\equiv\nabla u$ in the usual weak sense.

Proof.

We have $\nabla T_{k}(u)=\nabla u1_{\\{|u|<<\\}},$ so for all
$u\in\tau_{loc}^{1,1}(M)$ it exists a function $v:M\rightarrow\mathbb{R}^{N}$
measurable such as $v\equiv\nabla u$ a.e. and $v\in L_{loc}^{1}(M)$.

$v$ is unique in the sense almost everywhere, because :

for all $k,\varepsilon>0,$ we have
$T_{k}\left(T_{k+\varepsilon}(u)\right)=T_{k}(u).$ Therefore, we get in
$M_{k}=\\{|u|<k\\}$ legality $\nabla T_{k+\varepsilon}=\nabla T_{k}$ a.e.
hence the result, and so $v$ unique a.e.

It remains to show that $u\in W_{loc}^{1,1}(M)$ if $v\in L_{loc}^{1}(M).$
Indeed, in this case $\nabla T_{k}(u)\rightarrow v$ in $L_{loc}^{1}(M),$ so we
have to prove that $u\in L_{loc}^{1}(M).$ By contradiction, if $u\notin
L_{loc}^{1},$ there will be a closed ball $B\subset M$ such as

$t_{k}=\left\|T_{k}(u)\right\|_{L^{1}(B)}\rightarrow\infty\quad\text{ when
}\quad k\rightarrow\infty.$

by normalization, $v_{k}=\frac{T_{k}(u)}{t_{k}}.$ so $v_{k}\rightarrow 0$ a.e.
$\left\|v_{k}\right\|_{L^{1}(B)}=1$ and $\left\|\nabla
v_{k}\right\|_{L^{1}(B)}\rightarrow 0,$ contradiction with the compactness of
the injection of $W^{1,1}(B)$ in $L^{1}(B).$

###  2.3 Solutions in the sense of entropy

In this section we will develop the concept of the solution in the sense of
entropy which will allow us to study elliptic equations with second member in
$L^{1}(M)$.

suppose that $f\in L^{1}(M)$ and consider the next equation:

$\left\\{{\begin{array}[]{*{20}{c}}{-{\Delta_{p}}u}&=&{f(x)}&{{\rm{in}}}&M\\\
u&=&0&{{\rm{on}}}&{\partial M}\end{array}}\right.$ (2.2)

Let $u\in\tau_{0}^{1,p}(M)$ a solution of the equation (2) in $D^{\prime}(M),$
so for all $\phi\in C_{0}^{\infty}(M),$ we have

$\int_{M}|\nabla u|^{p-2}\nabla u\nabla\phi d\sigma_{g}=\int_{M}f\phi
d\sigma_{g}\quad\forall\phi\in C_{0}^{\infty}(M)$

note that $f\in L^{1}(M),$ so by density and if we posit conditions of the
type ”Dirichlet homogeneous”, so we can take $T_{k}(u-\phi),k>0,$ as a test
function in the previous equation we get

$\int_{M}|\nabla u|^{p-2}\nabla u\nabla
T_{k}(u-\phi)d\sigma_{g}=\int_{M}T_{k}(u-\phi)fd\sigma_{g},$

so

$\int_{\\{|u-\phi|<k\\}}|\nabla u|^{|p-2}\nabla
u\nabla(u-\phi)d\sigma_{g}=\int_{M}T_{k}(u-\phi)fd\sigma_{g}.$ (2.3)

Note that each term in (3) is well defined $:$ as $\phi\in L^{\infty}(M),$ so

$\displaystyle|u-\phi|<k$ $\displaystyle\Rightarrow|u|-|\phi|<|u-\phi|<k$
$\displaystyle\Rightarrow|u|<k+|\phi|$
$\displaystyle\Rightarrow|u|<k+\|\phi\|_{\infty}$
$\displaystyle\Rightarrow|u|<\overline{k}$

or $\overline{k}=k+\|\phi\|_{\infty}$.

as $|\nabla u|^{p-1}\in L^{1}(M),$ so

$\displaystyle\int_{\\{|u-\phi|<k\\}}$ $\displaystyle|\nabla u|^{p-2}\nabla
u\nabla(u-\phi)d\sigma_{g}$ $\displaystyle=\int_{\\{|u-\phi|<k\\}}|\nabla
u|^{p}d\sigma_{g}-\int_{\\{|u-\phi|<k\\}}|\nabla u|^{p-2}\nabla u\nabla\phi
d\sigma_{g}$ $\displaystyle\leq\int_{\\{|u|<\overline{k}\\}}|\nabla
u|^{p}d\sigma_{g}+\int_{\\{|u-\phi|<k\\}}|\nabla
u|^{p-1}|\nabla\phi|d\sigma_{g}$ $\displaystyle\leq\int_{M}\left|\nabla
T_{\overline{k}}(u)\right|^{p}d\sigma_{g}+c_{1}\int_{\\{|u-\phi|<k\\}}|\nabla
u|^{p}d\sigma_{g}+c_{2}\int_{\\{|u-\phi|<k\\}}|\nabla\phi|^{p}d\sigma_{g}$
$\displaystyle\leq\quad c_{3}\int_{M}\left|\nabla
T_{\overline{k}}(u)\right|^{p}d\sigma_{g}+c_{2}\int_{\\{|u-\phi|<k\\}}|\nabla\phi|^{p}d\sigma_{g}$
$\displaystyle\leq C\left(\int_{M}\left|\nabla
T_{\overline{k}}(u)\right|^{p}d\sigma_{g}+\int_{\\{|u-\phi|<k\\}}|\nabla\phi|^{p}d\sigma_{g}\right)$

So

$\int_{\\{|u-\phi|<k\\}}|\nabla u|^{p-2}\nabla u\nabla(u-\phi)d\sigma_{g}\leq
C\left(\int_{M}\left|\nabla
T_{\overline{k}}(u)\right|^{p}d\sigma_{g}+\int_{\\{|u-\phi|<k\\}}|\nabla\phi|^{p}d\sigma_{g}\right).$
(2.4)

Since $T_{\overline{k}}(u)\in W_{0}^{1,p}(M)$ ie, $u\in\tau_{0}^{1,p}(M)$ and
$\phi\in L^{\infty}(M)\cap W_{0}^{1,p}(M),$ the second member in $(4)$ is
bounded, so the first member of $(3)$ is well defined.

We are in a position to give the next definition

######  Definition 8 (Solution in the sense of entropy)

Let $f\in L^{1}(M),$ we say that $u\in\tau_{0}^{1,p}(M)$ is an entropy
solution of the problem $(1)$ if $(3)$ is checked for each $\phi\in
L^{\infty}(M)\cap W_{0}^{1,p}(M)$ and for all $k>0$.

Let’s start by demonstrating some properties of the entropy solutions.

###### Lemma 11

Si $u\in\tau_{0}^{1,p}(M)$ is an entropy solution of $(1)$ so for all $k>0$

$\frac{1}{k}\int_{\\{|u|<k\\}}|\nabla
u|^{p}d\sigma_{g}\leq\int_{M}|f|d\sigma_{g}=\|f\|_{1}.$

Therefore, we obtain the next estimate in $L^{p}(M)$

$\left\|\nabla T_{k}(u)\right\|_{p}^{p}\leq k\|f\|_{1}.$ (2.5)

Proof.

