# Magnetic field driven Lifshitz transition and one-dimensional Weyl nodes in
three-dimensional pentatellurides

Zhigang Cai  School of Science, Jiangnan University, Wuxi 214122, China  Yi-
Xiang Wang<EMAIL_ADDRESS>School of Science, Jiangnan
University, Wuxi 214122, China  School of Physics and Electronics, Hunan
University, Changsha 410082, China

###### Abstract

Recent experiments reported that the magnetic field can drive the Lifshitz
transition and one-dimensional (1D) Weyl nodes in the quantum limit of three-
dimensional pentatellurides, as they own low carrier densities and can achieve
the extreme quantum limit at a low magnetic field. In this paper, we will
investigate the conditions for the existence of the 1D Weyl nodes and their dc
transport properties. We find that in the strong topological insulator (TI)
phase of ZrTe5, the formation of the Weyl nodes depends heavily on the carrier
density; while in the weak TI phase of HfTe5, the Weyl nodes are more likely
to appear. These behaviors are attributed to the fact that in the strong and
weak TI phases, the zeroth Landau levels exhibit opposite evolutions with the
magnetic field. Moreover, the signatures of the critical fields that
characterize the distinct behaviors of the system can be directly captured in
the conductivities.

##  I Introduction

When a perpendicular magnetic field is applied on a two-dimensional (2D) or
three-dimensional (3D) electronic system, the electrons will be confined to
move on curved orbits due to the Lorentz force L.D.Landau , and, consequently,
a set of discrete energy levels, i.e., the Landau levels (LLs) will form. If
the magnetic field is strong and a few LLs are occupied, the system lies in
the quantum Hall regime K.V.Klitzing ; D.C.Tsui ; D.Yoshioka . Further
increasing the magnetic field, if all electrons occupy only the zeroth LLs,
the extreme quantum limit will be reached. For most semiconductor materials,
realizing the quantum limit seems impossible as the required magnetic field
strength would be inaccessible in the experiment.

In the past 20 years, the developments of topological materials have provided
insights to investigate the interactions between the magnetic field and
electronic systems M.Z.Hasan ; X.L.Qi ; Lv2021 ; J.A.Sobota . Among the
emerging topological materials, 3D pentatellurides Q.Li ; T.Liang2018
including ZrTe5 and HfTe5 are representatives and can exhibit a number of
desirable features: (i) the low-energy bands are well described by the
effective noninteracting models that are topologically nontrivial H.Weng ;
(ii) they have narrow gaps and small band masses, leading to the sizable
cyclotron frequencies even for a weak magnetic field E.Martino ; Z.G.Chen ;
and (iii) the crystal sample owns a high purity, with the electron mobility
reaching the order of $10^{5}$ cm2V-1s-1, and the low carrier density of about
$10^{16}\sim 10^{17}$ cm-3 at a low temperature F.Tang ; E.Martino ; P.Wang ;
S.Galeski2020 . Therefore, the quantum limit may be reached in 3D
pentatellurides at a weak magnetic field, which makes its study possible.

Recent experiments in pentatellurides reported that the system can indeed
reach the quantum limit at the magnetic field of several Tesla S.Galeski2022 ;
W.Wu . More importantly, there exists a magnetic field driven Lifshitz
transition to the 1D Weyl regime: the crossing points of the zeroth LLs can be
regarded as 1D Weyl nodes, because the band structure and spin texture are
analogous to those of the Weyl nodes that are formed by Bloch band crossings
X.Wan ; S.Y.Xu ; Lv2015 . In this regime, since the Fermi surface is perfectly
nested, the electronic states are unstable to the interactions G.Gruner ;
F.Qin , which makes it a good platform to investigate the strongly correlated
electronic states. In ZrTe5, the Lifshitz transition was characterized by the
combined dc electric transport and ultrasound measurements S.Galeski2022 .
Moreover, the chemical potential was found to meet and cross the Weyl nodes,
resulting in the enhancement of the hole band occupation and the weakening of
the electron band occupation S.Galeski2022 . In HfTe5, the Lifshitz transition
was demonstrated by the magneto-infrared spectroscopy W.Wu , where a highly
unusual reduction of optical activity and the variation of the accompanying
resonant peaks were observed when the chemical potential crosses the zeroth
LLs. Based on these observations, we propose the following questions: Do the
Weyl nodes always exist in pentatellurides through modulating the magnetic
field? If not, what are the conditions for their existence?

In this paper, we will systematically explore the above questions in theory.
Although the ground states of ZrTe5 and HfTe5 depend heavily on the specific
experimental conditions, such as the growth method, temperature B.Xu , and so
on, it is widely believed that their ground states are located close to the
phase boundary between a strong topological insulator (TI) and a weak TI
H.Weng . In the strong TI, the band inversions can occur in all three
directions; whereas in the weak TI, the band inversion occurs only in one
direction or in one plane. More importantly, the two phases can be
characterized by the Z2 topological invariant index L.Fu2007a ; L.Fu2007b .
Here we will take the ground state in ZrTe5 and HfTe5 as the strong and weak
TI, respectively, which are supported by many experimental studies Z.G.Chen ;
G.Manzoni ; Y.Jiang ; J.Wang ; W.Wu .

