\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 37, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2007/37\hfil Nonlinear variational inequalities]
{On a class of nonlinear variational inequalities:
High concentration of the graph of weak solution via its
fractional dimension and Minkowski content}
\author[L. Korkut, M. Pa\v{s}i\'{c} \hfil EJDE-2007/37\hfilneg]
{Luka Korkut, Mervan Pa\v{s}i\'{c}} % in alphabetical order
\address{Luka Korkut \newline
Department of Mathematics\\
Faculty of Electrical Engineering and Computing\\
University of Zagreb\\
Unska 3, 10000 Zagreb, Croatia}
\email{luka.korkut@fer.hr}
\address{Mervan Pa\v{s}i\'{c}\newline
Department of Mathematics\\
Faculty of Electrical Engineering and Computing\\
University of Zagreb\\
Unska 3, 10000 Zagreb, Croatia}
\email{mervan.pasic@fer.hr}
\thanks{Submitted November 19, 2006. Published March 1, 2007.}
\subjclass[2000]{35J85, 34B15, 28A75}
\keywords{Double obstacles; nonlinear p-Laplacian; graph; \hfill\break\indent
fractional dimension; Minkowski content; singularity of
derivative}
\begin{abstract}
Weak continuous bounded solutions of a class of nonlinear
variational inequalities associated to one-dimensional
$p$-Laplacian are studied. It is shown that a kind of boundary
behaviour of nonlinearity in the main problem produces a kind of
high boundary concentration of the graph of solutions. It is
verified by calculating lower bounds for the upper
Minkowski-Bouligand dimension and Minkowski content of the graph
of each solution and its derivative. Finally, the order of growth
for singular behaviour of the $L^{p}$ norm of derivative of
solutions is given.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{proposition}[theorem]{Proposition}
\section{Introduction}
Let $1
0$.
The main subject of the paper is the graph $G(u)$ of a continuous real
function $u$ defined on $[a,b]$, that is
\[
G(u)=\{(t,u(t)):a\leq t\leq b\}.
\]
In order to describe a kind of very high boundary concentration of $G(u)$
near the point $t=a$, where $u$ is any solution of \eqref{1eq}, we
associate to $G(u)$ the following two numbers:
\begin{itemize}
\item The upper Minkowski-Bouligand (box-counting) dimension of
$G(u)$,
\[
\dim _{M}G(u)=\limsup_{\varepsilon \to 0}\big( 2-
\frac{\log |G_{\varepsilon }(u)|}{\log \varepsilon }\big) ,
\]
where $G_{\varepsilon }(u)$ denotes the
$\varepsilon$-neighbourdhood of $G(u)$ and $|G_{\varepsilon }(u)|$
denotes the Lebesgue measure of $G_{\varepsilon }(u)$.
\item The $s$-dimensional upper Minkowski content of $G(u)$,
\[
M^{s}(G(u))=\limsup_{\varepsilon \to 0}(2\varepsilon
)^{s-2}|G_{\varepsilon }(u)|,
\]
where $s\in (1,2)$.
\end{itemize}
In Section 3, for arbitrarily given $s\in (1,2)$ we will find some
sufficient conditions on the obstacles $\varphi $ and $\psi $ and on the
nonlinearity $f(t,\eta ,\xi )$ such that each solution $u$ of \eqref{1eq}
satisfies
\begin{equation}
|G_{\varepsilon }(u)|\geq \frac{1}{2^{6}}(b-a)^{s}\varepsilon ^{2-s}>0\quad
\text{for each }\varepsilon \in (0,\varepsilon _{0}), \label{1Geu}
\end{equation}
where $\varepsilon _{0}>0$ will be precised too. According to the
definitions of $\dim _{M}G(u)$ and $M^{s}(G(u))$, the inequality
\eqref{1Geu} enables us to show that each solution $u$ of
\eqref{1eq} satisfies
\begin{equation}
\dim _{M}G(u)\geq s\quad \text{and}\quad M^{s}(G(u))\geq
\frac{1}{2^{7}} (b-a)^{s}>0. \label{1lbs}
\end{equation}
Since $\dim _{M}(A\cup B)=\max \{\dim _{M}A,\dim _{M}B\}$ and
$u\in W^{1,p}(a+\varepsilon ,b)$ for each $\varepsilon >0$, we
have that $u$ is an absolutely continuous function on
$[a+\varepsilon ,b]$ which together with \eqref{1lbs} gives us
\[
\dim _{Mloc}(G(u);a)\geq s\quad\text{and}\quad
\dim _{Mloc}(G(u);t)=1\quad \text{for each }t\in (a,b].
\]
Here $\dim _{Mloc}(G(u);t)$ denotes the locally upper Minkowski-Bouligand
dimension of $G(u)$ at a point $t\in [a,b]$, given by
\[
\dim _{Mloc}(G(u);t)=\limsup_{\varepsilon \to 0}\dim _{M}(G(u)\cap
B_{\varepsilon }(t,u(t))),
\]
where $B_{r}(t_{1},t_{2})$ denotes a ball with radius $r>0$
centered at the point $(t_{1},t_{2})\in \mathbb{R}^{2}$.
As an easy consequence, we derive that each solution $u$ of \eqref{1eq}
satisfies:
\begin{gather*}
u\notin W^{1,p}(a,b)\quad \text{and}\quad \mathop{\rm length}(G(u))=\infty , \\
u\in W^{1,p}(a+\varepsilon ,b)\quad \text{and}\quad
\mathop{\rm length}(G(u|_{[a+\varepsilon ,b]}))<\infty \quad
\text{for any }\varepsilon >0,
\end{gather*}
where $u|_{I}$ denotes the restriction of $u$ on $I$. Thus,
according to the previous statements, we may conclude that the
graph $G(u)$ of any solution $u $ of \eqref{1eq} is (in a sense)
highly concentrated at the boundary point $t=a$. Furthermore, the
statement $length(G(u))=\infty $ is precised in \eqref{1lbs}.
For arbitrarily given $s\in (1,2)$ and under the same hypotheses
on $\varphi $, $\psi $ and $f(t,\eta ,\xi )$ as in getting of
\eqref{1Geu}--\eqref{1lbs}, we will prove in Section 3 that each
solution $u$ of \eqref{1eq} satisfies
\begin{equation}
|G_{\varepsilon }(u|_{(a,c]})|\geq \frac{1}{2^{6}}(c-a)^{s}\varepsilon
^{2-s}>0\ \text{for any }c\in (a,b)\text{ and }\varepsilon \in
(0,\varepsilon _{c}), \label{3Geures}
\end{equation}
where the number $\varepsilon _{c}$ will be also determined. The preceding
inequality yields
\begin{equation}
M^{s}(G(u)\cap B_{r}(a,u(a)))\geq \frac{1}{2^{7}}(\frac{r}{\sqrt{5}})^{s}
\quad \text{ for any }r\in (0,b-a). \label{1Mslb}
\end{equation}
It completes the second inequality in \eqref{1lbs}.
Next, for arbitrarily given $s\in (1,2)$ and under the same
hypotheses on $\varphi $, $\psi $ and $f(t,\eta ,\xi )$ as in
getting of \eqref{1Geu}-\eqref{1Mslb},
we will show in Section 4 that each solution $u$ of \eqref{1eq}
such that $u\in C^{1}(a,b)$ satisfies
\begin{equation}
|G_{\varepsilon }(u')|\geq
\frac{\sqrt2}{2^{4}}(b-a)^{s/2}\varepsilon ^{1-s/2}>0\quad
\text{for each }\varepsilon \in (0,\varepsilon _{0}). \label{1derlb}
\end{equation}
where $\varepsilon _{0}>0$ will be also precised. Here $u'$
denotes the derivative of $u$ in the classical sense. According to
the definitions of $\dim _{M}G(u')$ and $M^{s}(G(u'))$, from
\eqref{1derlb} we get that each smooth enough solution $u$ od
\eqref{1eq} satisfies:
\begin{equation}
\dim _{M}G(u')\geq 1+\frac{s}{2}\quad \text{and}\quad
M^{1+s/2}(G(u'))\geq \frac{1}{2^{4}}(b-a)^{s/2}>0. \label{1derlbs}
\end{equation}
In \eqref{1derlbs} we have two estimations for singular
behaviour of $u'$ near the boundary point $t=a$. Much more
information about singular behaviour of $u'$ near the point $t=a$
can be obtained from asymptotic behaviour of
$\|u'\|_{L^{p}(a+\varepsilon ,b)}$ as $\varepsilon \approx 0$.
More precisely, from above observation we have in particular that
each solution $u$ of \eqref{1eq} satisfies
\[
\limsup_{\varepsilon \to
0}\|u'\|_{L^{p}(a+\varepsilon ,b)}=\infty .
\]
However, in Section 5 we will be able to precise this statement.
That is, for arbitrarily given $s\in (1,2)$ and under related
hypotheses on $\varphi $, $\psi $ and $f(t,\eta ,\xi )$ as in
getting of \eqref{1Geu}--\eqref{1derlbs}, we will prove that
each solution $u$ of \eqref{1eq} satisfies
\begin{equation}
\Big(\int_{a+\varepsilon }^{b}|u'|^{p}dt\Big)^{1/p}\geq
c\big(\frac{ 1}{\varepsilon }\big)^{s-1}\quad \text{for some }
\varepsilon \in (0,\varepsilon _{1}), \label{1Lpnorm}
\end{equation}
where $c>0$ and $\varepsilon _{1}>0$ will be also precised.
Immediately from \eqref{1Lpnorm} we obtain the lower bound for
the order of growth of the local singular behaviour of
$\|u'\|_{L^{p}(a+\varepsilon ,b)}$ as $\varepsilon \approx 0$,
that is
\begin{equation}
\limsup_{\varepsilon \to 0}\frac{\log
\big(\int_{a+\varepsilon }^{b}|u'|^{p}dt\big)^{1/p}}{\log
1/\varepsilon }\geq s-1. \label{1normlb}
\end{equation}
It is worth to mention that the local regular behaviour of
$\|u'\|_{L^{p}}$ is widely considered even in more dimensional
case, where $u$ is any solution of quasilinear elliptic equations
associated to $p$ -Laplacian. See for instance Rakotoson's paper
\cite{Rak90} and references therein.
Preceding results were obtained in the author's paper \cite{Pasic03}
but for the case of corresponding equation:
\begin{equation}
\begin{gathered}
-(|y'|^{p-2}y')'=f(t,y,y')\quad \text{in } (a,b), \\
y(a)=y(b)=0, \\
y\in W_{\rm loc}^{1,p}((a,b])\cap C([a,b]).
\end{gathered} \label{1ceq}
\end{equation}
In this paper we show how the methods presented in \cite{Pasic03} permit us
to obtain some new singular properties of the graph of solutions of the
variational inequality \eqref{1eq}. About some regular properties of
solutions of quasilinear elliptic variational inequalities, we refer reader
to \cite{FerPosRak99,Lio69,Pasic96,RakoTem87}.
About the fractal dimensions and their properties we refer to
\cite{Bou25,Falc99,Lem90,Matt95,Peit92,Trico82,Trico95}.