As $\quad u\in\tau_{0}^{1,p}(M)\Rightarrow T_{k}(u)\in
W_{0}^{1,p}(M)\Rightarrow T_{k}(u)\in L^{p}(M).\quad$ If $\phi=0$ and grace at
$(3)$ we will have

$\int_{\\{|u|<k\\}}|\nabla u|^{p-2}\nabla u\nabla ud\sigma_{g}=\int
T_{k}(u)fd\sigma_{g}=\int_{\\{|u|<k\\}}ufd\sigma_{g}\leq
k\int_{\\{|u|<k\\}}|f|d\sigma_{g}\leq k\int_{M}|f|d\sigma_{g},$

do $\displaystyle\int_{M}\left|\nabla T_{k}(u)\right|^{p}d\sigma_{g}\leq
k\|f\|_{1}.$

###  2.4 estimates

Before demonstrating the existence of the entropy solution, we will prove some
preliminary estimates based on the estimate $(5).$ These estimates will relate
to $u$ and $|\nabla u|$ in Marcinkiewicz spaces and we can consider them as
keys to demonstrate compactness results in $L^{q}(M)$ spaces with $q$ suitably
chosen. The first main result is the next lemma.

###### Lemma 12

Let $1<p<N$ and $(M,g)$ a Riemannian manifold of dimension $N$ Consider
$u\in\tau_{0}^{1,p}(M)$ such as

$\frac{1}{k}\int_{\\{|u|<k\\}}|\nabla u|^{p}d\sigma_{g}\leq\alpha,$ (2.6)

for all $k>0.$ So $u\in\mathcal{M}^{p_{1}}(M)$ with
$p_{1}=\frac{N(p-1)}{N-p}.$ More precisely, there exists $C=C(N,p)>0$ such as

$\operatorname{meas}\\{|u|>k\\}\leq C\alpha^{\frac{N}{N-p}}k^{-p_{1}}.$ (2.7)

Proof. Let $1<p<N$ and $u\in\tau_{0}^{1,p}(M),$ so $T_{k}(u)\in
W_{0}^{1,p}(M)$ for all $k>0,$ and according to the inequality of Sobolev we
have

$\left\|T_{k}(u)\right\|_{p^{*}}\leq c(N,p)\left\|\nabla
T_{k}(u)\right\|_{p}\quad\text{ or }\quad p^{*}=\frac{Np}{N-p},$

because of $(6),$ we have $\int_{M}\left|\nabla
T_{k}(u)\right|^{p}d\sigma_{g}\leq k\alpha,$ ie. $\left\|\nabla
T_{k}(u)\right\|_{p}^{p}\leq k\alpha$,

and consequently $\left\|\nabla
T_{k}(u)\right\|_{p}\leq(k\alpha)^{\frac{1}{p}},$ so
$\left\|T_{k}(u)\right\|_{p^{*}}\leq c(N,p)(k\alpha)^{\frac{1}{p}}$.

for $0<\varepsilon\leq k,$ We have
$\\{|u|>\varepsilon\\}=\left\\{\left|T_{k}(u)>\varepsilon\right|\right\\},$ so

$\operatorname{meas}\\{|u|>k\\}\leq\varepsilon^{-p^{*}}\left\|T_{k}(u)\right\|_{p^{*}}^{p^{*}}\leq
c_{1}(N,p)(k\alpha)^{\frac{p^{*}}{p}}\varepsilon^{-p^{*}}\leq
c_{1}(N,p)\alpha^{\frac{N}{N-p}}k^{\frac{N}{N-p}}\varepsilon^{-\frac{Np}{N-p}}.$

for $\varepsilon=k,\operatorname{meas}\\{|u|>k\\}\leq
c_{1}(N,p)\alpha^{\frac{N}{N-p}}k^{-\frac{N(v-1)}{N-p}},$ we obtain

$\operatorname{meas}\\{|u|>k\\}\leq Ck^{-p_{1}},$

or $C=c_{1}(N,p)\alpha^{\frac{N}{N-p}}\quad$ and $p_{1}=\frac{N(p-1)}{N-p}.$
So it results than $\phi_{u}(k)\leq Ck^{-p_{1}},$ and as a conclusion it
results than $u\in\mathcal{M}^{p_{1}}(M)$. $\blacksquare$

We now prove estimates on the gradient of $u$.

###### Lemma 13

Let $1<p<N$ and suppose that $u\in\tau_{0}^{1,p}(M)$ satisfied $(6)$ for all
$k$.
so for all $h>0$

$\operatorname{meas}\bigg{\\{}|\nabla u|>h\bigg{\\}}\leq
C(N,p)\alpha^{\frac{N}{N-1}}h^{-p_{2}},\quad p_{2}=\frac{N(p-1)}{N-1}.$

Proof

for $k,\lambda>0,$ we pose

$\Phi(k,\lambda)=\operatorname{meas}\left\\{|\nabla
u|^{p}>\lambda,|u|>k\right\\},$

according to the Lemma $(13)$ we have

$\Phi(k,0)\leq C(N,p)M^{\frac{N}{N-p}}h^{-p_{1}}$ (2.8)

As the function $\lambda\mapsto\Phi(k,\lambda)$ is decreasing, we get for
$k,\lambda>0$ and for $0\leq s\leq\lambda,\Phi(0,\lambda)\leq\Phi(0,s),$ so

$\displaystyle\Phi(0,\lambda)\leq\Phi(0,s)$
$\displaystyle\Rightarrow\int_{0}^{\lambda}\Phi(0,\lambda)ds\leq\int_{0}^{\lambda}\Phi(0,s)ds$
$\displaystyle\Rightarrow\Phi(0,\lambda)\leq\frac{1}{\lambda}\int_{0}^{\lambda}\Phi(0,s)ds$

and

$\displaystyle\frac{1}{\lambda}\int_{0}^{\lambda}\Phi(0,s)ds$
$\displaystyle=\frac{1}{\lambda}\int_{0}^{\lambda}\Phi(k,s)ds+\frac{1}{\lambda}\int_{0}^{\lambda}(\Phi(0,s)-\Phi(k,s))ds$
$\displaystyle\leq\frac{1}{\lambda}\int_{0}^{\lambda}\Phi(k,0)ds+\frac{1}{\lambda}\int_{0}^{\lambda}(\Phi(0,s)-\Phi(k,s))ds$
$\displaystyle\leq\Phi(k,0)+\frac{1}{\lambda}\int_{0}^{\lambda}(\Phi(0,s)-\Phi(k,s))ds$

so

$\Phi(0,\lambda)\leq\frac{1}{\lambda}\int_{0}^{\lambda}\Phi(0,s)ds\leq\Phi(k,0)+\int_{0}^{\lambda}(\Phi(0,s)-\Phi(k,s))ds$
(2.9)

Note that

$\Phi(0,s)-\Phi(k,s)=\operatorname{meas}\left\\{|u|<k,|\nabla
u|^{p}>s\right\\}$

as $(7)$, we will have

$\int_{0}^{\infty}(\Phi(0,s)-\Phi(k,s))ds=\int_{\\{|u|<k\\}}|\nabla
u|^{p}d\sigma_{g}\leq k\alpha.$ (2.10)

Finally from $(9)$ and using $(8)$ and $(10),$ we get to

$\Phi(0,\lambda)\leq\frac{\alpha
k}{\lambda}+C(N,p)\alpha^{\frac{N}{N-p}}k^{-p_{1}}.$

we pose $P(k)=\frac{\alpha k}{\lambda}+c\alpha^{\frac{N}{N-p}}k^{-p_{1}}$, so
minimizing $P(k),$ of $k,$ we will have to solve the equation
$P^{\prime}(k)=0,$ which implies that

$\frac{\alpha}{\lambda}-cp_{1}\alpha^{\frac{N}{N-p}}k^{-p_{1}-1}=0$

and so $k=\left(c\lambda
p_{1}\alpha^{\frac{p}{N-p}}\right)^{\frac{1}{p_{1}+1}}.$