Under the condition of fixed carrier density, we calculate the chemical
potential as a function of the magnetic field $B$. The chemical potential
variation, together with the LL movements, will lead to several critical
fields that characterize the distinct behaviors of the system. We also study
the dc transport property in the quantum limit by calculating the longitudinal
conductivity $\sigma_{xx}$ and Hall conductivity $\sigma_{xy}$ by using the
Kubo-Streda formula. We find that in the strong TI phase of ZrTe5, the
magnetic field can drive the zeroth LLs from crossing to be separated, thus
the appearance of the 1D Weyl nodes depends heavily on the carrier density;
whereas in the weak TI phase of HfTe5, the magnetic field drives the zeroth
LLs from separated to cross each other, which makes the 1D Weyl nodes more
likely to appear. The critical fields can be captured by the peaks of
$\sigma_{xx}$ and the vanishing $\sigma_{xy}$. Moreover, the linear dependence
of $\sigma_{xy}$ on the inverse magnetic field, $\sigma_{xy}\sim B^{-1}$,
holds in the quantum oscillation regime but does not in the quantum limit. We
attribute this observation to the $g_{2}$-spin Zeeman term that breaks the
particle-hole symmetry. Our paper can help one better understand the quantum
limit and the 1D Weyl nodes that are driven by the magnetic field in 3D
pentatelluride experiments.

##  II Model and Methods

We use the effective $\bm{k}\cdot\bm{p}$ model to describe the low-energy
excitations in 3D pentatellurides. In the four-component basis
$\begin{pmatrix}|+,\uparrow\rangle&|-,\uparrow\rangle&|+,\downarrow\rangle&|-,\downarrow\rangle\end{pmatrix}^{T}$,
the Hamiltonian is written as ($\hbar=1$) E.Martino ; Y.Jiang ; Z.Rukelj ;
J.Wang

$\displaystyle H(\bm{k})=$ $\displaystyle
v(k_{x}\sigma_{z}\otimes\tau_{x}+k_{y}I\otimes\tau_{y})+v_{z}k_{z}\sigma_{x}\otimes\tau_{x}$
$\displaystyle+[M-\xi(k_{x}^{2}+k_{y}^{2})-\xi_{z}k_{z}^{2}]I\otimes\tau_{z}.$
(1)

Here $\sigma$ and $\tau$ are the Pauli matrices acting on the spin and orbit
degrees of freedom, respectively. $v$ and $v_{z}$ are the Fermi velocities,
$\xi$ and $\xi_{z}$ are the band inversion parameters, and $M$ denotes the
Dirac mass. When taking $v_{z}=0$, $H(\bm{k})$ becomes decoupled, and the
upspin and downspin are good quantum numbers. In the 3D system, there exist
the inversion symmetry (IS) ${\cal I}^{-1}H(-\bm{k}){\cal I}=H(\bm{k})$ with
the operator ${\cal I}=\tau_{z}$, the time-reversal symmetry (TRS) ${\cal
T}^{-1}H_{0}(-\bm{k}){\cal T}=H_{0}(\bm{k})$ with ${\cal T}=i\sigma_{y}K$ and
$K$ being the complex conjugation, the particle-hole symmetry (PHS) ${\cal
P}^{-1}H(\bm{k}){\cal P}=-H(\bm{k})$ with ${\cal P}=i\sigma_{y}\tau_{x}$, as
well as the chiral symmetry (CS) ${\cal C}^{-1}H({\bm{k}}){\cal
C}=-H(-{\bm{k}})$ with ${\cal C}=\sigma_{y}\otimes\tau_{y}$. Thus according to
the Altland and Zirnbauer notations, the topological states described by
$H(\bm{k})$ belong to the chiral symplectic class CII A.P.Schnyner ; C.K.Chiu
.

When a uniform magnetic field $\bm{B}=B\bm{e}_{z}$ acts on the 3D system, the
1D Landau bands will form, with the dispersions along the magnetic field
direction. To solve the LLs, we choose the vector potential in the Landau
gauge as $\bm{A}=-By\bm{e}_{x}$ and make the Peierls substitution
$\bm{\pi}=\bm{k}-e\bm{A}$. Then we define the raising and lowering operators
$a^{\dagger}=\frac{l_{B}}{\sqrt{2}}(\pi_{x}+i\pi_{y})$ and
$a=\frac{l_{B}}{\sqrt{2}}(\pi_{x}-i\pi_{y})$, with $[a,a^{\dagger}]=1$ and the
magnetic length $l_{B}=\frac{1}{\sqrt{eB}}=\frac{25.6}{\sqrt{B}}$ nm. Besides
the orbital effect, the magnetic field can also cause the spin Zeeman
splitting, which is described as

$\displaystyle
H_{Z}=-\frac{1}{2}g_{1}\mu_{B}B\sigma_{z}-\frac{1}{2}g_{2}\mu_{B}B\sigma_{z}\tau_{z}.$
(2)

Here $\mu_{B}$ denotes the Bohr magneton, and $g_{1}$ and $g_{2}$ are the
Landé $g-$factors. Since we have ${\cal P}^{-1}\sigma_{z}{\cal P}=-\sigma_{z}$
and ${\cal P}^{-1}\sigma_{z}\tau_{z}{\cal P}=\sigma_{z}\tau_{z}$, the PHS is
preserved by the $g_{1}$ term but is broken by the $g_{2}$ term.