Finally, let us remark that the existence of at least one solution
$y$ of \eqref{1ceq} was discussed in
\cite[Appendix, p. 303-304]{Pasic03}, where the nonlinearity
$f(t,\eta ,\xi )$
satisfy related assumptions needed here to obtain
\eqref{1Geu}--\eqref{1normlb} (about the existence of continuous
solutions for the equations with singular nonlinearity see
\cite[Chapter 14]{ORegan}). Moreover, if for instance $\varphi
(a)=\psi (a)=0$ and if $\varphi $ is decreasing and convex on
$[a,b]$ and if $\psi $ is increasing and concave on $[a,b]$, and
if $f(t,\eta ,\xi )$ satisfies:
\begin{gather*}
f(t,\eta ,\xi )<0,\quad t\in (a,b),\; \eta >\psi (t) \text{
and }\xi \in \mathbb{R}, \\
f(t,\eta ,\xi )>0,\quad t\in (a,b),\; \eta <\varphi (t) \text{ and
}\xi \in \mathbb{R},
\end{gather*}
then each solution $y$ of the equation \eqref{1ceq} satisfies
$\varphi (t)\leq y(t)\leq \psi (t)$ in $[a,b]$. So in such case,
each solution of \eqref{1ceq} also satisfies the variational
inequality \eqref{1eq} and thus, the existence of solutions of
\eqref{1eq} in this case follows from the existence result of
the equation \eqref{1ceq}.
\section{Control of essential infimum and essential supremum
of solutions}
In this section, we present a method which plays an important role
in the proofs of the main results. It is so called the control of
$\mathop{\rm ess\,inf}$ and $\mathop{\rm ess\,sup}$ of
solutions introduced in \cite{KPZ99} and considered in
\cite{KPZ01} and \cite{KPZ02} to get some qualitative properties
of solutions of quasilinear elliptic equations and variational
inequalities. Here, we show that this method can be applied to
solutions of variational inequality \eqref{1eq} to derive
some consequences needed in the proofs of the main results.
\begin{lemma}[Control of ess sup] \label{lem2.1}
Let $(a_{2},b_{2})\subset \subset (a,b)$ be an open
interval. Let $\omega _{2}$ be an arbitrarily given real number
such that
\begin{equation}
\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\varphi <\omega _{2}<
\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\psi . \label{2thom}
\end{equation}
Let $J_{2}$ be a set defined by
$J_{2} = (\mathop{\rm ess\,inf} _{(a_{2},b_{2})}\varphi ,\omega
_{2})$ and let the Caratheodory function $f(t,\eta ,\xi )$
satisfy:
\begin{gather}
f(t,\eta ,\xi )\geq 0,\quad t\in (a_{2},b_{2}),\; \eta \in J_{2},\;
\xi \in \mathbb{R}, \label{2ftpoz} \\
\int_{A_{2}}\mathop{\rm ess\,inf}_{(\eta ,\xi )\in J_{2}\times
R}f(t,\eta ,\xi )dt
> \frac{c(p)}{(b_{2}-a_{2})^{p-1}}
\frac{(\mathop{\rm ess\,inf} _{(a_{2},b_{2})}\psi
-\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\varphi )^{p} }
{\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\psi -\omega _{2}}, \label{2ftstrog}
\end{gather}
where $c(p)=2[4(p-1)]^{p-1}$ and $A_{2}$ is a set defined by
\[
A_{2}=[a_{2}+\frac{1}{4}(b_{2}-a_{2}),b_{2}-\frac{1}{4}(b_{2}-a_{2})].
\]
Then for any solution $u$ of \eqref{1eq} there is a
$\sigma _{2}\in (a_{2},b_{2})$ such that
\begin{equation}
u(\sigma _{2})\geq \omega _{2}. \label{2con2}
\end{equation}
\end{lemma}
We will also need the dual result of Lemma \ref{lem2.1}.
\begin{lemma}[control of ess inf] \label{lem2.2}
Let $(a_{1},b_{1})\subset \subset (a,b)$ be an open
interval. Let $\theta _{1}$ be an arbitrarily given real number
such that
\begin{equation}
\mathop{\rm ess\,sup}_{(a_{1},b_{1})}\varphi <\theta _{1}<
\mathop{\rm ess\,sup}_{(a_{1},b_{1})}\psi .
\label{2thomdual}
\end{equation}
Let $J_{1}$ be a set defined by $J_{1}$ = $(\theta _{1},
\mathop{\rm ess\,sup}_{(a_{1},b_{1})}\psi )$ and let the
Caratheodory function $f(t,\eta ,\xi )$ satisfy:
\begin{gather}
f(t,\eta ,\xi )\leq 0,\quad t\in (a_{1},b_{1}),\; \eta \in J_{1},\;
\xi \in \mathbb{R}, \label{2ftneg} \\
\int_{A_{1}}\mathop{\rm ess\,sup}_{(\eta ,\xi )\in J_{1}\times
R}f(t,\eta ,\xi )dt
<-\frac{c(p)}{(b_{1}-a_{1})^{p-1}}\frac{(\mathop{\rm ess\,sup} _{(a_{1},b_{1})}
\psi -\mathop{\rm ess\,sup}_{(a_{1},b_{1})}\varphi )^{p} }{\theta
_{1}-\mathop{\rm ess\,sup}_{(a_{1},b_{1})}\varphi },
\label{2ftnegstrog}
\end{gather}
where $c(p)=2[4(p-1)]^{p-1}$ and $A_{1}$ is a set defined by
\[
A_{1}=[a_{1}+\frac{1}{4}(b_{1}-a_{1}),b_{1}-\frac{1}{4}(b_{1}-a_{1})].
\]
Then for any solution $u$ of \eqref{1eq} there is a $\sigma _{1}\in
(a_{1},b_{1})$ such that
\begin{equation}
u(\sigma _{1})\leq \theta _{1}. \label{2con1}
\end{equation}
\end{lemma}
Let us remark that the conditions \eqref{2thom} and \eqref{2thomdual}
will be easy fulfiled in Theorem \ref{thm3.3} below.
\begin{proof}[Proof of Lemma \ref{lem2.1}]
Let $t_{0}$ and $r$ be two real numbers defined as follows:
\[
t_{0}=\frac{a_{2}+b_{2}}{2},\quad r=\frac{1}{4}(b_{2}-a_{2}).
\]
Let $B_{r}=B_{r}(t_{0})$ denote a ball with radius $r>0$ centered at the
point $t_{0}$. Then we have
\[
B_{2r}=B_{2r}(t_{0})=(a_{2},b_{2}),\quad B_{r}=B_{r}(t_{0})=A_{2},
\]
where the set $A_{2}$ is appearing in \eqref{2ftstrog}. Since
$|B_{2r}|=4r=b_{2}-a_{2}$, where $|A|$ denotes the Lebesgue
measure of a set $A$, and using the preceding notations, the
hypotheses \eqref{2ftpoz} and \eqref{2ftstrog} can be
rewritten in the form:
\begin{gather}
f(t,\eta ,\xi )\geq 0,\quad t\in B_{2r},\; \eta \in J_{2},\; \xi \in
\mathbb{R}, \label{2ftpozproof}
\\
\int_{B_{r}}\mathop{\rm ess\,inf}_{(\eta ,\xi )\in J_{2}\times
R}f(t,\eta ,\xi )dt
>\frac{c(p)}{4^{p-1}}\frac{1}{r^{p-1}}\frac{(\mathop{\rm ess\,inf} _{B_{2r}}\psi -\mathop{\rm ess\,inf}_{B_{2r}}\varphi
)^{p}}{ \mathop{\rm ess\,inf}_{B_{2r}}\psi -\omega _{2}}.
\label{2ftstrogproof}
\end{gather}
Next, let $u$ be a solution of \eqref{1eq}. Let us suppose a
contrary statement to \eqref{2con2}, that is
\begin{equation}
u(t)<\omega _{2}\quad \text{for each }t\in B_{2r}. \label{2contr}
\end{equation}
Since $\varphi \leq u$ in $(a,b)$ and because of \eqref{2thom},
besides \eqref{2contr} we have also
\begin{equation}
\mathop{\rm ess\,inf}_{B_{2r}}\varphi \leq u(t)<\omega _{2}<
\mathop{\rm ess\,inf}_{B_{2r}}\psi \ \text{ for each }t\in
B_{2r}. \label{2contrth}
\end{equation}
Using $c(p)=2[4(p-1)]^{p-1}$ and $|B_{r}|=2r$, from
\eqref{2ftpozproof}--\ref{2contrth}, we get
\begin{gather}
f(t,u,u')\geq 0\quad \text{in }B_{2r}, \label{2ftpozbr}
\\
\int_{B_{r}}f(t,u,u')dt>(p-1)^{p-1}\frac{|B_{r}|}{r^{p}}\frac{(
\mathop{\rm ess\,inf}_{B_{2r}}\psi
-\mathop{\rm ess\,inf} _{B_{2r}}\varphi )^{p}}{\mathop{\rm ess\,inf}_{B_{2r}}
\psi -\omega _{2}} . \label{2ftstrogbr}
\end{gather}
Regarding \eqref{2ftpozbr} and \eqref{2ftstrogbr} we are
here in a very similar situation as in the proof
\cite[Theorem 5, p. 256]{KPZ01} or \cite[Theorem 4.1, p. 282]{Pasic03}.
In this direction, it is known that for any $c_{0}>1$ there exists a
function $\Phi \in C_{0}^{\infty }(\mathbb{R})$ , $0\leq \Phi \leq
1$ in $\mathbb{R}$ such that the following properties are
fulfilled, see \cite[Lemma 5, pp. 267]{KPZ01}:
\begin{equation}
\begin{gathered}
\Phi (t)=1,\; t\in B_{r}\quad \text{and}\quad
\Phi (t)=0,\; t\in \mathbb{R}\setminus B_{2r}, \\
\Phi (t)>0,\; t\in B_{2r}\quad \text{and}\quad
|\Phi '(t)|\leq \frac{c_{0}}{r },\; t\in \mathbb{R}.
\end{gathered} \label{2local}
\end{equation}
For any $c_{0}>1$, we take a test function
\[
v(t)=(\mathop{\rm ess\,inf}_{B_{2r}}\psi -u(t))\Phi
^{p}(t)+u(t),\quad t\in \mathbb{R}.
\]
With the help of \eqref{2contrth} we have that
$\mathop{\rm ess\,inf} _{B_{2r}}\psi -u(t)\geq 0$ in
$B_{2r}\,$and so, it is easy to check that
\[
v\in K(\varphi ,\psi )\quad \text{and}\quad
\mathop{\rm supp}(v-u)\subseteq B_{2r}\subset \subset (a,b),
\]
where the space $K(\varphi ,\psi )$ was defined in \eqref{1eq}.
Therefore, we may put in \eqref{1eq} this test function and we
obtain
\begin{align*}
-\int_{B_{2r}}|u'|^{p}\Phi ^{p}dt
&\geq -p\int_{B_{2r}}|u'|^{p-2}u'(\mathop{\rm ess\,inf}_{B_{2r}}\psi -u)\Phi
^{p-1}\Phi 'dt \\
&\quad + \int_{B_{2r}}f(t,u,u')(\mathop{\rm ess\,inf}_{B_{2r}}\psi
-u)\Phi ^{p}dt.
\end{align*}
Multiplying this inequality by $-1$ we get
\begin{equation}
\begin{aligned}
\int_{B_{2r}}|u'|^{p}\Phi ^{p}dt
&\leq p\int_{B_{2r}}|u'|^{p-1}\Phi ^{p-1}(\mathop{\rm ess\,inf}_{B_{2r}}
\psi -u)|\Phi '|dt \\
&\quad -\int_{B_{2r}}f(t,u,u')(\mathop{\rm ess\,inf}_{B_{2r}}\psi
-u)\Phi ^{p}dt.