Consequently,

$\displaystyle\Phi(0,\lambda)$ $\displaystyle\leq
k\left[\frac{\alpha}{\lambda}+c\alpha^{\frac{N}{N-p}}k^{-p_{1}-1}\right]\leq
k\left[\frac{\alpha}{\lambda}+\frac{\alpha}{\lambda
p_{1}}\alpha^{\frac{N}{N-p}}\alpha^{-\frac{N}{N-p}}\right]$ $\displaystyle\leq
k\frac{\alpha}{\lambda}\left[1+\frac{1}{p_{1}}\right]\leq\frac{\alpha}{\lambda}\left[1+\frac{1}{p_{1}}\right]\left(c\lambda
p_{1}\alpha^{\frac{p}{N-p}}\right)^{\frac{1}{p_{1}+1}}$
$\displaystyle\leq\frac{\alpha}{\lambda}\left[1+\frac{1}{p_{1}}\right]\left(c\lambda
p_{1}\alpha^{\frac{p}{N-p}}\right)^{\frac{N-p}{p(N-1)}}\leq\frac{\alpha}{\lambda}\left[1+\frac{1}{p_{1}}\right]\left(cp_{1}\right)^{\frac{N-p}{p(N-1)}}\lambda^{-\frac{N-p}{p(N-1)}}\alpha^{\frac{p}{N-p}\frac{N-p}{p(N-1)}}$
$\displaystyle\leq\left[1+\frac{1}{p_{1}}\right]\left(cp_{1}\right)^{\frac{N-p}{p(N-1)}}\lambda^{-\frac{N(p-1)}{p(N-1)}}\alpha^{\frac{N}{N-1}}$

so

$\Phi(0,\lambda)\leq
C(N,p)\alpha^{\frac{N}{N-1}}\lambda^{-\frac{N(p-1)}{p(N-1)}},\quad\text{ with
}\quad
C(N,p)=\left[1+\frac{1}{p_{1}}\right]\left(cp_{1}\right)^{\frac{N-p}{p(N-1)}}$

we pose $\lambda=h^{p},$ so

$\operatorname{meas}\left\\{|\nabla u|^{p}>h^{p}\right\\}\leq
C(N,p)\alpha^{\frac{N}{N-1}}h^{-\frac{N(p-1)}{(N-1)}}$

and consequently

$\operatorname{meas}\\{|\nabla u|>h\\}\leq
C(N,p)\alpha^{\frac{N}{N-1}}h^{-p_{2}}\quad\text{ with }\quad
p_{2}=\frac{N(p-1)}{(N-1)}$

$\blacksquare$

Hence the result.

###  2.5 Existence of the entropy solution

We are in a position to demonstrate the main result of this article, more
precisely we have the next theorem

###### Theorem 2.1

Let $1<p<N$ and Let $(M,g)$ a compact Riemannian manifold , so it exists $u$
an entropy solution of the problem $(2)$ with $u\in\tau_{0}^{1,p}(M).$
Furthermore

$u\in\mathcal{M}^{p_{1}}(M)\quad\text{ and }\quad|\nabla
u|\in\mathcal{M}^{p_{2}}(M),$

or $p_{1}=\dfrac{N(p-1)}{N-p}\quad$ and $\quad p_{2}=\dfrac{N(p-1)}{N-1}.$

In the case $p>2-\dfrac{1}{N}$ the solution $u\in W_{0}^{1,q}(M)$ for all
$q<p_{2}$.

Proof.

The main idea of the demonstration is to proceed by approximation.

Step $1.$

As $f\in L^{1}(M)$ there is a sequence of functions
$\left\\{f_{n}\right\\}\subset L^{\infty}(M)$ such as $f_{n}\longrightarrow f$
in $L^{1}(M)$.

for $f_{n}\in L^{\infty}(M)$ it exists $u_{n}\in W_{0}^{1,p}(M),$ the unique
weak solution of the problem

$\left\\{\begin{array}[]{ccc}{-\Delta_{p}u_{n}=f_{n}}&{\text{ in }}&{M}\\\
{u_{n}=0}&{\text{ on }}&{\partial M}\end{array}\right.$ (2.11)

note that $T_{k}\left(u_{n}\right)\in L^{1}(M)\cap L^{\infty}(M)$ for all
$k>0,$ so taking $T_{k}\left(u_{n}\right)$ as test function in$(11)$ we get
that

$\int_{M}\left|\nabla u_{n}\right|^{p-2}\nabla u_{n}\nabla
T_{k}\left(u_{n}\right)d\sigma_{g}=\int_{M}f_{n}T_{k}\left(u_{n}\right)d\sigma_{g}$

so

$\int_{\left\\{\left|u_{n}\right|<k\right\\}}\left|\nabla
u_{n}\right|^{p}d\sigma_{g}\leq kc$

and so

$\frac{1}{k}\int_{\left\\{\left|u_{n}\right|<k\right\\}}\left|\nabla
u_{n}\right|^{p}d\sigma_{g}\leq c$

ie,

$\int_{M}\left|\nabla T_{k}\left(u_{n}\right)\right|^{p}d\sigma_{g}\leq kc$

therefore it is concluded that $\left\\{\nabla
T_{k}\left(u_{n}\right)\right\\}$ is bounded in $L^{p}(M)$ for all $k>0$ . So
it exists $w_{k}$ tel que $T_{k}\left(u_{n}\right)\rightarrow w_{k}$ weakly in
$W_{0}^{1,p}(M)$ for each $k>0,$ and $T_{k}\left(u_{n}\right)\rightarrow
w_{k}$ a.e in $M.$ we pose $w_{k}\equiv T_{k}(u)$ in the set or
$\left|w_{k}\right|<k,$ it’s clear that $u$ is well defined because
$T_{k+h}\left(u_{n}\right)=T_{k}\left(T_{h}\left(u_{n}\right)\right)$ and
consequently $T_{k}\left(u_{n}\right)\rightarrow T_{k}(u)$ strongly in
$L^{q}(M)$ for all $q<p^{*}$.

According to Lemmas 12 and $13,$ we have

$u_{n}\in\mathcal{M}^{p_{1}}(M)\quad$ et $\quad\left|\nabla
u_{n}\right|\in\mathcal{M}^{p_{2}}(M)$,

with $p_{1}=\frac{N(p-1)}{N-p}$ and $p_{2}=\frac{N(p-1)}{N-1},\quad$ so

$\left\|u_{n}\right\|_{\mathcal{M}^{p_{1}(M)}}\leq C\quad$ and
$\quad\left\|\nabla u_{n}\right\|_{\mathcal{M}^{p_{2}}(M)}\leq\overline{C},$

and as $L^{q}(M)\subset\mathcal{M}^{q}(M)\subset L^{q-\varepsilon}(M)$ for all
$q,\varepsilon>0,$ so

$\left\|u_{n}\right\|_{L^{p_{1}-\varepsilon}(M)}\leq C\quad$ et
$\quad\left\|\nabla u_{n}\right\|_{L^{p_{2}-\varepsilon}(M)}\leq\overline{C}.$

Note that if $p>2-\frac{1}{N},$ so $p_{2}>1$ and consequently
$\left\\{u_{n}\right\\}$ will be bounded in $W_{0}^{1,p_{2}-\varepsilon}(M)$
for all $\varepsilon>0$ with $p_{2}-\varepsilon\geq 1,$ so $u_{n}\rightarrow
u$ weakly in $W_{0}^{1,p_{2}-\varepsilon}(M)$ .

