#  On the Strong Comparison Principle for
Degenerate Elliptic Problems with Convection

Jiří Benedikt  ,  Petr Girg  ,  Lukáš Kotrla  and  Peter Takáč  Jiří Benedikt,
Petr Girg, Lukáš Kotrla
Department of Mathematics and NTIS, Faculty of Applied Scences, University of
West Bohemia, Univerzitní 8, CZ–301 00 Plzeň, Czech Republic
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Peter Takáč
Institut für Mathematik, Universität Rostock, Ulmenstraße 69, D–18055 Rostock,
Germany<EMAIL_ADDRESS>
###### Abstract.

The weak and strong comparison principles (WCP and SCP, respectively) are
investigated for quasilinear elliptic boundary value problems with the
$p$-Laplacian in one space dimension,
$\Delta_{p}(u)\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}\frac{\mathrm{d}}{\mathrm{d}x}\left(|u^{\prime}|^{p-2}u^{\prime}\right)\,.$
We treat the “degenerate” case of $2<p<\infty$ and allow also for the
nontrivial convection velocity $b:\,[-1,1]\to\mathbb{R}$ in the underlying
domain $\Omega=(-1,1)$. We establish the WCP under a rather general, “natural
sufficient condition” on the convection velocity, $b(x)$, and the reaction
function, $\varphi(x,u)$. Furthermore, we establish also the SCP under a
number of various additional hypotheses. In contrast, with these hypotheses
being violated, we construct also a few rather natural counterexamples to the
SCP and discuss their applications to an interesting classical problem of
fluid flow in porous medium, “seepage flow of fluids in inclined bed”. Our
methods are based on a mixture of classical and new techniques.

###### Key words and phrases:

Quasilinear parabolic equation; degenerate $p$-Laplacian; examples and
counterexamples to strong comparison principle; Hopf’s boundary point lemma

###### 2010 Mathematics Subject Classification:

35B09, 35B51, 35K20, 35K55, 35K65, 35K92

##  1\. Introduction

The weak and strong comparison principles (WCP and SCP, respectively, for
brevity) play an important role in the theory of elliptic and parabolic
problems for partial differential equations. In our present work we focus on
such problems in one space dimension that “idealize” long, thin domains
$\Omega\subset\mathbb{R}^{N}$ of type $\Omega=\omega\times(a,b)$ where
solutions are expected to be rather homogeneous on the relatively small cross-
section $\omega\subset\mathbb{R}^{N-1}$ (a domain in $\mathbb{R}^{N-1}$) of
the relatively long domain $\Omega=\omega\times(a,b)$. Here, we assume that
$\mathrm{diameter}(\omega)\ll|b-a|$ meaning that the ratio
$\mathrm{diameter}(\omega)/|b-a|$ is a sufficiently small number ${}<1$.

In general, in an arbitrary domain $\Omega\subset\mathbb{R}^{N}$ of space
dimension $N$ ($N\geq 1$), we consider the elliptic partial differential
equation

(1.1) $\left\\{\hskip 5.69054pt\begin{aligned}
{}-\Delta_{p}u-\mathbf{b}(x)\cdot\nabla u+\varphi(x,u)&=f(x)&&\qquad\mbox{ for
}\,x\in\Omega\,;\\\ u&=0&&\qquad\mbox{ on
}\,\partial\Omega\,.\end{aligned}\right.\hfill$

Here, we assume that $2<p<\infty$,
$\Delta_{p}(u)\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}\mathrm{div}\left(|\nabla
u|^{p-2}\nabla u\right)$ stands for $p$-Laplacian,
$\mathbf{b}:\,\Omega\to\mathbb{R}^{N}$ is an essentially bounded measurable
vector field which stands for the convection velocity, often termed also drift
or advection velocity, the reaction function
$\varphi:\,\Omega\times\mathbb{R}\to\mathbb{R}$ is continuously partially
differentiable with respect to the second variable, $u$, with the partial
derivative $\frac{\partial\varphi}{\partial u}(x,u)$ $>0$ for all
$(x,u)\in\Omega\times(\mathbb{R}\setminus\\{0\\})$, and $\varphi(x,0)=0$ for
all $x\in\Omega$. Finally, $f:\,\Omega\to\mathbb{R}$ describes the volumetric
source density, $f\in L^{\infty}(\Omega)$; sometimes we will take $f\in
L^{\infty}(\Omega)\cap C(\Omega)$ or similar.

Under some reasonable hypotheses on the vector field $\mathbf{b}$ and the
function $\varphi$, a standard _weak comparison_ result for this Dirichlet
boundary value problem is established in the monograph by D. Gilbarg and N. S.
Trudinger [9, Sect. 10, pp. 263–271], Theorem 10.1 (p. 263) and Theorem 10.7
(p. 268). In contrast, the corresponding _strong comparison_ and _Hopf-type
comparison_ results are, in general, not valid under analogous hypotheses; cf.
M. Guedda and L. Véron [10], Prop. 2.1 (pp. 886–887) and Prop. 2.2 (p. 888),
and later work in M. Cuesta and P. Takáč [4, 5]. We begin this article by
giving a few simple counterexamples to the strong comparison principle. Our
counterexamples will be constructed in the one-dimensional domain
$\Omega=(-1,1)\subset\mathbb{R}$. In fact, they have motivated the entire work
presented in this article. Our main goal is to find reasonable sufficient
conditions on the (now) scalar field $\mathbf{b}\equiv
b:\,[-1,1]\to\mathbb{R}^{1}$ and the reaction function $\varphi$ that would
exclude such counterexamples; we will formulate sufficient conditions on $b$
and $\varphi$ that will guarantee the validity of the strong comparison
principle. While assuming $\varphi\equiv 0$, we formulate sufficient
conditions on the convection velocity $b$ in Section 3. The main purpose of
the counterexamples will be to “illustrate” the necessity of our sufficient
conditions.

###  1.1. Mathematical Problems of weak and strong Elliptic Comparison
Principles

The one-dimensional analogue of the elliptic partial differential equation
(1.1) has been investigated in the work by M. Cuesta and P. Takáč [4, 5]. More
precisely, this work treats the special case with $b(x)\equiv 0$ in an open
interval $\Omega=(-1,1)\subset\mathbb{R}^{1}$ and
$\varphi(x,u)=\lambda\,|u|^{p-2}u$ for all $x\in(-1,1)$ and all
$u\in\mathbb{R}=(-\infty,\infty)$, with a constant $\lambda\in(0,\infty)$
large enough, say, $\lambda\geq\lambda_{p}>0$. In this way, the original
problem (1.1) is reduced to the following two-point boundary value problem,

(1.2) $\left\\{\hskip 5.69054pt\begin{aligned}
{}-\left(|u^{\prime}|^{p-2}u^{\prime}\right)^{\prime}+\lambda\,|u|^{p-2}u&=f(x)\qquad\mbox{
for }\,x\in(-1,1)\,;\\\ u(\pm 1)&=0\,.\end{aligned}\right.$

This clearly is a variational problem on the Sobolev space $W_{0}^{1,p}(-1,1)$
with the left-hand side

$[\mathcal{A}u](x)\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}{}-\left(|u^{\prime}|^{p-2}u^{\prime}\right)^{\prime}+\lambda\,|u|^{p-2}u\,,\quad
x\in(-1,1)\,,$

being a monotone operator $\mathcal{A}:\,W_{0}^{1,p}(-1,1)\to
W^{-1,p^{\prime}}(-1,1)$ from its domain $W_{0}^{1,p}(-1,1)$ into the dual
space $W^{-1,p^{\prime}}(-1,1)=\left[W_{0}^{1,p}(-1,1)\right]^{\prime}$ where
$p^{\prime}=p/(p-1)$. We note the continuous imbedding
$L^{\infty}(-1,1)=\left[L^{1}(-1,1)\right]^{\prime}\hookrightarrow
W^{-1,p^{\prime}}(-1,1)$ that is dual to the Sobolev imbedding
$W_{0}^{1,p}(-1,1)\hookrightarrow L^{1}(-1,1)\,.$ The strict monotonicity of
$\mathcal{A}$ hinges upon the fact that $\frac{\partial\varphi}{\partial
u}(x,u)=\lambda\,(p-1)\,|u|^{p-2}>0$ is valid for all
$(x,u)\in\Omega\times(\mathbb{R}\setminus\\{0\\})$. It forces the WCP (weak
comparison principle, proved in Proposition 2.1), by a standard variational
argument that we will explain in Section 2 below. In this step, what matters
is the following observation:

> The larger the constant $\lambda\geq 0$, the “stronger” the strict
> monotonicity of $\mathcal{A}$.

For $\lambda=0$, even the SCP (strong comparison principle) for problem (1.2)
is established in [4, 5]. Nevertheless, the strong comparison principle fails
to hold for every $\lambda>0$ large enough, say, $\lambda\geq\lambda_{p}>0$,
as is shown in M. Cuesta and P. Takáč [5, Sect. 4, Example 4.1, pp. 740–741];
see Example 1.1 in §1.2 below.

This is precisely the problem we treat in the present article: Conditions that
are favorable for the validity of the WCP may have a negative effect on the
validity of the SCP. The latter may fail to hold under those conditions (in
Example 1.1).

Our strategy will be as follows: In Section 2 we establish the WCP under
rather simple, but still “reasonably” general conditions on the convection
velocity $b:\,\Omega\to\mathbb{R}$ in the interval
$\Omega=(-1,1)\subset\mathbb{R}$ and the reaction function
$\varphi:\,\Omega\times\mathbb{R}\to\mathbb{R}$.

In the remaining parts of this work we then assume the validity of the WCP and
try to formulate “reasonable” sufficient conditions that guarantee the
validity of the SCP. We also investigate the “sharpness” of our sufficient
conditions in Section 3. We provide an interesting new counterexample to the
SCP in Example 3.6 for the special case $\varphi\equiv 0$.

A classical treatment of the WCP and SCP can be found in the monographs by P.
Pucci and J. Serrin [13] (see also the article [12]) and M. H. Protter and H.
F. Weinberger [14]. Both these books use the Hopf boundary point lemma, due to
E. Hopf, in order to derive SCP from WCP, see e.g. [14, Chapt 2, Sect. 3],
Theorem 7 on p. 65. We will use a modification of this lemma adapted to the
$p$-Laplacian in the works by P. Pucci and J. Serrin [13, §5.4], Theorem 5.5.1
on p. 120; cf. also [13, §5.1], Example on p. 104, P. Tolksdorf [18, Prop.
3.2.1 and 3.2.2, p. 801], and J. L. Vázquez [19, Theorem 5, p. 200]. An
important role in our application of Hopf’s lemma is played by the result
(Hopf–Oleinik lemma) established recently in the article by D. E.
Apushkinskaya and A. I. Nazarov [1, Theorem 2.1, p. 681]. Last but not least,
we should mention that the first sharp results on the SCP for the
$p$-Laplacian, $\Delta_{p}$, were obtained in M. Guedda and L. Véron [10],
Prop. 2.1 (pp. 886–887) and Prop. 2.2 (p. 888).

Several engineering problems connected with the quasilinear “diffusion”
operator $p$-Laplacian, $\Delta_{p}$, have been mentioned and sketched in the
recent article by the present authors, J. Benedikt, P. Girg, L. Kotrla, and P.
Takáč [3]. The two primary interests are focused on (i) water supplies for
larger population communities and (ii) oil exploration and polution. For
interesting applications of our results in the real world, we refer the reader
to the recent work by R. K. Bansal [2] for engineering-type research on these
topics. There, a mathematical model of “seepage flow of fluids in inclined
bed” is presented and investigated. Problems under consideration in our paper
can be thought of being stationary cases of time-dependent problems studied in
[2]. In our case we consider the power law (with $p>2$ corresponding to “slow”
diffusion, see Paragraph 1.3 below) instead of Darcy’s law; cf. [3, Sect. 2]
for “historical” justification and numerous references. Indeed, it has been
observed experimentally for a very slow flow (in terms of Reynolds’ number) in
a porous medium, that Darcy’s law should be replaced by another law, namely,
the power law corresponding to the case $p>2$. We refer the reader to the
article by J. P. Soni, N. Islam, and P. Basak [15, p. 239] for a nice overview
of the values of the constant $p$ for various materials where $p$ ranges from
$2.12$ to $4.74$. Note that the constant $n=p-1$ is used in [15] in place of
$p$.