\end{aligned}\label{2first}
\end{equation}
For the record, with the help of \eqref{2contrth} we also have:
\begin{equation}
\begin{gathered}
\mathop{\rm ess\,inf}_{B_{2r}}\psi -u(t)\leq \mathop{\rm ess\,inf}
_{B_{2r}}\psi -\mathop{\rm ess\,inf}_{B_{2r}}\varphi ,\; t\in B_{2r},
\\
\mathop{\rm ess\,inf}_{B_{2r}}\psi -u(t)\geq
\mathop{\rm ess\,inf} _{B_{2r}}\psi -\omega _{2},\; t\in B_{2r}.
\end{gathered}
\label{2ess}
\end{equation}
Using $(p-1)p'=p$ and $\delta _{1}(p\delta _{2})\leq \frac{d}{
p'}\delta _{1}^{p'}+(\frac{p}{d})^{p-1}\delta _{2}^{p}$ especially
for
\[
\delta _{1}=|u'|^{p-1}\Phi ^{p-1},\quad
\delta _{2}=(\mathop{\rm ess\,inf}_{B_{2r}}\psi -u)|\Phi '|,\quad
d=p',
\]
with the help of \eqref{2ftpozbr}, \eqref{2first} and
\eqref{2ess} we obtain
\begin{equation}
\begin{aligned}
0 &=\bigl[1-\frac{d}{p'}\bigl]\int_{B_{2r}}|u'|^{p}\Phi
^{p}dt\leq \bigl(\frac{p}{p'}\bigl)^{p-1}(\mathop{\rm ess\,inf} _{B_{2r}}\psi -\mathop{\rm ess\,inf}_{B_{2r}}\varphi
)^{p}\int_{B_{2r}}|\Phi '|^{p}dt \\
&\quad -(\mathop{\rm ess\,inf}_{B_{2r}}\psi -\omega
_{2})\int_{B_{r}}f(t,u,u')\Phi ^{p}dt. \label{2pom}
\end{aligned}
\end{equation}
Now, by means of \eqref{2local} we derive
\begin{align*}
0 &\leq \bigl(\frac{p}{p'}\bigl)^{p-1}(\mathop{\rm ess\,inf}
_{B_{2r}}\psi -\mathop{\rm ess\,inf}_{B_{2r}}\varphi
)^{p}|B_{2r}\setminus B_{r}|\bigl(\frac{c_{0}}{r}\bigl)^{p} \\
&\quad -(\mathop{\rm ess\,inf}_{B_{2r}}\psi -\omega
_{2})\int_{B_{r}}f(t,u,u')dt.
\end{align*}
Since $|B_{2r}\setminus B_{r}|=|B_{r}|$ and passing to the limit
as $c_{0}\to 1$ we obtain
\[
\int_{B_{r}}f(t,y,y')dt\leq
(p-1)^{p-1}\frac{|B_{r}|}{r^{p}}\frac{( \mathop{\rm ess\,inf}_{B_{2r}}\psi -\mathop{\rm ess\,inf} _{B_{2r}}\varphi
)^{p}}{\mathop{\rm ess\,inf}_{B_{2r}}\psi -\omega _{2}} .
\]
But, this inequality contradicts the assumption \eqref{2ftstrogbr} and so
the hypothesis \eqref{2contr} is not possible. Thus, the desired statement
\eqref{2con2} is proved.
\end{proof}
Analogously we can obtain the proof of Lemma \ref{lem2.2}.
In Section 5, we need to use slightly different versions of preceding lemmas.
\begin{lemma}[A version of Lemma \ref{lem2.1}] \label{lem2.3}
Let $(a_{2},b_{2})\subset \subset (a,b)$ be an open interval. Let
$\tilde{\theta}_{0}$, $\tilde{\omega}_{0}$ and $\omega _{2}$ be
three arbitrarily given real numbers such that
\begin{equation}
\tilde{\theta}_{0}\leq \mathop{\rm ess\,inf}_{(a_{2},b_{2})}\varphi
<\omega _{2}<\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\psi
\leq \tilde{ \omega}_{0}.
\label{2omle23}
\end{equation}
Let $J_{2}$ be a set defined by
$J_{2} = (\tilde{\theta}_{0},\omega_{2})$ and let the Caratheodory
function $f(t,\eta ,\xi )$ satisfy:
\begin{equation}
\begin{gathered}
f(t,\eta ,\xi )\geq 0,\quad t\in (a_{2},b_{2}),\; \eta \in J_{2},\; \xi \in
\mathbb{R}, \\
\int_{A_{2}}\mathop{\rm ess\,inf}_{(\eta ,\xi )\in J_{2}\times
R}f(t,\eta ,\xi )dt
>\frac{c(p)}{(b_{2}-a_{2})^{p-1}}\frac{(\tilde{\omega}_{0}-\tilde{\theta}
_{0})^{p}}{\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\psi -\omega
_{2}},
\end{gathered} \label{2ftsle23}
\end{equation}
where $c(p)=2[4(p-1)]^{p-1}$ and $A_{2}$ is a set defined by
\[
A_{2}=[a_{2}+\frac{1}{4}(b_{2}-a_{2}),b_{2}-\frac{1}{4}(b_{2}-a_{2})].
\]
Then for any solution $u$ of \eqref{1eq} there is a
$\sigma _{2}\in (a_{2},b_{2})$ such that
\begin{equation}
u(\sigma _{2})\geq \omega _{2}. \label{2jumple23}
\end{equation}
\end{lemma}
We will also need the dual result of Lemma \ref{lem2.3}.
\begin{lemma}[A version of Lemma \ref{lem2.2}] \label{lem2.4}
Let $(a_{1},b_{1})\subset \subset (a,b)$ be an open interval.
Let $\tilde{\theta}_{0}$, $\tilde{\omega}_{0}$ and $\theta _{1}$ be
three arbitrarily given real numbers such that
\[
\tilde{\theta}_{0}\leq \mathop{\rm ess\,sup}_{(a_{1},b_{1})}\varphi
<\theta _{1}<\mathop{\rm ess\,sup}_{(a_{1},b_{1})}\psi
\leq \tilde{ \omega}_{0}.
\]
Let $J_{1}$ be a set defined by
$J_{1} = (\theta _{1},\tilde{\omega} _{0})$ and let the Caratheodory
function $f(t,\eta ,\xi )$ satisfy:
\begin{gather*}
f(t,\eta ,\xi )\leq 0,\quad t\in (a_{1},b_{1}),\; \eta \in J_{1},\; \xi \in
\mathbb{R}, \\
\int_{A_{1}}\mathop{\rm ess\,sup}_{(\eta ,\xi )\in J_{1}\times
R}f(t,\eta ,\xi )dt
<-\frac{c(p)}{(b_{1}-a_{1})^{p-1}}\frac{(\tilde{\omega}_{0}-\tilde{\theta}
_{0})^{p}}{\theta _{1}-\mathop{\rm ess\,sup}_{(a_{1},b_{1})}\varphi },
\end{gather*}
where $c(p)=2[4(p-1)]^{p-1}$ and $A_{1}$ is a set defined by
\[
A_{1}=[a_{1}+\frac{1}{4}(b_{1}-a_{1}),\; b_{1}-\frac{1}{4}(b_{1}-a_{1})].
\]
Then for any solution $u$ of \eqref{1eq} there is a $\sigma _{1}\in
(a_{1},b_{1})$ such that
\[
u(\sigma _{1})\leq \theta _{1}.
\]
\end{lemma}
The proof of Lemma \ref{lem2.3} can be done in analogous way as we did the
proof of Lemma \ref{lem2.1}. At the end of this section,
we give an example for the
nonlinearity $f(t,\eta ,\xi )$ which satisfies the assumptions of
Lemma \ref{lem2.1} and Lemma \ref{lem2.2} together.
\begin{example} \label{exa2.5} \rm
Let $a0$
such that for each $\varepsilon \in (0,\varepsilon _{0})$} \\
&\text{there is a $k(\varepsilon )\in \mathbf{N}$ such that
$a_{j-1}-a_{j}\leq \varepsilon /2$ for each $j\geq k(\varepsilon )$.}
\end{aligned} \label{3keps}
\end{equation}
Let $\theta (t)$ and $\omega (t)$ be two measurable and bounded real
functions on $[a,b]$, $\theta (t)\leq \omega (t)$, $t\in [a,b]$, such that
\begin{equation}
\begin{gathered}
\mathop{\rm ess\,inf}_{(a_{2k+2},a_{2k+1})}\theta \geq
\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta , \\
\mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\omega \leq
\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega ,\quad k\geq 1.
\end{gathered} \label{3to}
\end{equation}
Let $u$ be a continuous function on $(a,b]$ such that there is a
sequence $\sigma _{k}\in (a_{k},a_{k-1})$ satisfying
\[
u(\sigma _{2k})\geq \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega
\quad\text{and}\quad u(\sigma_{2k+1})
\leq \mathop{\rm ess\,inf}_{_{(a_{2k+1},a_{2k})}}\theta ,\ k\geq 1.
\]
Then
\begin{equation}
|G_{\varepsilon }(u)|\geq \int_{a}^{a_{k(\varepsilon )}}(\omega (t)-\theta
(t))dt\ \ \text{for each }\varepsilon \in (0,\varepsilon _{0}),
\label{3Geu}
\end{equation}
where $k(\varepsilon )$ and $\varepsilon _{0}$ are appearing
in \eqref{3keps}. Moreover, if for a real number $c\in (a,b)$
there is an $\varepsilon _{c}\in (0,\varepsilon _{0})$ such that
$a_{k(\varepsilon )-1}\in (a,c)$ for each
$\varepsilon \in (0,\varepsilon _{c})$ then we have
\begin{equation}
|G_{\varepsilon }(u|_{[a,c]})|\geq \int_{a}^{a_{k(\varepsilon )}}(\omega
(t)-\theta (t))dt\ \ \text{for each }\varepsilon \in (0,\varepsilon
_{c}). \label{3GeuR}
\end{equation}
\end{lemma}
Let us remark that the condition \eqref{3to} can be easy
satisfied if for instance $\theta (t)$ is decreasing and $\omega
(t)$ is increasing on $[a,b]$ . The proof of Lemma \ref{lem3.1} is omitted
because it is very similar to the proof of
\cite[Lemma 2.1, p. 271]{Pasic03}.
Next, we want to find some conditions on $f(t,\eta ,\xi )$ such that each
solution $u$ of \eqref{1eq} admits rapid oscillations in the sense of
Lemma \ref{lem3.1}.
\begin{lemma} \label{lem3.2}
Let $a_{k}$ be a decreasing sequence of real
numbers from interval $(a,b)$ satisfying \eqref{3keps}.
Let for each $k\geq 1$ the obstacles $\varphi (t)$ and $\psi
(t)$ satisfy:
\begin{equation}
\begin{gathered}
\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\varphi
<\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2<\mathop{\rm ess\,inf}
_{(a_{2k},a_{2k-1})}\psi , \\
\mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\psi
>\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\varphi
/2>\mathop{\rm ess\,sup} _{(a_{2k+1},a_{2k})}\varphi .
\end{gathered} \label{3thom}
\end{equation}
Let for each $k\geq 1$ the sets $J_{k}$ be defined by:
\begin{gather*}
J_{2k}=(\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\varphi ,
\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2), \\
J_{2k+1}=(\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\varphi
/2, \mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\psi ).
