\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 95, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/95\hfil Half-linear Euler equation]
{Nonoscillation criteria and energy functional
for even-order half-linear two-term \\ differential equations}

\author[O. Do\v{s}l\'y, V. R\accent23u\v{z}i\v{c}ka \hfil EJDE-2016/95\hfilneg]
{Ond\v{r}ej Do\v{s}l\'y, Vojt\v{e}ch R\accent23u\v{z}i\v{c}ka}

\address{Ond\v{r}ej Do\v{s}l\'y \newline
Department of Mathematics and Statistics,
Masaryk University,
Kotl\'a\v{r}sk\'a 2, CZ-611 37 Brno, Czech Republic}
\email{dosly@math.muni.cz}

\address{Vojt\v{e}ch R\accent23u\v{z}i\v{c}ka \newline
Department of Mathematics and Statistics,
 Masaryk University,
Kotl\'a\v{r}sk\'a 2,
 CZ-611 37 Brno,
Czech Republic}
\email{211444@mail.muni.cz}

\thanks{Submitted July 27, 2015. Published April 12, 2016.}
\subjclass[2010]{34C10}
\keywords{Even-order half-linear differential equation; energy functional;
\hfill\break\indent nonoscillation, Wirtinger inequality}

\begin{abstract}
 We investigate oscillatory properties of even-order half-linear differential
 equations and conditions for negativity of the associated energy functional.
 First, using the relationship between positivity of the functional and
 nonoscillation of the investigated equation, we prove Hille-Nehari type
 nonoscillation criteria which extend criteria known in the linear case.
 In the second part of the paper, we present conditions which guarantee
 that the energy functional attains a negative value, i.e., it is unbounded
 below.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

We consider the even-order half-linear two-term differential equation
\begin{equation} \label{HL}
(-1)^n\bigl(t^{\alpha}\Phi(y^{(n)})\bigr)^{(n)}+c(t)\Phi(y)=0,
\end{equation}
where $\Phi(y)=|y|^{p-2}y$, $p>1$, is the odd power function and 
$\alpha\in \mathbb{R}$.
If $p=2$, then \eqref{HL} reduces to the \emph{linear} even-order
Sturm-Liouville differential equation
\begin{equation} \label{LE}
(-1)^n\bigl(t^{\alpha}y^{(n)}\bigr)^{(n)}+c(t)y=0
\end{equation}
whose oscillation and spectral theory is relatively deeply developed. We refer
to the books \cite{Glazman,Weidmann}, 
the papers \cite{D-PRSE-91,D-MN-97,D-EJQTDE,E-K-Z,Fiedler,H-L,M-Pf-APM}, and
the references given therein.

Equation \eqref{HL} is a particular case of the general even-order
half-linear differential equation
\begin{equation} \label{HLG}
\sum_{k=0}^n (-1)^k \big(r_k(t)\Phi(y^{(k)})\big)^{(k)}=0
\end{equation}
which, in the linear case $p=2$, takes the form
\begin{equation} \label{LEG}
\sum_{k=0}^n (-1)^k \big(r_k(t)y^{(k)}\big)^{(k)}=0.
\end{equation}
The investigation of oscillatory properties of \eqref{LEG} is based on the
relationship between this equation and its quadratic energy
functional
\begin{equation} \label{QF2}
\mathcal{F}(y;a,b)=\int_a^b \Big[\sum_{k=0}^n r_k(t)\bigl(y_k(t)\bigr)^2\Big]\,dt
\end{equation}
and on the fact that using the substitution
\begin{equation*}
x=\begin{pmatrix} y\\y'\\ \vdots\\ y^{(n-1)}\end{pmatrix},\quad
u=\begin{pmatrix} \sum_{k=1}^n(-1)^{k-1}\bigl(r_ky^{(k)}\bigr)^{(k)}
\\ \vdots \\-\bigl(r_ny^{(n)}\bigr)'+r_{n-1}y^{(n-1)}\\ r_ny^{(n)}
\end{pmatrix}
\end{equation*}
equation \eqref{LEG} can be written as the linear Hamiltonian system
\begin{equation} \label{LHS}
x'=Ax+B(t)u,\quad u'=C(t)x-A^Tu
\end{equation}
with
\begin{gather*}
B(t)=\operatorname{diag}\big\{0,\dots,0,\frac{1}{r_n(t)}\big\},\quad
C(t)=\operatorname{diag}\{r_0(t),\dots,r_{n-1}(t)\}, \\
A=A_{i,j}=\begin{cases} 
1  &j=i+1,\; i=1\dots,n-1,\\
 0 & \text{elsewhere}.
 \end{cases}
\end{gather*}
In particular, using the so-called Reid Roundabout Theorem for
\eqref{LHS}
(see \cite[Theorem 6.3, p.~284]{Reid-book}),
it is proved that $\mathcal{F}(y;T,\infty)>0$ for every $0\not\equiv y
\in W^{n,2}_0[T,\infty)$ (the definition of this space is recalled
later) if and only if no nontrivial solution of \eqref{LEG} has
more than one zero point of multiplicity $n$ in $[T,\infty)$, i.e.,
there exists no pair of distinct points $t_1,t_2\in [T,\infty)$
such that
\begin{equation} \label{conj-points}
y^{(i)}(t_1)=0=y^{(i)}(t_2),\quad i=0,\dots,n-1.
\end{equation}
Following the linear case, equation \eqref{HLG} is said to be
\emph{nonoscillatory} if there exists
$T\in \mathbb{R}$ such that for any nontrivial solution of this equation there is no
pair of distinct points in $[T,\infty)$ such that \eqref{conj-points}
holds.
Points $t_1,t_2$ with this property are said to be \emph{conjugate points}
relative to \eqref{HLG}.


Equation \eqref{HLG} can be written as a Hamiltonian type system
\begin{equation} \label{LHS-1/2}
x'=Ax+B(t)\Phi^{-1}(u),\quad u'=C(t)\Phi(x)-A^Tu
\end{equation}
with
\begin{equation} \label{xu}
x=\begin{pmatrix} y\\y'\\ \vdots\\y^{(n-1)}\end{pmatrix},\quad
u=\begin{pmatrix} \sum_{k=1}^{n}
(-1)^{k-1}\bigl(r_k\Phi(y^{(k)})\bigr)^{(k)}\\\vdots\\
-\bigl(r_n\Phi(y^{(n)})\bigr)'+ r_{n-1}\Phi(y^{(n-1)})\\ r_n\Phi(y^{(n)})
 \end{pmatrix}.
\end{equation}
The functions $\Phi,\Phi^{-1}$ of a vector argument are
defined in a natural way as
\begin{equation*}
\Phi(x)=\begin{pmatrix}\Phi(x_1)\\ \Phi(x_2)\\ \vdots\\ \Phi(x_n)
 \end{pmatrix},\quad
\Phi^{-1}(u)=\begin{pmatrix}\Phi^{-1}(u_1)\\ \Phi^{-1}(u_2)\\ \vdots \\
\Phi^{-1}(u_n)\end{pmatrix}
\end{equation*}
for column vectors $x=(x_i)_{i=1}^n$ and $u=(u_i)_{i=1}^n$,
where the scalar function $\Phi^{-1}(y)=|y|^{q-2}y$ is the
inverse function of $\Phi$, i.e., $q$ can be expressed as $q=\frac{p}{p-1}$.
The number $q$ is called the \textit{conjugate exponent} of $p$ and satisfies
the equality $\frac{1}{p}+\frac{1}{q}=1$.

However, a Roundabout type theorem for \eqref{LHS-1/2} is missing,
so the theory of \eqref{LHS-1/2} and \eqref{HLG} is much less developed
than in the linear case. Concerning oscillatory properties of \eqref{HL}
and \eqref{HLG}, as far as we know, only the papers \cite{D-R-EJQTDE,
O-R-EJQTDE} and the book \cite[Sec. 9.4]{D-R-book} deal with this problem.