As far as applications of the two comparison principles, WCP and SCP, to these
engineering problems are concerned, the following difference between them is
of current interest:

> If the source term $f(x)$ on the right-hand side in problem (1.1) is
> increased to a function $g(x)$ ($g\geq f$ almost everywhere (a.e.) in
> $\Omega$) only locally near a point $x_{0}\in\Omega$, i.e., $g(x)>f(x)$
> holds for almost all $x\in B_{r}(x_{0})\subset\Omega$ in a small ball
> $B_{r}(x_{0})\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}\\{x\in\mathbb{R}^{N}:\,|x-x_{0}|<r\\}\subset\Omega$
> of radius $r>0$ centered at $x_{0}\in\Omega$, does the solution $u(x)$
> increase to $v(x)$ (for notation, see problems (1.3) in the next paragraph,
> §1.2)
>
>   * (i)
>
> only locally near the point $x_{0}\in\Omega$, i.e., $v(x)>u(x)$ holds for
> all $x\in\Omega$ near the point $x_{0}$, say, for all $x\in
> B_{R}(x_{0})\subset\Omega$, but $v(x)=u(x)$ holds for all
> $x\in\Omega\setminus B_{R}(x_{0})$, with some radius $R$, $r<R<\infty$, or
>
>   * (ii)
>
> the inequality $v(x)>u(x)$ holds for all $x\in\Omega$, that is, globally
> throughout the entire domain $\Omega$ ?
>
>

>
> The first statement, (i), means that the SCP fails to hold, whereas the
> second statement, (ii), is equivalent to the SCP.

###  1.2. A Simple Counterexample to the Strong Comparison Principle

We begin by recalling a special case from the work by M. Cuesta and P. Takáč
[5, Sect. 4, Example 4.1, pp. 740–741], where a counterexample to the SCP for
the two-point boundary value problem in eq. (1.2) above is constructed by
making the special choice $b(x)\equiv 0$ and
$\varphi(x,u)=\lambda\,|u|^{p-2}u$ for all $x\in(-1,1)$ and all
$u\in\mathbb{R}=(-\infty,\infty)$ in the following two-point boundary value
problems with $2<p<\infty$:

(1.3) $\left\\{\quad\begin{aligned}
{}-\left(|u^{\prime}|^{p-2}u^{\prime}\right)^{\prime}-b(x)\,u^{\prime}+\varphi(x,u)&=f(x)\,,\\\
{}-\left(|v^{\prime}|^{p-2}v^{\prime}\right)^{\prime}-b(x)\,v^{\prime}+\varphi(x,v)&=g(x)\,;\\\
\quad\mbox{ with }\quad u(\pm 1)\leq v(\pm 1)\quad\mbox{ and }\quad f(x)&\leq
g(x)\;\mbox{ for }\,x\in(-1,1)\,.\end{aligned}\right.$

The constant $\lambda\in(0,\infty)$ that appears in the reaction function
$\varphi(x,u)=\lambda\,u^{p-2}u$ in eq. (1.3) has to be chosen large enough,
say, $\lambda\geq\lambda_{p}>0$. As we work with nonnegative solutions
$u,v:\,[-1,1]\to\mathbb{R}$ to problems (1.3) only, it suffices to define the
reaction function $\varphi(x,u)=\lambda\,u^{p-1}$ for all $x\in(-1,1)$ and all
$u\in\mathbb{R}_{+}=[0,\infty)$ only.

As the boundary value problems in (1.3) in our present article allow for the
(additional) convection terms $b(x)\,u^{\prime}$ and $b(x)\,v^{\prime}$,
respectively, our counterexample below is a slight modification of that one
mentioned above [5, Example 4.1] (with $b(x)\equiv 0$).

######  Example 1.1 (A Counterexample to the SCP).

Let us consider the two-point boundary value problems (1.3) that have arised
from the following homogeneous Dirichlet boundary value problem:

(1.4) $\left\\{\hskip 5.69054pt\begin{aligned}
{}-\left(|u^{\prime}|^{p-2}u^{\prime}\right)^{\prime}-b(x)\,u^{\prime}+\varphi(u)&=f(x)\qquad\mbox{
for }\,x\in(-1,1)\,;\\\ u(\pm 1)&=0\,.\end{aligned}\right.$

Let the real numbers $p,\theta\in\mathbb{R}$ be given with $p>2$ and
$\theta>1$ arbitrary, and let us define the following functions, while still
allowing $b:\,[-1,1]\to\mathbb{R}$ to be arbitrary, but continuously
differentiable:

(1.5) $\displaystyle u_{\theta}(x)\stackrel{{\scriptstyle\rm
def}}{{=}}1-|x|^{\theta}\qquad\mbox{ for }\,x\in[-1,1]\qquad\mbox{ and }$
(1.6) $\displaystyle\begin{aligned} f_{\theta}(x)&\stackrel{{\scriptstyle\rm
def}}{{=}}{}-\left(|u_{\theta}^{\prime}|^{p-2}u_{\theta}^{\prime}\right)^{\prime}-b(x)\,u_{\theta}^{\prime}+\varphi(u_{\theta})\\\
&=(p-1)\theta^{p-1}(\theta-1)\,|x|^{(\theta-1)(p-1)-1}+\theta\,b(x)\cdot\mathrm{sgn}(x)\,|x|^{\theta-1}+\varphi\left(1-|x|^{\theta}\right)\,.\end{aligned}$

We notice that for $1<\theta_{1}<\theta_{2}<\infty$ we have
$u_{\theta_{1}}(x)<u_{\theta_{2}}(x)$ for all $x\in(-1,1)\setminus\\{0\\}$,
whereas $u_{\theta_{1}}(0)=u_{\theta_{2}}(0)=1$. In formula (1.10) below we
make a suitable choice of the convection velocity $b(x)$, take
$\varphi:\,\mathbb{R}\to\mathbb{R}$ linear, $\varphi(u)=\lambda\,u$ in formula
(1.11), and state sufficient conditions on the exponents
$\theta_{1}<\theta_{2}$ and the constant $\lambda>0$, such that the following
inequality

(1.7) $f_{\theta_{1}}(x)<f_{\theta_{2}}(x)\quad\mbox{ holds for all
}\,x\in(-1,1)\setminus\\{0\\}\,.$

Consequently, the strong comparison principle for problem (1.4) fails to hold
at the point $x=0$ owing to $u_{\theta_{1}}(0)=u_{\theta_{2}}(0)$.

When looking for a convection velocity $b(x)$ with some desired properties, it
is worth of noticing that the reflection transformation
$\widetilde{\phantom{x}}:\,[-1,1]\to[-1,1]:\,x\mapsto{\tilde{x}}\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}{}-x$
brings problem (1.4) into the following equivalent “reflection” form:

(1.4$\widetilde{\phantom{x}}$) $\left\\{\hskip 5.69054pt\begin{aligned}
{}-\left(|{\tilde{u}}^{\prime}|^{p-2}{\tilde{u}}^{\prime}\right)^{\prime}+b(-{\tilde{x}})\,{\tilde{u}}^{\prime}+\varphi({\tilde{u}})&=f(-{\tilde{x}})\qquad\mbox{
for }\,{\tilde{x}}\in(-1,1)\,;\\\ {}-{\tilde{u}}(\pm
1)&=0\,.\end{aligned}\right.$

Here, ${\tilde{u}}({\tilde{x}})=u(-{\tilde{x}})=u(x)$ is the new unknown
function of the variable ${\tilde{x}}={}-x$, so that its derivative equals to
${\tilde{u}}^{\prime}({\tilde{x}})=\frac{\mathrm{d}{\tilde{u}}}{\mathrm{d}{\tilde{x}}}({\tilde{x}})=\frac{\mathrm{d}}{\mathrm{d}{\tilde{x}}}\,u(-{\tilde{x}})={}-\frac{\mathrm{d}u}{\mathrm{d}x}(-{\tilde{x}})={}-u^{\prime}(-{\tilde{x}})={}-u^{\prime}(x)\,,$
by the chain rule. In particular, the convection velocity $b(x)$ is
transformed into the new convection velocity
${\tilde{b}}({\tilde{x}})={}-b(-{\tilde{x}})={}-b(x)$ in the new problem
(1.4$\widetilde{\phantom{x}}$) above, with the derivative
${\tilde{b}}^{\prime}({\tilde{x}})=b^{\prime}(x)$. Consequently, the sign of
the convection velocity $b(x)$, in the expression $\pm b(x)$, does not matter
in our counterexample. However, a sign change in $\pm b(x)$ may cause a change
of the constant $\lambda>0$ in our choice of the reaction function
$\varphi(u)=\lambda\,u$; cf. eq. (1.11) just below combined with the
calculations in eq. (1.18).

We verify inequality (1.7) by showing that for every
$\theta\in(\theta_{1},\theta_{2})$ we have

(1.8) $\frac{\partial f_{\theta}}{\partial\theta}(x)>0\quad\mbox{ for all
}\,x\in(-1,1)\setminus\\{0\\}\,;$

cf. inequalities (1.16). A reader who is interested in detailed calculations
is referred to [5, Example 4.1] and to our calculations below leading to
inequalities (1.16) and (1.19).

As a simple example, owing to ineq. (1.16) below, we may take

Example 1.1 continued: Let us take

(1.9) $\displaystyle
2<p<\infty\,,\quad\frac{p}{p-2}<\theta_{1}<\theta_{2}<\infty\,,$ (1.10)
$\displaystyle\begin{aligned}
b(x)&\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}(p-1)^{2}(\theta_{1}-1)\theta_{1}^{p-2}\,|x|^{(p-2)(\theta_{2}-1)-2}\,x\\\
&\equiv(p-1)^{2}(\theta_{1}-1)\theta_{1}^{p-2}\,|x|^{(p-2)(\theta_{2}-1)-1}\cdot\mathrm{sgn}(x)\quad\mbox{
for }\,x\in[-1,1]\,,\end{aligned}$ (1.11) $\displaystyle\mbox{ and
}\qquad\left\\{\hskip 5.69054pt\begin{aligned}
&\varphi(x,s)\equiv\varphi(s)\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}\lambda\,s\quad\mbox{
for }\,(x,s)\in[-1,1]\times\mathbb{R}\\\ &\quad\mbox{ with the constant
}\quad\lambda\geq
2(p-1)^{2}\,(\theta_{2}-1)\theta_{2}^{p-1}>0\,.\end{aligned}\right.$

We observe that for every $x\in[-1,1]$ we have

(1.12)
$b^{\prime}(x)=(p-1)^{2}(\theta_{1}-1)\theta_{1}^{p-2}\,[(p-2)(\theta_{2}-1)-1]\,|x|^{(p-2)(\theta_{2}-1)-2}\geq
0\,.$

We note that $\mathrm{sgn}(0)=0$, $\mathrm{sgn}(x)=1$ if $x>0$, and
$\mathrm{sgn}(x)={}-1$ if $x<0$.