\end{gather*}
Next, let for each $k\geq 1$ the Caratheodory function $f(t,\eta ,\xi) $
satisfy:
\begin{gather}
f(t,\eta ,\xi )\geq 0,\quad t\in (a_{2k},a_{2k-1}),\; \eta \in J_{2k},\;
\xi \in \mathbb{R}, \label{3ftpoz} \\
\begin{aligned}
&\int_{A_{2k}}\mathop{\rm ess\,inf}_{(\eta ,\xi )\in J_{2k}\times
R}f(t,\eta ,\xi )dt\\
&>\frac{c(p)}{(a_{2k-1}-a_{2k})^{p-1}}
\frac{(\mathop{\rm ess\,inf} _{(a_{2k},a_{2k-1})}\psi
-\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})} \varphi )^{p}}
{\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\psi
- \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2},
\end{aligned}\label{3ftstrogpoz}
\end{gather}
and
\begin{gather}
f(t,\eta ,\xi )\leq 0,\quad t\in (a_{2k+1},a_{2k}),\; \eta \in J_{2k+1},\;
\xi \in \mathbb{R}, \label{3ftneg} \\
\begin{aligned}
&\int_{A_{2k+1}}\mathop{\rm ess\,sup}_{(\eta ,\xi )\in J_{2k+1}\times
R}f(t,\eta ,\xi )dt\\
&<-\frac{c(p)}{(a_{2k}-a_{2k+1})^{p-1}}
\frac{(\mathop{\rm ess\,sup} _{(a_{2k+1},a_{2k})}\psi
-\mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})} \varphi )^{p}}
{\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\varphi /2
- \mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\varphi },
\end{aligned} \label{3ftstrogneg}
\end{gather}
where $c(p)=2[4(p-1)]^{p-1}$ and $A_{k}$ is a family of sets
defined by
\[
A_{k}=[a_{k}+\frac{1}{4}(a_{k-1}-a_{k}),\,
a_{k-1}-\frac{1}{4} (a_{k-1}-a_{k})],\ k\geq 1.
\]
Then for any solution $u$ of \eqref{1eq} there is a sequence $\sigma
_{k}\in (a_{k},a_{k-1})$ which satisfies
\begin{equation}
u(\sigma _{2k})\geq \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2
\quad\text{and}\quad
u(\sigma _{2k+1})\leq \mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}
\varphi /2,\ k\geq 1. \label{3jump}
\end{equation}
\end{lemma}
\begin{proof} Let $k$ be a fixed natural number, $k\geq 1$, and let
\[
\omega _{2}=\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2
\quad \text{and} \quad
\theta _{1}= \mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\varphi /2.
\]
Regarding to the hypotheses \eqref{3thom} and
\eqref{3ftpoz}--\eqref{3ftstrogneg}, it is clear that the
assumptions of Lemma \ref{lem2.1} and Lemma \ref{lem2.2}
are satisfied on the
intervals $[a_{2k},a_{2k-1}]$ and $[a_{2k+1},a_{2k}]$
respectively. Therefore, we may use these two lemmas and so, there
is a $\sigma _{2}=\sigma _{2k}\in (a_{2k},a_{2k-1})$ and $\sigma
_{1}=\sigma _{2k+1}\in (a_{2k+1},a_{2k})$ satisfying
\eqref{2con2} and \eqref{2con1} respectively. Since $k$ is
arbitrarily fixed, it implies the existence of a sequence $\sigma
_{k}\in (a_{k+1},a_{k})$ which satisfies the desired
condition \eqref{3jump}.
\end{proof}
Combining the preceding two lemmas we derive some new metric properties for
solutions of \eqref{1eq}. It is the subject of the following result.
\begin{theorem} \label{thm3.3}
For arbitrarily given real number $s\in (1,2)$, let the
sequence $a_{k}$ and the obstacles $\varphi $ and $\psi $ be given by:
\begin{equation}
\begin{gathered}
a_{k}=a+\frac{b-a}{2}(\frac{1}{k})^{1/\beta },\quad k\geq 1, \\
\varphi (t)=-2(t-a)\quad \text{and}\quad
\psi (t)=2(t-a),\; t\in (a,b),
\end{gathered} \label{3akto}
\end{equation}
where $\beta $ satisfies $1<\beta <\infty$ and $\beta =\frac{s}{2-s}$.
If the Caratheodory function $f(t,\eta ,\xi )$ satisfies
\eqref{3ftpoz}--\eqref{3ftstrogneg}
in respect to such $(\varphi ,\psi ,a_{k})$, then each solution $u$
of \eqref{1eq} satisfies:
\begin{gather}
|G_{\varepsilon }(u)|\geq \frac{1}{2^{6}}(b-a)^{s}\varepsilon ^{2-s}\quad
\text{for each } \varepsilon \in (0,\varepsilon _{0}=\frac{b-a}{\beta }),
\label{3GeuTh} \\
|G_{\varepsilon }(u|_{[a,c]})|\geq \frac{1}{2^{6}}(c-a)^{s}\varepsilon
^{2-s}\quad \text{for each } \varepsilon \in (0,\varepsilon _{c}),
\label{3GeuRTh} \\
\dim _{M}G(u)\geq s\quad \text{and}\quad
M^{s}(G((y))\geq \frac{1}{2^{7}} (b-a)^{s}, \label{3DimTh} \\
M^{s}(G(u)\cap B_{r}(a,u(a)))\geq \frac{1}{2^{7}}(\frac{r}{\sqrt{5}})^{s}
\quad \text{for any } r\in (0,b-a), \label{3MsTh} %\label{3DsTh}
\end{gather}
where $\varepsilon _{c}=\min \{\varepsilon _{0},\frac{(c-a)^{\beta +1}
}{\beta (b-a)^{\beta }}\}$.
\end{theorem}
\begin{proof} The proof is done in a few steps.
\emph{Proof of \eqref{3GeuTh}}.
It is not difficult to check
see the proof of \cite[Corollary 5.2, p. 289]{Pasic03}, that the
sequence $a_{k}$ given in \eqref{3akto} satisfies the hypothesis
\eqref{3keps} in respect to $\varepsilon _{0}$ and
$k(\varepsilon )$ determined by
\begin{equation}
c_{0}\left( \frac{1}{\varepsilon }\right) ^{\frac{\beta }{\beta +1}}\leq
k(\varepsilon )\leq 2c_{0}\left( \frac{1}{\varepsilon }\right) ^{\frac{\beta
}{\beta +1}}\ \ \text{ for each }\varepsilon \in (0,\varepsilon _{0}), \\
\label{3eps0}
\end{equation}
where $c_{0}=2\left( \frac{b-a}{\beta }\right) ^{\frac{\beta}{\beta +1 }}$
and $\varepsilon _{0}=\frac{b-a}{\beta }$.
Let us remark that double inequalities in \eqref{3eps0} is
needed to ensure $k(\varepsilon )\in \mathbf{N}$. Also, it is
clear that the obstacles $\varphi $ and $\psi $ given in
\eqref{3akto} satisfy the hypothesis \eqref{3thom}. Thus, the
assumptions of Lemma \ref{lem3.2} are fulfilled and therefore, we have that
each solution $u$ of \eqref{1eq} has rapid oscillations in the
sense of \eqref{3jump}. Moreover, it implies that each solution
$u$ of \eqref{1eq} satisfies the main assumption of Lemma \ref{lem3.1},
where $\omega =\psi /2$ and $\theta =\varphi /2$. So, we obtain
\[
|G_{\varepsilon }(u)|\geq \frac{1}{2}\int_{a}^{a_{k(\varepsilon )}}(\psi
(t)-\varphi (t))dt\ \ \text{ for each }\varepsilon \in (0,\varepsilon _{0}).
\]
Putting the data from \eqref{3akto} in the right hand side of the
preceding inequality, we get
\begin{equation}
|G_{\varepsilon }(u)|\geq \int_{a}^{a_{k(\varepsilon
)}}2(t-a)dt=\big( \frac{b-a}{2}\big) ^{2}\big(
\frac{1}{k(\varepsilon )}\big) ^{2/ \beta }\quad
\text{for each }\varepsilon \in (0,\varepsilon _{0}). \label{3start}
\end{equation}
Let us remark that from the left inequality in \eqref{3eps0} we have in
particular
\[
\frac{1}{k(\varepsilon )}\geq \frac{1}{4}\big( \frac{\beta
}{b-a}\big) ^{ \frac{\beta }{\beta +1}}\varepsilon ^{\frac{\beta
}{\beta +1}}\ \text{ for each }\varepsilon \in (0,\varepsilon
_{0}).
\]
Putting this inequality in \eqref{3start}, for any $\varepsilon
\in (0,\varepsilon _{0})$, we get
\[
|G_{\varepsilon }(u)|\geq \frac{(b-a)^{2}}{4}\left(
\frac{1}{4}\right) ^{ \frac{2}{\beta }}\frac{\beta
^{\frac{2}{\beta +1}}}{(b-a)^{\frac{2}{\beta +1} }}\varepsilon
^{\frac{2}{\beta +1}}\geq \frac{1}{2^{6}}(b-a)^{s}\varepsilon
^{2-s},
\]
where we have used that $\beta >1$ and $2\beta /(\beta +1)=s$.
Thus, we have
proved the inequality \eqref{3GeuTh}.
\emph{Proof of \eqref{3GeuRTh}}. For $c\in (a,b)$,
let
\begin{equation}
\varepsilon _{c}=\min \{\varepsilon _{0}=\frac{b-a}{\beta
},\frac{1}{\beta } \frac{(c-a)^{\beta +1}}{(b-a)^{\beta }}\}.
\label{3epsc}
\end{equation}
It is easy to check that for any $c\in (a,b)$ the number
$\varepsilon _{c}$ given by \eqref{3epsc} satisfies
\[
a_{k(\varepsilon )-1}\in (a,c)\quad \text{for each }
\varepsilon \in (0,\varepsilon _{c}),
\]
where the sequence $a_{k}$ is given in \eqref{3akto} and the
number $k(\varepsilon )$ is given in \eqref{3eps0}. Therefore,
we may apply Lemma \ref{lem3.1} again and so, for each $c\in (a,b)$ and for
any solution $u$ of \eqref{1eq} we have
\[
|G_{\varepsilon }(u|_{[a,c]})|\geq \frac{1}{2}\int_{a}^{a_{k(\varepsilon
)}}(\psi (t)-\varphi (t))dt\ \ \text{ for each }\varepsilon \in
(0,\varepsilon _{c}).
\]
Putting the data from \eqref{3akto} in this inequality and using
the same calculation as in the proof of \eqref{3GeuTh} we prove
\eqref{3GeuRTh}.
\emph{Proof of \eqref{3DimTh}}.
According to the definition of $\dim_{M}G(u)$, from \eqref{3GeuTh}
immediately follows that
\begin{align*}
\dim _{M}G(u) &=\limsup_{\varepsilon \to 0}\big(
2-\frac{\log |G_{\varepsilon }(u)|}{\log \varepsilon }\big) \\
&\geq \limsup_{\varepsilon \to 0}\big( 2-\frac{
\log [\varepsilon ^{2-s}(b-a)^{s}/2^{6}]}{\log \varepsilon }]\big) \\
&=\limsup_{\varepsilon \to 0}\big( 2-(2-s)\frac{ \log
\varepsilon }{\log \varepsilon }-\frac{\log
[(b-a)^{s}/2^{6}]}{\log \varepsilon }\big) =s.
\end{align*}
It proves the first inequality in \eqref{3DimTh}. Also, according
to the definition of $M^{s}(G(u))$, from \eqref{3GeuTh} we get:
\begin{align*}
M^{s}(G(u))
&=\limsup_{\varepsilon \to 0}(2\varepsilon
)^{s-2}|G_{\varepsilon }(u)|\geq \limsup_{\varepsilon \to
0}(2\varepsilon )^{s-2}[\frac{(b-a)^{s}}{2^{6}}\varepsilon ^{2-s}] \\
&=2^{s-2}\frac{(b-a)^{s}}{2^{6}}\limsup_{\varepsilon \to
0}(\varepsilon ^{s-2}\varepsilon ^{2-s})\geq \frac{1}{2^{7}}(b-a)^{s},
\end{align*}
which proves the second inequality in \eqref{3DimTh}.