So for all $\varphi\in W^{1,\infty}(M),$ we have

$\int_{M}\left|\nabla u_{n}\right|^{p-2}\nabla u_{n}\nabla\varphi
d\sigma_{g}=\int_{M}f_{n}\varphi d\sigma_{g},$

as $\left|\nabla u_{n}\right|^{p_{2}-\varepsilon}\equiv\left(\left|\nabla
u_{n}\right|^{p-1}\right)^{\frac{N}{N-1}-\varepsilon}$ and $\quad p-1<p_{2},$
so for $\quad p-1<p_{2}-\varepsilon$ $\left|\nabla u_{n}\right|^{p-1}\in
L^{\frac{N}{N-1}-\varepsilon}(M)$.

as $|\nabla\varphi|\in L^{\infty}(M)$ and $f_{n}\rightarrow f\quad$ in
$L^{1}(M),$ so going to the limit when $n\rightarrow\infty,$ we find that

$\int_{M}|\nabla u|^{p-2}\nabla u\nabla\varphi d\sigma_{g}=\int_{M}f\varphi
d\sigma_{g}$

It’s clear that $u_{n}\rightarrow u$ strongly in $L^{\overline{q}}(M)$ such as
$1\leq\overline{q}<\overline{p}^{*}$ with
$\overline{p}^{*}=\frac{N\overline{p}}{N-\overline{p}}>1$ or
$\overline{p}=p_{2}-\varepsilon$

Step $2.$ To analyze the general case $1<p,$ we start by demonstrating that
$u\in\tau_{0}^{1,p}(M).$

We pose $\nabla T_{k}(u)=\nabla w_{k},$ is clear that $\nabla T_{k}(u)$ is
well defined because $w_{k}\in W_{0}^{1,p}(M),$ to go to the limit in $k$ we
will start by show that $\nabla u_{n}$ converges to $\nabla u$ locally in
measure. To prove it we show that $\left\\{\nabla u_{n}\right\\}$ is a Cauchy
sequence in measure.

Let $t$ and $\varepsilon>0,$ so

$\displaystyle\big{\\{}\left|\nabla u_{n}-\nabla u_{m}\right|>t\big{\\}}$
$\displaystyle\subset\big{\\{}\left|\nabla
u_{n}\right|>A\big{\\}}\cup\big{\\{}\left|\nabla
u_{m}\right|>A\big{\\}}\cup\big{\\{}\left|u_{n}-u_{m}\right|>k\big{\\}}$
(2.12) $\displaystyle\cup\big{\\{}\left|u_{n}-u_{m}\right|\leq k,\left|\nabla
u_{n}\right|\leq A,\left|\nabla u_{m}\right|\leq A,\left|\nabla u_{n}-\nabla
u_{m}\right|>t\big{\\}}.$

We choose $A$ big enough as

$\operatorname{meas}\left\\{\left|\nabla
u_{n}\right|>A\right\\}\leq\varepsilon\quad\text{ for all }\quad
n\in\mathbb{N},$

(this is possible by Lemma 13).

To estimate the last term in $(12),$ we use the next algebraic inequalities.

for all $\xi,\eta\in\mathbb{R}^{N},$ we have

$\left\langle|\xi|^{p-2}\xi-|\eta|^{p-2}\eta,\xi-\eta\right\rangle\geq 0,$

again if $\ xi\ neq\ eta$ then

$\left\langle|\xi|^{p-2}\xi-|\eta|^{p-2}\eta,\xi-\eta\right\rangle>0$

and if $|\xi|<A,|\eta|<A$ and $|\xi-\eta|>t,$ so it exists $\mu>0$ such that

$\left\langle|\xi|^{p-2}\xi-|\eta|^{p-2}\eta,\xi-\eta\right\rangle\geq\mu.$

Knowing that $-\Delta_{p}u_{n}=f_{n}\quad$ and $\quad-\Delta_{p}u_{m}=f_{m},$
so by subtracting and using $T_{k}\left(u_{n}-u_{m}\right)$as a test function,
we get

$\displaystyle\int_{\left\\{\left|u_{n}-u_{m}\right|\leq
k\right\\}}\left\langle\left|\nabla u_{n}\right|^{p-2}\nabla u_{n}\right.$
$\displaystyle-\left|\nabla u_{m}\right|^{p-2}\nabla u_{m},\nabla u_{n}-\nabla
u_{m}\rangle d\sigma_{g}$
$\displaystyle=\int_{M}\left(f_{n}-f_{m}\right)T_{k}\left(u_{n}-u_{m}\right)d\sigma_{g}\leq
2ck.$

According to the Lemma $12,$ we have

$\displaystyle\quad\operatorname{meas}\left\\{\left|u_{n}-u_{m}\right|\leq
k,\left|\nabla u_{n}\right|\leq A,\left|\nabla u_{m}\right|\leq A,\left|\nabla
u_{n}-\nabla u_{m}\right|>t\right\\}$ $\displaystyle\leq$
$\displaystyle\quad\operatorname{meas}\left\\{\left|u_{n}-u_{m}\right|\leq
k,\quad\left(\left|\nabla u_{n}\right|^{p-2}u_{n}-\left|\nabla
u_{m}\right|^{p-2}u_{m}\right)\cdot\left(\nabla u_{n}-\nabla
u_{m}\right)\geq\mu\right\\}$ $\displaystyle\leq$
$\displaystyle\quad\frac{1}{\mu}\int_{\left\\{\left|u_{n}-u_{m}\right|\leq
k\right\\}}\left\langle\left|\nabla u_{n}\right|^{p-2}u_{n}-\left|\nabla
u_{m}\right|^{p-2}u_{m},\nabla u_{n}-\nabla u_{m}\right\rangle d\sigma_{g}$
$\displaystyle\leq$ $\displaystyle\quad\frac{1}{\mu}2ck\leq\varepsilon,$

if $k$ is small enough, as $k\leq\dfrac{\mu\varepsilon}{2c}$.

So we fix $A$ and $k,$ if $n_{0}$ big enough, we have to $n,m\geq
n_{0},\operatorname{mes}\big{\\{}\left|u_{n}-u_{m}\right|>k\big{\\}}\leq\varepsilon,$
and so

$\operatorname{meas}\left\\{\left|\nabla u_{n}-\nabla
u_{m}\right|>k\right\\}\leq 2\varepsilon.$

So $\left\\{\nabla u_{n}\right\\}$ converges locally to a $v$ function and as
a consequence a.e. in $M.$ Since $\left\\{\nabla
T_{k}\left(u_{n}\right)\right\\}$ is bounded in $L^{p}(M)$ for all $k>0$ and
$\nabla T_{k}\left(u_{n}\right)\rightharpoonup$ $\nabla T_{k}(u)$ weakly in
$L^{p}(M),$ we deduce that $v=\nabla u$ a.e. Note that in general
$v\notin\left(L^{1}(M)\right)^{N}.$ It is clear that if $p>2-\frac{1}{N},$
then $v\notin\left(L^{1}(M)\right)^{N},$ and so $u\in W_{0}^{1,1}(M)$ and from
Lemma 10 we deduce $\nabla u=v\quad a.e.$

And consequently $u\in\tau_{0}^{1,1}(M)$.

To see that $u\in\tau_{0}^{1,p}(M),$ we consider $\phi_{n}\in
C_{0}^{\infty}(M)$ such that

$\left\|\nabla\phi_{n}-\nabla
T_{k}\left(u_{n}\right)\right\|_{L^{p}(\Omega)}\leq\frac{1}{n}\quad\text{ et
}\quad\left\|\phi_{n}-T_{k}\left(u_{n}\right)\right\|_{L^{p^{*}}(\Omega)}\leq\frac{1}{n}.$

We have then

$\nabla\phi_{n}\longrightarrow\nabla T_{k}(u)\quad\text{ fortement dans }\quad
L^{p}(M)$

and

$\phi_{n}\longrightarrow T_{k}(u)\quad\text{ fortement in }\quad
L_{loc}^{q}(M)\quad\text{ for }\quad q<p^{*}.$

As a conclusion we get that $\phi_{n}$ converges strongly to $T_{k}(u)$ and
consequently $u\in\tau_{0}^{1,p}(M).$

Step $3.$ In this step we will demonstrate the strong convergence of
truncations in $W_{0}^{1,p}(M),$ ie for $k>0$ fixed on a
$T_{k}\left(u_{n}\right)\rightarrow T_{k}(u)$ strongly in $W_{0}^{1,p}(M).$

Note that $T_{k}\left(u_{n}\right)\rightarrow T_{k}(u)$ weakly in
$W_{0}^{1,p}(M)$ for all $k>0$.