With the trial wavefunction
$\psi_{n}=(c_{n}^{1}|n\rangle,c_{n}^{2}|n-1\rangle,c_{n}^{3}|n-1\rangle,c_{n}^{4}|n\rangle)^{T}$,
where the harmonic oscillator state $|n\rangle$ is defined by
$a^{\dagger}a|n\rangle=n|n\rangle$ and $c_{n}^{1,\cdots,4}$ are the
coefficients, we obtain the energies for the zeroth and $n\geq 1$ LLs L.You

$\displaystyle\varepsilon_{0\lambda}(k_{z})=-\lambda\big{(}M-\xi_{z}k_{z}^{2}-\frac{\xi}{l_{B}^{2}}+\frac{1}{2}g_{1}\mu_{B}B\big{)}+\frac{1}{2}g_{2}\mu_{B}B,$
(3)
$\displaystyle\varepsilon_{ns\lambda}(k_{z})=s\Big{[}\big{(}M-\frac{2n\xi}{l_{B}^{2}}-\xi_{z}k_{z}^{2}-\frac{\lambda}{2}g_{2}\mu_{B}B\big{)}^{2}+\frac{2nv^{2}}{l_{B}^{2}}\Big{]}^{\frac{1}{2}}$
$\displaystyle\qquad\qquad\quad+\lambda\big{(}\frac{\xi}{l_{B}^{2}}-\frac{1}{2}g_{1}\mu_{B}B\big{)},$
(4)

respectively. Here the index $s=\pm 1$ denotes the conduction/valence band,
and $\lambda=\pm 1$ characterizes the upspin/downspin branch.

With the density of states (DOS) $D(\varepsilon)$, the chemical potential
$\mu$ is determined by the carrier density $n_{0}$ as B.Fu ; C.Wang

$\displaystyle n_{0}=$ $\displaystyle\int_{0}^{\infty}d\varepsilon
D(\varepsilon)f(\varepsilon-\mu)+\int_{-\infty}^{0}d\varepsilon
D(\varepsilon)[f(\varepsilon-\mu)-1],$ (5)

where $f(x)=\frac{1}{\text{exp}(\beta x)+1}$ is the Fermi-Dirac distribution
function with $\beta=\frac{1}{k_{B}T}$ the inverse temperature, and the charge
neutrality is taken at the zero energy.

The contribution of the zeroth LL to the DOS is given analytically as

$\displaystyle
D_{0\lambda}(\varepsilon)=\frac{g}{2\pi}\Big{[}\xi_{z}\big{(}C+\lambda\varepsilon-\frac{\lambda}{2}g_{2}\mu_{B}B\big{)}\Big{]}^{-\frac{1}{2}},$
(6)

where $g=\frac{1}{2\pi l_{B}^{2}}$ is the LL degeneracy in the $x$-$y$ plane
and can be denoted as the uniform DOS,
$C=M-\frac{\xi}{l_{B}^{2}}+\frac{1}{2}g_{1}\mu_{B}B$. We see that
$D_{0\lambda}$ exhibits the square-root singularity, leading to the asymmetric
peaks at $\varepsilon=-\lambda C+\frac{1}{2}g_{2}\mu_{B}B$.

We further study the dc transport properties of the system, as the
conductivities or resistivities can be measured directly in the experiments to
help judge the electronic states. The conductivity tensors are calculated by
using the Kubo-Streda formula L.Smrcka ; G.D.Mahan ,

$\displaystyle\sigma_{\alpha\beta}=$ $\displaystyle\frac{1}{2\pi
V}\sum_{\bm{k}}\int_{-\infty}^{\infty}d\varepsilon
f(\varepsilon-\mu)\Big{[}\text{Tr}\Big{(}J_{\alpha}\frac{dG^{R}}{d\varepsilon}J_{\beta}(G^{A}-G^{R})$
$\displaystyle-
J_{\alpha}(G^{A}-G^{R})J_{\beta}\frac{dG^{A}}{d\varepsilon}\Big{)}\Big{]},$
(7)

where $V$ is the volume of the 3D system, $J_{\alpha}=e\frac{\partial
H}{\partial k_{\alpha}}$ is the current density operator along the $\alpha$
direction, and $G^{R/A}(\varepsilon,\eta)=(\varepsilon-H\pm i\eta)^{-1}$ is
the retarded/advanced Green’s function, with $\eta$ representing the LL
linewidth broadening that is introduced phenomenologically to represent the
impurity scatterings and will be taken as a constant for simplicity. In the
following, we focus on the zero temperature.

With the help of the LL energies and wavefunctions, the longitudinal
conductivity $\sigma_{xx}$ and Hall conductivity $\sigma_{xy}$ can be derived
directly. The selection rules $n\rightarrow n\pm 1$ are determined from the
nonvanishing matrix element of the current density, and there is no limit on
the $s$ and $\lambda$ index. We note that the conductivity components satisfy
the following relations:

$\displaystyle\sigma_{xx}(ns\lambda\rightarrow
n+1,s^{\prime}\lambda^{\prime})=\sigma_{xx}(n+1,s^{\prime}\lambda^{\prime}\rightarrow
ns\lambda),$ (8) $\displaystyle\sigma_{xy}(ns\lambda\rightarrow
n+1,s^{\prime}\lambda^{\prime})=-\sigma_{xy}(n+1,s^{\prime}\lambda^{\prime}\rightarrow
ns\lambda),$ (9)

meaning that the contributions to $\sigma_{xx}$ from the LL transition
$ns\lambda\rightarrow(n+1,s^{\prime}\lambda^{\prime})$ and from
$(n+1,s^{\prime}\lambda^{\prime})\rightarrow ns\lambda$ are equal, while those
to $\sigma_{xy}$ are opposite. The expressions of $\sigma_{xx}$ and
$\sigma_{xy}$ are obtained as Y.X.Wang2023