This article consists essentially of two parts. The first one can be
regarded as a continuation of \cite{D-R-EJQTDE}. In our paper we prove
Hille-Nehari nonoscillation criteria for \eqref{HL} which extend
previously proved (in \cite{D-MN-97,D-O-CZMJ}) nonoscillation criteria
for \eqref{LE}.
The second one is devoted to the investigation of conditions which imply
that the $p$-degree energy functional associated with \eqref{HL}
attains a negative value.


\section{Preliminary results}


In our investigation, an important role is played by the test functions
from certain Sobolev spaces which are defined as follows. We denote
\begin{align*}
W^{n,p}_0[T,\infty)
=\Big\{& y \colon [T,\infty) \to \mathbb{R} :
 y^{(n-1)}\in\mathcal{AC}[T,\infty);\
 y^{(n)}\in\mathcal{L}^p(T,\infty);\\
& \text{there exists $T_1>T$ such that $y(t) = 0$ for $t\geq T_1$}\\
&\text{and } y^{(i)}(T)=0 \text{ for } i=0,\dots,n-1 \Big\}
\end{align*}
and
\begin{align*}
W^{n,p}_0(\mathbb{R})=\Big\{& y \colon \mathbb{R} \to \mathbb{R} :
y^{(n-1)}\in\mathcal{AC}(\mathbb{R});\ y^{(n)}\in\mathcal{L}^p(\mathbb{R}); \\
&\text{and there exists $T_1\in \mathbb{R}$
 such that $y(t)= 0$ for } |t|\geq T_1 \Big\}.
\end{align*}
We  use the following variational lemma which is proved e.g. in
\cite[Sec. 9.4]{D-R-book}.

\begin{lemma} \label{L:variational}
Suppose that there exists $T\in \mathbb{R}$ such that
\begin{equation} \label{EF-n}
{\mathcal F}(y;T,\infty)=\int_T^\infty \Big[\sum_{k=0}^n r_{k}(t)
|y^{(k)}|^p\Big]\,dt>0
\end{equation}
for every nontrivial $y\in W^{n,p}_0[T,\infty)$. Then equation \eqref{HLG}
is nonoscillatory, i.e., no solution of \eqref{HLG} has more than
one zero point of multiplicity $n$ in $[T,\infty)$.
\end{lemma}

Another principal tool we use is the Wirtinger type inequality which we
will apply in the following form, see \cite[Lemma 2.1.1]{D-R-book}.

\begin{lemma} \label{L:0}
Let $M$ be a positive continuously differentiable function for which
$M'(t)\ne 0$ in $[T,\infty)$ and let $y \in W^{1,p}_0[T,\infty)$.
Then
\begin{equation} \label{wirt}
\int_T^\infty |M'(t)||y|^p\,dt
\leq p^{\,p}\int_T^\infty \frac{M^p(t)}{|M'(t)|^{p-1}}|y'|^p\,dt.
\end{equation}
\end{lemma}
If we take  $(-\infty,\infty)$ instead of $[T,\infty)$
and $W^{1,p}_0(\mathbb{R})$ instead of $W^{1,p}_0[T,\infty)$ in Lemma \ref{L:0},
then the corresponding statement also holds.

The previous inequality, with $M^p(t)/|M'(t)|^{p-1}=t^{\alpha}$ and $\alpha\ne p-1$,
applied to $y\in W^{1,p}_0[T,\infty),$ reduces to the inequality
\begin{equation} \label{alpha-ne-p-1}
\int_T^\infty t^{\alpha}|y'|^p\,dt \geq \gamma_{p,\alpha}\int_T^\infty
t^{\alpha-p}|y|^p\,dt,\quad
\gamma_{p,\alpha}=\Big(\frac{|p-1-\alpha|}{p}\Big)^p.
\end{equation}
If $\alpha=p-1$, then we have the inequality
\begin{equation} \label{alpha=p-1}
\int_T^\infty t^{p-1}|y'|^p\,dt \geq \gamma_p\int_T^\infty
\frac{|y|^p}{t\log^p t}\,dt,\quad
\gamma_p=\gamma_{p,0}=\Big(\frac{p-1}{p}\Big)^p.
\end{equation}

We will also use the following auxiliary inequality.

\begin{lemma} \label{L:1}
Let $\beta\in \mathbb{R}$ and $y\in W^{1,p}_0[T,\infty)$, then
\begin{equation} \label{1}
\int_T^\infty \frac{|y|^p}{t^{p\beta+1}\log^p t}\,dt\leq
\frac{1}{\gamma_p}\int_T^\infty t^{p-1}\big|
\big(\frac{y}{t^\beta}\big)'\big|^p\,dt.
\end{equation}
\end{lemma}

\begin{proof}
For $y\in W^{1,p}_0[T,\infty)$, we denote $z=y/t^\beta$ and by 
using integration by parts and the H\"older inequality we have
\begin{align*}
\int_T^\infty \frac{|y|^p}{t^{p\beta+1}\log^p t}\,dt
&= \int_T^\infty \frac{|z|^p}{t\log^p t}\,dt
\\
&= \frac{1}{1-p} \cdot \frac{|z|^p}{\log^{p-1}t}\Big|_T^\infty
 -\frac{p}{1-p}\int_T^\infty \frac{\Phi(z)}{t^{1/q}\log^{p-1}t} \cdot \frac{z'}{t^{-\frac{1}{q}}}\,dt
\\
&\leq \frac{p}{p-1} \Big(\int_T^\infty \frac{|z|^p}{t\log^p t}\,dt\Big)^{1/q}
 \Big(\int_T^\infty t^{p-1}|z'|^p\,dt\Big)^{1/p}
\\
&\leq \gamma_p^{-\frac{1}{q}} \big(\frac{p}{p-1}\big)
 \Big(\int_T^\infty t^{p-1}|z'|^p\,dt\Big)^{1/q}
 \Big(\int_T^\infty t^{p-1}|z'|^p\,dt\Big)^{1/p}
\\
&= \frac{1}{\gamma_p} \int_T^\infty t^{p-1}\big|\big(\frac{y}{t^\beta}\big)'
\big|^p\,dt,
\end{align*}
where between the third and the forth line of the previous computation
inequality \eqref{alpha=p-1} has been used.
\end{proof}

The proof of the next lemma can be found e.g. in \cite{D-MN-97}.

\begin{lemma} \label{L:2}
Let $m\in \{0,\dots,n-1\}$, then 
\begin{equation*}
y^{(n)}=\Big\{\frac{1}{t}\big[t^{m+1}\big(\frac{y}{t^m}\big)'\big]^{(m)}
\Big\}^{(n-m-1)}.
\end{equation*}
\end{lemma}

\section{Nonoscillation criteria} \label{S3}


In this section we formulate and prove Hille-Nehari type nonoscillation
criteria for \eqref{HL}. As we have pointed out in \cite{D-R-EJQTDE}, an
important role in the investigation of oscillatory properties of \eqref{HL}
plays the fact whether or not
$
\alpha\in \{p-1,2p-1,\dots,np-1\}=:{\mathcal M}_p,
$
the case $\alpha\not\in {\mathcal M}_p$ being easier than
the other one. The next theorem deals with the case $\alpha\in {\mathcal M}_p$.