Let $k,h>0$ such that $h>k>0,$ we assume

$w_{n}=T_{2k}\left(u_{n}-T_{h}\left(u_{n}\right)T_{k}\left(u_{n}\right)-T_{k}(u)\right)$

Taking $w_{n}$ as a test function in $(11),$ it results

$\int_{M}\left|\nabla u_{n}\right|^{p-2}\nabla u_{n}\nabla
w_{n}d\sigma_{g}=\int_{M}f_{n}w_{n}d\sigma_{g},$

we pose $I=\int_{M}\left|\nabla u_{n}\right|^{p-2}\nabla u_{n}\nabla w_{n}dx,$
when $k\rightarrow\infty$ and $h\rightarrow\infty$ we have
$\int_{M}f_{n}w_{n}dx\rightarrow 0,$ so $\int_{M}\left|\nabla
u_{n}\right|^{p-2}\nabla u_{n}\nabla w_{n}dx\rightarrow 0.$

We pose $\alpha=4k+h.\quad$ if $\left|u_{n}\right|>\alpha,\nabla w_{n}=0.$ So

$\displaystyle I$
$\displaystyle=\int_{\left\\{\left|u_{n}\right|<\alpha\right\\}}\left|\nabla
u_{n}\right|^{p-2}\nabla u_{n}\nabla w_{n}d\sigma_{g}$
$\displaystyle=\int_{\left\\{\left|u_{n}\right|<k\right\\}}\left|\nabla
u_{n}\right|^{p-2}\nabla u_{n}\nabla
w_{n}d\sigma_{g}+\int_{\left\\{k<\left|u_{n}\right|<\alpha\right\\}}\left|\nabla
u_{n}\right|^{p-2}\nabla u_{n}\nabla w_{n}d\sigma_{g}$
$\displaystyle=\int_{M}\left|\nabla T_{k}\left(u_{n}\right)\right|^{p-2}\nabla
T_{k}\left(u_{n}\right)\nabla\left(T_{k}\left(u_{n}\right)-T_{k}(u)\right)d\sigma_{g}$
$\displaystyle+\int_{\left\\{k<\left|u_{n}\right|<\alpha\right\\}}\left|\nabla
u_{n}\right|^{p-2}\nabla u_{n}\nabla w_{n}d\sigma_{g}.$

note that $1-$ if $k<\left|u_{n}\right|\leq h$ so $\nabla w_{n}=\nabla
T_{k}(u)$

and

$2-$ if $h<\left|u_{n}\right|<\alpha$ so $\nabla w_{n}=\nabla T_{k}(u)$,

so

$\displaystyle\int_{\left\\{k<\left|u_{n}\right|<\alpha\right\\}}\left|\nabla
u_{n}\right|^{p-2}\nabla u_{n}\nabla w_{n}d\sigma_{g}$
$\displaystyle=\int_{\left\\{\left|u_{n}\right|>k\right\\}}\left|\nabla
T_{\alpha}\left(u_{n}\right)\right|^{p-2}\nabla
T_{\alpha}\left(u_{n}\right)\nabla T_{k}(u)d\sigma_{g}$
$\displaystyle\geq-\int_{\left\\{\left|u_{n}\right|>k\right\\}}\left|\nabla
T_{\alpha}\left(u_{n}\right)\right|^{p-1}\left|\nabla
T_{k}(u)\right|d\sigma_{g}$

we obtain

$\displaystyle I$ $\displaystyle\geq\int_{M}\left|\nabla
T_{k}\left(u_{n}\right)\right|^{p-2}\nabla
T_{k}\left(u_{n}\right)\nabla\left(T_{k}\left(u_{n}\right)-T_{k}(u)\right)d\sigma_{g}$
$\displaystyle-\int_{\left\\{\left|u_{n}\right|>k\right\\}}\left|\nabla
T_{\alpha}\left(u_{n}\right)\right|^{p-1}\left|\nabla
T_{k}(u)\right|d\sigma_{g}.$

As $\left\\{\left|\nabla T_{\alpha}\left(u_{n}\right)\right|^{p-1}\right\\}$
is bounded in $L^{\frac{p}{p-1}}(M),\left|\nabla
T_{k}(u)\right|1_{\left\\{\left|u_{n}\right|>k\right\\}}$ is bounded in
$L^{p}(M)$ and $\left|\nabla
T_{k}(u)\right|\mathbb{1}_{\left\\{\left|u_{n}\right|>k\right\\}}\rightarrow
0$ strongly in $L^{p}(M)$ when $k\rightarrow\infty,$ we get that

$\int_{\left\\{\left|u_{n}\right|>k\right\\}}\left|\nabla
T_{\alpha}\left(u_{n}\right)\right|^{p-1}\left|\nabla
T_{k}(u)\right|d\sigma_{g}\rightarrow 0\quad\text{ when }\quad
k\rightarrow\infty.$

So

$\displaystyle J$ $\displaystyle=\int_{M}\left|\nabla
T_{k}\left(u_{n}\right)\right|^{p-2}\nabla
T_{k}\left(u_{n}\right)\nabla\left(T_{k}\left(u_{n}\right)-T_{k}(u)\right)dx$
$\displaystyle=\int_{M}\left(\left|\nabla
T_{k}\left(u_{n}\right)\right|^{p-2}\nabla
T_{k}\left(u_{n}\right)-\left|\nabla T_{k}(u)\right|^{p-2}\nabla
T_{k}(u)\right)\left(\nabla T_{k}\left(u_{n}\right)-\nabla
T_{k}(u)\right)d\sigma_{g}$ $\displaystyle+\int_{M}\left|\nabla
T_{k}(u)\right|^{p-2}\nabla T_{k}(u)\left(\nabla
T_{k}\left(u_{n}\right)-\nabla T_{k}(u)\right)d\sigma_{g}$

as

$\int_{M}\left|\nabla T_{k}(u)\right|^{p-2}\nabla T_{k}(u)\left(\nabla
T_{k}\left(u_{n}\right)-\nabla T_{k}(u)\right)d\sigma_{g}\rightarrow
0\quad\text{ when }\quad n\rightarrow\infty,$

so

$J=\int_{M}\left(\left|\nabla T_{k}\left(u_{n}\right)\right|^{p-2}\nabla
T_{k}\left(u_{n}\right)-\left|\nabla T_{k}(u)\right|^{p-2}\nabla
T_{k}(u)\right)\left(\nabla T_{k}\left(u_{n}\right)-\nabla
T_{k}(u)\right)d\sigma_{g}+o(1).$

So

$\displaystyle I$ $\displaystyle\geq\int_{M}\left(\left|\nabla
T_{k}\left(u_{n}\right)\right|^{p-2}\nabla
T_{k}\left(u_{n}\right)-\left|\nabla T_{k}(u)\right|^{p-2}\nabla
T_{k}(u)\right)\left(\nabla T_{k}\left(u_{n}\right)-\nabla
T_{k}(u)\right)d\sigma_{g}+o(1)$ $\displaystyle\geq c\int_{M}\left|\nabla
T_{k}\left(u_{n}\right)-\nabla
T_{k}(u)\right|^{p}d\sigma_{g}+o(1)\quad\text{if }\quad p\geq 2$

and

$I\geq C(p)\int_{M}\frac{\left|\nabla T_{k}\left(u_{n}\right)-\nabla
T_{k}(u)\right|^{2}}{\left(\left|\nabla T_{k}(u)\right|+\left|\nabla
T_{k}\left(u_{n}\right)\right|\right)^{2-p}}d\sigma_{g}+o(1)\quad\text{ if
}\quad p<2.$