\emph{Proof of \eqref{3MsTh}}.
At the first, since $u\in K(\varphi ,\psi )$
we have in particular that
\[
\varphi (t)\leq u(t)\leq \psi (t),t\in [a,b]\quad \text{and}\quad
u(a)=0.
\]
Making intersections of $\varphi (t)=-2(t-a)$ and
$\psi (t)=2(t-a)$ with $B_{r}(a,0)$, it is easy to see that
\[
G(u|_{[a,a+\frac{r}{\sqrt{5}}]})\subseteq G(u)\cap B_{r}(a,0)\quad
\text{for any }r\in (0,\sqrt{5}(b-a)),
\]
and so, we have
\begin{equation}
M^{s}(G(u|_{[a,a+\frac{r}{\sqrt{5}}]}))\leq M^{s}(G(u)\cap
B_{r}(a,0)) \quad \text{for any }r\in (0,(b-a)). \label{3MsBr}
\end{equation}
On the other hand, using \eqref{3GeuRTh} for
$c=a+\frac{r}{\sqrt{5}}$, we get
\[
M^{s}(G(u|_{[a,a+\frac{r}{\sqrt{5}}]}))\geq
\frac{1}{2^{7}}(\frac{r}{\sqrt{5} })^{s}\quad
\text{ for any }r\in (0,b-a).
\]
Combining this inequality with \eqref{3MsBr} we get the proof of
\eqref{3MsTh}.
Thus, we have proved all statements of Theorem \ref{thm3.3}.
\end{proof}
At the end of this section, we give an example of such a class of the
nonlinearity $f(t,\eta ,\xi )$ which satisfies the assumptions of
Theorem \ref{thm3.3}.
\begin{example} \label{exa3.4} \rm
In order to simplify the notation, let
$\tilde{\theta}_{2k+1}$, $\tilde{\omega}_{2k+1}$,
$\tilde{\theta}_{2k}$, $\tilde{\omega}_{2k}$, $\theta _{2k+1}$,
and $\omega _{2k}$ be defined by:
\begin{gather*}
\tilde{\theta}_{2k}=\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\varphi,\quad
\tilde{\omega}_{2k}=\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\psi ,\quad
\omega _{2k}=\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})} \psi /2, \\
\tilde{\theta}_{2k+1}=\mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})} \varphi ,\quad
\tilde{\omega}_{2k+1}=\mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\psi ,\quad
\theta_{2k+1}=\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\varphi /2.
\end{gather*}
Let $f=f(t,\eta ,\xi )$ be a Caratheodory function
\begin{align*}
f &=\frac{\pi c(p)}{\sin \frac{\pi }{4}}\sum_{k=1}^{\infty
}\bigl[(\tilde{ \omega}_{2k}-\tilde{\theta}_{2k})^{p}\frac{(\eta
-\tilde{\omega}_{2k})^{-}}{( \tilde{\omega}_{2k}-\omega
_{2k})^{2}}\frac{\sin (\frac{\pi }{a_{2k-1}-a_{2k}
}(t-a_{2k}))}{(a_{2k-1}-a_{2k})^{p}}K_{[a_{2k},a_{2k-1}]}(t)
\\
&\quad -(\tilde{\omega}_{2k+1}-\tilde{\theta}_{2k+1})^{p}\frac{(\eta
-\tilde{ \theta}_{2k+1})^{+}}{(\theta
_{2k+1}-\tilde{\theta}_{2k+1})^{2}}\frac{\sin ( \frac{\pi
}{a_{2k}-a_{2k+1}}(t-a_{2k-1}))}{(a_{2k}-a_{2k+1})^{p}}
K_{[a_{2k+1},a_{2k}]}(t)\bigl],
\end{align*}
where $c(p)$ is appearing in \eqref{3ftstrogpoz} and
\eqref{3ftstrogneg} , and where $K_{A}(t)$ denotes as usually
the characteristic function of a set $A$. Also, $\eta ^{-}=\max
\{0,-\eta \}$ and $\eta ^{+}=\max \{0,\eta \}$. It is not
difficult to check that $f(t,\eta ,\xi )$ is continuous in all its
variables and that $f(t,\eta ,\xi )$ satisfies the
hypotheses \eqref{3ftpoz}--\eqref{3ftstrogneg}.
\end{example}
\section{Lower bounds for $\dim _{M}G(u')$ and $M^{s}(G(u'))$}
In this section, the inequalities \eqref{1derlb} and \eqref{1derlbs}
will be verified. As the first, we give a discrete version of
Lemma \ref{lem3.1},
which is a modification of \cite[Lemma 6.3, p. 291]{Pasic03}.
\begin{lemma} \label{lem4.1}
Let $\sigma _{k}$ be a decreasing sequence of real numbers
from interval $(a,b)$ satisfying
\begin{equation}
\begin{aligned}
&\text{$\sigma _{k}\searrow a$ and there is an $\varepsilon _{0}>0$
such that for each $\varepsilon \in (0,\varepsilon _{0})$}\\
&\text{there is a $k(\varepsilon )\in \mathbf{N}$ such that
$\sigma _{j-1}-\sigma _{j}\leq \varepsilon /2$
for each $j\geq k(\varepsilon )$.}
\end{aligned} \label{4sigk}
\end{equation}
Let $\delta _{k}$ be a sequence of real numbers such that
\[
\delta _{2k+1}>0\text{ and }\delta _{2k}<0,\ k\geq 1.
\]
Let $z$ be a continuous function on $(a,b]$ for which there is a
sequence $s_{k}\in (\sigma _{k},\sigma _{k-1})$ such that
\[
z(s_{2k+1})\geq \delta _{2k+1}\quad \text{and}\quad
z(s_{2k})\leq \delta _{2k},\quad k\geq 1.
\]
Then there holds true
\[
|G_{\varepsilon }(z)|\geq \sum_{k=k(\varepsilon )}^{\infty }\delta
_{2k+1}(\sigma _{2k}-\sigma _{2k+1})\quad
\text{for each}\quad \varepsilon \in (0,\varepsilon _{0}),
\]
where $k(\varepsilon )$ and $\varepsilon _{0}$ are appearing in
\eqref{4sigk}.
\end{lemma}
The proof of the lemma above is exactly the same as the proof of
\cite[Lemma 2.1, p. 271]{Pasic03}.
As a basic result, we need the following lemma on the asymptotic
behaviour of $|G_{\varepsilon }(u')|$ as $\varepsilon \approx 0$,
where $u'$ is the derivative in the classical sense of any smooth
enough real function $u$.
\begin{lemma} \label{lem4.2}
Let $a_{k}$ be a decreasing sequence of real
numbers from interval $(a,b)$ satisfying \eqref{3keps}. Let
$\omega _{2k}$ and $\theta _{2k+1}$ be two sequences of real
numbers satisfying
\begin{equation}
\omega _{2k}>\min \{\theta _{2k+1},\theta _{2k-1}\},\quad k\geq 1.
\end{equation}
Let $u$ be a real function, $u\in C((a,b])\cap C^{1}(a,b)$,
for which there is a sequence $\sigma _{k}\in (a_{k},a_{k-1})$
such that
\begin{equation}
u(\sigma _{2k})\geq \omega _{2k}\quad\text{and}\quad
u(\sigma_{2k+1})\leq \theta _{2k+1},\quad k\geq 1. \label{4jump}
\end{equation}
Then
\begin{equation}
|G_{\varepsilon }(u')|\geq \sum_{k=k(\varepsilon /2)}^{\infty
}(\omega _{2k}-\theta _{2k+1})\quad\text{for each }
\varepsilon \in (0,\varepsilon _{0}), \label{4Geud}
\end{equation}
where $k(\varepsilon )$ and $\varepsilon _{0}$ are defined in
\eqref{3keps}.
\end{lemma}
\begin{proof}
Lagrange's mean value theorem, applied on
the interval $(\sigma _{k},\sigma _{k-1})$, where the sequence
$\sigma _{k}$ is defined in \eqref{4jump}, we get the existence
of a sequence $s_{k}\in (\sigma _{k},\sigma _{k-1})$, $k\geq 1$
such that
\begin{equation}
u'(s_{2k+1})=\frac{u(\sigma _{2k})-u(\sigma _{2k+1})}{\sigma
_{2k}-\sigma _{2k+1}}, \quad
u'(s_{2k})=\frac{u(\sigma
_{2k-1})-u(\sigma _{2k})}{\sigma _{2k-1}-\sigma _{2k}}.
\label{4Lagrang}
\end{equation}
Using \eqref{4jump} and the notation:
\[
z(t)=u'(t),\ t\in (a,b), \quad
\delta _{2k+1}=\frac{\omega _{2k}-\theta _{2k+1}}{\sigma _{2k}-\sigma
_{2k+1}},\quad
\delta _{2k}=\frac{\theta _{2k-1}-\omega _{2k}}{\sigma_{2k-1}-\sigma _{2k}},
\]
the statement \eqref{4Lagrang} can be rewritten in the form:
there is $s_{k}\in (\sigma _{k},\sigma _{k-1})$, $k\geq 1$, such
that
\begin{equation}
z(s_{2k+1})\geq \delta _{2k+1}>0\quad \text{and}\quad
z(s_{2k})\leq \delta _{2k}<0,\quad k\geq 1. \label{4pom1}
\end{equation}
On the other hand, it is easy to see that the sequence
$\sigma_{k}$ just like $a_{k}$ satisfies a very similar condition to
\eqref{3keps}; that is,
\begin{equation}
\sigma _{k}\searrow a\quad\text{and}\quad
\sigma _{j-1}-\sigma _{j}\leq \varepsilon/2\quad
\text{for each } j\geq k(\frac{\varepsilon }{2}),\;
\varepsilon \in (0,\varepsilon _{0}), \label{4pom2}
\end{equation}
where $k(\varepsilon )$ and $\varepsilon _{0}$ are exactly the
same as in \eqref{3keps}. Now, by means of \eqref{4pom1} and
\eqref{4pom2}, we have that the function $z$ satisfies the
assumptions of Lemma \ref{lem4.1} and so, we get:
\begin{align*}
|G_{\varepsilon }(u')|
&=|G_{\varepsilon }(z)|
\geq \sum_{k=k(\varepsilon /2)}^{\infty }\delta _{2k+1}(\sigma
_{2k}-\sigma_{2k+1}) \\
&=\sum_{k=k(\varepsilon /2)}^{\infty }\frac{(\omega _{2k}-\theta _{2k+1})}{
\sigma _{2k}-\sigma _{2k+1}}(\sigma _{2k}-\sigma _{2k+1}) \\
&=\sum_{k=k(\varepsilon /2)}^{\infty }(\omega _{2k}-\theta _{2k+1})\quad
\text{for each }\varepsilon \in (0,\varepsilon _{0}).
\end{align*}
Thus, Lemma \ref{lem4.2} is proved.