\begin{theorem} \label{T:1}
Suppose that $\alpha=jp-1$ for some $j\in\{1,\dots,n\}$ and
\begin{equation} \label{2}
\liminf_{t\to\infty} \log^{p-1} t \int_t^\infty c_{-}(s)s^{p(n-j)}\,ds>K
\end{equation}
where $c_{-}(t)=\min\{0,c(t)\}$ and
\begin{equation*}
K=-\frac{1}{p}\big(\frac{p-1}{p}\big)^{p-1}[(j-1)!(n-j)!]^p.
\end{equation*}
Then equation \eqref{HL} is nonoscillatory.
\end{theorem}

\begin{proof}
Denote $k=\frac{np-1-\alpha}{p}=n-j\in \mathbb{N}$ and for $y\in W^{n,p}_0[T,\infty)$
denote $z=\frac{y}{t^k}$.
Let $T$ be so large that the limited expression in \eqref{2} is greater than 
$K$ for $t\geq T$.
Using Lemma \ref{L:1} (to obtain the last line from the previous one), we have
\begin{align*}
\int_T^\infty c(t)|y|^p\,dt
&\geq \int_T^\infty c_{-}(t)|y|^p\,dt
= \int_T^\infty c_{-}(t)t^{pk}\big|\big(\frac{y}{t^k}\big)\big|^p\,dt
\\
&= p\int_T^\infty c_{-}(t)t^{pk}\Big(\int_T^t \Phi(z)z'\,ds \Big)dt
\\
&=p\int_T^\infty \Phi(z)z'\frac{1}{\log^{p-1} t}\log^{p-1} t\int_t^\infty
c_{-}(s)s^{pk}\,ds\,dt
\\
&> pK\int_T^\infty \Phi(z)z'\frac{1}{\log^{p-1}t}\,dt
\geq pK\int_T^\infty \frac{|\Phi(z)|}{t^{1/q}\log^{p-1}t}
\cdot t^{1/q}|z'|\,dt
\\
&\geq pK \Big(\int_T^\infty \frac{|y|^p}{t^{pk+1}\log^{q(p-1)}t}\,dt\Big)^{1/q}
\Big(\int_T^\infty t^{\frac{p}{q}}|z'|^p\,dt\Big)^{1/p}
\\
&= pK\Big(\int_T^\infty \frac{|y|^p}{t^{pk+1}\log^p t}\,dt\Big)^{1/q}
\Big(\int_T^\infty t^{p-1}\big|\big(\frac{y}{t^k}\big)'\big|^p\,dt\Big)^{1/p}
\\
&\geq \frac{pK}{\gamma_p^{1/q}}\int_T^\infty t^{p-1}
\big|\big(\frac{y}{t^k}\big)'\big|^p\,dt
\end{align*}
for nontrivial $y\in W^{n,p}_0[T,\infty)$. The second line of the previous
computation comes from the equality $(|z|^p)'=p\Phi(z)z'$
by integrating over $[T,t]$ and using the definition of $z$.
To obtain the fifth line the H\"older
inequality is used together with the equality $|\Phi(z)|^q=|z|^p$.

Next we apply Lemma \ref{L:2} to $\int_T^\infty t^{\alpha}|y^{(n)}|^p\,dt$.
We put $m=k$ in Lemma \ref{L:2}, i.e., $n-m-1=(n-k)-1=j-1$.
Further, denote
\begin{equation*}
u(t)=t^{k+1}\Big(\frac{y(t)}{t^k}\Big)', \quad 
v(t)=\frac{1}{t}\Big[t^{k+1}\Big(\frac{y(t)}{t^k}\Big)'\Big]^{(k)}
= \frac{1}{t}\left[u(t)\right]^{(n-j)}.
\end{equation*}
Then using repeated application of the Wirtinger inequality
\eqref{alpha-ne-p-1} we have
\begin{align*}
\int_T^\infty t^{\alpha}|y^{(n)}|^p\,dt
&= \int_T^\infty t^{jp-1}|v^{(j-1)}|^p\,dt\\
&\geq \left[(j-1)!\right]^p \int_T^\infty t^{p-1}|v|^p\,dt
\\
&=[(j-1)!]^p \int_T^\infty t^{-1}|u^{(n-j)}|^p\,dt
\\
&\geq  [(j-1)!(n-j)!]^p\int_T^\infty
t^{-1-(n-j)p}|u|^p\,dt
\\
&=[(j-1)!(n-j)!]^p \int_T^\infty t^{-1-(n-j)p}t^{(n-j+1)p}
\big|\big(\frac{y}{t^k}\big)'\big|^p\,dt
\\
&=[(j-1)!(n-j)!]^p\int_T^\infty t^{p-1}\big|\big(\frac{y}{t^k}\big)'\big|^p\,dt
\end{align*}
for $y\in W^{n,p}_0[T,\infty)$. Summarizing the previous computations,
\begin{align*}
&\int_T^\infty \big\{t^{\alpha}|y^{(n)}|^p+c(t)|y|^p\big\}\,dt\\
&>\Big\{[(j-1)!(n-j)!]^p+\frac{pK}{\gamma_p^{1/q}}\Big\}
\int_T^\infty t^{p-1}\big|\big(\frac{y}{t^k}\big)'\big|^p\,dt=0
\end{align*}
for nontrivial $y\in W^{n,p}_0[T,\infty)$. This means, 
by Lemma \ref{L:variational}, that \eqref{HL} is nonoscillatory.
\end{proof}

The next example illustrates the nonoscillation criterion in Theorem
\ref{T:1} and shows that the constant $K$ in \eqref{2} cannot be improved.

\begin{example} \label{E:1} \rm
Consider the equation
\begin{equation} \label{equation1}
(-1)^n\bigl( t^{jp-1}\Phi(y^{(n)})\bigr)^{(n)}+
\frac{\gamma}{t^{(n-j)p+1}\log^2 t}\Phi(y)=0
\end{equation}
for some $j\in\{1,\dots,n\}$. Then
\begin{equation*}
\log^{p-1} t \int_t^\infty \frac{\gamma s^{(n-j)p}}{s^{(n-j)p+1}\log^p s}=
\frac{\gamma}{p-1}.
\end{equation*}
Hence, by Theorem \ref{T:1}, equation \eqref{equation1} is nonoscillatory
if
$$
\gamma>-\big(\frac{p-1}{p}\big)^p[(j-1)!(n-j)!]^p.
$$
In particular, if $n=1$ in \eqref{equation1}, then $j=1$ and the 
criterion from Theorem \ref{T:1} complies with the known result
that the second order equation
\begin{equation*}
-\bigl(t^{p-1}\Phi(y')\bigr)'+\frac{\gamma}{t\log^p t}\Phi(y)=0
\end{equation*}
is nonoscillatory if and only if $\gamma\geq -\big(\frac{p-1}{p}\big)^p$.
Note also that we cannot apply Theorem \ref{T:1} if the limit in \eqref{2}
equals the constant $K$ as shows the example of the second order
Riemann-Weber type equation
\begin{equation*}
-\bigl(t^{p-1}\Phi(y')\bigr)'+\Big[-\big(\frac{p-1}{p}\big)^p
\frac{1}{t\log^p t}+
\frac{\mu}{t\log^p t\log^2(\log t)}\Big]\Phi(y)=0
\end{equation*}
which is nonoscillatory if
$\mu\geq-\frac{1}{2}\big(\frac{p-1}{p}\big)^{p-1}$
and oscillatory in the opposite case, see \cite{E-Sch}.
\end{example}

The fundamental role in the proof of the next theorem is played
by a nonoscillation criterion for the \emph{second order} half-linear
differential equations. To formulate it, consider the pair of
second order differential equations
\begin{equation} \label{c}
-\bigl(r(t)\Phi(x')\bigr)'+c(t)\Phi(x)=0
\end{equation}
and its perturbation
\begin{equation} \label{cd}
-\bigl(r(t)\Phi(x')\bigr)'+[c(t)+d(t)]\Phi(x)=0,
\end{equation}
where $r,c,d$ are continuous functions with $r(t)>0$.
The following nonoscillation criterion is proved in \cite[Theorem 3]{D-L}.

\begin{proposition} \label{P:1}
Suppose that \eqref{c} is nonoscillatory and possesses a positive
solution $h$ satisfying
\begin{itemize}
\item[(i)] $h'(t)\ne 0$ for large $t$;
\item[(ii)]
\begin{equation*}
\int^\infty \frac{dt}{r(t)h^2(t)|h'(t)|^{p-2}}=\infty;
\end{equation*}
\item[(iii)] There exists a finite limit
\begin{equation*}
\lim_{t\to\infty} r(t)h(t)\Phi(h'(t))=:L\ne 0.
\end{equation*}
\end{itemize}
Moreover, suppose that the integral $\int^\infty d(t)h^p(t)\,dt$
is convergent.
Then equation \eqref{cd} is nonoscillatory provided
\begin{gather} \label{limsup}
\liminf_{t\to\infty} G(t)\int_t^\infty d(s)h^p(s)\,ds>- \frac{1}{2q},\\
 \label{liminf}
\limsup_{t\to\infty} G(t)\int_t^\infty d(s)h^p(s)\,ds<\frac{3}{2q},
\end{gather}
where $G(t)=\int^t r^{-1}(s)h^{-2}(s)|h'(s)|^{2-p}\,ds$ and
$q$ is the conjugate exponent of $p$.
\end{proposition}

Note that the previous proposition is proved in \cite{D-L}
under the assumption $h'(t)>0$, but a straightforward modification
of the proof shows that it extends also to the case when $h'(t)<0$
for large $t$.