So

$\int_{M}\left|\nabla T_{k}\left(u_{n}\right)-\nabla
T_{k}(u)\right|^{p}d\sigma_{g}\leq
o(1)+\int_{M}f_{n}w_{n}d\sigma_{g}\quad\text{ if }\quad p\geq 2$

and

$C(p)\int_{M}\frac{\left|\nabla T_{k}\left(u_{n}\right)-\nabla
T_{k}(u)\right|^{2}}{\left(\left|\nabla T_{k}(u)\right|+\left|\nabla
T_{k}\left(u_{n}\right)\right|\right)^{2-p}}d\sigma_{g}\leq
o(1)+\int_{M}f_{n}w_{n}\quad\text{ if }\quad p<2.$

Consequently

$\int_{M}\left|\nabla T_{k}\left(u_{n}\right)-\nabla
T_{k}(u)\right|^{p}d\sigma_{g}\rightarrow 0\quad\text{ if }\quad p\geq 2,$

and

$\int_{M}\frac{\left|\nabla T_{k}\left(u_{n}\right)-\nabla
T_{k}(u)\right|^{2}}{\left(\left|\nabla T_{k}(u)\right|+\left|\nabla
T_{k}\left(u_{n}\right)\right|\right)^{2-p}}d\sigma_{g}\rightarrow
0\quad\text{ if }\quad p<2.$

For the second case we have

$\begin{array}[]{c}{\displaystyle\int_{M}\left|\nabla
T_{k}\left(u_{n}\right)-\nabla
T_{k}(u)\right|^{p}d\sigma_{g}=\displaystyle\int_{M}\frac{\left|\nabla
T_{k}\left(u_{n}\right)-\nabla T_{k}(u)\right|^{p}}{\left(|\nabla
T_{k}(u)|+|\nabla
T_{k}\left(u_{n}\right)|\right)^{\frac{p(2-p)}{2}}}\left(\left|\nabla
T_{k}(u)\right|+\left|\nabla
T_{k}\left(u_{n}\right)\right|\right)^{\frac{p(2-p)}{2}}d\sigma_{g}}\\\
{\quad\leq\left(\displaystyle\int_{M}\frac{\left|\nabla
T_{k}\left(u_{n}\right)-\nabla T_{k}(u)\right|^{2}}{\left(\left|\nabla
T_{k}(u)\right|+\left|\nabla
T_{k}\left(u_{n}\right)\right|\right)^{2-p}}d\sigma_{g}\right)^{\frac{p}{2}}\left(\displaystyle\int_{M}\left(\left|\nabla
T_{k}\left(u_{n}\right)\right|^{p}+\left|\nabla
T_{k}(u)\right|^{p}\right)d\sigma_{g}\right)^{\frac{2-p}{2}}}.\end{array}$

So for $p<2$,

$\int_{M}\left|\nabla T_{k}\left(u_{n}\right)-\nabla
T_{k}(u)\right|^{p}d\sigma_{g}\rightarrow 0\text{ for }n\rightarrow\infty$

As a conclusion we obtain that $T_{k}\left(u_{n}\right)\rightarrow T_{k}(u)$
strongly in $W_{0}^{1,p}(M)$ for all $k>0.$

Step $4.$ To complete the proof it remains to show that $u$ is an entropy
solution.

recall that

$\left\\{\begin{array}[]{ccc}{-\Delta_{p}u_{n}=f_{n}}&{\text{ in }}&{M}\\\
{u_{n}=0}&{\text{ on}}&{\partial M}\end{array}\right.$

Let $v\in L^{\infty}(M)\cap W_{0}^{1,p}(M),$ for all $k$ fixed $>0,$ we have

$\int_{M}\left|\nabla u_{n}\right|^{p-2}\nabla u_{n}\nabla
T_{k}\left(u_{n}-v\right)d\sigma_{g}=\int_{M}f_{n}T_{k}\left(u_{n}-v\right)d\sigma_{g}$

As $u_{n}\longrightarrow u\quad a.e.$ in $M,$ and $f_{n}\rightarrow f\quad$ in
$L^{1}(M).$

So
$\int_{M}f_{n}T_{k}\left(u_{n}-v\right)d\sigma_{g}\rightarrow\int_{M}fT_{k}(u-v)d\sigma_{g}$
for $n\rightarrow\infty$.

As $v\in L^{\infty}(M)\cap W_{0}^{1,p}(M),$ so it exists a positive constant
$c>0$ such that

$\int_{M}\left|\nabla u_{n}\right|^{p-2}\nabla u_{n}\nabla
T_{k}\left(u_{n}-v\right)d\sigma_{g}=\int_{\left\\{\left|u_{n}\right|\leq
c\right\\}}\left|\nabla T_{c}\left(u_{n}\right)\right|^{p-2}\nabla
T_{c}\left(u_{n}\right)\nabla T_{k}\left(u_{n}-v\right)d\sigma_{g}$

Note that it is sufficient to take $c\geq k+\|v\|_{\infty}.$ As
$T_{k}\left(u_{n}\right)\rightarrow T_{k}(u)$ strongly in $W_{0}^{1,p}(M),$ so
we conclude that

$\int_{\left\\{\left|u_{n}\right|\leq c\right\\}}\left|\nabla
T_{c}\left(u_{n}\right)\right|^{p-2}\nabla T_{c}\left(u_{n}\right)\nabla
T_{k}\left(u_{n}-v\right)d\sigma_{g}\rightarrow\int_{\\{|u|\leq
c\\}}\left|\nabla T_{c}(u)\right|^{p-2}\nabla T_{c}(u)\nabla
T_{k}(u-v)d\sigma_{g}$

for $n\rightarrow\infty.$ Consequently and for $n\rightarrow\infty$ we get
that

$\int_{M}|\nabla u|^{p-2}\nabla u\nabla
T_{k}(u-v)d\sigma_{g}=\int_{M}fT_{k}(u-v)d\sigma_{g}$

So $u$ is an entropy solution of the problem $(1,2).$ $\blacksquare$

###  2.6 Uniqueness of the solution in the sense of entropy

We deal here with the question of the uniqueness of entropy solutions
$u\in\tau_{0}^{1,p}(M)$ for the problem $(2),$ note that $u$ checks $(3)$ for
all $\phi\in L^{\infty}(M)\cap W_{0}^{1,p}(M)$ and for all $k>0$

The main result of this section is the next theorem

###### Theorem 2.2

Let $u_{1}$ et $u_{2}$ dtwo functions in $\tau_{0}^{1,p}(M),$ such as $u_{1}$
and $u_{2}$ are entropy solutions to the problem

$-\Delta_{p}u=f(x)$

so $u_{1}=u_{2}$.

Proof.

Note that $f\in L^{1}(M),$ substitute in the relation $(3)$ with test
functions $T_{h}\left(u_{1}\right)$ and $T_{h}\left(u_{2}\right)$ and by
addition gets that

$\int_{\left\\{\left|u_{1}-T_{h}\left(u_{2}\right)\right|<k\right\\}}\left|\nabla
u_{1}\right|^{p-2}\nabla
u_{1}\nabla\left(u_{1}-T_{h}\left(u_{2}\right)\right)d\sigma_{g}=\int_{M}fT_{k}\left(u_{1}-T_{h}\left(u_{2}\right)\right)d\sigma_{g},$

$\int_{\left\\{\left|u_{2}-T_{h}\left(u_{1}\right)\right|<k\right\\}}\left|\nabla
u_{2}\right|^{p-2}\nabla
u_{2}\nabla\left(u_{2}-T_{h}\left(u_{1}\right)\right)d\sigma_{g}=\int_{M}fT_{k}\left(u_{2}-T_{h}\left(u_{1}\right)\right)d\sigma_{g}.$

By combining the two results we get

$\displaystyle\int_{\left\\{\left|u_{1}-T_{h}\left(u_{2}\right)\right|<k\right\\}}\left|\nabla
u_{1}\right|^{p-2}\nabla
u_{1}\nabla\left(u_{1}-T_{h}\left(u_{2}\right)\right)d\sigma_{g}$