\end{proof}
Next, we give the main result of this section.
\begin{theorem} \label{thm4.3}
Let the hypotheses of Theorem \ref{thm3.3} be still
assumed; that is: for arbitrarily given real number $s\in (1,2)$,
let the sequence $a_{k}$ and the obstacles $\varphi $ and $\psi $
be given by \eqref{3akto}, and let the Caratheodory function
$f(t,\eta ,\xi )$ satisfy \eqref{3ftpoz}--\eqref{3ftstrogneg} in
respect to such $(\varphi ,\psi ,a_{k})$. Then each solution $u$
of \eqref{1eq} satisfies:
\begin{gather}
|G_{\varepsilon }(u')|\geq \frac{\sqrt{2}}{2^{4}}(b-a)^{s/2}\varepsilon
^{1-s/2}\quad\text{for each } \varepsilon \in
(0,\varepsilon _{0}=\frac{b-a}{ \beta }), \label{4GeudTh} \\
\dim _{M}G(u')\geq 1+\frac{s}{2}\quad \text{and}\quad
M^{1+s/2}(G(u'))\geq \frac{1}{2^{4}}(b-a)^{s/2}. \label{4DimTh}
\end{gather}
\end{theorem}
The proof of the above theorem can be done with similar arguments as
in \cite[Theorem 3.4 and Corollary 3.5]{Pasic-Zupan}.
\section{Full control of ess inf and
ess sup of solutions}
In contrast to the method of control of $\mathop{\rm ess\,inf}$
and $\mathop{\mathrm{ess\sup}}$ of solutions of
\eqref{1eq} which was presented in Section 2, here we involve on
the nonlinearity $f(t,\eta ,\xi )$ slightly stronger conditions
than \eqref{2ftstrog} and \eqref{2ftnegstrog} to obtain some
stronger conclusions than \eqref{2con2} and \eqref{2con1}.
More precisely, for any solution $u$ of \eqref{1eq} we need to
estimate from below the measure of sets where
$\mathop{\rm ess\,inf}u$ and $\mathop{\mathrm{ess\sup}}u$ are
exceeded. It will play an important role in the following
section, where the inequality \eqref{1Lpnorm} and
\eqref{1normlb} will be proved. The so called full control of
$\mathop{\rm ess\,inf}$ and $\mathop{\mathrm{ess\sup}}$ of
solutions of corresponding equation \eqref{1ceq} was considered
in \cite[Section 4] {Pasic03}.
Here, it is the subject of the following two lemmas.
\begin{lemma} \label{lem5.1}
Let $(a_{2},b_{2})\subset \subset (a,b)$ be an open
interval. Let $\omega _{2}$ be an arbitrarily given real number such that
\begin{equation}
\mathop{\rm ess\,sup}_{(a_{2},b_{2})}\varphi <\omega _{2}<
\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\psi . \label{5omle}
\end{equation}
Let $J_{2}$ be a set defined by
$J_{2} =(\mathop{\rm ess\,inf} _{(a_{2},b_{2})}\varphi ,\omega _{2})$
and let the Caratheodory function $f(t,\eta ,\xi )$
satisfy:
\begin{gather}
f(t,\eta ,\xi )\geq 0,\quad t\in (a_{2},b_{2}),\; \eta \in J_{2},\;
\xi \in \mathbb{R}, \label{5fple} \\
\mathop{\rm ess\,inf}_{t\in A_{2}}f(t,\eta ,\xi )
>\frac{c(p)}{(b_{2}-a_{2})^{p}}\frac{(\mathop{\rm ess\,sup}
_{(a_{2},b_{2})}\psi -\mathop{\rm ess\,inf}_{(a_{2},b_{2})}
\varphi )^{p} }{\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\psi -\omega _{2}},\quad
\eta \in J_{2},\; \xi \in \mathbb{R}, \label{5fsle}
\end{gather}
where $c(p)=2(16^{p})(p-1)^{p-1}$ and $A_{2}$ is a set defined by
\[
A_{2}=[a_{2}+\frac{1}{16}(b_{2}-a_{2}),\, b_{2}-\frac{1}{16}(b_{2}-a_{2})].
\]
Then for any solution $u$ of \eqref{1eq} we have
\begin{equation}
u(t)\geq \omega _{2}\quad \text{for each }
t\in [a_{2}+\frac{1}{4} (b_{2}-a_{2}),b_{2}-\frac{1}{4}(b_{2}-a_{2})].
\label{5jump2}
\end{equation}
\end{lemma}
The dual result of Lemma \ref{lem5.1} is the following.
\begin{lemma} \label{lem5.2}
Let $(a_{1},b_{1})\subset \subset (a,b)$ be an open
interval. Let $\theta _{1}$ be an arbitrarily given real number such that
\[
\mathop{\rm ess\,sup}_{(a_{1},b_{1})}\varphi <\theta _{1}<
\mathop{\rm ess\,inf}_{(a_{1},b_{1})}\psi .
\]
Let $J_{1}$ be a set defined by $J_{1}$ = $(\theta _{1},
\mathop{\rm ess\,sup}_{(a_{1},b_{1})}\psi )$ and let the
Caratheodory function $f(t,\eta ,\xi )$ satisfy:
\begin{gather*}
f(t,\eta ,\xi )\leq 0,\quad t\in (a_{1},b_{1}),\;
\eta \in J_{1},\; \xi \in \mathbb{R}, \\
\mathop{\rm ess\,sup}_{t\in A_{1}}f(t,\eta ,\xi )
<-\frac{c(p)}{(b_{1}-a_{1})^{p}}
\frac{(\mathop{\rm ess\,sup} _{(a_{1},b_{1})}\psi
-\mathop{\rm ess\,inf}_{(a_{1},b_{1})}\varphi )^{p} }{\theta
_{1}-\mathop{\rm ess\,sup}_{(a_{1},b_{1})}\varphi },\quad
\eta \in J_{1},\; \xi \in \mathbb{R},
\end{gather*}
where $c(p)=2(16^{p})(p-1)^{p-1}$ and $A_{1}$ is a set defined by
\[
A_{1}=[a_{1}+\frac{1}{16}(b_{1}-a_{1}),\, b_{1}-\frac{1}{16}(b_{1}-a_{1})].
\]
Then for any solution $u$ of \eqref{1eq} we have
\begin{equation}
u(t)\leq \theta _{1}\quad \text{for each }
t\in [a_{1}+\frac{1}{4} (b_{1}-a_{1}),b_{1}-\frac{1}{4}(b_{1}-a_{1})].
\label{5jump1}
\end{equation}
\end{lemma}
The above lemma can be proved analogously as in
the proof of Lemma \ref{lem5.1} to be shown below.
For the proof we use the following two
propositions that will be shown later.
\begin{proposition} \label{prop5.2}
Let $(c,d)\subseteq (a_{2},b_{2})$ be an open
interval. Let $\omega _{2}$ be an arbitrarily given real number such that
\begin{equation}
\mathop{\rm ess\,sup}_{(c,d)}\varphi <\omega _{2}
<\mathop{\rm ess\,inf}_{(c,d)}\psi . \label{5omcd}
\end{equation}
Let $J_{2}$ be a set defined by
$J_{2}=(\mathop{\rm ess\,inf} _{(c,d)}\varphi ,\omega _{2})$ and
let the Caratheodory function $f(t,\eta ,\xi )$ satisfy
\begin{equation}
f(t,\eta ,\xi )\geq 0,\quad t\in (c,d),\; \eta \in J_{2},\; \xi \in \mathbb{R}.
\label{5fpcd}
\end{equation}
Then for any solution $u$ of \eqref{1eq} such that
$u(c)=u(d)=\omega _{2}$ there is a $t^{*}\in (c,d)$ satisfying
\begin{equation}
u(t^{*})\geq \omega _{2}. \label{5jumpom2}
\end{equation}
\end{proposition}
The condition $u(c)=u(d)=\omega _{2}$ can be avoided as follows.
\begin{proposition} \label{prop5.3}
Let $(c,d)\subseteq (a_{2},b_{2})$ be an open
interval such that
\[
N(c,d)\subseteq (a_{2},b_{2}),\quad
\text{where }N(c,d)=(c-\frac{d-c}{2},d+ \frac{d-c}{2}).
\]
Let $\omega _{2}$ be an arbitrarily given real number such that
\begin{equation}
\mathop{\rm ess\,inf}_{N(c,d)}\varphi <\omega _{2}
<\mathop{\rm ess\,inf}_{N(c,d)}\psi . \label{5omN2}
\end{equation}
Let $J_{2}$ be a set defined by
$J_{2} =(\mathop{\rm ess\,inf} _{(a_{2},b_{2})}\varphi ,\omega_{2})$ and
let the Caratheodory function $f(t,\eta ,\xi )$ satisfy:
\begin{gather}
f(t,\eta ,\xi )\geq 0,\quad t\in N(c,d),\; \eta \in J_{2},\;
\xi \in \mathbb{R} , \label{5fpN2} \\
\mathop{\rm ess\,inf}_{t\in (c,d)}f(t,\eta ,\xi )
>2^{p+1}\frac{(p-1)^{p-1}}{(d-c)^{p}}
\frac{(\mathop{\rm ess\,sup} _{(a_{2},b_{2})}\psi
-\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\varphi )^{p} }
{\mathop{\rm ess\,inf}_{N(c,d)}\psi -\omega _{2}},
\label{5fsN2}
\end{gather}
for $ \eta \in J_{2}$ and $\xi \in \mathbb{R}$.
Then for any solution $u$ of \eqref{1eq} there is a
$t^{*}\in N(c,d) $ satisfying
$u(t^{*})\geq \omega _{2}$.
\end{proposition}
The proof of these two propositions will be presented later;
meanwhile we proceed with the proof of Lemma \ref{lem5.1}.
\begin{proof}[Proof of Lemma \ref{lem5.1}]
Since for any $(c,d)\subseteq (a_{2},b_{2})$ and a function $g=g(t)$
we have
\[
\mathop{\rm ess\,inf}_{(a_{2},b_{2})}g\leq
\mathop{\rm ess\,inf} _{(c,d)}g\quad\text{and}\quad
\mathop{\rm ess\,sup}_{(c,d)}g\leq \mathop{\rm ess\,sup}_{(a_{2},b_{2})}g,
\]
one can show that the main hypotheses
\eqref{5omle}--\eqref{5fple} guarantee that the conditions
\eqref{5omcd}--\eqref{5fpcd} and \eqref{5omN2}--\eqref{5fpN2}
are satisfied, where $(c,d)\subseteq a_{2},b_{2})$ such
that $N(c,d)\subseteq (a_{2},b_{2})$. Thus, Proposition \ref{prop5.2} may be
used here as well as Proposition \ref{prop5.3} provided the hypothesis
\eqref{5fsN2} is satisfied too.
Next, we claim that:
\begin{equation}
\begin{aligned}
&\text{for any $(c,d)\subseteq A_{2}$ such that
$d-c=(b_{2}-a_{2})/8$ }
\\
&\text{there is $t^{*}\in (c-\frac{b_{2}-a_{2}}{16},d+\frac{b_{2}-a_{2}}{16})$
such that $u(t^{*})\geq \omega _{2}$},
\end{aligned} \label{5claim}
\end{equation}
where
\[
A_{2}=[a_{2}+\frac{b_{2}-a_{2}}{16},\, b_{2}-\frac{b_{2}-a_{2}}{16}].
\]
To prove \eqref{5claim}, let $(c,d)$ be an open
interval such that $(c,d)\subseteq A_{2}$ and
$d-c=(b_{2}-a_{2})/8$. It is clear that
\[
N(c,d)=(c-\frac{b_{2}-a_{2}}{16},\, d+\frac{b_{2}-a_{2}}{16})\subseteq
(a_{2},b_{2}),
\]
where $N(c,d)=(c-\frac{d-c}{2},d+\frac{d-c}{2})$. Putting
$b_{2}-a_{2}=8(d-c) $ in \eqref{5fsle} and using
$c(p)=2(16^{p})(p-1)^{p-1}$ we get
\begin{align*}
&\mathop{\rm ess\,inf}_{t\in (c,d)}f(t,\eta ,\xi )\geq
\mathop{\rm ess\,inf}_{t\in A_{2}}f(t,\eta ,\xi ) \\
&>2^{p+1}\frac{(p-1)^{p-1}}{(d-c)^{p}}\frac{(\mathop{\rm ess\,sup}
_{(a_{2},b_{2})}\psi -\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\varphi )^{p}
}{\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\psi -\omega _{2}} \\
&\geq 2^{p+1}\frac{(p-1)^{p-1}}{(d-c)^{p}}
\frac{(\mathop{\rm ess\,sup} _{(a_{2},b_{2})}\psi
-\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\varphi )^{p} }
{\mathop{\rm ess\,inf}_{N(c,d)}\psi -\omega _{2}},\quad
\eta \in J_{2},\; \xi \in \mathbb{R}.