The energy functional on an interval $[T,\infty)$ associated with
\eqref{cd} is
\begin{equation*}
\int_T^\infty \left[r(t)|y'|^p +(c(t)+d(t))|y|^p\right]\,dt
\end{equation*}
and this functional is positive for every
$0\not\equiv y\in W_0^{1,p}[T,\infty)$ if and only if \eqref{cd}
is nonoscillatory and $T$ is sufficiently large, see \cite{D-R-book}.

In the next theorem we use the notation
\begin{equation*} % \label{gamma}
\gamma_{n,p,\alpha}:=\prod_{j=1}^n \Big(
\frac{|jp-1-\alpha|}{p}\Big)^p
\end{equation*}
and we investigate \eqref{HL} as a perturbation of the Euler type
half-linear differential equation
\begin{equation*}
(-1)^n \left(t^\alpha\Phi(y^{(n)})\right)^{(n)}-
\frac{\gamma_{n,p,\alpha}}{t^{np-\alpha}}\Phi(y)=0.
\end{equation*}

\begin{theorem} \label{T:2}
Suppose that $\alpha \not \in \{p-1,2p-1,\dots,np-1\}$ and
the integral
$$
\int^\infty \left(c(t)+\frac{\gamma_{n,p,\alpha}}{ t^{\alpha-np}}\right)
t^{np-1-\alpha}\,dt
$$
is convergent. Equation \eqref{HL} is nonoscillatory provided
\begin{gather} \label{C-1}
\liminf_{t\to\infty} \log t\int_t^\infty \left[c(s)+
\frac{\gamma_{n,p,\alpha}}{s^{np-\alpha}}\right]s^{np-1-\alpha}\,ds>
-\frac{p(p-1)}{2(np-1-\alpha)^2}\gamma_{n,p,\alpha},\\
 \label{C-2}
\limsup_{t\to\infty} \log t\int_t^\infty \left[c(s)+
\frac{\gamma_{n,p,\alpha}}{s^{np-\alpha}}\right]s^{np-1-\alpha}\,ds<
\frac{3p(p-1)}{2(np-1-\alpha)^2}\gamma_{n,p,\alpha}.
\end{gather}
\end{theorem}

\begin{proof}
Denote $d_0(t):=(c(t)+\gamma_{n,p,\alpha}t^{\alpha-np})$.
The energy functional on $[T,\infty)$ associated with \eqref{HL} is
\begin{align*}
\mathcal{F}(y)&= \int_T^\infty \big[t^\alpha |y^{(n)}|^p +c(t)|y|^p\big]\,dt\\
&=\int_T^\infty \left(
t^\alpha|y^{(n)}|^p-\gamma_{n,p,\alpha}t^{\alpha-np}|y|^p\right)\,dt
+ \int_T^\infty d_0(t)|y|^p\,dt.
\end{align*}
The first term in the first integral on the previous line
can be estimated using the Wirtinger inequality as follows
\begin{equation*}
\int_T^\infty t^{\alpha} |y^{(n)}|^p\,dt \geq
\gamma_{n-1,p,\alpha}\int_T^\infty
t^{\alpha-(n-1)p}|y'|^p\,dt.
\end{equation*}
Using this inequality,
\begin{align*}
\mathcal{F}(y)
&=\gamma_{n-1,p,\alpha} \Big\{ \int_T^\infty 
 \Big[ \frac{t^{\alpha}}{\gamma_{n-1,p,\alpha}}|y^{(n)}|^p
 +\Big( \frac{d_0(t)}{\gamma_{n-1,p,\alpha}} 
  -\frac{\gamma_{n,p,\alpha}t^{\alpha-np}}{\gamma_{n-1,p,\alpha}} \Big)|y|^p\Big] 
  \,dt \Big\} \\
&\geq \gamma_{n-1,p,\alpha} 
\Big\{ \int_T^\infty \Big[ \frac{|y'|^p}{t^{(n-1)p-\alpha}}
+\Big( \frac{d_0(t)}{\gamma_{n-1,p,\alpha}}
 -\Big( \frac{|np-1-\alpha|}{p}\Big)^p \frac{1}{t^{np-\alpha}} \Big)|y|^p\Big]\,dt
 \Big\}.
\end{align*}
The last integral is the energy functional associated with the second
order half-linear differential equation
\begin{equation} \label{d}
-\Big(t^{\alpha-(n-1)p}\Phi(x')\Big)'+
\Big[-\Big(\frac{|np-1-\alpha|}{p}\Big)^pt^{\alpha-np}+
\frac{d_0(t)}{\gamma_{n-1,p,\alpha}}\Big]\Phi(x)=0
\end{equation}
and this functional is positive for every $0\not\equiv y\in W_0^{1,p}
[T,\infty)$ if and only if \eqref{d} is nonoscillatory and $T$ is sufficiently
large.

Next, we apply Proposition \ref{P:1} to \eqref{d} with
$$
r(t)=t^{\alpha-(n-1)p}, \quad c(t)= -\Big(\frac{|np-1-\alpha|}{p}\Big)^p
t^{\alpha-np} \quad \text{and} \quad 
d(t)=\frac{d_0(t)}{\gamma_{n-1,p,\alpha}}.
$$
The equation
\begin{equation*}
-\left(t^{\alpha-(n-1)p}\Phi(x')\right)'-
\Big(\frac{|np-1-\alpha|}{p}\Big)^pt^{\alpha-np}\Phi(x)=0
\end{equation*}
has a solution $h(t)=t^{(np-1-\alpha)/p}$ (i.e. nonoscillatory)
for which $h'(t)\ne 0$ for $t>0$.
By a direct computation we have
\begin{gather*}
r(t)h(t)\Phi(h'(t))=\Phi\Big(\frac{np-1-\alpha}{p}\Big)\ne 0, \\
r(t)h^2(t)|h'(t)|^{p-2}=\Big(\frac{|np-1-\alpha|}{p}\Big)^{p-2}t,
\end{gather*}
hence (ii) and (iii) of Proposition \ref{P:1} are satisfied.
Moreover,
\begin{equation*}
G(t)=\int^t r^{-1}(s)h^{-2}(s)|h'(s)|^{2-p}\,ds
= \Big(\frac{p}{|np-1-\alpha|}\Big)^{p-2}\log t.
\end{equation*}
Then \eqref{limsup} reads as follows
(note that $q=p/(p-1)$)
\begin{equation*}
\liminf_{t\to\infty} \Big(\frac{p}{|np-1-\alpha|}\Big)^{2-p}
\log t \int_t^\infty \frac{d_0(s)}{\gamma_{n-1,p,\alpha}}s^{np-1-\alpha}\,ds
>-\frac{1}{2}\big(\frac{p-1}{p}\big)
\end{equation*}
and substituting for $d_0(s)$ we have
\begin{align*}
\liminf_{t\to\infty} \log t\int_t^\infty
\left(c(s)+\frac{\gamma_{n,p,\alpha}}{s^{np-\alpha}}\right)
s^{np-1-\alpha}\,ds
&> -\frac{p-1}{2p}\Big(\frac{|np-1-\alpha|}{p}\Big)^{p-2}\gamma_{n-1,p,\alpha}
\\
&=-\frac{p(p-1)\gamma_{n,p,\alpha}}{2(np-1-\alpha)^2}.
\end{align*}
Similarly, \eqref{liminf} reduces to
\begin{equation*}
\limsup_{t\to\infty} \log t\int_t^\infty
\left(c(s)+\frac{\gamma_{n,p,\alpha}}{s^{np-\alpha}}\right)
s^{np-1-\alpha}\,ds<
\frac{3p(p-1)\gamma_{n,p,\alpha}}{2(np-1-\alpha)^2}.
\end{equation*}
Hence, if \eqref{C-1}, \eqref{C-2} hold, equation \eqref{d} is
nonoscillatory and the functional
\begin{equation*}
\int_T^\infty \Big[ \frac{|y'|^p}{t^{(n-1)p-\alpha}}
-\Big(\frac{|np-1-\alpha|}{p}\Big)^p\frac{|y|^p}{t^{np-\alpha}}+
\frac{d_0(t)}{\gamma_{n-1,p,\alpha}}|y|^p\Big]\,dt>0
\end{equation*}
if $T$ is sufficiently large what we needed to prove.
\end{proof}