$\displaystyle\sigma_{xx}=$
$\displaystyle\frac{\sigma_{0}\eta^{2}}{\pi^{2}l_{B}^{2}}\int_{-\infty}^{\infty}dk_{z}\sum_{n\geq
0}\sum_{s,s^{\prime}}\sum_{\lambda}\frac{M_{ns,\lambda;n+1,s^{\prime},\lambda}^{2}}{[(\mu-\varepsilon_{ns,\lambda})^{2}+\eta^{2}][(\mu-\varepsilon_{n+1,s^{\prime},\lambda})^{2}+\eta^{2}]},$
(10) $\displaystyle\sigma_{xy}=$ $\displaystyle\frac{\sigma_{0}}{\pi
l_{B}^{2}}\int_{-\infty}^{\infty}dk_{z}\sum_{n\geq
0}\sum_{s,s^{\prime}}\sum_{\lambda}\frac{[(\varepsilon_{ns,\lambda}-\varepsilon_{n+1,s^{\prime},\lambda})^{2}-\eta^{2}]M_{ns,\lambda;n+1,s^{\prime},\lambda}^{2}}{[(\varepsilon_{ns,\lambda}-\varepsilon_{n+1,s^{\prime},\lambda})^{2}+\eta^{2}]^{2}}[\theta(\mu-\varepsilon_{ns})\theta(\varepsilon_{n+1,s^{\prime},\lambda}-\mu)$
$\displaystyle-\theta(\mu-\varepsilon_{n+1,s^{\prime},\lambda})\theta(\varepsilon_{ns,\lambda}-\mu)],$
(11)

where $\sigma_{0}=\frac{e^{2}}{2\pi}$ is the unit of the quantum conductivity,
and $\theta(x)$ is the step function. Explicitly, the matrix elements are
given as

$\displaystyle
M_{ns,1;n+1,s^{\prime},1}=-\frac{\sqrt{2n}\xi}{l_{B}}c_{ns,1}^{1}c_{n+1,s^{\prime},1}^{1}+\frac{\sqrt{2(n+1)}\xi}{l_{B}}c_{ns,1}^{2}c_{n+1,s^{\prime},1}^{2}+vc_{ns,1}^{2}c_{n+1,s^{\prime},1}^{1},$
(12) $\displaystyle
M_{ns,-1;n+1,s^{\prime},-1}=-\frac{\sqrt{2(n+1)}\xi}{l_{B}}c_{ns,-1}^{3}c_{n+1,s^{\prime},-1}^{3}+\frac{\sqrt{2n}\xi}{l_{B}}c_{ns,-1}^{4}c_{n+1,s^{\prime},-1}^{4}-vc_{ns,-1}^{3}c_{n+1,s^{\prime},-1}^{4}.$
(13)

Due to the complicated form of the matrix elements
$M_{ns,\lambda;n+1,s^{\prime},\lambda}$, the integrations over $k_{z}$ in Eqs.
(10) and (11) may not be completed analytically and need to be solved with
numerics.

##  III Strong topological insulator in ZrTe5

Figure 1: (Color online) (a) The LL spectra and the chemical potential $\mu$
in the strong TI, with the LL index $(ns\lambda)$ being labeled. The carrier
density is fixed at $n_{0}=6.76\times 10^{16}$ cm-3 and the critical fields
are $B_{s}^{0}=2$ T, $B_{s}^{1}=3.85$ T, $B_{s}^{2}=7.25$ T, $B_{s}^{3}=11.75$
T and $B_{s}^{4}=13$ T, as indicated by the dotted lines. The dashed blue line
helps to judge $B_{s}^{2}$ and the inset shows the enlarged plot around zero
energy. (b)$-$(f) The LL dispersions at the critical fields
$B_{s}^{0,1,2,3,4}$, respectively, in which the dotted red lines denote the
positions of $\mu$.

In this section, we study ZrTe5 and take the model parameters from the
experiment Y.Jiang : $M=5$ meV, $(v,v_{z})=(6,0)\times 10^{5}$ m/s,
$(\xi,\xi_{z})=(100,200)$ meV nm2, $g_{1}=-8$ and $g_{2}=10$. In fact, when
the Fermi velocity $v_{z}$ is nonvanishing, a gap may be opened with the
magnitude Y.X.Wang2021

$\displaystyle\Delta=2\sqrt{\frac{Mv_{z}^{2}}{\xi_{z}}-\frac{v_{z}^{4}}{4\xi_{z}^{2}}}.$
(14)

Here to observe the 1D Weyl nodes, we simply take $v_{z}=0$, so there is no
gap opening and the system behaves as a semimetal. Since the features of band
inversions in the $x$-$y$ plane as well as the $z$ direction are kept and a
finite $v_{z}$ will not affect the main conclusions of this paper (see
Appendix A), we still regard the ground state of the system as a strong TI. At
a weak magnetic field, the band inversion in the $z$ direction leads to the
zeroth LLs crossing at the momentum $k_{z}=\pm k_{c}$, with

$\displaystyle k_{c}=\Big{(}\frac{M}{\xi_{z}}-\frac{e\xi
B}{\xi_{z}}+\frac{g_{1}\mu_{B}B}{2\xi_{z}}\Big{)}^{\frac{1}{2}}.$ (15)

We fix the carrier density at $n_{0}=6.76\times 10^{16}$ cm-3 and calculate
the chemical potential $\mu$ through solving Eq. (5) self-consistently. The
results are displayed in Fig. 1(a) as a function of the magnetic field $B$, in
which $\mu$ exhibits a non-monotonous variation. In fact, $\mu$ is determined
by the interplay between the local DOS due to the 1D LL dispersion and the
uniform DOS $g$ that is proportional to $B$. When $B$ is weak, $\mu$ lies
above the $n\geq 1$ LLs and the quantum oscillations are clearly visible in
$\mu$. According to the Onsager’s relation, the oscillation period is closely
related to the Fermi surface area D.Shoenberg , as analyzed by us in a
previous study Y.X.Wang2023 . When $B$ increases, the critical fields that are
caused by the combined effects of the chemical potential variation and the LL
movements will appear successively [Fig. 1(a)].