$\displaystyle+\int_{\left\\{\left|u_{2}-T_{h}\left(u_{1}\right)\right|<k\right\\}}\left|\nabla
u_{2}\right|^{p-2}\nabla
u_{2}\nabla\left(u_{2}-T_{h}\left(u_{1}\right)\right)d\sigma_{g}$ (2.13)
$\displaystyle=\int_{M}f\left(T_{k}\left(u_{1}-T_{h}\left(u_{2}\right)\right)+T_{k}\left(u_{2}-T_{h}\left(u_{1}\right)\right)\right)d\sigma_{g}.$

The conclusion $u_{1}=u_{2}$ will be reached after going to the limit $h\
rightarrow\ infty$ in this formula. Let

$\displaystyle I$
$\displaystyle=\int_{\left\\{\left|u_{1}-T_{h}\left(u_{2}\right)\right|<k\right\\}}\left|\nabla
u_{1}\right|^{p-2}\nabla
u_{1}\nabla\left(u_{1}-T_{h}\left(u_{2}\right)\right)d\sigma_{g}$
$\displaystyle+\int_{\left\\{\left|u_{2}-T_{h}\left(u_{1}\right)\right|<k\right\\}}\left|\nabla
u_{2}\right|^{p-2}\nabla
u_{2}\nabla\left(u_{2}-T_{h}\left(u_{1}\right)\right)d\sigma_{g}.$

we pose

$A_{0}=\big{\\{}x\in
M:\left|u_{1}-u_{2}\right|<k,\left|u_{1}\right|<h,\left|u_{2}\right|<h\big{\\}}.$

In $A_{0}$ the first member of $(13)$ is reduced to the next term

$I_{0}=\int_{A_{0}}\left(\left|\nabla u_{1}\right|^{p-2}\nabla
u_{1}-\left|\nabla u_{2}\right|^{p-2}\nabla u_{2}\right)\left(\nabla
u_{1}-\nabla u_{2}\right)d\sigma_{g}.$

Let

$A_{1}=\left\\{x\in
M:\left|u_{1}-T_{h}\left(u_{2}\right)\right|<k,\left|u_{2}\right|\geq
h\right\\},$

so

$\int_{A_{1}}\left|\nabla u_{1}\right|^{p-2}\nabla
u_{1}\nabla\left(u_{1}-T_{h}\left(u_{2}\right)\right)d\sigma_{g}=\int_{A_{1}}\left|\nabla
u_{1}\right|^{p}d\sigma_{g}\geq 0,$

and on set

$A_{2}=\left\\{x\in
M:\left|u_{1}-T_{h}\left(u_{2}\right)\right|<k,\left|u_{2}\right|<h,\left|u_{1}\right|\geq
h\right\\},$

we are getting

$\displaystyle\int_{A_{2}}\left|\nabla u_{1}\right|^{p-2}\nabla
u_{1}\nabla\left(u_{1}-T_{h}\left(u_{2}\right)\right)d\sigma_{g}$
$\displaystyle=\int_{A_{2}}\left|\nabla u_{1}\right|^{p-2}\nabla
u_{1}\left(\nabla u_{1}-\nabla u_{2}\right)d\sigma_{g}$
$\displaystyle\geq-\int_{A_{2}}\left|\nabla u_{1}\right|^{p-2}\nabla
u_{1}\nabla u_{2}d\sigma_{g}.$

In the same way, we can define all $A_{1}^{\prime}$ and $A_{2}^{\prime}$ as

$A_{1}^{\prime}=\left\\{x\in
M:\left|u_{2}-T_{h}\left(u_{1}\right)\right|<k,\left|u_{1}\right|\geq
h\right\\},$

and

$A_{2}^{\prime}=\left\\{x\in
M:\left|u_{2}-T_{h}\left(u_{1}\right)\right|<k,\left|u_{1}\right|<h,\left|u_{2}\right|\geq
h\right\\}$

Then the second term of $(13)$ can be written as a sum of

$\int_{A_{1}^{\prime}}\left|\nabla u_{2}\right|^{p-2}\nabla u_{2}\left(\nabla
u_{2}-\nabla
T_{h}\left(u_{1}\right)\right)d\sigma_{g}=\int_{A_{1}^{\prime}}\left|\nabla
u_{2}\right|^{p}d\sigma_{g}\geq 0$

and

$\displaystyle\int_{A_{2}^{\prime}}\left|\nabla u_{2}\right|^{p-2}\nabla
u_{2}\left(\nabla u_{2}-\nabla T_{h}\left(u_{1}\right)\right)d\sigma_{g}$
$\displaystyle=\int_{A_{2}^{\prime}}\left|\nabla u_{2}\right|^{p-2}\nabla
u_{2}\left(\nabla u_{2}-\nabla u_{1}\right)d\sigma_{g}$
$\displaystyle\geq-\int_{A_{2}^{\prime}}\left|\nabla u_{2}\right|^{p-2}\nabla
u_{2}\nabla u_{1}d\sigma_{g}$

Therefore we conclude that

$\displaystyle I$ $\displaystyle\geq I_{0}+\int_{A_{1}}\left|\nabla
u_{1}\right|^{p}d\sigma_{g}-\int_{A_{2}}\left|\nabla u_{1}\right|^{p-2}\nabla
u_{1}\nabla u_{2}d\sigma_{g}$ $\displaystyle+\int_{A_{1}^{\prime}}\left|\nabla
u_{2}\right|^{p}d\sigma_{g}-\int_{A_{2}^{\prime}}\left|\nabla
u_{2}\right|^{p-2}\nabla u_{2}\nabla u_{1}d\sigma_{g}$ $\displaystyle\geq
I_{0}-\left(\int_{A_{2}}\left|\nabla u_{1}\right|^{p-2}\nabla u_{1}\nabla
u_{2}d\sigma_{g}+\int_{A_{2}^{\prime}}\left|\nabla u_{2}\right|^{p-2}\nabla
u_{2}\nabla u_{1}d\sigma_{g}\right)$ $\displaystyle\geq I_{0}-I_{3}$

or

$I_{3}=\int_{A_{2}}\left|\nabla u_{1}\right|^{p-2}\nabla u_{1}\nabla
u_{2}d\sigma_{g}+\int_{A_{2}^{\prime}}\left|\nabla u_{2}\right|^{p-2}\nabla
u_{2}\nabla u_{1}d\sigma_{g}$

The first term of $I_{3}$ can be estimated by

$\displaystyle\int_{A_{2}}\left|\nabla u_{1}\right|^{p-2}\nabla u_{1}\nabla
u_{2}d\sigma_{g}$ $\displaystyle\leq\int_{A_{2}}\left|\nabla
u_{1}\right|^{p-1}\left|\nabla u_{2}\right|d\sigma_{g}$
$\displaystyle\leq\left(\int_{A_{2}}\left|\nabla
u_{1}\right|^{p}d\sigma_{g}\right)^{\frac{p-1}{p}}\left(\int_{A_{2}}\left|\nabla
u_{2}\right|^{p}d\sigma_{g}\right)^{\frac{1}{p}}$
$\displaystyle\leq\left\|\nabla
u_{1}\right\|_{L^{p}\left(\left\\{h\leq\left|u_{1}\right|\leq
h+k\right\\}\right)}^{p-1}\left\|\nabla
u_{2}\right\|_{L^{p}\left(\left\\{h-k\leq\left|u_{2}\right|\leq
h\right\\}\right)}.$

as $\left\|\nabla
u_{1}\right\|_{L^{p}\left(\left\\{h\leq\left|u_{1}\right|\leq
h+k\right\\}\right)}^{p-1}\left\|\nabla
u_{2}\right\|_{L^{p}\left(\left\\{h-k\leq\left|u_{2}\right|\leq
h\right\\}\right)}\rightarrow 0$ when $h\rightarrow\infty$for all $k>0,$ it
results than $\displaystyle\int_{A_{2}}\left|\nabla u_{1}\right|^{p-2}\nabla
u_{1}\nabla u_{2}d\sigma_{g}$ converges to $0$ when $h\rightarrow\infty$for
tout $k>0$.