\begin{remark} \rm
If $d(t)\leq 0$ in \eqref{cd}, then, of course, condition \eqref{liminf}
is redundant. If $d(t)\geq 0$, then \eqref{cd} is a minorant to
\eqref{c} and its nonoscillation follows from the half-linear
Sturmian theory, see \cite{D-R-book}.
\end{remark}

\begin{corollary} \label{C:2}
Consider the higher order Riemann-Weber type half-linear differential
equation
\begin{equation} \label{RW}
(-1)^n \bigl(t^{\alpha}\Phi(y^{(n)})\bigr)^{(n)}
-\Big[\frac{\gamma_{n,p,\alpha}}{t^{np-\alpha}}
+\frac{\mu}{t^{np-\alpha}\log^2 t}\Big]\Phi(y)=0
\end{equation}
with $\alpha\not\in {\mathcal M}_p$.
Then \eqref{RW} is nonoscillatory if

\begin{equation} \label{inequality-mu}
\mu<\frac{p(p-1)\gamma_{n,p,\alpha}}{2(np-1-\alpha)^2}.
\end{equation}
\end{corollary}

\begin{proof}
We denote $c(t)=-[\frac{\gamma_{n,p,\alpha}}{t^{np-\alpha}}
+\frac{\mu}{t^{np-\alpha}\log^2 t}]$ and we show that assumptions of
Theorem~\ref{T:2} are satisfied.
We have
$$
\int_t^\infty \left(c(s)+\frac{\gamma_{n,p,\alpha}}{ s^{\alpha-np}}\right)
s^{np-1-\alpha}\,ds
 = -\int_t^\infty \frac{\mu}{s\log^2 s}\,ds =
-\frac{\mu }{\log t}.
$$
Condition \eqref{C-2} is obvious (see proof of Theorem~\ref{T:2} and
Remark~1) and condition \eqref{C-1} is reduced to the condition
$$
\mu<\frac{p(p-1)\gamma_{n,p,\alpha}}{2(np-1-\alpha)^2}.
$$
\end{proof}

\begin{example} \rm
Consider the case $n=1$ in the previous corollary.
Then equation \eqref{RW} reduces to the second order Riemann-Weber
type equation
\begin{equation} \label{RW2}
\bigl(t^\alpha \Phi(y')\bigr)'+\Big[\Big(\frac{|p-1-\alpha|}{p}\Big)^p
t^{\alpha-p}+\frac{\mu}{t^{p-\alpha}\log^2 t}\Big]\Phi(y)=0.
\end{equation}
It is known, see \cite{D-Y}, that this equation is nonoscillatory
if
$$
\mu\leq \mu_{p,\alpha},\quad 
\mu_{p,\alpha}:=\frac{p-1}{2p}
\Big(\frac{|p-1-\alpha|}{p}\Big)^{p-2}
$$
and oscillatory in the opposite case.
This result shows that inequality in \eqref{inequality-mu} is exact
since in the case $n=1$
$$
\frac{p(p-1)\gamma_{n,p,\alpha}}{2(np-1-\alpha)^2}=
\frac{p(p-1)}{2(p-1-\alpha)^2}\Big(\frac{|p-1-\alpha|}{p}\Big)^p
=\mu_{p,\alpha}.
$$
This result also shows that the constant in the right-hand side of
inequality \eqref{C-1} cannot be improved.
\end{example}

\section{Negativity of the energy functional} \label{sec4}


As a motivation, let us consider the second order half-linear differential
equation
\begin{equation} \label{HLL}
-\bigl(r(t)\Phi(y')\bigr)'+c(t)\Phi(x)=0
\end{equation}
with continuous functions $c,r$ and $r(t)>0$.
It was proved in \cite{D-MS}
that if
$$
\int_{-\infty}r^{1-q}(t)\,dt=\infty=\int^\infty r^{1-q}(t)\,dt,\quad
\frac{1}{p}+\frac{1}{q}=1,
$$
and
$$
\int_{-\infty}^{\infty} c(t)\,dt\leq 0, \quad c(t)\not\equiv 0,
$$
then \eqref{HLL} is \emph{conjugate} on ${\mathbb R}$, i.e., there exists a
nontrivial solution with at least two different zeros on $\mathbb{R}$.
Conjugacy of \eqref{HLL} is equivalent to the existence of a
nontrivial function $y\in W_0^{1,p}(\mathbb{R})$ for which the energy functional
associated with \eqref{HLL}
\begin{equation}
{\mathcal F}(y;\mathbb{R})
=\int_{-\infty}^{\infty} \left[r(t)|y'|^p+c(t)|y|^p\right]\,dt
\end{equation}
attains a negative value. %and this implies that it is unbounded below.
In the terminology of linear equations, see \cite{G-Z}, a differential
operator with the property that there exists a function from a suitable
Sobolev space for which the associated energy functional is negative is called
\emph{supercritical}.

Concerning the $2n$-order linear differential equation
\begin{equation} \label{SL-n}
(-1)^n \left(r(t)y^{(n)}\right)^{(n)}+c(t)y=0
\end{equation}
a similar statement was proved first in \cite{M-Pf-PRSE} for a fourth
order linear equation and later it was extended to general
$2n$-order equation \eqref{SL-n} in \cite{D-PRSE-89}. This result
says that if there exists an integer $m$, $0\leq m\leq n-1$, such that
$$
\int_{-\infty} t^{2m}r^{-1}(t)dt=\infty
= \int^{\infty} t^{2m}r^{-1}(t)dt
$$
and there exists a polynomial $Q(t)=a_kt^k+\dots+a_1t+a_0$ of the
degree $0\leq k\leq n-m-1$ such that
$$
\int_{-\infty}^{\infty} Q^2(t)c(t)\,dt<0,
$$
then  \eqref{SL-n} is conjugate
on $\mathbb{R}$, i.e., there exists a nontrivial solution of \eqref{SL-n}
having two different zeros of multiplicity $n$ in $\mathbb{R}$. Again, this
statement is equivalent to the fact that the associated energy functional
$$
\int_{-\infty}^{\infty}\left[r(t)|y^{(n)}|^2+c(t)|y|^2\right]\,dt
$$
attains a negative value for some $y\in W^{n,p}_0(\mathbb{R})$.
In the next theorem we present a partial extension of these results to
\eqref{HL} with $\alpha=0$.