First, at the critical field $B_{s}^{0}$, the system enters into the quantum
limit, with all electrons confined to the zeroth LLs [Fig. 1(b)]. When $B$
increases, the $0+$ and $0-$ LLs move upwards and downwards, respectively.
Since the uniform DOS gaining surpasses the local DOS dropping, $\mu$
decreases with $B$.

At the critical field $B_{s}^{1}$, $\mu$ intersects the $0-$ LL [Fig. 1(c)],
with

$\displaystyle B_{s}^{1}=\frac{M-\mu}{e\xi-(g_{1}+g_{2})\mu_{B}/2}.$ (16)

Then the Fermi surface varies from incorporating two points to four points.
Correspondingly, the crossing points at $k_{z}=\pm k_{c}$ own opposite
chiralities and behave as two 1D Weyl nodes. The effective Hamiltonian ${\cal
H}^{\text{eff}}$ is written as

$\displaystyle{\cal H}^{\text{eff}}=v_{F}(k_{z}\mp k_{c})\sigma_{z},$ (17)

where the Fermi velocity is

$\displaystyle v_{F}=2\xi_{z}^{\frac{1}{2}}\Big{(}M-e\xi
B+\frac{g_{1}\mu_{B}B}{2}\Big{)}^{\frac{1}{2}}.$ (18)

Thus in the strong TI phase of ZrTe5, the magnetic field drives the Lifshitz
transition and the system lies in the 1D Weyl regime.

When $B$ further increases, $\mu$ decreases steadily, meaning that the
magnetic field can effectively modulate the position of $\mu$ with respect to
the Weyl nodes. We see that $\mu$ will meet the Weyl nodes at the critical
field $B_{s}^{2}$ [Fig. 1(d)], with

$\displaystyle B_{s}^{2}=\frac{2\mu}{g_{2}\mu_{B}},$ (19)

in which the $g_{2}$-spin Zeeman term plays a decisive role in driving
$B_{s}^{2}$; without $g_{2}$, $\mu$ cannot meet the Weyl nodes. In the
experiment, $B_{s}^{2}$ is detectable only when $g_{2}$ is strong enough. The
decreasing $\mu$ is followed by an upturn [Fig. 1(a), inset]. This is because
the $\Gamma$ point of the $0+$ LL moves to be above the zero energy, leading
to the carriers turning from holes to electrons.

Figure 2: (Color online) The zeroth LLs in the quantum limit, and the chemical
potential $\mu$ for a set of carrier densities
$n_{0}=(2.751,2.143,1.122)\times 10^{17}$ cm-3. The inset plots the
characteristic carrier density $n_{s}$, with the horizontal lines denoting
different $n_{0}$.  Figure 3: (Color online) The DOS (a), longitudinal
conductivity $\sigma_{xx}$ (b) and Hall conductivity $\sigma_{xy}$ (c) versus
the magnetic field $B$ in the strong TI, with the parameters the same as Fig.
1(a) and the critical fields labeled by the dotted lines. (b) and (c) are
plotted for different linewidths $\eta$. The arrow in (c) marks the kink
structure in $\sigma_{xy}$. The inset in (c) plots $\sigma_{xy}$ versus the
inverse magnetic field $B^{-1}$ at $\eta=0$, where the red dashed line shows
the linear relation of $\sigma_{xy}\sim B^{-1}$, with the slope extracted as
$k=0.108$ m$\Omega^{-1}$cm-1T.

After that, $\mu$ will be close to the $\Gamma$ point of the $0+$ LL and
crosses it at the critical field $B_{s}^{3}$ [Fig. 1(e)], with

$\displaystyle B_{s}^{3}=\frac{M+\mu}{e\xi-(g_{1}-g_{2})\mu_{B}/2}.$ (20)

When $B>B_{s}^{3}$, only the $0-$ LL is occupied and the Weyl nodes do not
exist.

Finally, at the critical field $B_{s}^{4}$, with

$\displaystyle B_{s}^{4}=\frac{M}{e\xi-g_{1}\mu_{B}/2},$ (21)

the zeroth LLs touch each other at the $\Gamma$ point [Fig. 1(f)]. Further
increasing $B$, the zeroth LLs will be separated, and the system becomes a
trivial insulator.

According to the above analysis, in the strong TI phase, the magnetic field
can drive the zeroth LLs from crossing to be separated. Correspondingly, the
system turns from topological nontrivial to trivial. Based on this
observation, we suggest that the 1D Weyl nodes do not always appear in the
strong TI phase; their appearance depends heavily on the carrier density of
the system. To further clarify this point, we calculate the chemical potential
$\mu$ for a set of the carrier density $n_{0}$ and plot the results in Fig. 2.
Since a gap is opened when $B>B_{s}^{4}$, if $\mu$ lies in the gap, only the
$0-$ LL is occupied. The characteristic carrier density $n_{s}$ is determined
as

$\displaystyle
n_{s}(B)=\int_{0}^{\mu=\varepsilon_{0-}(\Gamma)}D_{0-}(\varepsilon)d\varepsilon=\frac{g}{\pi}\sqrt{\frac{C+g_{2}\mu_{B}B/2}{\xi_{z}}}.$
(22)

The inset of Fig. 2 shows that $n_{s}$ increases with $B$ monotonously.