\end{align*}
Therefore, the assumption \eqref{5fsN2} is satisfied too and so,
by Proposition \ref{prop5.3} there is $t^{*}\in N(c,d)$ such that
$u(t^{*})\geq \omega _{2}$. Thus, the assertion \eqref{5claim}
is verified.
Next, we define two intervals $(c_{1},d_{1})$ and $(c_{2},d_{2})$ by
\begin{gather*}
(c_{1},d_{1}) =(a_{2}+\frac{1}{16}(b_{2}-a_{2}),a_{2}+\frac{3}{16}
(b_{2}-a_{2})) \\
(c_{2},d_{2})
=(b_{2}-\frac{3}{16}(b_{2}-a_{2}),b_{2}-\frac{1}{16}
(b_{2}-a_{2})).
\end{gather*}
It is easy to check that
\[
(c_{i},d_{i})\subseteq A_{2}\quad \text{and}\quad
d_{i}-c_{i}=(b_{2}-a_{2})/8, \text{ for }i=1,2.
\]
So, applying \eqref{5claim} to both interval $[c_{1},d_{1}]$ and
$[c_{2},d_{2}]$ we get two points $t_{1}^{*}$ and $t_{2}^{*}$ such
that
\begin{equation}
t_{i}^{*}\in (c_{i}-\frac{1}{16}(b_{2}-a_{2}),d_{i}+\frac{1}{16}
(b_{2}-a_{2}))\quad\text{and}\quad
u(t_{i}^{*})\geq \omega _{2},\quad \text{for }i=1,2. \label{5gore}
\end{equation}
It is clear that
\begin{equation}
\big[a_{2}+\frac{1}{4}(b_{2}-a_{2}),b_{2}-\frac{1}{4}(b_{2}-a_{2})\big]
\subseteq [t_{1}^{*},t_{2}^{*}]\subseteq (a_{2},b_{2}).
\label{5sub}
\end{equation}
Next, we claim that
\begin{equation}
u(t)\geq \omega _{2}\quad \text{for each }t\in [t_{1}^{*},t_{2}^{*}].
\label{5jumpzv}
\end{equation}
On the contrary, if there is a point
$t_{0}\in [t_{1}^{*},t_{2}^{*}]$ satisfying $u(t_{0})<\omega _{2}$ then by
means of \eqref{5gore} we can construct an open interval
$(c,d)\subseteq (t_{1}^{*},t_{2}^{*})$ such that
$u(c)=u(d)=\omega_{2}$ and $u(t)<\omega _{2}$ in $(c,d)$.
For example, we can choose $c$ and $d$ by
\[
c=\max \{t\in [t_{1}^{*},t_{0}]:u(t)=\omega _{2}\}\quad\text{and}\quad
d=\min \{t\in [t_{0},t_{2}^{*}]:u(t)=\omega _{2}\}.
\]
But, by Proposition \ref{prop5.2} it is not possible and so, the assertion
\eqref{5jumpzv} holds true. Because of \eqref{5sub}, it
gives us the
desired conclusion \eqref{5jump2}. Thus, Lemma \ref{lem5.1} is proved.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop5.2}]
Let us suppose the opposite claim to \eqref{5jumpom2}; that is,
\begin{equation}
u(c)=u(d)=\omega _{2}\quad\text{and}\quad
u(t)<\omega _{2}\quad \text{for each }t\in (c,d).
\label{5contrary}
\end{equation}
We are going to prove that \eqref{5contrary} is not possible. In this
direction, let $v$ be a test function defined by
\[
v(t)=\begin{cases}
\omega _{2} & \text{in }(c,d), \\
u(t) & \text{otherwise.}
\end{cases}
\]
Since $u\in K(\varphi ,\psi )$ and because of \eqref{5omcd} and
\eqref{5contrary}, we have also that $v\in K(\varphi ,\psi )$ and
\[
v(t)-u(t)=\begin{cases}
\omega _{2}-u(t)>0 & \text{in }(c,d), \\
0 & \text{otherwise.}
\end{cases}
\]
Hence, this test function can be applied in \eqref{1eq} and so, we obtain
\[
0\leq \int_{c}^{d}|u'|^{p}dt\leq -\int_{c}^{d}f(t,u,u')(\omega
_{2}-u(t))dt\leq 0,
\]
where the main assumption \eqref{5fpcd} is used. So, we get
$u'=0$ in $(c,d)$. But, it contradicts \eqref{5contrary}.
Thus, \eqref{5contrary} is not possible and the desired
conclusion \eqref{5jumpom2} is proved.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop5.3}]
Let $(c,d)\subseteq (a_{2},b_{2})$
be an interval such that $N(c,d)\subseteq (a_{2},b_{2})$,
where $N(c,d)=(c-\frac{ d-c}{2},d+\frac{d-c}{2})$.
Let $\omega_{2}$ be an arbitrarily given real number satisfying
\eqref{5omN2} and let the Caratheodory function
$f(t,\eta ,\xi)$ satisfy \eqref{5fpN2} and \eqref{5fsN2}.
Immediately from \eqref{5fsN2} we get
\begin{equation}
\int_{c}^{d}\mathop{\rm ess\,inf}_{(\eta ,\xi )\in J_{2}\times
R}f(t,\eta ,\xi )dt
> 2^{p}\frac{(p-1)^{p-1}}{(d-c)^{p-1}}
\frac{(\mathop{\rm ess\,sup} _{(a_{2},b_{2})}\psi
-\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\varphi )^{p} }
{\mathop{\rm ess\,inf}_{N(c,d)}\psi -\omega _{2}}. \label{5ftcd}
\end{equation}
Let the numbers $c_{2}$ and $d_{2}$ and the set $A_{2}$ be defined by
\[
c_{2}=c-\frac{d-c}{2},\quad
d_{2}=d+\frac{d-c}{2}, \quad A_{2}=[c,d].
\]
Then
\begin{gather*}
N(c,d)=(c_{2},d_{2}),\\
A_{2}=[c_{2}+\frac{1}{4}(d_{2}-c_{2}),d_{2}-\frac{1 }{4}(d_{2}-c_{2})], \\
2(d-c)=d_{2}-c_{2}.
\end{gather*}
Therefore, from the inequalities \eqref{5omN2}, \eqref{5fpN2} and
\eqref{5ftcd}, we get
\begin{gather*}
\tilde{\theta}_{0}\leq \mathop{\rm ess\,inf}_{(c_{2},d_{2})}\varphi
<\omega _{2}<\mathop{\rm ess\,inf}_{(c_{2},d_{2})}\psi
\leq \tilde{ \omega}_{0},
\\
f(t,\eta ,\xi )\geq 0,\quad t\in (c_{2},d_{2}),\; \eta \in J_{2},\;
\xi \in \mathbb{R},
\\
\int_{A_{2}}\mathop{\rm ess\,inf}_{(\eta ,\xi )\in
J_{2}\times R}f(t,\eta ,\xi
)dt>\frac{c(p)}{(d_{2}-c_{2})^{p-1}}\frac{(\tilde{\omega}
_{0}-\tilde{\theta}_{0})^{p}}{\mathop{\rm ess\,inf}_{(c_{2},d_{2})}\psi
-\omega _{2}},
\end{gather*}
where $J_{2}=(\tilde{\theta}_{0},\omega _{2})$,
$\tilde{\theta}_{0}= \mathop{\rm ess\,inf}_{(a_{2},b_{2})}\varphi $,
$\tilde{\omega}_{0}=\mathop{\rm ess\,sup}_{(a_{2},b_{2})}\psi $, and
$c(p)=2[4(p-1)]^{p-1}$. Hence, the assumptions of Lemma \ref{lem2.3} are
satisfied especially on the open interval
$(c_{2},d_{2})\subset \subset (a,b)$, it implies the existence of
a $t^{*}\in (c_{2},d_{2})$ such that $u(t^{*})\geq \omega _{2}$. Thus,
Proposition \ref{prop5.3} is shown.
\end{proof}
\section{The asymptotic behaviour of $\|u'\|_{L^{p}}$ as
$\varepsilon \approx 0$}
In this section, we will study the asymptotic behaviour of
$\|u'\|_{L^{p}}$ as $\varepsilon \approx 0$ which was presented by
the inequalities \eqref{1Lpnorm} and \eqref{1normlb}. It will
be made for such continuous functions which satisfy a ''jumping''
condition in the sense
of \eqref{5jump2} and \eqref{5jump1}, as follows.
\begin{lemma} \label{lem6.1}
Let $a_{k}$ be a decreasing sequence of real numbers from
interval $(a,b)$ satisfying
\begin{equation}
\begin{aligned}
&\text{$a_{k}\searrow a$ and $a_{k}-a_{k+1}\leq a_{k-1}-a_{k}$,
$k\geq 1$ and}\\
& \text{there is an $\varepsilon _{2}>0$ such that
for each $\varepsilon \in (0,\varepsilon _{2})$}\\
&\text{there is a $j(\varepsilon )\in \mathbf{N}$ such
that $a_{j(\varepsilon )}>a+\varepsilon$}.
\end{aligned} \label{6jeps}
\end{equation}
Let $u$ be a real function defined on $[a,b]$ such that
$u\in W_{\rm loc}^{1,p}((a,b])\cap C([a,b])$ and
\begin{equation}
\begin{gathered}
u(t)>0\quad \text{for each } t\in \Lambda _{2k}, \\
u(t)<0\quad \text{for each } t\in \Lambda _{2k+1},\; k\geq 1,
\end{gathered} \label{6jump}
\end{equation}
where
\[
\Lambda _{k}=[a_{k}+\frac{1}{4}(a_{k-1}-a_{k}),a_{k-1}-\frac{1}{4}
(a_{k-1}-a_{k})],\quad k\geq 1.
\]
Then there is a sequence $x_{k}\in (a,b)$, $k\in \mathbf{N}$
and a constants $c$ only depending on given data such that each
solution $u$ of \eqref{1eq} satisfies
\begin{equation}
\int_{a+\varepsilon }^{b}|u'(t)|^{p}dt\geq
c\sum_{k=3}^{j(\varepsilon )}\frac{(\max_{\Lambda _{k}}|u|)^{p}}{
(a_{k-2}-a_{k-1})^{p-1}}\quad \text{for each }
\varepsilon \in (0,\varepsilon _{2}), \label{6main}
\end{equation}
where $j(\varepsilon )$ is appearing in \eqref{6jeps}.