Proof of Ineq. (1.8). For $u(x)\equiv u_{\theta}(x)$ with $1<\theta<\infty$ we
make use of eq. (1.5) to calculate

$\displaystyle u^{\prime}(x)\equiv
u_{\theta}^{\prime}(x)=\frac{\mathrm{d}}{\mathrm{d}x}\left(1-|x|^{\theta}\right)={}-\theta\,|x|^{\theta-2}\,x={}-\theta\,|x|^{\theta-1}\cdot\mathrm{sgn}(x)\,,$
(1.13) $\displaystyle\frac{\partial
u_{\theta}}{\partial\theta}(x)={}-|x|^{\theta}\,\log|x|\geq 0\quad\mbox{ for
all }\,x\in[-1,1]\setminus\\{0\\}\,,\quad\mbox{ and }\quad\frac{\partial
u_{\theta}}{\partial\theta}(0)=0\,.$

It follows that

$\displaystyle|u^{\prime}(x)|^{p-2}u^{\prime}(x)$
$\displaystyle={}-\theta^{p-1}\,|x|^{(\theta-1)(p-2)+\theta-2}x={}-\theta^{p-1}\,|x|^{(\theta-1)(p-1)-1}x\,,$
$\displaystyle{}-\left(|u^{\prime}|^{p-2}u^{\prime}\right)^{\prime}(x)$
$\displaystyle\equiv{}-\genfrac{}{}{}{1}{\mathrm{d}}{\mathrm{d}x}\left(|u^{\prime}|^{p-2}u^{\prime}\right)(x)=\theta^{p-1}(\theta-1)(p-1)|x|^{(\theta-1)(p-1)-1}\,.$

Similarly, we use eq. (1.6) to get

(1.14) $\displaystyle\begin{aligned} \frac{\partial
f_{\theta}}{\partial\theta}(x)&{}=(p-1)\left[p\theta^{p-1}-(p-1)\theta^{p-2}\right]|x|^{(\theta-1)(p-1)-1}\\\
&{}+(p-1)^{2}\theta^{p-1}(\theta-1)|x|^{(\theta-1)(p-1)-1}\,\log|x|\\\
&{}+b(x)\cdot\mathrm{sgn}(x)\,|x|^{\theta-1}+\theta\,b(x)\cdot{\rm
sgn}(x)\,|x|^{\theta-1}\,\log|x|\\\
&{}-\varphi^{\prime}\left(1-|x|^{\theta}\right)\,|x|^{\theta}\,\log|x|\,.\end{aligned}$

In order to determine the sign of ${\partial f_{\theta}}/{\partial\theta}$, we
take into account that $\varphi(x,s)\equiv\varphi(s)=\lambda\,s$, by eq.
(1.11), and thus simplify the right-hand side of eq. (1.14) as follows:

(1.15) $\displaystyle\frac{\partial f_{\theta}}{\partial\theta}(x)$
$\displaystyle{}=|x|^{\theta-1}\left\\{(p-1)\left[p\theta^{p-1}-(p-1)\theta^{p-2}\right]\,|x|^{(\theta-1)(p-2)-1}+b(x)\cdot\mathrm{sgn}(x)\right\\}$

We observe that the desired inequality (1.7) is satisfied provided the two
expressions in braces on the right-hand side of equation (1.15) above have the
correct sign, whenever $0<|x|<1$, that is to say,

$\displaystyle(p-1)\left[p\theta^{p-1}-(p-1)\theta^{p-2}\right]\,|x|^{(\theta-1)(p-2)-1}+b(x)\cdot\mathrm{sgn}(x)>0\qquad\mbox{
and }$
$\displaystyle(p-1)^{2}\theta^{p-1}(\theta-1)\,|x|^{(\theta-1)(p-2)-1}+\theta\,b(x)\cdot\mathrm{sgn}(x)-\lambda\,|x|\leq
0\,.$

The last two inequalities, respectively, are equivalent with

(1.16)
$\displaystyle{}-(p-1)\left[p\theta^{p-1}-(p-1)\theta^{p-2}\right]\,|x|^{(\theta-1)(p-2)-2}<b(x)/x$
$\displaystyle\leq$
$\displaystyle{}-(p-1)^{2}\,\theta^{p-2}(\theta-1)\,|x|^{(\theta-1)(p-2)-2}+\theta^{-1}\,\lambda\quad\mbox{
whenever }\,0<|x|<1\,.$

Hence, the validity of the desired inequality (1.8) is derived from our choice
of the convection velocity $b(x)$ in (1.10), with a help from eq. (1.11) as
follows:

First, the inequality on the left-hand side of (1.16) is justified by
combining the inequalities in (1.9) with $0<|x|\leq 1$ in the calculations
that follow:

$\displaystyle\frac{b(x)}{x}+(p-1)\left[p\theta^{p-1}-(p-1)\theta^{p-2}\right]\,|x|^{(\theta-1)(p-2)-2}$
$\displaystyle\begin{aligned}
&{}=(p-1)^{2}(\theta_{1}-1)\theta_{1}^{p-2}\,|x|^{(p-2)(\theta_{2}-1)-2}\\\
&{}+(p-1)\,[p\theta-(p-1)]\,\theta^{p-2}\,|x|^{(\theta-1)(p-2)-2}\end{aligned}$
(1.17) $\displaystyle\begin{aligned}
&{}\geq(p-1)^{2}(\theta_{1}-1)\theta_{1}^{p-2}\,|x|^{(p-2)(\theta_{2}-1)-2}\\\
&{}+(p-1)\,[p\theta_{1}-(p-1)]\,\theta_{1}^{p-2}\,|x|^{(\theta_{2}-1)(p-2)-2}\end{aligned}$
$\displaystyle=(p-1)\,\theta_{1}^{p-2}\,[(p-1)(\theta_{1}-1)+p\,(\theta_{1}-1)+1]\,|x|^{(p-2)(\theta_{2}-1)-2}$
$\displaystyle=(p-1)\,\theta_{1}^{p-2}\,[(2p-1)(\theta_{1}-1)+1]\,|x|^{(p-2)(\theta_{2}-1)-2}>0\,.$

Similarly, the inequality on the right-hand side of (1.16) is justified by

(1.18)
$\displaystyle\frac{b(x)}{x}+(p-1)^{2}\,\theta^{p-2}(\theta-1)\,|x|^{(\theta-1)(p-2)-2}$
$\displaystyle\begin{aligned}
&{}=(p-1)^{2}(\theta_{1}-1)\theta_{1}^{p-2}\,|x|^{(p-2)(\theta_{2}-1)-2}\\\
&{}+(p-1)^{2}\,\theta^{p-2}(\theta-1)\,|x|^{(\theta-1)(p-2)-2}\end{aligned}$
$\displaystyle<2(p-1)^{2}\,\theta_{2}^{p-2}(\theta_{2}-1)<2(p-1)^{2}\,(\theta_{2}-1)\theta_{2}^{p-1}\,\theta^{-1}\leq\theta^{-1}\,\lambda\,,$

owing to $\lambda\geq 2(p-1)^{2}\,(\theta_{2}-1)\theta_{2}^{p-1}>0$.

The “counterexample claims” in Example 1.1 thus follow from (1.8).
$\sqcap$$\sqcup$

We remark that the inequality “lower bound $\,<\,$ upper bound” in ineq.
(1.16) above is satisfied thanks to

${}-(p-1)\left[p\theta^{p-1}-(p-1)\theta^{p-2}\right]\,\leq{}-(p-1)^{2}\theta^{p-2}(\theta-1)$

combined with $\lambda=\varphi^{\prime}\left(1-|x|^{\theta}\right)>0$ whenever
$0<|x|<1$. Indeed, we have

(1.19)
$\displaystyle\left[p\theta^{p-1}-(p-1)\theta^{p-2}\right]-(p-1)\theta^{p-2}(\theta-1)$
$\displaystyle{}=(p-1)\theta^{p-2}\left(\frac{p}{p-1}\,\theta-1-\theta+1\right)$
$\displaystyle{}=(p-1)\theta^{p-1}\,\frac{1}{p-1}=\theta^{p-1}>1\,.$

As far as our counterexample is concerned, this observation allows us to
choose any continuous function $b:\,[-1,1]\to\mathbb{R}$ satisfying ineq.
(1.16) in Example 1.1 above. Our choice of $b(x)$ in eq. (1.10) is even
continuously differentiable and its derivative $b^{\prime}(x)$ vanishes at
zero, $b^{\prime}(0)=\lim_{x\to 0}b(x)/x=0\,,$ owing to the inequalities in
(1.9).

Our counterexample (Example 1.1) allows for the following slight modification
/ generalization.

######  Example 1.2 (A modification of Example 1.1).

In addition to eqs. (1.9) and (1.11), instead of eq. (1.10) in Example 1.1
above, another choice of the convection velocity $b(x)$ can be any
continuously differentiable function $b:\,[-1,1]\to\mathbb{R}$ satisfying the
following conditions:

$\displaystyle\left\\{\hskip 5.69054pt\begin{aligned}
&b(0)=b^{\prime}(0)=0\quad\mbox{ and }\quad b^{\prime}(x)\geq 0\quad\mbox{ for
every }\,x\in[-1,1]\,,\\\
&0\leq\frac{b(x)}{x}\leq(p-1)^{2}(\theta_{1}-1)\theta_{1}^{p-2}\,|x|^{(p-2)(\theta_{2}-1)-2}\quad\mbox{
whenever }\,0<|x|\leq 1\,.\end{aligned}\right.$

Then the desired inequalities in (1.16) are obtained in much the same way as
in Example 1.1, namely, from inequalities very similar to (1.17) and (1.18).
$\sqcap$$\sqcup$

We remark that in both our counterexamples, Example 1.1 and 1.2, the SCP fails
to hold precisely at the point $x=0$,

$\\{0\\}=\\{x\in(-1,1):\,b(x)=0\\}\,,$

where the convection velocity $b(x)$ vanishes. In contrast with this
observation, in Example 3.6 (in §3.2 below) we will see another entirely
different counterexample to the SCP with $b(x)\equiv b_{0}=\mathrm{const}<0$
being a negative constant and $\lambda=0$. Moreover, the equality $u(x)=v(x)$
will hold throughout the interval $[-1,\,-1/2]\subset[-1,1]$ whereas
$u(x)<v(x)$ holds for every $x\in(-1/2,\,1)$.

It is also necessary to stress that the “natural sufficient condition” on the
functions $b$ and $\varphi$, stated in ineq. (2.1) in Paragraph §2.1, which
guarantees the validity of the WCP, is trivially fulfilled in both our
Examples 1.1 and 1.2, thanks to

$\displaystyle\frac{1}{2}\,b^{\prime}(x)+\frac{\partial\varphi}{\partial
u}(x,s)\geq\frac{1}{2}\,b^{\prime}(x)+\inf_{s^{\prime}\in\mathbb{R}}\frac{\partial\varphi}{\partial
u}(x,s^{\prime})=\frac{1}{2}\,b^{\prime}(x)+\lambda\geq\lambda>0$
$\displaystyle\quad\mbox{ for }\,(x,s)\in[-1,1]\times\mathbb{R}\,,$

where $\varphi:\,[-1,1]\times\mathbb{R}\to\mathbb{R}$ is given by
$\varphi(x,s)\equiv\varphi(s)=\lambda\,s$ for
$(x,s)\in[-1,1]\times\mathbb{R}$, by eq. (1.11). Under this condition, ineq.
(2.1), the WCP will be established in Proposition 2.1 below.

###  1.3. Interpretation for non-Newtonian fluid flows (e.g., of a pollutant)

The case of vanishing convection velocity
$\mathbf{b}:\,\Omega\to\mathbb{R}^{N}$ in the general $N$-dimensional elliptic
problem (1.1) leads to a very popular variational problem for the critical
points of the corresponding energy functional

(1.20)
$\displaystyle\mathcal{E}(u)\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}\frac{1}{p}\int_{\Omega}|\nabla
u|^{p}\,\mathrm{d}x+\int_{\Omega}\Phi(x,u)\,\mathrm{d}x\,,\quad u\in
W_{0}^{1,p}(\Omega)\,;$ $\displaystyle\quad\mbox{ with
}\quad\Phi(x,s)\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}\int_{0}^{s}\varphi(x,t)\,\mathrm{d}t\quad\mbox{
for }\,(x,s)\in\Omega\times\mathbb{R}\,,$

defined on the Sobolev space $W_{0}^{1,p}(\Omega)$. This variational problem
has been investigated in numerous articles during at least the last seven
decades. A classical reference for this quasilinear elliptic problem is the
monograph by J.-L. Lions [11]. More recent results can be found in P. Drábek
[6], P. Drábek, A. Kufner, and F. Nicolosi [7], and P. Takáč [17], and in the
numerous references therein ([6, 7, 17]).