In Fig. 2, we see that there exist three cases for the 1D Weyl nodes: (i) When
the carrier density $n_{0}<n_{s}(B_{s}^{4})$, e.g., $n_{0}=1.122\times
10^{17}$ cm-3, the chemical potential $\mu$ intersects the $0-$ LL before the
zeroth LLs getting separated, which enables the formation of the Weyl nodes.
This is similar to the case in Fig. 1(a). (ii) When
$n_{s}(B_{s}^{4})<n_{0}<n_{s}(B_{m})$, e.g., $n_{0}=2.143\times 10^{17}$ cm-3,
$\mu$ can only meet the $\Gamma$ point of the $0+$ LL after a gap is opened in
the system. Here $B_{m}$ represents the maximum magnetic field that would be
accessible in the experiment and is taken as $B_{m}=30$ T. Then $\mu$ will
drop across the gap and lies below the $\Gamma$ point of the $0-$ LL, leading
to the absence of the Weyl nodes. (iii) When $n_{0}>n_{s}(B_{m})$, e.g.,
$n_{0}=2.751\times 10^{17}$ cm-3, $\mu$ cannot intersect the $0-$ LLs even
after a gap is opened, and no Weyl nodes exist. Therefore, in the strong TI
phase, the carrier density of the crystal sample directly determines the Weyl
nodes that are driven by the magnetic field. For comparison, in Ref. W.Wu , a
schematic plot of the chemical potential variation with $B$ is presented for a
fixed carrier density in the strong TI phase, which is just case (iii).

To find the signatures of the critical fields in the quantum limit, we
calculate the DOS as well as the conductivity. With the parameters chosen the
same as Fig. 1(a), we plot the results in Fig. 3 as functions of the magnetic
field $B$, in which the critical fields are labeled by the dotted lines.

In Fig. 3(a), in the quantum limit, three asymmetric peaks are exhibited at
the critical fields $B_{s}^{0}$, $B_{s}^{1}$ and $B_{s}^{3}$, where $\mu$
crosses the $(1+-)$, $0-$, and $0+$ LL, respectively. Explicitly, the peaks at
$B_{s}^{0}$, $B_{s}^{1}$, and $B_{s}^{3}$ have long tails towards
$B<B_{s}^{0}$, $B>B_{s}^{1}$, and $B<B_{s}^{3}$, respectively. Moreover, the
DOS remains finite when $\mu$ meets the Weyl nodes at $B_{s}^{2}$. This is
different from the Weyl nodes that are formed by breaking the TRS or IS in a
Dirac semimetal X.Wan ; S.Y.Xu ; Lv2015 , where the DOS vanishes at the Weyl
nodes.

Figure 4: (Color online) (a) The LL spectra and the chemical potential $\mu$
in the weak TI, with the LL index $(ns\lambda)$ being labeled. The carrier
density is fixed at $n_{0}=1.25\times 10^{17}$ cm-3 and the critical fields
are $B_{w}^{0}=4$ T, $B_{w}^{1}=7$ T, $B_{w}^{2}=8.6$ T, and $B_{w}^{3}=10.85$
T, as labeled by the dotted lines. The dashed blue line helps to judge
$B_{w}^{3}$, and the inset shows the enlarged plot around zero energy.
(b)$-$(e) The LL dispersions at the critical fields $B_{w}^{0,1,2,3}$,
respectively, in which the red blue lines denote the positions of $\mu$.

In Figs. 3(b) and 3(c), the conductivities are shown, with the linewidth
$\eta$ included to represent the effect of the impurity scatterings. In the
quantum limit, we observe that: (i) The longitudinal conductivity
$\sigma_{xx}$ increases with $\eta$, whereas the Hall conductivity
$\sigma_{xy}$ shows certain robustness to $\eta$. These results are consistent
with our previous studies Y.X.Wang2020 . (ii) In $\sigma_{xx}$, corresponding
to the DOS, three peaks are exhibited at the critical fields
$B_{s}^{0},B_{s}^{1}$ and $B_{s}^{3}$. Such peaks are distinguishable when
$\eta$ is weak but would be smeared at a strong $\eta=10$ meV. (iii)
$\sigma_{xy}$ decreases smoothly with $B$ except a kink as marked by the arrow
[Fig. 3(c)], which is attributed to the upturn behavior of $\mu$. (iv)
$\sigma_{xy}$ vanishes at the critical field $B_{s}^{2}$ and will reverse its
sign when the carriers change from electrons to holes. The vanishing
$\sigma_{xy}$ shows certain robustness to $\eta$, which favors the
experimental observations. Thus the signatures of $B_{s}^{0,1,3}$ and
$B_{s}^{2}$ can be captured by the peaks of $\sigma_{xx}$ and the zero value
of $\sigma_{xy}$, respectively. But no signatures of $B_{s}^{4}$ are found in
the conductivities, since the conductivities reflect the properties of the
Fermi surface or Fermi sea in the system, and $\mu$ cannot meet the touching
$\Gamma$ point of the zeroth LLs unless the carrier density is exactly
$n_{0}=n_{0}^{s}(B_{s}^{4})$. In the experiment, the signatures of $B_{s}^{1}$
and $B_{s}^{2}$ have been reported in the dc transport measurements of ZrTe5
S.Galeski2022 .

