\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 $10$. 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 $$|G_{\varepsilon }(u)|\geq \frac{1}{2^{6}}(b-a)^{s}\varepsilon ^{2-s}>0\quad \text{for each }\varepsilon \in (0,\varepsilon _{0}), \label{1Geu}$$ 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 $$\dim _{M}G(u)\geq s\quad \text{and}\quad M^{s}(G(u))\geq \frac{1}{2^{7}} (b-a)^{s}>0. \label{1lbs}$$ 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 $$|G_{\varepsilon }(u|_{(a,c]})|\geq \frac{1}{2^{6}}(c-a)^{s}\varepsilon ^{2-s}>0\ \text{for any }c\in (a,b)\text{ and }\varepsilon \in (0,\varepsilon _{c}), \label{3Geures}$$ where the number $\varepsilon _{c}$ will be also determined. The preceding inequality yields $$M^{s}(G(u)\cap B_{r}(a,u(a)))\geq \frac{1}{2^{7}}(\frac{r}{\sqrt{5}})^{s} \quad \text{ for any }r\in (0,b-a). \label{1Mslb}$$ 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 $$|G_{\varepsilon }(u')|\geq \frac{\sqrt2}{2^{4}}(b-a)^{s/2}\varepsilon ^{1-s/2}>0\quad \text{for each }\varepsilon \in (0,\varepsilon _{0}). \label{1derlb}$$ 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: $$\dim _{M}G(u')\geq 1+\frac{s}{2}\quad \text{and}\quad M^{1+s/2}(G(u'))\geq \frac{1}{2^{4}}(b-a)^{s/2}>0. \label{1derlbs}$$ 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 $$\Big(\int_{a+\varepsilon }^{b}|u'|^{p}dt\Big)^{1/p}\geq c\big(\frac{ 1}{\varepsilon }\big)^{s-1}\quad \text{for some } \varepsilon \in (0,\varepsilon _{1}), \label{1Lpnorm}$$ 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 $$\limsup_{\varepsilon \to 0}\frac{\log \big(\int_{a+\varepsilon }^{b}|u'|^{p}dt\big)^{1/p}}{\log 1/\varepsilon }\geq s-1. \label{1normlb}$$ 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: $$\begin{gathered} -(|y'|^{p-2}y')'=f(t,y,y')\quad \text{in } (a,b), \\ y(a)=y(b)=0, \\ y\in W_{\rm loc}^{1,p}((a,b])\cap C([a,b]). \end{gathered} \label{1ceq}$$ 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 $$\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\varphi <\omega _{2}< \mathop{\rm ess\,inf}_{(a_{2},b_{2})}\psi . \label{2thom}$$ 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 $$u(\sigma _{2})\geq \omega _{2}. \label{2con2}$$ \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 $$\mathop{\rm ess\,sup}_{(a_{1},b_{1})}\varphi <\theta _{1}< \mathop{\rm ess\,sup}_{(a_{1},b_{1})}\psi . \label{2thomdual}$$ 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 $$u(\sigma _{1})\leq \theta _{1}. \label{2con1}$$ \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 $$u(t)<\omega _{2}\quad \text{for each }t\in B_{2r}. \label{2contr}$$ Since $\varphi \leq u$ in $(a,b)$ and because of \eqref{2thom}, besides \eqref{2contr} we have also $$\mathop{\rm ess\,inf}_{B_{2r}}\varphi \leq u(t)<\omega _{2}< \mathop{\rm ess\,inf}_{B_{2r}}\psi \ \text{ for each }t\in B_{2r}. \label{2contrth}$$ 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}: $$\begin{gathered} \Phi (t)=1,\; t\in B_{r}\quad \text{and}\quad \Phi (t)=0,\; t\in \mathbb{R}\setminus B_{2r}, \\ \Phi (t)>0,\; t\in B_{2r}\quad \text{and}\quad |\Phi '(t)|\leq \frac{c_{0}}{r },\; t\in \mathbb{R}. \end{gathered} \label{2local}$$ 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{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} For the record, with the help of \eqref{2contrth} we also have: $$\begin{gathered} \mathop{\rm ess\,inf}_{B_{2r}}\psi -u(t)\leq \mathop{\rm ess\,inf} _{B_{2r}}\psi -\mathop{\rm ess\,inf}_{B_{2r}}\varphi ,\; t\in B_{2r}, \\ \mathop{\rm ess\,inf}_{B_{2r}}\psi -u(t)\geq \mathop{\rm ess\,inf} _{B_{2r}}\psi -\omega _{2},\; t\in B_{2r}. \end{gathered} \label{2ess}$$ 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{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} 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 $$\tilde{\theta}_{0}\leq \mathop{\rm ess\,inf}_{(a_{2},b_{2})}\varphi <\omega _{2}<\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\psi \leq \tilde{ \omega}_{0}. \label{2omle23}$$ 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: $$\begin{gathered} f(t,\eta ,\xi )\geq 0,\quad t\in (a_{2},b_{2}),\; \eta \in J_{2},\; \xi \in \mathbb{R}, \\ \int_{A_{2}}\mathop{\rm ess\,inf}_{(\eta ,\xi )\in J_{2}\times R}f(t,\eta ,\xi )dt >\frac{c(p)}{(b_{2}-a_{2})^{p-1}}\frac{(\tilde{\omega}_{0}-\tilde{\theta} _{0})^{p}}{\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\psi -\omega _{2}}, \end{gathered} \label{2ftsle23}$$ 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 $$u(\sigma _{2})\geq \omega _{2}. \label{2jumple23}$$ \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} 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 $$\begin{gathered} \mathop{\rm ess\,inf}_{(a_{2k+2},a_{2k+1})}\theta \geq \mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta , \\ \mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\omega \leq \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega ,\quad k\geq 1. \end{gathered} \label{3to}$$ 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 $$|G_{\varepsilon }(u)|\geq \int_{a}^{a_{k(\varepsilon )}}(\omega (t)-\theta (t))dt\ \ \text{for each }\varepsilon \in (0,\varepsilon _{0}), \label{3Geu}$$ 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 $$|G_{\varepsilon }(u|_{[a,c]})|\geq \int_{a}^{a_{k(\varepsilon )}}(\omega (t)-\theta (t))dt\ \ \text{for each }\varepsilon \in (0,\varepsilon _{c}). \label{3GeuR}$$ \end{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: $$\begin{gathered} \mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\varphi <\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2<\mathop{\rm ess\,inf} _{(a_{2k},a_{2k-1})}\psi , \\ \mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\psi >\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\varphi /2>\mathop{\rm ess\,sup} _{(a_{2k+1},a_{2k})}\varphi . \end{gathered} \label{3thom}$$ 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 $$u(\sigma _{2k})\geq \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2 \quad\text{and}\quad u(\sigma _{2k+1})\leq \mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})} \varphi /2,\ k\geq 1. \label{3jump}$$ \end{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: $$\begin{gathered} a_{k}=a+\frac{b-a}{2}(\frac{1}{k})^{1/\beta },\quad k\geq 1, \\ \varphi (t)=-2(t-a)\quad \text{and}\quad \psi (t)=2(t-a),\; t\in (a,b), \end{gathered} \label{3akto}$$ 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 $$c_{0}\left( \frac{1}{\varepsilon }\right) ^{\frac{\beta }{\beta +1}}\leq k(\varepsilon )\leq 2c_{0}\left( \frac{1}{\varepsilon }\right) ^{\frac{\beta }{\beta +1}}\ \ \text{ for each }\varepsilon \in (0,\varepsilon _{0}), \\ \label{3eps0}$$ 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 $$|G_{\varepsilon }(u)|\geq \int_{a}^{a_{k(\varepsilon )}}2(t-a)dt=\big( \frac{b-a}{2}\big) ^{2}\big( \frac{1}{k(\varepsilon )}\big) ^{2/ \beta }\quad \text{for each }\varepsilon \in (0,\varepsilon _{0}). \label{3start}$$ 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 $$\varepsilon _{c}=\min \{\varepsilon _{0}=\frac{b-a}{\beta },\frac{1}{\beta } \frac{(c-a)^{\beta +1}}{(b-a)^{\beta }}\}. \label{3epsc}$$ 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 $$M^{s}(G(u|_{[a,a+\frac{r}{\sqrt{5}}]}))\leq M^{s}(G(u)\cap B_{r}(a,0)) \quad \text{for any }r\in (0,(b-a)). \label{3MsBr}$$ 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{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} 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 $$\omega _{2k}>\min \{\theta _{2k+1},\theta _{2k-1}\},\quad k\geq 1.$$ 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 $$u(\sigma _{2k})\geq \omega _{2k}\quad\text{and}\quad u(\sigma_{2k+1})\leq \theta _{2k+1},\quad k\geq 1. \label{4jump}$$ Then $$|G_{\varepsilon }(u')|\geq \sum_{k=k(\varepsilon /2)}^{\infty }(\omega _{2k}-\theta _{2k+1})\quad\text{for each } \varepsilon \in (0,\varepsilon _{0}), \label{4Geud}$$ 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 $$u'(s_{2k+1})=\frac{u(\sigma _{2k})-u(\sigma _{2k+1})}{\sigma _{2k}-\sigma _{2k+1}}, \quad u'(s_{2k})=\frac{u(\sigma _{2k-1})-u(\sigma _{2k})}{\sigma _{2k-1}-\sigma _{2k}}. \label{4Lagrang}$$ 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 $$z(s_{2k+1})\geq \delta _{2k+1}>0\quad \text{and}\quad z(s_{2k})\leq \delta _{2k}<0,\quad k\geq 1. \label{4pom1}$$ 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, $$\sigma _{k}\searrow a\quad\text{and}\quad \sigma _{j-1}-\sigma _{j}\leq \varepsilon/2\quad \text{for each } j\geq k(\frac{\varepsilon }{2}),\; \varepsilon \in (0,\varepsilon _{0}), \label{4pom2}$$ 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 $$\mathop{\rm ess\,sup}_{(a_{2},b_{2})}\varphi <\omega _{2}< \mathop{\rm ess\,inf}_{(a_{2},b_{2})}\psi . \label{5omle}$$ 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 $$u(t)\geq \omega _{2}\quad \text{for each } t\in [a_{2}+\frac{1}{4} (b_{2}-a_{2}),b_{2}-\frac{1}{4}(b_{2}-a_{2})]. \label{5jump2}$$ \end{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 $$u(t)\leq \theta _{1}\quad \text{for each } t\in [a_{1}+\frac{1}{4} (b_{1}-a_{1}),b_{1}-\frac{1}{4}(b_{1}-a_{1})]. \label{5jump1}$$ \end{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 $$\mathop{\rm ess\,sup}_{(c,d)}\varphi <\omega _{2} <\mathop{\rm ess\,inf}_{(c,d)}\psi . \label{5omcd}$$ 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 $$f(t,\eta ,\xi )\geq 0,\quad t\in (c,d),\; \eta \in J_{2},\; \xi \in \mathbb{R}. \label{5fpcd}$$ Then for any solution $u$ of \eqref{1eq} such that $u(c)=u(d)=\omega _{2}$ there is a $t^{*}\in (c,d)$ satisfying $$u(t^{*})\geq \omega _{2}. \label{5jumpom2}$$ \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 $$\mathop{\rm ess\,inf}_{N(c,d)}\varphi <\omega _{2} <\mathop{\rm ess\,inf}_{N(c,d)}\psi . \label{5omN2}$$ 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{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} 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 $$t_{i}^{*}\in (c_{i}-\frac{1}{16}(b_{2}-a_{2}),d_{i}+\frac{1}{16} (b_{2}-a_{2}))\quad\text{and}\quad u(t_{i}^{*})\geq \omega _{2},\quad \text{for }i=1,2. \label{5gore}$$ It is clear that $$\big[a_{2}+\frac{1}{4}(b_{2}-a_{2}),b_{2}-\frac{1}{4}(b_{2}-a_{2})\big] \subseteq [t_{1}^{*},t_{2}^{*}]\subseteq (a_{2},b_{2}). \label{5sub}$$ Next, we claim that $$u(t)\geq \omega _{2}\quad \text{for each }t\in [t_{1}^{*},t_{2}^{*}]. \label{5jumpzv}$$ 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, $$u(c)=u(d)=\omega _{2}\quad\text{and}\quad u(t)<\omega _{2}\quad \text{for each }t\in (c,d). \label{5contrary}$$ 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 $$\int_{c}^{d}\mathop{\rm ess\,inf}_{(\eta ,\xi )\in J_{2}\times R}f(t,\eta ,\xi )dt > 2^{p}\frac{(p-1)^{p-1}}{(d-c)^{p-1}} \frac{(\mathop{\rm ess\,sup} _{(a_{2},b_{2})}\psi -\mathop{\rm ess\,inf}_{(a_{2},b_{2})}\varphi )^{p} } {\mathop{\rm ess\,inf}_{N(c,d)}\psi -\omega _{2}}. \label{5ftcd}$$ 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{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} 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 $$\begin{gathered} u(t)>0\quad \text{for each } t\in \Lambda _{2k}, \\ u(t)<0\quad \text{for each } t\in \Lambda _{2k+1},\; k\geq 1, \end{gathered} \label{6jump}$$ 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 $$\int_{a+\varepsilon }^{b}|u'(t)|^{p}dt\geq c\sum_{k=3}^{j(\varepsilon )}\frac{(\max_{\Lambda _{k}}|u|)^{p}}{ (a_{k-2}-a_{k-1})^{p-1}}\quad \text{for each } \varepsilon \in (0,\varepsilon _{2}), \label{6main}$$ 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 $p>N$, $$\sup_{\Omega }|u|\leq c_{p}|\Omega |^{1/N-1/p}\|\nabla u\|_{p}. \label{6sob}$$ 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: $$\begin{gathered} u(x_{k})=0,\quad x_{k}\in (a_{k}-\frac{1}{4}(a_{k}-a_{k+1}),a_{k}+\frac{1}{4} (a_{k-1}-a_{k})), \\ \Lambda _{k}\subseteq (x_{k},x_{k-1}),\ k\geq 2\quad\text{and}\quad |x_{k}-x_{k-1}|\leq \frac{3}{2}(a_{k-2}-a_{k-1}),\quad k\geq 3. \end{gathered} \label{6xk}$$ 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 $$\|u'\|_{L^{p}(x_{k},x_{k-1})}^{p}\geq c^{p}\frac{1}{ |x_{k}-x_{k-1}|^{p-1}}\big(\sup_{(x_{k},x_{k-1})}|u|\big)^{p},\quad k\geq 2, \label{6start}$$ 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: $$\begin{gathered} \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\varphi <\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\psi /2<\mathop{\rm ess\,inf} _{(a_{2k},a_{2k-1})}\psi , \\ \mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\psi >\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\varphi /2>\mathop{\rm ess\,sup} _{(a_{2k+1},a_{2k})}\varphi . \end{gathered}$$ 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} \begin{thebibliography}{99} \bibitem{Bou25} G. 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