The variational approach does not allow for the convection term
$\mathbf{b}(x)\cdot\nabla u$ in problem (1.1). It can be treated by methods of
pseudo-monotone operators ([11, Chapt. 2, §2.4 – §2.5, pp. 179–182]) or fixed
points for the variational problem ([11, Chapt. 2, §2.6, pp. 182–190]). A more
recent approach to the non-variational problem (1.1) with a convection term
can be found in J. García-Melián, J. C. Sabina de Lis, and P. Takáč [8].

Finally, if the WCP is valid (as stated in Proposition 2.1 below), then also
monotone methods can be applied to the non-variational problem (1.1).

> From the engineer’s point of view (cf. [3]), the WCP means that increasing
> the content of certain substance (e.g., a pollutant such as oil or nitrate)
> in the model described by the elliptic partial differential equation (1.1)
> by means of the source function $f(x)$ leads to an increase of the substance
> density $u(x)$ throughout the entire domain $\Omega\subset\mathbb{R}^{N}$.
> The point here is that the increase from $u(x)$ to, say, $v(x)$ does not
> have to be strict at every point $x\in\Omega$, that is, the strict
> inequality $u(x)<v(x)$ might not hold in spite of the increase of $f(x)$ to
> $g(x)$ with $f(x)\leq g(x)$ for every $x\in\Omega$ and $f\not\equiv g$ in
> $\Omega$. Here, we have replaced the pair $(f,u)$ by the new pair $(g,v)$ in
> eq. (1.1); cf. eqs. (1.3).
>
> In particular, if pollution by the substance with the density $u(x)$ in the
> model is caused locally by the source function $f(x)$, by increasing the
> source function $f(x)$ to $g(x)$ with $f(x)\leq g(x)$ and $f\not\equiv g$, a
> global pullution increase from the substance density $u(x)$ to $v(x)$ with
> $u(x)<v(x)$ (throughout the entire domain $\Omega$), occurs if and only if
> the SCP is valid.

As usual, given a pair of functions $f,g\in L^{\infty}(\Omega)$, the relation
$f\not\equiv g$ in $\Omega$ means that the set $\\{x\in\Omega:\,f(x)\neq
g(x)\\}$ has positive $N$-dimensional Lebesgue measure in $\mathbb{R}^{N}$. If
$f,g\in C(\Omega)$, then $f\not\equiv g$ in $\Omega$ means that the set
$\\{x\in\Omega:\,f(x)\neq g(x)\\}$ contains an open ball
$B_{r}(x_{0})\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}\\{x\in\mathbb{R}^{N}:\,|x-x_{0}|<r\\}\subset\Omega$
of radius $r>0$.

##  2\. The Weak Comparison Principle

In this section we focus on the WCP, a weaker form of the Comparison Principle
that we wish to investigate. To this end, let us consider the two-point
boundary value problems (1.3) stated in the Introduction (Section 1, §1.2). In
the next paragraph, §2.1, we provide a simple, natural sufficient condition on
the functions $b:\,[-1,1]\to\mathbb{R}$ and
$\varphi:\,[-1,1]\times\mathbb{R}\to\mathbb{R}$ which guarantee the validity
of the WCP, that is to say, $u(x)\leq v(x)$ for all $x\in[-1,1]$.

###  2.1. A Natural Sufficient Condition

We assume that the functions $b(x)$, $\varphi(x,u)$, $f(x)$, and $g(x)$
satisfy the following hypothesis:

  * (M)

$b:\,[-1,1]\to\mathbb{R}$ is continuously differentiable, both
$\varphi,\;\frac{\partial\varphi}{\partial
u}:\,[-1,1]\times\mathbb{R}\to\mathbb{R}$ are continuous, and the following
“monotonicity” inequality is satisfied:

(2.1) $\frac{1}{2}\,b^{\prime}(x)+\frac{\partial\varphi}{\partial u}(x,s)\geq
0\quad\mbox{ for all }\,(x,s)\in(-1,1)\times\mathbb{R}\,.$

In addition, we assume that $f,g\in L^{\infty}(-1,1)$.

In order to obtain a weak comparison principle for problem (1.3) analogous to
Theorem 10.7 in [9, Sect. 10, §10.4, p. 268], we establish the following
rather general result:

######  Proposition 2.1 (Weak Comparison Principle).

Assume Hypothesis (M). If $f,g\in L^{\infty}(-1,1)$ satisfy $f\leq g$ a.e. in
$(-1,1)$, and $u(-1)\leq v(-1)$ and $u(+1)\leq v(+1)$, then $u\leq v$ holds
everywhere in $[-1,1]$.

Proof. We follow similar ideas as does the proof of Theorem 10.7 (under
condition (iii)) in [9, pp. 270–271]. Subtracting the differential equations
for $u$ and $v$ in (1.3) from each other, we arrive at

$\displaystyle{}-\frac{\mathrm{d}}{\mathrm{d}x}\left(|u^{\prime}|^{p-2}u^{\prime}-|v^{\prime}|^{p-2}v^{\prime}\right)-b(x)\,(u^{\prime}-v^{\prime})+\left[\varphi(x,u)-\varphi(x,v)\right]$
$\displaystyle{}=f(x)-g(x)\leq 0\quad\mbox{ for }\,x\in(-1,1)\,.$

We multiply this inequality by the function
$w\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(u-v)^{+}$, $w\in
W_{0}^{1,p}(-1,1)$, which has the vanishing boundary values $w(\pm 1)=0$, and
integrate with respect to $x\in(-1,1)$, thus arriving at

(2.2)
$\displaystyle\int_{-1}^{1}\left(|u^{\prime}|^{p-2}u^{\prime}-|v^{\prime}|^{p-2}v^{\prime}\right)w^{\prime}(x)\,\mathrm{d}x-\int_{-1}^{1}b(x)\,(u^{\prime}-v^{\prime})\,w(x)\,\mathrm{d}x$
$\displaystyle{}+\int_{-1}^{1}\left[\varphi(x,u)-\varphi(x,v)\right]w(x)\,\mathrm{d}x=\int_{-1}^{1}(f-g)\,w(x)\,\mathrm{d}x\leq
0\,.$

Since the boundary conditions $u(-1)\leq v(-1)$ and $u(+1)\leq v(+1)$
guarantee $w(\pm 1)=0$, the second integral on the left-hand side of eq.
(LABEL:e:(u-v)^+) becomes

(2.3)
$\displaystyle\int_{-1}^{1}b(x)\,(u^{\prime}-v^{\prime})\,w(x)\,\mathrm{d}x=\frac{1}{2}\int_{-1}^{1}b(x)\,\frac{\mathrm{d}}{\mathrm{d}x}\,[w(x)]^{2}\,\mathrm{d}x={}-\frac{1}{2}\int_{-1}^{1}b^{\prime}(x)\,[w(x)]^{2}\,\mathrm{d}x\,.$

In order to treat the first and third integrals on the left-hand side of eq.
(LABEL:e:(u-v)^+), we use the standard formulas

(2.4) $\displaystyle|u^{\prime}|^{p-2}u^{\prime}-|v^{\prime}|^{p-2}v^{\prime}$
$\displaystyle{}=(p-1)\left[\int_{0}^{1}\left|(1-\theta)\,u^{\prime}(x)+\theta\,v^{\prime}(x)\right|^{p-2}\,\mathrm{d}\theta\right](u^{\prime}-v^{\prime})\,,$
(2.5) $\displaystyle\varphi(x,u)-\varphi(x,v)$
$\displaystyle{}=\left[\int_{0}^{1}\frac{\partial\varphi}{\partial
u}\bigl{(}x,\,(1-\theta)\,u(x)+\theta\,v(x)\bigr{)}\,\mathrm{d}\theta\right](u-v)\,.$

We insert eqs. (LABEL:e:b(u'-v')(u-v)^+), (2.4), and (2.5) into eq.
(LABEL:e:(u-v)^+), thus obtaining

$\displaystyle(p-1)\int_{-1}^{1}\left[\int_{0}^{1}\left|(1-\theta)\,u^{\prime}(x)+\theta\,v^{\prime}(x)\right|^{p-2}\,\mathrm{d}\theta\right][w^{\prime}(x)]^{2}\,\mathrm{d}x+\frac{1}{2}\int_{-1}^{1}b^{\prime}(x)\,[w(x)]^{2}\,\mathrm{d}x$
$\displaystyle{}+\int_{-1}^{1}\left[\int_{0}^{1}\frac{\partial\varphi}{\partial
u}\bigl{(}x,\,(1-\theta)\,u(x)+\theta\,v(x)\bigr{)}\,\mathrm{d}\theta\right][w(x)]^{2}\,\mathrm{d}x=\int_{-1}^{1}(f-g)\,w(x)\,\mathrm{d}x\leq
0$

or, equivalently,

(2.6)
$\displaystyle(p-1)\int_{-1}^{1}\left[\int_{0}^{1}\left|(1-\theta)\,u^{\prime}(x)+\theta\,v^{\prime}(x)\right|^{p-2}\,\mathrm{d}\theta\right][w^{\prime}(x)]^{2}\,\mathrm{d}x$
$\displaystyle{}+\int_{-1}^{1}\left[\int_{0}^{1}\left(\frac{1}{2}\,b^{\prime}(x)+\frac{\partial\varphi}{\partial
u}\Bigl{(}x,\,(1-\theta)\,u(x)+\theta\,v(x)\Bigr{)}\right)\,\mathrm{d}\theta\right][w(x)]^{2}\,\mathrm{d}x$
$\displaystyle{}=\int_{-1}^{1}(f-g)\,w(x)\,\mathrm{d}x\leq 0\,.$

Finally, we take $s=(1-\theta)\,u(x)+\theta\,v(x)$ in inequality (2.1) and
insert it into (LABEL:ineq:(u-v)^+); it forces $w^{\prime}(x)=0$ almost
everywhere throughout $(-1,1)$ as the first integral on the left-hand side
must be non-positive. The boundary conditions $w(\pm 1)=0$ then yield
$w(x)=(u(x)-v(x))^{+}=0$ for all $x\in[-1,1]$, that is, we have proved that
$u\leq v$ in $[-1,1]$, as claimed.

Although the following two corollaries are concerned with an “interior”-type
SCP, they are direct consequences of the WCP that we have just proved above in
Proposition 2.1. This is the reason why we have decided to state them right
after our proof of Proposition 2.1. In both corollaries that follow next, we
assume that the reaction function $\varphi(x,s)\equiv\varphi(x)$ is
independent from the dummy variable $s\in\mathbb{R}$. In this case, the
monotonicity inequality (2.1) is reduced to $b^{\prime}(x)\geq 0$ for all
$x\in(-1,1)$. We may take $\varphi(x,s)\equiv 0$ without loss of generality by
replacing the pair of functions $f(x)$ and $g(x)$ in eqs. (1.3) by
$f(x)-\varphi(x)$ and $g(x)-\varphi(x)$, respectively, thus arriving at

(2.7) $\left\\{\quad\begin{aligned}
&{}-\left(|u^{\prime}|^{p-2}u^{\prime}\right)^{\prime}-b(x)\,u^{\prime}=f(x)\,,\qquad{}-\left(|v^{\prime}|^{p-2}v^{\prime}\right)^{\prime}-b(x)\,v^{\prime}=g(x)\,;\\\
&\quad\mbox{ with }\quad u(\pm 1)\leq v(\pm 1)\quad\mbox{ and }\quad f(x)\leq
g(x)\;\mbox{ for }\,x\in(-1,1)\,.\end{aligned}\right.$

######  Corollary 2.2 (“Interior” Strong Comparison Principle).

In the situation of Proposition 2.1, with $\varphi(x,\,\cdot\,)\equiv 0$, we
have the following two (mutually exclusive) alternatives:

  * (a)

The set

$\displaystyle
P_{1}\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}\\{x\in(-1,1):\,u(x)<v(x)\\}$

is not empty. Then there is a pair of points $-1<a_{-1}<a_{1}<1$ such that
$u(a_{-1})<v(a_{-1})$ and $u(a_{1})<v(a_{1})$. Given any such a pair, set

$\eta=\min\\{v(a_{-1})-u(a_{-1}),\,v(a_{1})-u(a_{1})\\}>0\,.$

Furthermore, in problem (2.7) above we have $v(x)\geq u(x)+\eta>u(x)$ for
every $x\in[a_{-1},\,a_{1}]$. Finally, $P_{1}$ is an open interval in
$\mathbb{R}$.