In the inset of Fig. 3(c), we plot $\sigma_{xy}$ as a function of the inverse
magnetic field $B^{-1}$ in the clean case $\eta=0$. We see that in the quantum
oscillation regime, the classical linear dependence of $\sigma_{xy}$ on
$B^{-1}$ A.A.Abrikosov ; V.Konye ,

$\displaystyle\sigma_{xy}=\frac{n_{0}e}{B},$ (23)

is retrieved. With the extracted slope $k=0.108$ m$\Omega^{-1}$ cm-1 T, the
carrier density is obtained as $n_{0}=6.75\times 10^{16}$ cm-3, which agrees
well with the chosen value. But in the quantum limit, $\sigma_{xy}$ shows
evident deviations from the linear relation. Intuitively, if the linear
relation holds in the quantum limit, $\sigma_{xy}$ would vanish at an infinite
magnetic field; now since $\sigma_{xy}$ vanishes at the critical field
$B_{s}^{2}$, the linear relation can no longer hold.

To further understand the above behaviors, we investigate the role of the PHS
in $\sigma_{xy}$ by choosing a set of the Zeeman splittings $g_{1}$ and
$g_{2}$ and plot the results in the inset of Fig. 3(c). When $g_{1}=g_{2}=0$,
the system owns the PHS and the Weyl nodes are located at the zero energy. We
see that the linear dependence of $\sigma_{xy}$ on $B^{-1}$ holds in the
quantum limit. When $g_{1}=-8$, the $0+$ and $0-$ LLs will move upwards and
downwards, respectively, but keep crossing each other. As the PHS is
preserved, the Weyl nodes remain located at the zero energy and the linear
dependence holds in the quantum limit. However, at a finite $g_{2}=10$, the
PHS is broken and thus the Weyl nodes are shifted in energy. Consequently, the
linear dependence will be destroyed in the quantum limit. In a recent
magnetotransport experiment of ZrTe5, the Hall resistivity $\rho_{xy}$
exhibited a similar dependence on the magnetic field S.Galeski2021 , which
supports our theoretical analysis.

Actually, when the chemical potential $\mu>0$ lies between the $n$th and
$(n+1)$th LLs, the dominant contributions to $\sigma_{xy}$ come from the LL
transition $(n,1,\lambda)\rightarrow(n+1,1,\lambda)$. This is reminiscent of
the 2D Dirac fermion behavior in graphene V.P.Gusynin . When the PHS is
broken, the $0+$ and $0-$ LLs will move asymmetrically in energy. On one hand,
such movements will not change the topological property of the LLs, which can
explain the vanishing $\sigma_{xy}$ at the Weyl nodes. On the other hand, in
the quantum oscillation regime, $\sigma_{xy}$ is related to the transition of
$n\geq 1$ LLs, thus the LL movements will not affect $\sigma_{xy}$ and the
linear relation; whereas in the quantum limit, $\sigma_{xy}$ is related to the
transition of the zeroth LL to the $n=1$ LL, and the LL movements will change
$\sigma_{xy}$ and destroy the linear relation.

##  IV Weak topological insulator in HfTe5

In this section, we study HfTe5 and take the model parameters from the
experiments W.Wu : $M=2.5$ meV, $(v,v_{z})=(4.5,0)\times 10^{5}$ m/s,
$(\xi,\xi_{z})=(120,-200)$ meV nm2, $g_{1}=-6$ and $g_{2}=10$. Now the ground
state of the system is a weak TI that features the band inversion only in the
$x-y$ plane. Under a weak magnetic field, the topological trivial bands in the
$z$ direction will lead to a gap between the zeroth LLs and the system behaves
as a gapped insulator. At a fixed carrier density $n_{0}=1.25\times 10^{17}$
cm-3, the calculated chemical potential $\mu$ is displayed in Fig. 4. The
increasing magnetic field $B$ will drive the emergence of several critical
fields [Fig. 4(a)].

Figure 5: (Color online) The DOS (a), longitudinal conductivity $\sigma_{xx}$
(b) and Hall conductivity $\sigma_{xy}$ (c) versus the magnetic field $B$ in
the weak TI, with the parameters the same as Fig. 4(a) and the critical fields
labeled by the dotted lines. (b) and (c) are plotted for different linewidths
$\eta$. The inset in (c) plots $\sigma_{xy}$ versus the inverse magnetic field
$B^{-1}$ at $\eta=0$, where the red dashed line shows the linear relation of
$\sigma_{xy}\sim B^{-1}$, with the slope extracted as $k=0.2055$
m$\Omega^{-1}$cm-1T.

Firstly, the quantum limit is achieved at the critical field $B_{w}^{0}$ [Fig.
4(b)]. We see that the $0+/-$ LL lies in the valence/conduction band, which is
different from the strong TI. With increasing $B$, the $0+$ and $0-$ LLs will
move upwards and downwards, respectively. Since both the uniform DOS and local
DOS increase with $B$, $\mu$ also decreases.

At the critical field $B_{w}^{1}$, the zeroth LLs touch each other at the
$\Gamma$ point so that the gap is closed [Fig. 4(c)]. Then the zeroth LLs
cross each other at $k_{z}=\pm k_{c}$, with the gap being persistently closed.
Note that $k_{c}$ has the same expression as Eq. (15). When $B$ further
increases, the decreasing $\mu$ will meet the $0+$ LL at the critical field
$B_{w}^{2}$ [Fig. 4(d)]. Meantime, the Lifshitz transition occurs in the weak
TI phase, where the Fermi surface varies from incorporating two points to four
points. Correspondingly, the crossing points act as two 1D Weyl nodes. When
$B>B_{w}^{2}$, $\mu$ shows an upturn and then decreases slowly. This is
because the local DOS drops when $\mu$ crosses the $\Gamma$ point of the $0+$
LL. Finally, $\mu$ will meet the Weyl nodes at the critical field $B_{w}^{3}$
[Fig. 4(e)].