\begin{theorem} \label{T:conjugacy}
Suppose that
\begin{equation} \label{positivity}
\int_{-\infty}^{\infty} c(t)\,dt<0
\end{equation}
and $c(t)\leq 0$ for $t$ close to $-\infty$ and $\infty$.
Then the energy functional
\begin{equation} \label{QFF}
{\mathcal F}_n(y;\mathbb{R})=\int_{-\infty}^{\infty} \left[|y^{(n)}|^p+
c(t)|y|^p\right]\,dt
\end{equation}
associated with the equation
\begin{equation} \label{alpha=0}
(-1)^n \bigl(\Phi(y^{(n)})\bigr)^{(n)}+c(t)\Phi(y)=0
\end{equation}
attains a negative value over $W^{n,p}_0(\mathbb{R})$.
\end{theorem}

\begin{proof}
According to \eqref{positivity}, there exist $t_1<t_2$ such that
\begin{equation*} \label{epsilon}
\int_{t_1}^{t_2} c(t)\,dt=:-\varepsilon<0\quad\text{and}\quad
c(t)\leq 0, \ t\in (-\infty,t_1]\cup[t_2,\infty).
\end{equation*}
Let $t_0<t_1<t_2<t_3$ (the values $t_0,t_3$ will be specified later) and
define the test function as follows
$$
y(t)=\begin{cases} 
0 & t\in (-\infty,t_0],\\
 f(t) & t\in [t_0,t_1],\\
 1 & t\in [t_1,t_2],\\
 g(t) & t\in [t_2,t_3],\\
 0 & t\in [t_3,\infty).
\end{cases}
$$
The function $f$ is defined using the following construction
(the construction of $g$ will be
specified later). To simplify the notation,
we denote $\delta:=q-1$ ($q$ is the conjugate exponent of $p$). Let
\begin{gather*}
y_1(t)=(t-t_0)^n,\ y_2(t)= (t-t_0)^{\delta +n}, %\ y_3=(t-t_0)^{2\delta +n},
\dots,
y_n(t)=(t-t_0)^{(n-1)\delta+n}.
\end{gather*}
These functions are solutions of $\Phi(y^{(n)})=C_k(t-t_0)^k$, $k=0,\dots,n-1$
for suitable constants $C_k$ (i.e., of $\left(\Phi(y^{(n)})\right)^{(n)}=0$)
for $t\geq t_0$.
We define $f$ as a linear combination $f=c_1y_1+\dots+c_ny_n$ where the
constants $c_1,\dots,c_n$ we define in such a way that $f$ satisfies the
conditions
$$
f(t_1)=1,\ f^{(i)}(t_1)=0,\ i=1,\dots,n-1,
$$
(because we need $y\in W_0^{n,p}(\mathbb{R})$). This means that the 
constants $c_1,\dots,c_n$
form a solution of the linear system (where we denote $T:=t_1-t_0$)
\begin{align*}
1& =T^n c_1 + T^{\delta+n}c_2+\dots+ T^{(n-1)\delta +n}c_n,\\
0& =nT^{n-1}c_1+ (\delta+n)T^{\delta+n-1}c_2+\dots + [(n-1)\delta+n]T^{(n-1)\delta+n-1}c_n,\\
 &\quad \dots\\
0&= \Delta_{i,1}T^{n-i}c_1+ \Delta_{i,2} T^{\delta+n-i}c_2+\dots+
 \Delta_{i,n} T^{(n-1)\delta+n-i}c_n,\\
 &\quad  \dots\\
0&= n! Tc_1+ \Delta_{n,2}T^{\delta+1} c_2+\dots + \Delta_{n,n} T^{(n-1)\delta+1}c_n,
\end{align*}
where we have used the notation $\Delta_{i,j}:=[(j-1)\delta+n]\dots[(j-1)\delta+n-i+2]$.
The determinant of the matrix of this linear system can be expressed as follows.
We factor out $T^{(j-1)\delta}$ from the $j$-th column
and then $T^{n-i+1}$ from the $i$-th row.
Then it remains to calculate the determinant (where we explicitly write
the quantities $\Delta_{i,j}$) $\det \Delta_n$, where
\begin{equation*}
\Delta_n:=\begin{pmatrix}
1 & 1 & \dots & 1\\
n & (\delta+n) & \dots &(n-1)\delta +n\\
n(n-1) & (\delta\!+\!n)(\delta\!+\!n\!-\!1)
&\dots &[(n\!-\!1)\delta\!+\!n][(n\!-\!1)\delta\!+\!n\!-\!1]\\
 & & \vdots & \\
\prod_{l=1}^i(n\!-\!l\!+\!1) & \prod_{l=1}^i(\delta\!+\!n\!-\!l\!+\!1) &
\dots &\prod_{l=1}^i[(n\!-\!1)\delta\! +\!n\!-\!l\!+\!1]\\
& & \vdots & \\
n! & (\delta\!+\!n)\cdots(\delta\!+\!2) & \dots & [(n\!-\!1)
\delta\!+\!n]\cdots[(n\!-\!1)\delta\!+\!2]
\end{pmatrix}.
\end{equation*}

In the last section we show that this determinant is nonzero,
so the determinant of the linear
system for $c_k$ is $D:= T^{\frac{n(n+1)}{2}+ \frac{n(n-1)\delta}{2}} \det \Delta_n \ne 0$. By the Cramer rule
we find that the coefficients $c_k=c_k(T)$ can be expressed as
\begin{equation} \label{c-formula}
c_k(T)= h_k T^{-n-\frac{n(n-1)\delta}{2}}T^{\frac{n(n-1)\delta}{2}-(k-1)\delta}
=h_kT^{-n-(k-1)\delta}
\end{equation}
the constants $h_k$ can be expressed explicitly, but their values are
not important for our computations at this moment. Consequently,
\begin{align*}
f^{(n)}(t)
&=c_1(T)y^{(n)}(t)+\dots + c_n(T)y^{(n)}_n(t)
= c_1(T)n! + c_2(T)\tilde h_2(t-t_0)^\delta+ \dots
\\
&\dots+ c_k(T)\tilde h_k (t-t_0)^{(k-1)\delta}+\dots+ c_n(T)\tilde h_n(t-t_0)^{(n-1)\delta},
\end{align*}
where $\tilde h_k=[(k-1)\delta +n]\cdots[(k-1)\delta +1]$.
Consequently, in view of \eqref{c-formula} and using the Jensen inequality
for the function $x \mapsto |x|^p$,  we have
\begin{align*}
\int_{t_0}^{t_1} |f^{(n)}(t)|^p\,dt
& =\int_{t_0}^{t_1}\big| \sum_{k=1}^n h_k\tilde h_k T^{-n-(k-1)\delta}
(t-t_0)^{(k-1)\delta}\big|^p \,dt \\
&\leq \int_{t_0}^{t_1} n^{p-1}\sum_{k=1}^n \big|h_k\tilde h_k T^{-n-(k-1)\delta}
T^{(k-1)\delta} \big|^p \,dt
\\
&=CT^{-pn+1}\to 0\quad \text{as } T\to\infty, \text{ i.e., as } t_0\to-\infty,
\end{align*}
where $C=n^{p-1}\sum_{k=1}^n \left|h_k\tilde h_k\right|^p$.