  * (b)

The set $P_{1}$ is empty. This is the case if and only if all $u(-1)=v(-1)$,
$u(1)=v(1)$, and $f=g$ a.e. in $(-1,1)$ hold in the boundary value problems in
(2.7).

Proof of Corollary 2.2. Let us begin with Alternative (a), i.e., let
$P_{1}\not=\emptyset$. Thanks to the continuity of both functions
$u,v:\,[-1,1]\to\mathbb{R}$, $P_{1}$ is a nonempty open subset of the interval
$(-1,1)$. Consequently, there are two points $-1<a_{-1}<a_{1}<1$ such that

$\eta=\min\\{v(a_{-1})-u(a_{-1}),\,v(a_{1})-u(a_{1})\\}>0\,.$

We apply the WCP from Proposition 2.1 on the interval $[a_{-1},\,a_{1}]$ in
place of $[-1,1]$ and for the pair of functions $u(x)+\eta$, $v$ instead of
$u(x)$, $v(x)$, respectively. By these arguments, the inequalities
$u(a_{-1})+\eta\leq v(a_{-1})$ and $u(a_{1})+\eta\leq v(a_{1})$ imply
$u(x)+\eta\leq v(x)$ for every $x\in[a_{-1},\,a_{1}]$. Consequently, $P_{1}$
is an open and connected subset of $(-1,1)$ and, thus, an open interval
itself.

In contrast, Alternative (b) forces $u\equiv v$ throughout the entire interval
$[-1,1]$. Problems (2.7) then yield $f=g$ a.e. in $(-1,1)$.

Our corollary is proved.

##  3\. The Strong Comparison Principle near a Critical Point

In this section we focus on some rather technical issues and open questions
related to the SCP for problems (2.7) (that is to say, for problems (1.3) with
the vanishing reaction function $\varphi(x,s)\equiv 0$) in case when the two
solutions, $u,v:\,[-1,1]\to\mathbb{R}$, respectively, satisfy the weak
comparison principle and they meet in an interior point $x_{0}\in(-1,1)$, that
is, $u(x)\leq v(x)$ for all $x\in[-1,1]$ and $u(x_{0})=v(x_{0})$. With regard
to Corollary 2.2, Alt. (b), above, we will focus our attention on the most
problematic case when

  * (i)

$u(x)=v(x)$ holds for every $x\in[-1,x_{0}]$ whereas $u(x)<v(x)$ for every
$x\in(x_{0},a_{1})$ in a nonempty open interval $(x_{0},a_{1})$, for some
$a_{1}\in(x_{0},1]$, i.e., $P_{1}=(x_{0},a_{1})\neq\emptyset$.

The symmetric “mirror” case below can be treated analogously,

  * (i’)

$u(x)=v(x)$ holds for every $x\in[x_{0},1]$ whereas $u(x)<v(x)$ for every
$x\in(a_{-1},x_{0})$ in a nonempty open interval $(a_{-1},x_{0})$, for some
$a_{-1}\in[-1,x_{0})$, i.e., $P_{1}=(a_{-1},x_{0})\neq\emptyset$.

Throughout this entire section we focus on the interaction between the
diffusion and convection effects, that is, between the terms
$\left(|u^{\prime}|^{p-2}u^{\prime}\right)^{\prime}$ and $b(x)\,u^{\prime}$
$\bigl{(}\left(|v^{\prime}|^{p-2}v^{\prime}\right)^{\prime}$ and
$b(x)\,v^{\prime}$, respectively$\bigr{)}$ in eqs. (2.7), while completely
ignoring the dependence on the dummy variable $s\in\mathbb{R}$ in the reaction
function $\varphi(x,s)$ that stands for the value $s=u(x)$ ($s=v(x)$,
respectively). In other words, we take $\varphi(x,s)\equiv\varphi(x)$
independent from $s\in\mathbb{R}$ and include it in the “source” function
$f(x)$ on the right-hand side of eq. (1.4) by replacing $f(x)$ by the
difference $f(x)-\varphi(x)$. This reduction hypothesis thus allows us to
simply set $\varphi(x,s)\equiv 0$ in eq. (1.4); cf. problem (2.7) above.

###  3.1. The solutions $u$ and $v$ near a Common Interior Meeting Point

We begin by recalling a special case from the work by M. Cuesta and P. Takáč
[5, Sect. 4, Example 4.1, pp. 740–741] that we have adapted to our model in
eq. (1.4) in Example 1.1 above. In this counterexample to the SCP we have
$x_{0}=0$.

Recalling our notation from Corollary 2.2, we denote by $P_{1}$ and $P_{0}$
the set of all points $x\in(-1,1)$ where the SCP is valid, i.e., $u(x)<v(x)$,
and is not valid, i.e., $u(x)=v(x)$, respectively:

$\displaystyle
P_{1}\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}\\{x\in(-1,1):\,u(x)<v(x)\\}\quad\mbox{
and }\quad
P_{0}\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}\\{x\in(-1,1):\,u(x)=v(x)\\}\,;$

hence, $P_{1}\cup P_{0}=\\{x\in(-1,1):\,u(x)\leq v(x)\\}=(-1,1),$ by our
hypothesis on the validity of the WCP. Clearly, $P_{1}\cap P_{0}=\emptyset$.
From this WCP, that is, $u(x)\leq v(x)$ for every $x\in[-1,1]$, we derive
easily

$x_{0}\in
P_{0}\subset(-1,1)\;\Longrightarrow\;u^{\prime}(x_{0})=v^{\prime}(x_{0})\quad\mbox{
in addition to }\quad u(x_{0})=v(x_{0})\,.$

As usual, we denote by $\overline{P}_{1}$ the closure in
$[-1,1]\subset\mathbb{R}$ of the set $P_{1}$.

In the remaining part of this paragraph, §3.1, we will investigate the
behavior of the two functions, $u$ and $v$, in the vicinity of a point
$x_{0}\in P_{0}\cap\overline{P}_{1}$ with the property (3.1) below:

######  Lemma 3.1 (Weak Comparison Principle at $x_{0}$).

Assume that $x_{0}\in P_{0}\cap\overline{P}_{1}$ satisfies the following

> Hypothesis: There is a number $\delta>0$ such that
>
> (3.1) $(x_{0}-\delta,x_{0})\subset P_{1}\quad\mbox{ or
> }\quad(x_{0},x_{0}+\delta)\subset P_{1}\,.$

Then we have $u(x_{0})=v(x_{0})$ and $u^{\prime}(x_{0})=v^{\prime}(x_{0})=0$.

We postpone the proof until after Remark 3.4.

######  Corollary 3.2 (WCP at $x_{0}$).

Let $x_{0}\in P_{0}\cap\overline{P}_{1}$ satisfy Hypothesis (3.1) and assume
that both functions $f,g:\,(-1,1)\subset\mathbb{R}\to\mathbb{R}$ are
continuous at the point $x_{0}$. Then we have also

(3.2)
$\frac{\mathrm{d}}{\mathrm{d}x}\,\left(|u^{\prime}|^{p-2}u^{\prime}\right)\Big{|}_{x=x_{0}}={}-f(x_{0})\geq{}-g(x_{0})=\frac{\mathrm{d}}{\mathrm{d}x}\,\left(|v^{\prime}|^{p-2}v^{\prime}\right)\Big{|}_{x=x_{0}}\,.$

Recalling that $b:\,[-1,1]\to\mathbb{R}$ is continuously differentiable, by
our Hypothesis (M) with Ineq. (2.1), from Corollary 3.2 above we are able to
derive the following stronger result:

######  Corollary 3.3 (Vanishing Convection/Drift at $x_{0}$).

In the situation of Corollary 3.2 above, with $x_{0}\in
P_{0}\cap\overline{P}_{1}$ satisfying Hypothesis (3.1) and with both functions
$f,g:\,(-1,1)\subset\mathbb{R}\to\mathbb{R}$ being continuous at the point
$x_{0}$, we must have the equality

(3.3)
${}-\frac{\mathrm{d}}{\mathrm{d}x}\,\left(|u^{\prime}|^{p-2}u^{\prime}\right)\Big{|}_{x=x_{0}}=f(x_{0})=g(x_{0})={}-\frac{\mathrm{d}}{\mathrm{d}x}\,\left(|v^{\prime}|^{p-2}v^{\prime}\right)\Big{|}_{x=x_{0}}\,.$

The following remark on Hypothesis (3.1) is important for the results in Lemma
3.1 and Corollaries 3.2 and 3.3. Namely, if the following stronger condition

$(x_{0}-\delta,x_{0})\subset P_{1}\quad\mbox{ and
}\quad(x_{0},x_{0}+\delta)\subset P_{1}$

is valid, then we must have $u(x_{0})<v(x_{0})$ as well, by Corollary 2.2,
Alt. (a) (cf. also Corollary 2.2, Alt. (b)).

######  Remark 3.4.

Assume that

(3.4) $(x_{0}-\delta,x_{0})\cup(x_{0},x_{0}+\delta)\subset P_{1}\,.$

Then also $u(x_{0})<v(x_{0})$ holds, by Alt. (a) of Corollary 2.2, i.e.,
$x_{0}\not\in P_{0}$ and, consequently, we get
$(x_{0}-\delta,x_{0}+\delta)\subset P_{1}\,.$ $\sqcap$$\sqcup$

> In contrast, our counterexample to the SCP (Example 1.1) in the Introduction
> (Section 1), §1.1, shows that the equality $u(x_{0})=v(x_{0})$ is still
> possible in the more general model governed by the boundary value problem
> (1.4) which allows also for the reaction function $\varphi(u)$ taken to be
> equal to $\varphi(x,u)=\lambda\,|u|^{p-2}u$ in [5, Example 4.1] and to
> $\varphi(x,u)=\lambda\,u$ in Examples 1.1 and 1.2 above.
>
> From the engineer’s point of view (cf. [3]), in this counterexample the
> reaction of the reaction function $\varphi(u)$ is so strong near the point
> $x_{0}=0$ that at the point $x_{0}=0$ it eliminates an increase of the
> source function $f(x)$ throughout the two neighboring intervals in condition
> (3.4) above. Thus, the equality $u(x_{0})=v(x_{0})$ holds in this
> counterexample at $x_{0}=0$.
>
> In particular, if pollution by the substance with the density $u(x)$ in the
> model is caused globally in the region
>
> $(-1,0)\cup(0,1)=(-1,1)\setminus\\{0\\}\,,$
>
> increasing the source function $f(x)$ to, say, $g(x)$ globally throughout
> this entire region, $(-1,1)\setminus\\{0\\}$, except for the point
> $x_{0}=0$, i.e., $f(x)<g(x)$ whenever $0<|x|<1$ and $f(0)=g(0)$, does not
> mean that an increase of pollution must be felt also at the point $x_{0}=0$.