\end{lemma}
\begin{proof}
First, it is well known (see for instance
in \cite[Theorem 9.12 pp.166]{Bre83}) that in the space
$W_{0}^{1,p}(\Omega )$, $\Omega \subseteq \mathbb{R}^{N}$, there
is a constant $c_{p}>0$ such that for
$u\in W_{0}^{1,p}(\Omega )$ and $p>N$,
\begin{equation}
\sup_{\Omega }|u|\leq c_{p}|\Omega |^{1/N-1/p}\|\nabla u\|_{p}. \label{6sob}
\end{equation}
Next, let $u$ be a real function satisfying \eqref{6jump}. Then
there is a sequence $x_{k}$ of the zero-points of $u$ such that:
\begin{equation}
\begin{gathered}
u(x_{k})=0,\quad x_{k}\in (a_{k}-\frac{1}{4}(a_{k}-a_{k+1}),a_{k}+\frac{1}{4}
(a_{k-1}-a_{k})), \\
\Lambda _{k}\subseteq (x_{k},x_{k-1}),\ k\geq 2\quad\text{and}\quad
|x_{k}-x_{k-1}|\leq \frac{3}{2}(a_{k-2}-a_{k-1}),\quad k\geq 3.
\end{gathered} \label{6xk}
\end{equation}
In particular for $N=1$ and $\Omega =(x_{k},x_{k-1})$ we have
$u\in W_{0}^{1,p}(x_{k},x_{k-1})$ and so, from \eqref{6sob} follows
\[
\sup_{(x_{k},x_{k-1})}|u|\leq
c_{p}|x_{k}-x_{k-1}|^{1-1/p}\|u'\|_{L^{p}(x_{k},x_{k-1})};
\]
that is to say
\begin{equation}
\|u'\|_{L^{p}(x_{k},x_{k-1})}^{p}\geq c^{p}\frac{1}{
|x_{k}-x_{k-1}|^{p-1}}\big(\sup_{(x_{k},x_{k-1})}|u|\big)^{p},\quad
k\geq 2, \label{6start}
\end{equation}
where the constant $c>0$ does not depend on $k$, only on $p$.
Now, according to \eqref{6jump}, \eqref{6xk} and
\eqref{6start} we calculate that
\begin{align*}
\|u'\|_{L^{p}(a+\varepsilon ,b)}^{p}
&\geq \sum_{k=2}^{j(\varepsilon )}\|u'\|_{L^{p}(x_{k},x_{k-1})}^{p}\geq
c^{p}\sum_{k=2}^{j(\varepsilon )}\frac{1}{|x_{k}-x_{k-1}|^{p-1}}\big(
\sup_{(x_{k},x_{k-1})}|u|\big)^{p} \\
&\geq c^{p}(\frac{2}{3})^{p}\sum_{k=3}^{j(\varepsilon )}\frac{
(\max_{\Lambda _{k}}|u|)^{p}}{(a_{k-2}-a_{k-1})^{p-1}}\ \ \text{
for each}\ \varepsilon \in (0,\varepsilon _{2}).
\end{align*}
Thus, Lemma \ref{lem6.1} is proved.
\end{proof}
Combining Lemmas \ref{lem5.1} and \ref{lem5.2},
we are able to derive a kind of rapid
oscillations for solutions of \eqref{1eq} in the sense of \eqref{5jump2}
and \eqref{5jump1}.
\begin{lemma} \label{lem6.2}
Let $a_{k}$ be a decreasing sequence of real
numbers from interval $(a,b)$ satisfying \eqref{3keps}. Let for
each $k\geq 1$ the obstacles $\varphi (t)$ and $\psi (t)$
satisfy:
\begin{equation}
\begin{gathered}
\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\varphi <\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2<\mathop{\rm ess\,inf}
_{(a_{2k},a_{2k-1})}\psi , \\
\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\psi
>\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\varphi
/2>\mathop{\rm ess\,sup} _{(a_{2k+1},a_{2k})}\varphi .
\end{gathered}
\end{equation}
Let the sets $J_{k}$ be defined by:
\begin{gather*}
J_{2k}=(\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\varphi ,
\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2), \\
J_{2k+1}=(\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\varphi
/2, \mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\psi ),\quad k\geq 1.
\end{gather*}
Next, let for each $k\geq 1$ the Caratheodory function
$f(t,\eta ,\xi) $ satisfy
\begin{gather}
f(t,\eta ,\xi )\geq 0,\ \ t\in (a_{2k},a_{2k-1}),\ \eta \in J_{2k},\ \xi
\in \mathbb{R}, \label{6fpoz} \\
\mathop{\rm ess\,inf}_{t\in A_{2k}}f(t,\eta ,\xi )
>\frac{c(p)}{(a_{2k-1}-a_{2k})^{p}}\,
\frac{(\mathop{\rm ess\,sup} _{(a_{2k},a_{2k-1})}\psi
-\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})} \varphi )^{p}}
{\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\psi
- \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2}, \label{6fspoz}
\end{gather}
where $\eta \in J_{2k}$, $\xi \in \mathbb{R}$ and:
\begin{gather}
f(t,\eta ,\xi )\leq 0,\quad t\in (a_{2k+1},a_{2k}),\; \eta \in J_{2k+1},\;
\xi \in \mathbb{R}, \label{6fneg} \\
\mathop{\rm ess\,sup}_{t\in A_{2k+1}}f(t,\eta ,\xi )dt
<-\frac{c(p)}{(a_{2k}-a_{2k+1})^{p}}\frac{(\mathop{\rm ess\,sup}
_{(a_{2k+1},a_{2k})}\psi -\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}
\varphi )^{p}}{\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\varphi /2
- \mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\varphi }, \label{6fsneg}
\end{gather}
where $\eta \in J_{2k+1}$, $\xi \in \mathbb{R}$ and
$c(p)=2(16^{p})(p-1)^{p-1}$ and $A_{k}$ is a family of sets
defined by
\[
A_{k}=[a_{k}+\frac{1}{16}(a_{k-1}-a_{k}),a_{k-1}-\frac{1}{16}
(a_{k-1}-a_{k})],\ k\geq 1.
\]
Then for any solution $u$ of \eqref{1eq} we have:
\begin{gather}
u(t)\geq \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2\quad
\text{for each }t\in \Lambda _{2k}, \label{6rapid2} \\
u(t)\leq \mathop{\rm ess\,inf}_{_{(a_{2k+1},a_{2k})}}\varphi /2\quad
\text{for each }t\in \Lambda _{2k+1},\; k\geq 1, \label{6rapid1}
\end{gather}
where $\Lambda _{k}$ is a family of sets defined by
\[
\Lambda _{k}=[a_{k}+\frac{1}{4}(a_{k-1}-a_{k}),a_{k-1}-\frac{1}{4}
(a_{k-1}-a_{k})],\quad k\geq 1.
\]
\end{lemma}
\begin{proof}
It is clear that the assumptions of Lemmas \ref{lem5.1} and
\ref{lem5.2} are fulfilled on the intervals
$[a_{2},b_{2}]=[a_{2k},a_{2k-1}]$ and
$[a_{1},b_{1}]=[a_{2k+1},a_{2k}]$ respectively, where
$\omega_{2}= \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2$ and
$\theta _{1}= \mathop{\rm ess\,inf}_{_{(a_{2k+1},a_{2k})}}\varphi /2$.
Therefore, from
\eqref{5jump2} and \eqref{5jump1} immediately follows
\eqref{6rapid2} and \eqref{6rapid1}).
\end{proof}
Regarding Example \ref{exa3.4} above, it is easy to construct a class of
Caratheodory functions $f(t,\eta ,\xi )$ which satisfies the assumptions of
Lemma \ref{lem6.2}.
Next, we give the main result of the section.
\begin{theorem} \label{thm6.3}
For arbitrarily given real number $s\in (1,2)$, let the
sequence $a_{k}$ and the obstacles $\varphi $ and $\psi $ be given by
\eqref{3akto}. If the Caratheodory function $f(t,\eta ,\xi )$ satisfies
\eqref{6fpoz}--\eqref{6fsneg} in respect to such
$(\varphi ,\psi ,a_{k})$ then there are two positive constants $c$
and $\varepsilon _{2}$ depending only on given data
such that each solution $u$ of \eqref{1eq} satisfies
\begin{gather*}
\big(\int_{a+\varepsilon }^{b}|u'|^{p}dt\big)^{1/p}\geq c\big(
\frac{1}{\varepsilon }\big)^{s-1}\quad \text{for each }
\varepsilon \in (0,\min \{\varepsilon _{2},1\}), \\
\limsup_{\varepsilon \to 0}\frac{\log
\big(\int_{a+\varepsilon }^{b}|u'|^{p}dt\big)^{1/p}}{\log
1/\varepsilon }\geq s-1.
\end{gather*}
\end{theorem}
\begin{proof}
It is easy to see that $\varphi $, $\psi $ and
$a_{k}$ given by \eqref{3akto} satisfy the assumptions of
Lemma \ref{lem6.2}. It implies that each solution $u$ of \eqref{1eq} satisfies
the assumptions of Lemma \ref{lem6.1}, where
$j(\varepsilon )=k(\varepsilon)$, and $k(\varepsilon ) $ is given
in \eqref{3eps0}, and
\[
\varepsilon _{2}=\min \{\frac{b-a}{\beta},
\big(\frac{b-a}{2}(\frac{1}{2c_{0}})^{\frac{1}{\beta }}\big)
^{\frac{\beta +1}{\beta }}\},
\]
where $c_{0}$ is appearing in \eqref{3eps0}.
For the record, in order to prove that
$a_{k}$ given in \eqref{3akto} satisfies \eqref{6jeps} in
respect to $\varepsilon _{2}$, it is used the following elementary
inequalities
\[
\frac{1}{\beta }\big(\frac1k\big) ^{1+1/\beta }
\leq \big(\frac{1}{ k-1}\big) ^{1/\beta }-\big(\frac1k\big)
^{1/\beta }
\leq \frac{1}{ \beta }\big( \frac{1}{k-1}\big)^{1+1/\beta }
\leq \frac{2^{1+1/\beta }}{ \beta }\big(\frac1k\big) ^{1+1/\beta },
\]
where $k\geq 2$ and $\beta >0$. Putting such
$(\varphi ,\psi,a_{k})$ into \eqref{6main}, we obtain
\[
\int_{a+\varepsilon }^{b}|u'(t)|^{p}dt\geq
c\sum_{k=3}^{k(\varepsilon
)}(\frac{a_{k}+a_{k-1}}{2}-a)^{p}\frac{1}{
(a_{k-2}-a_{k-1})^{p-1}}\quad \text{ for each }\varepsilon \in
(0,\varepsilon _{2}).
\]
Now, with the help of the same technical details as in the proof of
\cite[Theorem 8.1, p. 298-299]{Pasic03},
from \eqref{3akto} and previous inequality easy follows that
\[
\|u'\|_{L^{p}(a+\varepsilon ,b)}^{p}\geq
c_{1}\sum_{k=3}^{k(\varepsilon )}k^{(1+\frac{1}{\beta
})(p-1)-\frac{p}{\beta }}\geq c_{1}(k(\varepsilon
))^{(1+\frac{1}{\beta })(p-1)-\frac{p}{\beta } +1},\quad
\varepsilon \in (0,\varepsilon _{2}).
\]
Taking the $p$-root in the preceding inequality and using
\eqref{3eps0}, we obtain
\begin{align*}
\|u'\|_{L^{p}(a+\varepsilon ,b)}\geq c_{1}(k(\varepsilon ))^{(1+
\frac{1}{\beta })(1-\frac{1}{p})-\frac{1}{\beta }+\frac{1}{p}}
&\geq c_{1}
\big(\frac{1}{\varepsilon }\big)^{\frac{2\beta }{\beta +1}-\frac{1}{p}}\big(
\frac{1}{\varepsilon }\big)^{(\frac{1}{p}-1)\frac{\beta }{\beta +1}} \\
&\geq c_{1}\big(\frac{1}{\varepsilon }\big)^{s-1},\ \varepsilon \in (0,\min
\{\varepsilon _{2},1\}).
\end{align*}
It proves Theorem \ref{thm6.3}
\end{proof}
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\end{document}