The construction of the function $g$ is similar. It is a function satisfying
the boundary condition $g(t_2)=1$, $g^{(i)}(t_2)=0$, $i=1,\dots, n-1$,
$g^{(i)}(t_3)=0$, $i=0,\dots,n-1$. This function we construct as a linear
combination of the functions
$$
\tilde y_k(t)=(t_3-t)^{(k-1)\delta +n},\quad k=1,\dots,n.
$$
Similarly as for the function $f$, we have
\begin{equation} \label{g-convergence}
\int_{t_2}^{t_3}|g^{(n)}(t)|^p\,dt\to 0\quad\text{as}\ t_3\to\infty.
\end{equation}


Summarizing the previous computations, we see that $t_0,t_3$
can be chosen in such a way
that
\begin{equation*}
\int_{t_0}^{t_1}|f^{(n)}(t)|^p\,dt<\frac{\varepsilon}{4},\quad
\int_{t_2}^{t_3}|g^{(n)}(t)|^p\,dt<\frac{\varepsilon}{4}.
\end{equation*}
Then we have
\begin{align*}
\int_{t_0}^{t_3} \left[|y^{(n)}(t)|^p+\,c(t)|y(t)|^p\right]\,dt 
&= \int_{t_0}^{t_1} |f^{(n)}(t)|^p\,dt
+\int_{t_0}^{t_1} c(t)|f(t)|^p\,dt -\int_{t_1}^{t_2} c(t)\,dt
\\
& + \int_{t_2}^{t_3} |g^{(n)}(t)|^p\,dt +\int_{t_2}^{t_3} c(t)|g(t)|^p\,dt
\\
< & \frac{\varepsilon}{4} -\varepsilon + \frac{\varepsilon}{4}<0,
\end{align*}
where we have used that $c(t)\leq 0$ for $t\in (-\infty,t_1]\cup [t_2,\infty)$.
\end{proof}

The formulation of the statement and construction of the test function
in the proof of the next theorem is a modification of Theorem
\ref{T:conjugacy}.
The meaning of the this theorem from the point of view of the oscillation theory
of higher order half-linear differential equations is discussed at the
end of this section.

\begin{theorem} \label{T:oscillation}
Suppose that $c(t)\leq 0$ for large $t$. If
\begin{equation} \label{q-divergence}
\int^{\infty} c(t)\,dt=-\infty,
\end{equation}
then there exists $T\in \mathbb{R}$ such that the energy functional
\begin{equation} \label{QFFF}
\int_{T}^\infty \big[\big|y^{(n)}\big|^p+c(t)|y|^p\big]\,dt
\end{equation}
associated with equation \eqref{alpha=0} attains a negative value
over $W^{n,p}_0[T,\infty)$.
\end{theorem}

\begin{proof}
Let $T\in \mathbb{R}$ be arbitrarily large and $T<t_0<t_1<t_2<t_3$.
Define the function $y$ essentially in the same way as in the previous proof,
only comparing with that proof, the function
$f$ may be arbitrary function satisfying at $t_0$ and $t_1$
the boundary condition $f^{(i)}(t_0)=0$, $i=0,\dots,n-1$,
$f(t_1)=1$, $f^{(i)}(t_1)=0$, $i=1,\dots,n-1$. We denote
$K=\int_{t_0}^{t_1} |f^{(n)}(t)|^p\,dt+\int_{t_0}^{t_1} c(t)|f^p(t)|\,dt$.
Now, we take $t_2$
so large that $c(t)\leq 0$ for $t\geq t_2$ and
$$
\int_{t_1}^{t_2} c(t)\,dt< -3K.
$$
The function $g$ is then the same as in the previous proof
with $t_3$ so large that $\int_{t_2}^{t_3} |g^{(n)}(t)|^p\,dt<K.$
Then, for the function $y$ constructed in this way, we have
\begin{align*}
\int_{T}^\infty \big[|y^{(n)}|^p\,dt + c(t)|y|^p\big]\,dt
&= \int_{t_0}^{t_1}|f^{(n)}(t)|^p\,dt+\int_{t_0}^{t_1}c(t)|f(t)|^p\,dt
\\
&\quad +\int_{t_1}^{t_2} c(t)\,dt+\int_{t_2}^{t_3}|g^{(n)}(t)|^p\,dt+
\int_{t_2}^{t_3} c(t)|g(t)|^p\,dt
\\
&\leq  K-3K+K<0,
\end{align*}
what we needed to prove.
\end{proof}

\begin{remark} \rm
(a) If $p=2$ in Lemma \ref{L:variational}, i.e., we consider linear equation
\eqref{LEG} and the associated quadratic functional \eqref{QF2}, we have
\emph{equivalence} in Lemma \ref{L:variational}. This equivalence is based
on the so-called Reid Roundabout theorem for associated linear 
Hamiltonian differential systems.

An analogue of the Roundabout theorem is missing for half-linear
Hamiltonian type system \eqref{LHS-1/2}, so we only have one implication
in Lemma \ref{L:variational} at this moment. Nevertheless, we
conjecture that the equivalence holds also in the half-linear case,
this problem is a subject of the present investigation (note
that this conjecture is true for second order equations \eqref{c},
see \cite[Chap.~2]{D-R-book}).
Having proved this conjecture, the construction of the test
function in the proofs of Theorem \ref{T:conjugacy} and Theorem
\ref{T:oscillation} can be used to establish various oscillation criteria
for \eqref{HL} similarly as in the linear case in
\cite{D-MN-94,D-MN-97,D-O-CZMJ,H-L,M-Pf-APM}.

(b) Since the energy functionals associated with half-linear equations
are homogeneous (of degree $p$), the fact that these functionals attain
a negative value also means that they are unbounded below.
\end{remark}

\section{A technical result}

In this section we prove that the determinant of the matrix $\Delta_n$
from the previous section is really nonzero, so the constants $c_k(T)$,
 $k=1,\dots,n,$
can be computed using the Cramer rule. This result may be known for people
working in the linear algebra, but we have not found it in the literature, so
we present it here.

Recall that we consider the matrix (with $\delta>0$)
\begin{equation*}
\Delta_n:=\begin{pmatrix}
1 & 1 & \dots & 1\\
n & (\delta+n) & \dots &(n-1)\delta +n\\
n(n-1) & (\delta\!+\!n)(\delta\!+\!n\!-\!1)
&\dots &[(n\!-\!1)\delta\!+\!n][(n\!-\!1)\delta\!+\!n\!-\!1]\\
 & & \vdots & \\
\prod_{l=1}^{i-1}(n\!-\!l\!+\!1) & \prod_{l=1}^{i-1}(\delta\!+\!n\!-\!l\!+\!1) &
\dots &\prod_{l=1}^{i-1}[(n\!-\!1)\delta\! +\!n\!-\!l\!+\!1]\\
& & \vdots & \\
n! & (\delta\!+\!n)\cdots(\delta\!+\!2) & \dots & [(n\!-\!1)\delta\!+\!n]\cdots[(n\!-\!1)\delta\!+\!2]
%\end{array}\right|.
\end{pmatrix}.
\end{equation*}

\begin{lemma} Let $\delta>0$ and $n \in \mathbb{N}$. Then
$$
\det(\Delta_n)= \delta^{\frac{n(n-1)}{2}}
\prod_{k=1}^n \left(k-1\right)!.
$$
\end{lemma}

\begin{proof}
Let $n \in \mathbb{N}$ be arbitrary but fixed in the following considerations.
Denote $A:=\Delta_n$, where $A=\left( a_{i,j} \right)_{i,j=1}^n$. Hence
$$
a_{i,j}=\prod_{l=1}^{i-1} \left[(j-1)\delta+n-l+1 \right].
$$

Using elementary row operations, we will find a triangular matrix with
the determinant equal to that of the original matrix $A$. For this purpose,
we will construct a finite sequence of square matrices
$A^{[1]},\dots,A^{[n]}$ such that
$A^{[1]}=( a_{i,j}^{[1]})_{i,j=1}^n=A$ and the matrix
$A^{[k]} = ( a_{i,j}^{[k]} )_{i,j=1}^n$ will be obtained from the
matrix $A^{[k-1]}$ by applying $n-1-(k-2)$ elementary row operations
for $k=2,\dots,n$. More precisely, we obtain the matrix $A^{[k]}$ by
subtracting a suitable multiple of $(i-1)$-th row of the matrix $A^{[k-1]}$
from the $i$-th row of the matrix $A^{[k-1]}$, and we repeat this for each
$i \geq k$ (for $i<k$, the rows $( a_{i,j}^{[k]})_{j=1,\dots, n}$
will be the same as in matrix $A^{[k-1]}$). As a suitable multiple of $(i-1)$-th
row, we consider such multiple, which after subtracting from $i$-th row
gives the zero on the first nonzero position of this $i$-th row.