Proof of Lemma 3.1. The two unknown functions
$u,v:\,(-1,1)\subset\mathbb{R}\to\mathbb{R}$ satisfy the following two-point
boundary value problems on $(-1,1)$, respectively; cf. (2.7):

(3.5) $\left\\{\hskip 5.69054pt\begin{aligned}
&{}-\left(|u^{\prime}|^{p-2}u^{\prime}\right)^{\prime}-b(x)u^{\prime}=f(x)\,,\qquad{}-\left(|v^{\prime}|^{p-2}v^{\prime}\right)^{\prime}-b(x)v^{\prime}=g(x)\,;\\\
&u(\pm 1)=v(\pm 1)=0\,,\quad\mbox{ and }\quad f\leq g\;\mbox{ a.e.\ in
}\,(-1,1)\,,\quad f\not\equiv g\;\mbox{ in }(-1,1)\,.\end{aligned}\right.$

By contradiction to the claim $u^{\prime}(x_{0})=v^{\prime}(x_{0})=0$, suppose
that either $u^{\prime}(x_{0})=v^{\prime}(x_{0})>0$ or else
$u^{\prime}(x_{0})=v^{\prime}(x_{0})<0$. Subtracting the latter from the
former equation in problems (3.5) above, we arrive at the linear “elliptic”
differential equation

(3.6) $\left\\{\hskip 14.22636pt\begin{aligned}
{}-[D(u^{\prime},v^{\prime})(x)w^{\prime}(x)]^{\prime}-b(x)\,w^{\prime}(x)&=g(x)-f(x)\
({}\geq 0)\quad\mbox{ in }\,(-1,1)\,;\\\ w(\pm 1)&=0\,,\end{aligned}\right.$

for the difference $w=v-u:\,(-1,1)\subset\mathbb{R}\to\mathbb{R}$ on $(-1,1)$.
This equation is regular at and near the point $x_{0}\in(-1,1)$, meaning that
the diffusion coefficient $D(u^{\prime},v^{\prime})(x)$ given by

(3.7) $\displaystyle D(u^{\prime},v^{\prime})(x)$
$\displaystyle{}=(p-1)\int_{0}^{1}\left|(1-\theta)u^{\prime}(x)+\theta
v^{\prime}(x)\right|^{p-2}\,\mathrm{d}\theta$ $\displaystyle{}\geq
c\left(\max_{0\leq\theta\leq 1}|(1-\theta)u^{\prime}(x)+\theta
v^{\prime}(x)|\right)^{p-2}\geq 0\quad\mbox{ on }\,[-1,1]$

satisfies $D(u^{\prime},v^{\prime})(x)>0$ for every $x\in(-1,1)$ near the
point $x_{0}\in(-1,1)$. Here, $c>0$ is some constant independent of
$u^{\prime}(x)$, $v^{\prime}(x)$, cf. [16, Ineq. (A.1), p. 233]. In general,
problem (3.6) is not uniformly elliptic. However, since the inequality
$D(u^{\prime},v^{\prime})(x)>0$ holds strictly provided at least one of the
two inequalities $u^{\prime}(x)\neq 0$ and $v^{\prime}(x)\neq 0$ is valid, the
problem for the function $w=v-u$ is, in fact, strictly elliptic.

Since also $x_{0}\in\overline{P}_{1}\cap(-1,1)$ and there is a number
$\delta>0$ such that Hypothesis (3.1) is valid, we may apply the boundary
point principle (Hopf’s lemma) established recently in the article by D. E.
Apushkinskaya and A. I. Nazarov [1, Theorem 2.1, p. 681]. Owing to
$w(x)=v(x)-u(x)>0$ for all $x\in(x_{0}-\delta,x_{0})$ or for all
$x\in(x_{0},x_{0}+\delta)$, and to $w(x_{0})=v(x_{0})-u(x_{0})=0$, we thus
conclude that $w^{\prime}(x_{0})<0$. This inequality clearly contradicts the
obvious inequality (the WCP established in Proposition 2.1)
$w(x)=v(x)-u(x)\geq 0$ for all points $x\in(-1,1)$ near $x_{0}$, where
$w(x_{0})=v(x_{0})-u(x_{0})=0$.

We have proved that $u^{\prime}(x_{0})=v^{\prime}(x_{0})=0$.

Proof of Corollary 3.2. We insert the conclusion of Lemma 3.1, particularly
$u^{\prime}(x_{0})=v^{\prime}(x_{0})=0$, into eqs. (3.5) in order to obtain
ineq. (3.2) in Corollary 3.2.

Proof of Corollary 3.3. Let us suppose that, by contradiction to eq. (3.3), we
have the sharp inequality in ineq. (3.2), that is, ${}-f(x_{0})>{}-g(x_{0}).$
Set

$\eta\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}\genfrac{}{}{}{1}{1}{2}\left({}-f(x_{0})+g(x_{0})\right)>0\,.$

All three functions $b,f,g:\,(-1,1)\subset\mathbb{R}\to\mathbb{R}$ being
continuous at the point $x_{0}$, we conclude from ineq. (3.2) that there is a
number $\delta_{1}\in(0,\delta)$ such that the inequality

(3.8)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\,\left(|u^{\prime}|^{p-2}u^{\prime}-|v^{\prime}|^{p-2}v^{\prime}\right)\Big{|}_{x}={}-b(x)\,[u^{\prime}(x)-v^{\prime}(x)]-f(x)+g(x)\geq\eta>0$
$\displaystyle\quad\mbox{ holds for every
}\,x\in(x_{0}-\delta_{1},\,x_{0}+\delta_{1})\,.$

We first recall that $u^{\prime}(x_{0})=v^{\prime}(x_{0})=0$, by Lemma 3.1,
then integrate the inequality in (3.8) above from $x_{0}$ to
$x\in(x_{0}-\delta_{1},\,x_{0}+\delta_{1})$, thus arriving at

(3.9)
$\displaystyle\left(|u^{\prime}|^{p-2}u^{\prime}-|v^{\prime}|^{p-2}v^{\prime}\right)\Big{|}_{x}$
$\displaystyle\leq{}-\eta\,|x-x_{0}|$ $\displaystyle\quad\mbox{ for every
}\,x\in(x_{0}-\delta_{1},\,x_{0}]\,,$ (3.10)
$\displaystyle\left(|u^{\prime}|^{p-2}u^{\prime}-|v^{\prime}|^{p-2}v^{\prime}\right)\Big{|}_{x}$
$\displaystyle\geq\eta\,|x-x_{0}|$ $\displaystyle\quad\mbox{ for every
}\,x\in[x_{0},\,x_{0}+\delta_{1})\,.$

Consequently, we have $u^{\prime}(x)-v^{\prime}(x)<0$ throughout the interval
$(x_{0}-\delta_{1},\,x_{0})$. When combined with $u(x_{0})=v(x_{0})$ and
$u^{\prime}(x_{0})=v^{\prime}(x_{0})$, this inequality forces

$\displaystyle
u(x)-v(x)=[u(x)-v(x)]-[u(x_{0})-v(x_{0})]=[u^{\prime}(\xi)-v^{\prime}(\xi)](x-x_{0})>0$
$\displaystyle\quad\mbox{ for every }\,x\in(x_{0}-\delta_{1},\,x_{0})\,\mbox{
and some }\,\xi\in(x,x_{0})\subset(x_{0}-\delta_{1},\,x_{0})\,,$

by the mean value theorem. But already this inequality, i.e., $u(x)-v(x)>0$
for all $x\in(x_{0}-\delta_{1},\,x_{0})$, contradicts the weak comparison
principle $u\leq v$ in $[-1,1]$.

Alternatively to ineq. (3.9), we may take advantage of the inequality in
(3.10) in a similar way: it yields $u^{\prime}(x)-v^{\prime}(x)>0$ on the
interval $(x_{0},\,x_{0}+\delta_{1})$. When combined with $u(x_{0})=v(x_{0})$
and $u^{\prime}(x_{0})=v^{\prime}(x_{0})$, this inequality forces

$\displaystyle
u(x)-v(x)=[u(x)-v(x)]-[u(x_{0})-v(x_{0})]=[u^{\prime}(\xi)-v^{\prime}(\xi)](x-x_{0})>0$
$\displaystyle\quad\mbox{ for every }\,x\in(x_{0},\,x_{0}+\delta_{1})\,\mbox{
and some }\,\xi\in(x_{0},x)\subset(x_{0},\,x_{0}+\delta_{1})\,,$

by the mean value theorem. Thus, we get $u(x)-v(x)>0$ for all
$x\in(x_{0},\,x_{0}+\delta_{1})$ which contradicts the weak comparison
principle $u\leq v$ in $[-1,1]$ again.

The corollary is proved.

The following proposition is a slight extension of Corollary 2.2 (“Interior”
Strong Comparison Principle):

######  Proposition 3.5 (“Interior” Strong Comparison Principle).

In the situation of Proposition 2.1 and Corollary 2.2, with
$\varphi(x,\,\cdot\,)\equiv 0$, the set

$\displaystyle P_{1}=\\{x\in(-1,1):\,u(x)<v(x)\\}\quad\mbox{ is an open
interval in $\mathbb{R}$. }$

If it is empty, then $u\equiv v$ and $f\equiv g$ hold throughout the interval
$P_{0}=(-1,1)$. If $P_{1}\neq\emptyset$, then we have $P_{1}=(a_{-1},\,a_{1})$
with a pair of points $-1\leq a_{-1}<a_{1}\leq 1$; moreover, only the
following three possibilities may occur:

  * (i)

$P_{1}=(-1,1)$ in which case the strong comparison principle (SCP) is valid:
$u(x)<v(x)$ holds for all $x\in(-1,1)$.

  * (ii)

$-1<a_{-1}<a_{1}\leqq 1$ in which case we have $u(x)=v(x)$ and $f(x)=g(x)$ for
all $x\in(-1,\,a_{-1}]$, together with the vanishing derivatives
$u^{\prime}(a_{-1})=v^{\prime}(a_{-1})=0$.

  * (iii)

$-1\leqq a_{-1}<a_{1}<1$ in which case we have $u(x)=v(x)$ and $f(x)=g(x)$ for
all $x\in[a_{1},\,1)$, together with the vanishing derivatives
$u^{\prime}(a_{1})=v^{\prime}(a_{1})=0$.

Proof of Proposition 3.5. Corollary 2.2 guarantees that the set $P_{1}$ is an
open interval in $\mathbb{R}$, possibly empty. If $P_{1}=\emptyset$, then
obviously $P_{0}=(-1,1)$. The complementary alternative, $P_{1}\neq\emptyset$,
forces $P_{1}=(a_{-1},\,a_{1})$ with a pair of points $-1\leqq
a_{-1}<a_{1}\leqq 1$, by Alt. (b) of Corollary 2.2. Case (i) is obvious.

If Case (i) is false, i.e., $P_{0}\neq\emptyset$, then at least one of the two
Cases (ii) and (iii) must occur: The conclusion in both these cases follows
from Corollary 2.2, Alt. (a), combined with Lemma 3.1.

###  3.2. The solutions $u$ and $v$ near a Common Critical Meeting Point

In this paragraph we “refine” the counterexample to the SCP presented in
Example 1.1 in the Introduction (Section 1), §1.2. We will construct a pair of
functions $f,g\in L^{\infty}(-1,1)$ such that

$f\leq g\;\mbox{ a.e.\ in }\,(-1,1)\quad\mbox{ and }\quad f\not\equiv
g\;\mbox{ in }\,(-1,1)\,,$

and the corresponding pair of solutions $u,v:\,[-1,1]\to\mathbb{R}$ to the
two-point boundary value problems (3.5) stated in the previous paragraph,
§3.1, such that in Hypothesis (3.1) we have

(3.11) $P_{0}=(-1,x_{0}]\quad\mbox{ and }\quad P_{1}=(x_{0},1)\quad\mbox{ with
}\,x_{0}={}-\genfrac{}{}{}{1}{1}{2}\in(-1,1)\,.$

######  Example 3.6 (Another Counterexample to the SCP).

In this counterexample we continue using the notation introduced in
Proposition 3.5 from the previous paragraph (§3.2). Here, we will construct an
example of Part (ii) in Proposition 3.5 with the pair of points
$a_{-1}={}-1/2<a_{1}=1$, that is to say,

  * (ii)

$-1<a_{-1}={}-1/2<a_{1}=1$ in which case we have $u(x)=v(x)$ and $f(x)=g(x)$
for all $x\in(-1,\,a_{-1}]$, together with the vanishing derivatives
$u^{\prime}(a_{-1})=v^{\prime}(a_{-1})=0$, and $u(x)<v(x)$ for all
$x\in(a_{-1},1)$ together with $u(\pm 1)=v(\pm 1)=0$.