In the same way we obtain the same conclusion for the second term of $I_{3}$ .

So we conclude that $I_{3}$ tends to 0 when $h\rightarrow\infty$ .

Regarding the second member of $(13),$ knowing that

$T_{k}\left(u_{1}-T_{h}\left(u_{2}\right)\right)+T_{k}\left(u_{2}-T_{h}\left(u_{1}\right)\right)\rightarrow
0$ a.e. in $in$ for $h\rightarrow\infty$

$\left|T_{k}\left(u_{1}-T_{h}\left(u_{2}\right)\right)+T_{k}\left(u_{2}-T_{h}\left(u_{1}\right)\right)\right|\leq
2k$

and that $f\ inL^{1}(M),$ so using the dominated Convergence Theorem we get
that

$\int_{M}f\left(T_{k}\left(u_{1}-T_{h}\left(u_{2}\right)\right)+T_{k}\left(u_{2}-T_{h}\left(u_{1}\right)\right)\right)d\sigma_{g}\rightarrow
0\text{ quand }h\rightarrow\infty\text{ for all }k>0.$

Combining previous estimates it results than

$\int_{A_{0}(h,k)}\left(\left|\nabla u_{1}\right|^{p-2}\nabla
u_{1}-\left|\nabla u_{2}\right|^{p-2}\nabla u_{2}\right)\left(\nabla
u_{1}-\nabla u_{2}\right)d\sigma_{g}\leq\varepsilon(h),$

or $\varepsilon(h)\rightarrow 0$ when $h\rightarrow\infty$ for all $k$ fixed
$>0.$ Since $A_{0}(h,k)$ converges to

$\left\\{x\in M:\left|u_{1}-u_{2}\right|<k\right\\},$

we conclude that

$\int_{\left\\{\left|u_{1}-u_{2}\right|<k\right\\}}\left(\left|\nabla
u_{1}\right|^{p-2}\nabla u_{1}-\left|\nabla u_{2}\right|^{p-2}\nabla
u_{2}\right)\left(\nabla u_{1}-\nabla u_{2}\right)d\sigma_{g}\leq 0.$

As

$\lambda\left\|\nabla u_{1}-\nabla
u_{2}\right\|_{L^{p}\left(\left\\{\left|u_{1}-u_{2}\right|<k\right\\}\right)}^{p}\leq\int_{\left\\{\left|u_{1}-u_{2}\right|<k\right\\}}\left(\left|\nabla
u_{1}\right|^{p-2}\nabla u_{1}-\left|\nabla u_{2}\right|^{p-2}\nabla
u_{2}\right)\left(\nabla u_{1}-\nabla u_{2}\right)d\sigma_{g}$

if $p>2$ and

$\int_{\left\\{\left|u_{1}-u_{2}\right|<k\right\\}}\frac{\left|\nabla
u_{1}-\nabla u_{2}\right|^{2}}{\left(\left|\nabla u_{1}\right|+\left|\nabla
u_{2}\right|\right)^{2-p}}d\sigma_{g}\leq$

$\int_{\left\\{\left|u_{1}-u_{2}\right|<k\right\\}}\left(\left|\nabla
u_{1}\right|^{p-2}\nabla u_{1}-\left|\nabla u_{2}\right|^{p-2}\nabla
u_{2}\right)\left(\nabla u_{1}-\nabla u_{2}\right)d\sigma_{g}$

for $p<2,$ then $\nabla u_{1}-\nabla u_{2}=0$ a.e. and consequently
$T_{k}\left(u_{2}\right)=T_{k}\left(u_{2}\right)$ for all $k>0.$ It is clear
that $u_{1}-u_{2}=c,$ using the fact that $u_{1}=u_{2}=0$on$\partial M,$ then
we concludes that $u_{1}=u_{2}$ a.e. Hence the result. $\blacksquare$

###  2.7 Some generalizations

The notion of the entropy solution can be defined for a very large class of
nonlinear elliptic operators, for example if we consider the next problem

$-\operatorname{div}(a(x,u,\nabla u))=F(x,u)$

$\operatorname{with}a(x,s,\xi):\mathbb{R}^{N}\times\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$
is a function of Carathéodory verifying
$(\mathbf{H}\mathbf{1}):|a(x,s,\xi)|\leq
c\left(|\xi|^{p-1}+|s|^{p-1}+k(x)\right),p.p.\quad x\in
M,(s,\xi)\in\mathbb{R}^{N}\times\mathbb{R},$

$(\mathbf{H}2):a(x,s,\xi)\xi\geq\lambda|\xi|^{p},p.p.\quad x\in
M,\quad(s,\xi)\in\mathbb{R}^{N}\times\mathbb{R},$

$(\mathbf{H}3):F$ is a Carateodory function, continuous and decreasing in $u$
for $x$ fixed and measurable in $x$ for $u$ fixed. Furthermore, $F(x,0)\in
L^{1}(M)$ et $F(x,c)\in L_{loc}^{1}(M)$ if $c\neq 0$ and $\mathrm{si}$

$G_{c}(x)=\sup_{|u|\leq c}|F(x,u)|,$

$G_{c}\in L_{loc}^{1}(M)$ for all $c>0.$

So under the conditions $(\mathbf{H}\mathbf{1}),(\mathbf{H}2)$ et
$(\mathbf{H}3)$ we can define the notion of the solution in the sense of
entropy. Regarding the uniqueness of the solution, in general the result is
not true but if $div(a(x,u,\nabla u))=\Delta_{p}u,$ then we can demonstrate
the uniqueness of the solution in the sense of entropy.

Keywords :  Quasi-linear elliptic equations, variational methods, functional
spaces, entropy solution, Riemannian manifold, space Marcinkiewicz .

## References

  * (1) P. Benilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, J. Vazquez, An $L^{1}$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4th serie, vol 22,n 2, p. 241 - 273, (1995).
  * (2) L. Boccardo, T. Gallouet, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), no. 1, 149$-$169\.
  * (3) G. Stampacchia, Elliptic Equations of the Second Order with Discontinuous Coefficients, The Press of the Montreal Univeersite, 1966.
  * (4) Serrin, J. Pathological solutions of elliptic differential equations. Ann. Scuola Normale Sup. Pisa Cl. Sci. 1964, 18 (3), 385$-$387\.
  * (5) Brezis, H. Kamin, S. Sublinear equations in IRN . Mauscripta Math. 1992, 74 (1),87$-$106\.
  * (6) B. Abdellaoui , I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p$-$Laplacian with a critical potential, Annali di Mathematica 182, p. 247 $-$ 270, (2003).
  * (7) E. Azroul, A. Abnoune and M.T.K. Abbassi, Study of a Second-Order Nonlinear Elliptic Problem Generated by a Divergence Type Operator on a Compact Riemannian Manifold,Filomat 32:14 (2018), 4811–4820.
  * (8) E.Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes, vol. 5, American Mathematical Society, 2000.
  * (9) P. Lindqvist,On the equation $\nabla p(u)+\lambda|u|^{p-2}u=0$ . Proc. Amer. Math. Soc. Vol.109, no. 1 (1990), 157$-$164\.
  * (10) I. Shafrir, Asymptotic behaviour of minimizing sequences for Hardy inequality.Commun. Contemp. Math. no. 2 (2000), 151$-$189\.
  * (11) M. Struwe, Variational Methods : Applications to Nonlinear Partial Differential Equations and Hamiltnian Systems, Springer, Berlin, 1996.
  * (12) Vazquez, J.L. A Strong Maximum Principle for Some Quasilinear Elliptic Equations. Applied Math. and Optimization 1984, 12 (3), 191$-$202\.