Before constructing such a sequence of matrices, note that the first row of the
matrix $A^{[1]}$ will be the same as first rows of the matrices
$A^{[1]},\dots,A^{[n]}$, then, in particular, we have
$$
a_{1,1}^{[m]} = a_{1,1}^{[1]}=1
$$
for $m=1,\dots,n$. \par
Now let us construct the matrix $A^{[2]}$. Let $i\in\{2,\dots,n\}$
be arbitrary, but fixed and let us consider the rows
$( a_{i-1,j}^{[1]} )_{j=1,\dots, n}$ and
$( a_{i,j}^{[1]})_{j=1,\dots, n}$, i.e., the submatrix written as
$$
\begin{pmatrix}
\prod_{l=1}^{i-2} \left(n-l+1\right) &\dots 
&\prod_{l=1}^{i-2} \left[(j-1)\delta+n-l+1 \right] &\dots  \\
&&\\
\prod_{l=1}^{i-1} \left(n-l+1\right) &\dots 
&\prod_{l=1}^{i-1} \left[(j-1)\delta+n-l+1 \right] &\dots 
\end{pmatrix}.
$$
After subtracting the $(i-1)$-th row multiplied by $n-(i-1)+1$
from the $i$-th row of the matrix $A^{[1]}$, we obtain entries
of the $i$-th row of the matrix $A^{[2]}$, i.e.,
\begin{align*}
a_{i,j}^{[2]} &= \prod_{l=1}^{i-1} [(j-1)\delta+n-l+1 ]-(n-i+2)
 \prod_{l=1}^{i-2} [(j-1)\delta+n-l+1 ] \\
 &= \prod_{l=1}^{i-2} [(j-1)\delta+n-l+1 ] \cdot 
 \{(j-1)\delta+n-(i-1)+1-(n-i-2)\}
\\
 &= (j-1)\delta \prod_{l=1}^{i-2} [(j-1)\delta+n-l+1 ]
\end{align*}
for $j=1,\dots,n$.

Note that the second row of the matrix $A^{[2]}$ will be again the same as
second rows of the matrices $A^{[3]},\dots,A^{[n]}$, then, in particular,
we have
$$
a_{2,2}^{[m]} = a_{2,2}^{[2]}=
(2-1)\delta \prod_{l=1}^0 [(2-1)\delta +n-l+1]=\delta
$$
for $m=2,\dots,n$. 

We obtain similar relations for the $i$-th row of the matrix $A^{[2]}$,
where $i\in\{3,\dots,n\}$. Then we have
$$
a_{i,j}^{[3]} = (j-2)(j-1)\delta^2 \prod_{l=1}^{i-3}
[(j-1)\delta+n-l+1]
$$
for $j=1,\dots,n$. 

Now we describe the general procedure of constructing of the matrix $A^{[k]}$
from the matrix $A^{[k-1]}$. Again, let $i\in\{k,k+1,\dots,n\}$ be arbitrary,
but fixed and let us consider rows 
$( a_{i-1,j}^{[k-1]})_{j=1,\dots, n}$,
$( a_{i,j}^{[k-1]})_{j=1,\dots, n}$, which are of the form
$$
\left(
\begin{array}{lllll}
0 & \dots & 0 & (k-2)(k-3)\cdot \dots \cdot 2 \delta^{k-2}
\prod_{l=1}^{i-k} \left[(k-2)\delta+n-l+1 \right] &
\dots\phantom{\,}\phantom{\,}\phantom{\,}\phantom{\,}\phantom{\,}\phantom{\,}
\\
0 & \dots & 0 & (k-2)(k-3)\cdot \dots \cdot 2 \delta^{k-2}
\prod_{l=1}^{i-(k-1)} \left[(k-2)\delta+n-l+1 \right] &
\dots\phantom{\,}\phantom{\,}\phantom{\,}\phantom{\,}\phantom{\,}\phantom{\,}
\end{array}
\right.
$$
$$
\left.
\begin{array}{ll}
\phantom{\,}\phantom{\,}\phantom{\,}\phantom{\,}\dots & (j-1)(j-2)\cdot \dots \cdot (j-(k-2))\delta^{k-2}
\prod_{l=1}^{i-k} \left[(j-1)\delta+n-l+1 \right]
\\
\phantom{\,}\phantom{\,}\phantom{\,}\phantom{\,}\dots & (j-1)(j-2)\cdot \dots \cdot (j-(k-2))\delta^{k-2}
\prod_{l=1}^{i-(k-1)} \left[(j-1)\delta+n-l+1 \right] \\
\end{array}
\right),
$$
with the $(k-1)$-th column containing first nonzero coefficients.
Therefore, we
subtract the
$(i-1)$-th row multiplied by $(k-2)\delta+n-(i-(k-1))+1$
from the $i$-th row of the matrix $A^{[k-1]}$ to obtain
entries of the $i$-th row of the matrix $A^{[k]}$, i.e.
\begin{align*}
a_{i,j}^{[k]}
&=(j-1)(j-2) \dots (j-(k-2))\delta^{k-2}\prod_{l=1}^{i-(k-1)} 
\left[(j-1)\delta+n-l+1 \right]
\\
 &\quad-\left\{(k-2)\delta+n-(i-(k-1))+1\right\} 
  \cdot (j-1)(j-2)\cdot \dots \cdot (j-(k-2))
\\
 &\quad \times \delta^{k-2}\prod_{l=1}^{i-k} \left[(j-1)\delta+n-l+1 \right]
\\
 &=(j-1)(j-2)\dots (j-(k-2))\delta^{k-2}\prod_{l=1}^{i-k} 
  \left[(j-1)\delta+n-l+1 \right]
\\
 &\quad \times \left\{(j-1)\delta+n-(i-(k-1))+1-((k-2)\delta+n-(i-(k-1))+1) \right\}
\\
 &=(j-1)(j-2)\cdot \dots \cdot (j-(k-1))\delta^{k-1}\prod_{l=1}^{i-k} 
  \left[(j-1)\delta+n-l+1 \right]
\end{align*}
for $j=1,\dots,n$. Such $a_{i,j}^{[k]}$ correspond to the expected form,
which had to be proved. 

The $k$-th rows of the matrices $A^{[k]},A^{[k+1]},\dots,A^{[n]}$ are equal,
and it follows
\begin{align*}
a_{k,k}^{[m]}=a_{k,k}^{[k]} 
&= (k-1)(k-2)\cdot \dots \cdot(k-(k-1))\delta^{k-1}
\prod_{l=1}^{0} \left[(k-1)\delta+n-l+1 \right] \\
&= (k-1)!\delta^{k-1}
\end{align*}
for $m=k,k+1\dots,n$. 

From this construction, it is clear, that $A^{[n]}$ is an upper triangular
matrix and that the diagonal elements of $A^{[n]}$ are
$$
\Big( a_{1,1}^{[n]},a_{2,2}^{[n]},\dots,a_{n,n}^{[n]} \Big)
=\Big( a_{1,1}^{[1]},a_{2,2}^{[2]},\dots,a_{n,n}^{[n]} \Big).
$$
Obviously, the determinant of the matrix $A$ has not changed by performed
operations, therefore, it holds
$$
\det(A)=\det(A^{[n]})=\prod_{k=1}^n a_{k,k}^{[n]}
= \prod_{k=1}^n \left[ (k-1)! \delta^{k-1} \right]
= \delta^{\frac{n(n-1)}{2}}\prod_{k=1}^n(k-1)!,
$$
which proves the lemma.
\end{proof}

\subsection*{Acknowledgements}
The research was supported by the Grant      
GA16-00611S of the Czech Grant Foundation and 
and by the Research Project MUNI/A/1154/2015 of Masaryk University.


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\end{document}