We feel that our construction will be more understandable when we start from
the graphs of the solutions $u,v:\,[-1,1]\to\mathbb{R}$ to the two-point
boundary value problems (3.5) sketched below in Figure 1 where we have
introduced the abbreviation

$x_{\mathrm{s}}\stackrel{{\scriptstyle{\mathrm{def}}}}{{=}}1-3/2^{\frac{p}{p-1}}=1-3/2^{p^{\prime}}\in\left(-\genfrac{}{}{}{1}{1}{2}\,,\,0\right)\quad\mbox{
with the conjugate exponent
}\;p^{\prime}=\genfrac{}{}{}{1}{p}{p-1}\in(1,2)\,.$

$u$$v$$u(x)=v(x)$$-1$$-1/2$$1/2$$1$$x_{\mathrm{s}}$$0$$1/2$$1$$x$$y$ Figure 1.
Counterexample to the strong comparison principle for $p=4$. Here,
$a_{-1}=-1/2$, $x_{\mathrm{s}}=1-3/2^{\frac{4}{3}}$ and the solutions $u$ and
$v$ are given by the respective formulas (3.12) and (3.13).

Indeed, we will show that the functions

(3.12) $\displaystyle u(x)\stackrel{{\scriptstyle\rm
def}}{{=}}\left\\{\begin{aligned} &{}-4x(x+1)\,,&&\quad
x\in\left[-1\,,\,-\frac{1}{2}\right)\,,\\\\[5.0pt]
&1-3^{\frac{1-p}{p-2}}(2x+1)^{\frac{p-1}{p-2}}\,,&&\quad
x\in\left[-\frac{1}{2}\,,\,1\right]\,,\end{aligned}\right.$

and

(3.13) $\displaystyle v(x)\stackrel{{\scriptstyle\rm
def}}{{=}}\left\\{\begin{aligned} &{}-4x(x+1)\,,&&\quad
x\in\left[-1\,,\,-\frac{1}{2}\right)\,,\\\\[5.0pt] &\qquad 1\,,&&\quad
x\in\left[-\frac{1}{2}\,,\,x_{\mathrm{s}}\right]\,,\\\\[5.0pt]
&1-2^{\frac{p}{p-2}}\cdot
3^{\frac{1-p}{p-2}}\cdot\left(x-x_{\mathrm{s}}\right)^{\frac{p-1}{p-2}}\,,&&\quad
x\in\left(x_{\mathrm{s}}\,,\,1\right]\,,\end{aligned}\right.$

satisfy (3.5) with $p>2$,

$\displaystyle
b=-\genfrac{(}{)}{}{0}{3}{2}^{1-p}\genfrac{(}{)}{}{0}{p-2}{p-1}^{1-p}<0\quad\mbox{
(constant {\it convection velocity\/}), }$ $\displaystyle
f(x)\stackrel{{\scriptstyle\rm def}}{{=}}\left\\{\begin{aligned}
&2^{p-1}\left(2^{p}(p-1)[{}-(2x+1)]^{p-2}-12\genfrac{(}{)}{}{0}{p-2}{p-1}^{1-p}\,(2x+1)\right)\,,\hskip
5.0pt&&x\in\left(-1\,,\,-\frac{1}{2}\right)\,,\\\\[5.0pt] &\qquad
0\,,&&x\in\left[-\frac{1}{2}\,,\,1\right)\,,\end{aligned}\right.$

and

$\displaystyle g(x)\stackrel{{\scriptstyle\rm
def}}{{=}}\left\\{\begin{aligned}
&2^{p-1}\left(2^{p}(p-1)[{}-(2x+1)]^{p-2}-12\genfrac{(}{)}{}{0}{p-2}{p-1}^{1-p}\,(2x+1)\right)\,,\hskip
5.0pt&&x\in\left(-1\,,\,-\frac{1}{2}\right)\,,\\\\[5.0pt] &\qquad
0\,,&&x\in\left[-{}\frac{1}{2}\,,\,x_{\mathrm{s}}\right]\,,\\\\[5.0pt]
&2^{\frac{(p-1)^{2}+1}{p-2}}\cdot
3^{{}-\frac{(p-1)^{2}}{p-2}}\cdot\genfrac{(}{)}{}{0}{p-1}{p-2}^{p}\left(x-x_{\mathrm{s}}\right)^{\frac{1}{p-2}}\,,\,&&x\in\left(x_{\mathrm{s}}\,,\,1\right)\,.\end{aligned}\right.$

Note that, thanks to $p>2$, both functions $f$ and $g$ are bounded on
$[-1,1]$, and thus $f,g\in L^{\infty}(-1,1)$. It is easily seen that
$f(x)=g(x)\geq 0$ in $\left(-1,\,-1/2\right)$, $f(x)=g(x)=0$ in
$\left[-1/2,\,x_{\mathrm{s}}\right]$, and $g(x)>f(x)=0$ in
$(x_{\mathrm{s}},1)$. We conlude that $f(x)\leq g(x)$ in $(-1,1)$.

By our definition of functions $u$, $v$, $f$, and $g$, we easily verify the
following equalities,

$f(x)={}-\left(|u^{\prime}(x)|^{p-2}u^{\prime}(x)\right)^{\prime}-b\,u^{\prime}(x)\quad\mbox{
and }\quad
g(x)={}-\left(|v^{\prime}(x)|^{p-2}v^{\prime}(x)\right)^{\prime}-b\,v^{\prime}(x)\,.$

The reader can verify these two relations by investigating the particular
cases of $x\in(-1,1)$ suggested by the partition of the interval $(-1,1)$
sketched in Figure 1 above, that is to say,

(3.14)
$(-1,1)=\left(-1,\,-1/2\right)\cup\left[-1/2,\,x_{\mathrm{s}}\right]\cup\left(x_{\mathrm{s}},\,0\right]\cup(0,1)\,.$

This is a matter of a direct calculation in each of the subintervals indicated
above.

By our definition of functions $u$ and $v$, we have $u\equiv v$ in
$(-1,\,-1/2)$. The strict inequality $u(x)<v(x)$ for every $x\in(-1/2,\,1)$ is
verified again by investigating the various subintervals of the partition
(3.14); cf. Figure 1.  $\sqcap$$\sqcup$

In order to prevent the equality $u(x)\equiv v(x)$ on the interval
$(-1,\,-1/2)$ in our Example 3.6 above, we make the following natural
hypothesis:

Hypothesis $\mathbf{H}_{\pm 1}$. Let $f,g\in L^{\infty}(-1,1)$ and $0\leq
f(x)\leq g(x)$ hold for almost all $x\in(-1,1)$. In addition, we assume that
for every $\delta>0$ small enough we have $f\not\equiv g$ in both intervals
$(-1,-1+\delta)$ and $(1-\delta,1)$, that is to say, both sets

$\\{x\in(-1,1):\,f(x)<g(x)\\}\cap(-1,-1+\delta)\quad\mbox{ and
}\quad\\{x\in(-1,1):\,f(x)<g(x)\\}\cap(1-\delta,1)$

have positive Lebesgue measure for every $\delta>0$ small enough.

Consequently, Hypothesis $\mathbf{H}_{\pm 1}$ is valid with any number
$\delta\in\left(0,\,1/2\right)$ in Lemma 3.7 below. This lemma will then yield
$v^{\prime}(1)<u^{\prime}(1)\leq 0\leq u^{\prime}(-1)<v^{\prime}(-1)$.

As an immediate consequence we obtain the following boundary point lemma of
Hopf type.

######  Lemma 3.7 (Hopf’s Maximum Principle at $x=\pm 1$).

Assume that Hypothesis $\mathbf{H}_{\pm 1}$ is valid. Then the following
boundary point lemma holds: $v^{\prime}(-1)<0<v^{\prime}(1)$.

In particular, the linear elliptic equation (3.6) for the difference
$w\stackrel{{\scriptstyle\mathrm{def}}}{{=}}v-u$ on $(-1,1)$ is regular in
$(-1,1)$ near the endpoints $x=\pm 1$, i.e., there is some $\delta>0$ small
enough such that the inequality
$D(u^{\prime},v^{\prime})(x)\geq\mathrm{const}>0$ holds strictly for every
$x\in(-1,-1+\delta)\cup(1-\delta,1)\,.$

Consequently, we have even the _Hopf boundary point comparison_ :
$v^{\prime}(1)<u^{\prime}(1)\leq 0\leq u^{\prime}(-1)<v^{\prime}(-1)$.

Proof of Lemma 3.7. We apply the Hopf boundary point lemma from P. Tolksdorf
[18, Prop. 3.2.1 and 3.2.2, p. 801] or J. L. Vázquez [19, Theorem 5, p. 200].
This result, due to E. Hopf, is described and proved in all details in the
monograph by P. Pucci and J. Serrin [13, §5.4], Theorem 5.5.1 on p 120; cf.
also [13, §5.1], Example on p. 104.

Owing to $g\geq 0$ in $(-1,1)$ and $g\not\equiv 0$ in both intervals
$(-1,-1+\delta)$ and $(1-\delta,1)$, for every $\delta>0$ small enough, we
thus get $v^{\prime}(-1)<0<v^{\prime}(1)$. In particular, the linear elliptic
equation (3.6) for the difference
$w\stackrel{{\scriptstyle\mathrm{def}}}{{=}}v-u$ on $(-1,1)$ being regular, we
arrive at $w^{\prime}(-1)<0<w^{\prime}(1)$ by the new version of the Hopf
boundary point lemma established recently in the article by D. E.
Apushkinskaya and A. I. Nazarov [1].

Hence, the desired inequalities $v^{\prime}(1)<u^{\prime}(1)\leq 0\leq
u^{\prime}(-1)<v^{\prime}(-1)$ follow.

######  Theorem 3.8 (Hopf’s “Global” Strong Comparison Principle).

Let us consider problem (3.5) for $u$ and $v$ on $(-1,1)$. If also $f,g\in
L^{\infty}(-1,1)$ with $0\leq f\leq g$ a.e. in $(-1,1)$ and Hypothesis
$\mathbf{H}_{\pm 1}$ is valid, then Hopf’s (“Global”) Strong Comparison
Principle holds:

$v^{\prime}(1)<u^{\prime}(1)\leq 0\leq
u^{\prime}(-1)<v^{\prime}(-1)\quad\mbox{ and }\quad 0\leq u(x)<v(x)\quad\mbox{
for all }\,x\in(-1,1)\,.$

Proof of Theorem 3.8. Let us recall that both functions
$u,v:\,[-1,1]\to\mathbb{R}$ are continuously differentiable. We begin by
applying Proposition 3.5 to conclude that the set

$P_{1}=\\{x\in(-1,1):\,u(x)<v(x)\\}\quad\mbox{ is an open interval in
$\mathbb{R}$, i.e., }\ $

$P_{1}=(a_{-1},\,a_{1})$ holds with a pair of points $-1\leq a_{-1}<a_{1}\leq
1$; hence, the complementary set in $(-1,1)$ equals to

$\displaystyle P_{0}=\\{x\in(-1,1):\,u(x)=v(x)\\}=(-1,a_{-1}]\cup[a_{1},1)\,.$

On the other hand, by Lemma 3.7 above, if we choose $\delta>0$ small enough,
such that both inequalities, $u^{\prime}(\xi)<v^{\prime}(\xi)$ and
$v^{\prime}(\eta)<u^{\prime}(\eta)$, are valid for all $\xi\in(-1,-1+\delta)$
and $\eta\in(1-\delta,1)$, then we must have also $u(\xi)<v(\xi)$ and
$u(\eta)<v(\eta)$ again for all $\xi\in(-1,-1+\delta)$ and
$\eta\in(1-\delta,1)$, thanks to the Dirichlet boundary conditions
$u(-1)=v(-1)$ and $u(1)=v(1)$. But this result forces
$(-1,-1+\delta)\cup(1-\delta,1)\subset P_{1}$ which clearly contradicts the
statement

$(-1,1)\setminus P_{1}=P_{0}=(-1,a_{-1}]\cup[a_{1},1)\quad\mbox{ unless
}\,P_{0}=\emptyset\,,$

i.e., $P_{1}=(-1,1)$ holds as desired. Lemma 3.7 yields also
$v^{\prime}(1)<u^{\prime}(1)\leq 0\leq u^{\prime}(-1)<v^{\prime}(-1)\,.$ The
remaining part of Hopf’s Comparison Principle follows directly from
$P_{1}=(-1,1)$.

The theorem is proved.

###  3.3. The case of constant convection velocity

In this paragraph we will assume that the convection velocity
$b:\,(-1,1)\to\mathbb{R}$ is a given constant, i.e., $b(x)\equiv
b_{0}\in\mathbb{R}$ for all $x\in(-1,1)$. Hence, Proposition 2.1 applies and,
thus, the WCP is valid. In this simplified model, we will be able to prove a
weaker form of Theorem 3.8, Theorem 3.9 below, under the following hypothesis,

Hypothesis $\mathbf{H}_{0}$. Let $f,g\in L^{\infty}(-1,1)$ and $0\leq f(x)\leq
g(x)$ hold for almost all $x\in(-1,1)$ together with $f\not\equiv g$ in the
interval $(-1,1)$, i.e., the set

$\\{x\in(-1,1):\,f(x)<g(x)\\}$

has positive Lebesgue measure. In addition, we assume that
$\varphi(x,\,\cdot\,)\equiv 0$ and the convection velocity
$b(x)=b_{0}\equiv\mathrm{const}$ is constant.

We note that $\mathbf{H}_{0}$ is weaker than Hypothesis $\mathbf{H}_{\pm 1}$
used in Lemma 3.7 and Theorem 3.8. This weaker result, in Theorem 3.9 below,
is motivated and closely related to our results above, in Paragraph §3.1.

######  Theorem 3.9 (Another Hopf’s “Global” Comparison Principle).

Let us consider problem (3.5) for $u$ and $v$ on $(-1,1)$, with
$b(x)=b_{0}\equiv\mathrm{const}$ being constant on $(-1,1)$. Assume that
$f,g\in L^{\infty}(-1,1)$ satisfy Hypothesis $\mathbf{H}_{0}$ and
$u(x_{0})=v(x_{0})$ holds at some point $x_{0}\in(-1,1)$. Then we have also
$u^{\prime}(x_{0})=v^{\prime}(x_{0})=0$ together with the following two
alternatives:

  * Alt. 1:

If $b_{0}\leq 0$ then we must have $u(x)=v(x)$ for all $x\in(-1,x_{0}]$
together with $f(x)=g(x)$ for almost every $x\in(-1,x_{0})$, that is, both,
$u\equiv v$ and $f\equiv g$ in $(-1,x_{0})$.

  * Alt. 2:

If $b_{0}\geq 0$ then we must have $u(x)=v(x)$ for all $x\in[x_{0},1)$
together with $f(x)=g(x)$ for almost every $x\in(x_{0},1)$, that is, both,
$u\equiv v$ and $f\equiv g$ in $(x_{0},1)$.

######  Remark 3.10.

The result in Alt. 1 reflects the scenario in Example 3.6 with
$b_{0}=-\genfrac{(}{)}{}{1}{3}{2}^{1-p}\genfrac{(}{)}{}{1}{p-2}{p-1}^{1-p}\hfil\break<0$
and $x_{0}=-1/2$. $\sqcap$$\sqcup$

Proof of Theorem 3.9. Let us recall that both functions
$u,v:\,[-1,1]\to\mathbb{R}$ are continuously differentiable. We begin by
applying Proposition 3.5 to conclude that the set

$P_{1}=\\{x\in(-1,1):\,u(x)<v(x)\\}\quad\mbox{ is an open interval in
$\mathbb{R}$, i.e., }\ P_{1}=(a_{-1},\,a_{1})$

with a pair of points $-1\leq a_{-1}<a_{1}\leq 1$. Hence, its complementary
set in $(-1,1)$ equals to

$\displaystyle P_{0}=\\{x\in(-1,1):\,u(x)=v(x)\\}=(-1,a_{-1}]\cup[a_{1},1)\,.$

We rewrite the general problem (3.5) in the special form with
$b(x)=b_{0}\equiv\mathrm{const}$ being a real constant, $b_{0}\in\mathbb{R}$,

(3.15) $\left\\{\quad\begin{aligned}
&{}-\frac{\mathrm{d}}{\mathrm{d}x}\,\left(|u^{\prime}|^{p-2}u^{\prime}+b_{0}\,u\right)=f(x)\,,\qquad{}-\frac{\mathrm{d}}{\mathrm{d}x}\,\left(|v^{\prime}|^{p-2}v^{\prime}+b_{0}\,v\right)=g(x)\,;\\\
&\quad u(\pm 1)=v(\pm 1)=0\,,\quad\mbox{ and }\quad\\\ &\quad 0\leq f\leq
g\quad\mbox{ a.e.\ in }\,(-1,1)\,,\quad f\not\equiv g\quad\mbox{ in
}(-1,1)\,.\end{aligned}\right.$

In analogy with our proof of Theorem 3.8 above, we wish to show that either
$P_{1}=(x_{0},\,1)$ or else $P_{1}=(-1,\,x_{0})$ holds, this time without
assuming the validity of Hypothesis $\mathbf{H}_{\pm 1}$. The desired
equivalent result states that either $P_{0}=(-1,\,x_{0}]$ or else
$P_{0}=[x_{0},\,1)$, respectively.

Thus, let $x_{0}\in(-1,1)$ be a point satisfying $u(x_{0})=v(x_{0})$. Then the
“local” inequality $u(x)\leq v(x)$ for all
$x\in(x_{0}-\delta,\,x_{0}+\delta)\subset(-1,1)$ forces also
$u^{\prime}(x_{0})=v^{\prime}(x_{0})$. Since $u\not\equiv v$ in $(-1,1)$, we
can choose $x_{0}\in(-1,1)$ above in such a way that at least one of the
following two statements is valid:

> Ineq. 1: $u(x)<v(x)$ holds for every $x\in(x_{0}-\delta,\,x_{0})$. Ineq. 2:
> $u(x)<v(x)$ holds for every $x\in(x_{0},\,x_{0}+\delta)$.

(We note that these inequalities may hold simultaneously.)

Furthermore, the linear elliptic equation (3.6) for the difference
$w\stackrel{{\scriptstyle\mathrm{def}}}{{=}}v-u$ on $(-1,1)$ is uniformly
elliptic unless $u^{\prime}(x_{0})=v^{\prime}(x_{0})=0$ is valid. However, if
$u^{\prime}(x_{0})=v^{\prime}(x_{0})\neq 0$, then the Hopf boundary point
lemma would force either $u^{\prime}(x_{0})>v^{\prime}(x_{0})$ or else
$u^{\prime}(x_{0})<v^{\prime}(x_{0})$, respectively. This would contradict the
equality $u^{\prime}(x_{0})=v^{\prime}(x_{0})$. Thus, we have a point
$x_{0}\in(-1,1)$ such that $u(x_{0})=v(x_{0})$ and
$u^{\prime}(x_{0})=v^{\prime}(x_{0})=0$; cf. Lemma 3.1 (WCP at $x_{0}$).

Next, we use $x_{0}\in(-1,1)$ as the initial point for our “shooting method”
applied to the special problem (3.15) above, with the initial values
$u(x_{0})=v(x_{0})$ and $u^{\prime}(x_{0})=v^{\prime}(x_{0})=0$:

$\left\\{\quad\begin{aligned}
|u^{\prime}(x)|^{p-2}u^{\prime}(x)+b_{0}\,u(x)&=|u^{\prime}(x_{0})|^{p-2}u^{\prime}(x_{0})+b_{0}\,u(x_{0})\textstyle-\int_{x_{0}}^{x}f(s)\,\mathrm{d}s\\\
&=b_{0}\,u(x_{0})\textstyle-\int_{x_{0}}^{x}f(s)\,\mathrm{d}s\quad\mbox{ for
all }\,x\in(-1,1)\,,\\\
|v^{\prime}(x)|^{p-2}v^{\prime}(x)+b_{0}\,v(x)&=|v^{\prime}(x_{0})|^{p-2}v^{\prime}(x_{0})+b_{0}\,v(x_{0})\textstyle-\int_{x_{0}}^{x}g(s)\,\mathrm{d}s\\\
&=b_{0}\,v(x_{0})\textstyle-\int_{x_{0}}^{x}g(s)\,\mathrm{d}s\quad\mbox{ for
all }\,x\in(-1,1)\,;\\\ &u(\pm 1)=v(\pm 1)=0\,,\quad\mbox{ and }\quad\\\ 0\leq
f\leq g&\quad\mbox{ a.e.\ in }\,(-1,1)\,,\quad f\not\equiv g\quad\mbox{ in
}(-1,1)\,.\end{aligned}\right.$

Subtracting the first differential equation (for $u$) from the second one (for
$v$) we arrive at

(3.16) $\left\\{\quad\begin{aligned}
&|v^{\prime}(x)|^{p-2}v^{\prime}(x)-|u^{\prime}(x)|^{p-2}u^{\prime}(x)={}-b_{0}(v(x)-u(x))\\\
&\textstyle{}-\int_{x_{0}}^{x}(g(s)-f(s))\,\mathrm{d}s\quad\mbox{ for all
}\,x\in(-1,1)\,,\\\ &u(\pm 1)=v(\pm 1)=0\,,\quad\mbox{ and }\quad\\\ &0\leq
f\leq g\quad\mbox{ a.e.\ in }\,(-1,1)\,,\quad f\not\equiv g\quad\mbox{ in
}(-1,1)\,.\end{aligned}\right.$

Case $b_{0}\leq 0$: We have ${}-b_{0}(v(x)-u(x))\geq 0$ and thus eq. (3.16)
yields

$\displaystyle|v^{\prime}(x)|^{p-2}v^{\prime}(x)-|u^{\prime}(x)|^{p-2}u^{\prime}(x)\textstyle\geq{}-\int_{x_{0}}^{x}(g(s)-f(s))\,\mathrm{d}s$
$\displaystyle\textstyle=\int_{x}^{x_{0}}(g(s)-f(s))\,\mathrm{d}s\geq
0\quad\mbox{ for all }\,x\in(-1,x_{0}]\,.$

Consequently, we must have $(v-u)^{\prime}(x)\geq 0$ for all $x\in(-1,x_{0}]$
which forces $v(x)-u(x)=0$ for all $x\in(-1,x_{0}]$, thanks to $v(-1)-u(-1)=0$
and $v(x_{0})-u(x_{0})=0$. This is possible only if $f(x)=g(x)$ holds for
almost every $x\in(-1,x_{0})$, as described in alternative “Alt. 1”.

Case $b_{0}\geq 0$: We have ${}-b_{0}(v(x)-u(x))\leq 0$ and thus eq. (3.16)
yields

$|v^{\prime}(x)|^{p-2}v^{\prime}(x)-|u^{\prime}(x)|^{p-2}u^{\prime}(x)\textstyle\leq{}-\int_{x_{0}}^{x}(g(s)-f(s))\,\mathrm{d}s\leq
0\quad\mbox{ for all }\,x\in[x_{0},1)\,.$

Consequently, we have $(v-u)^{\prime}(x)\leq 0$ for all $x\in[x_{0},1)$ whence
$v(x)-u(x)=0$ for all $x\in[x_{0},1)$, thanks to $v(x_{0})-u(x_{0})=0$ and
$v(1)-u(1)=0$. This entails $f(x)=g(x)$ for almost every $x\in(x_{0},1)$, as
claimed in alternative “Alt. 2”.

The proof is finished.

Acknowledgments: The research of Jiří Benedikt, Petr Girg, and Lukáš Kotrla
was partially supported by the Grant Agency of the Czech Republic (GAČR) under
Project No. 18–03253S. Ing. Lukáš Kotrla was partially supported also by the
Ministry of Education, Youth, and Sports (MŠMT, Czech Republic) under the
program NPU I, Project No. LO1506 (PU–NTIS).

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