According to the above analysis, in the weak TI phase, the magnetic field
drives the zeroth LLs from being gapped to cross each other. Correspondingly,
the system turns from topological trivial to nontrivial, which is opposite to
the strong TI. This is also seen from the fact that the critical fields
$B_{w}^{1}$, $B_{w}^{2}$, and $B_{w}^{3}$ have the same expressions as
$B_{s}^{4}$, $B_{s}^{3}$ and $B_{s}^{2}$, respectively. Note that chemical
potential variations with $B$ in the weak TI phase are consistent with Ref.
W.Wu .

It is interesting to ask whether there exists the critical field $B_{w}^{4}$
at which $\mu$ crosses the $0-$ LL. When it happens, $B_{w}^{4}$ will have the
same expression as $B_{s}^{1}$. Now, since only the $0+$ LL is occupied, the
characteristic carrier density $n_{w}^{1}$ is determined as

$\displaystyle n_{w}^{1}(B)$
$\displaystyle=\int_{0}^{\mu=\varepsilon_{0-}(\Gamma)}D_{0+}(\varepsilon)d\varepsilon$
$\displaystyle=\frac{g}{\pi}\Big{(}\sqrt{\frac{C-g_{2}\mu_{B}B/2}{\xi_{z}}}-\sqrt{\frac{2C}{\xi_{z}}}\Big{)},$
(24)

which decreases with $B$. Clearly, $B_{w}^{4}$ will appear if the carrier
density satisfies $n_{0}<n_{w}^{1}(B)$ at a certain $B$. For the magnetic
field $B_{w}^{1}<B<B_{m}$, the calculations show that $n_{w}^{1}$ lies in the
range $(0.96\sim 5.45)\times 10^{16}$ cm-3, which is far below the chosen
carrier density, thus $B_{w}^{4}$ would not appear.

We investigate the conditions for the Weyl nodes in the weak TI phase. When
the zeroth LLs get crossed, if the chemical potential $\mu$ meets the $\Gamma$
point of the $0+$ LL, the characteristic carrier density is

$\displaystyle n_{w}^{2}(B)=$
$\displaystyle\int_{0}^{\mu=\varepsilon_{0+}(\Gamma)}[D_{0+}(\varepsilon)+D_{0-}(\varepsilon)]d\varepsilon$
$\displaystyle=$
$\displaystyle\frac{g}{\pi}\Big{(}\sqrt{\frac{C-g_{2}\mu_{B}B/2}{\xi_{z}}}+\sqrt{\frac{2C}{\xi_{z}}}\Big{)},$
(25)

which increases with the magnetic field. For the magnetic field
$B_{w}^{1}<B<B_{m}$, $n_{w}^{2}$ is given in the range $(5.45\sim
133.87)\times 10^{16}$ cm-3, which is quite broad. In the weak TI phase, the
condition for the Weyl nodes is that the carrier density satisfies
$n_{w}^{1}(B)<n_{0}<n_{w}^{2}(B)$ at a certain $B$. Since this condition is
easily satisfied, the Weyl nodes are more likely to appear in the weak TI
phase.

Next we study the DOS and conductivities of the system. The results are
displayed in Fig. 5, where the critical fields are labeled by the dotted
lines. At the critical fields $B_{w}^{0}$ and $B_{w}^{2}$, the chemical
potential crosses the $(1+-)$ and $0+$ LL, respectively, which leads to the
asymmetric peaks in the DOS as well as in the longitudinal conductivity
$\sigma_{xx}$ [Figs. 5(a) and 5(b)]. The Hall conductivity $\sigma_{xy}$
decreases with $B$ smoothly and vanishes at the critical field $B_{w}^{3}$
[Fig. 5(c)]. Moreover, the effects of the linewidth $\eta$ on the
conductivities are the same as those in Fig. 3. Therefore, the signatures of
the critical field $B_{w}^{0,2,3}$ are clearly seen in the conductivities. In
the inset of Fig. 5(c), we plot $\sigma_{xy}$ as a function of the inverse
magnetic field $B^{-1}$ at $\eta=0$. In the quantum oscillation regime, the
linear dependence of $\sigma_{xy}$ on the inverse magnetic field $B^{-1}$ is
seen. With the extracted slope $k=0.1993$ m$\Omega^{-1}$cm-1T, the carrier
density is obtained as $n_{0}=\frac{k}{e}=1.244\times 10^{17}$ cm-3, which
agrees well with the chosen carrier density. But in the quantum limit,
$\sigma_{xy}$ shows evident deviations from the linear dependence. This
observation can also be attributed to the broken PHS caused by the
$g_{2}$-spin Zeeman term in the system, which is similar to the above strong
TI phase analysis. In Ref. W.Wu  of the longitudinal resistivity $R_{xx}$
measurements, a prominent linear behavior before saturating at high fields was
observed but there were no evident peaks for the critical fields, thus more dc
transport measurements in HfTe5 are expected in the future.

##  V Discussions and Conclusions

Figure 6: (Color online) The LL dispersions versus $k_{z}$ when the magnetic
field $B=2$ T in (a) and $B=7$ T in (b), with the LL index $(ns\lambda)$ being
labeled. The Fermi velocity $v_{z}=5\times 10^{4}$ m/s, and the other
parameters are the same as Fig. 1(a). The saddle points of the zeroth LL,
$\Gamma$ and $\zeta_{0}$, are marked as asterisks.

In this paper, we focus on fixed carrier density $n_{0}$. Now we discuss the
case of fixed chemical potential $\mu_{0}$, since it can provide more insights
to understand the exotic transport behavior in pentatellurides, such as the 3D
quantum Hall effect S.Galeski2021 ; F.Xiong ; Y.X.Wang2023 . In both strong
and weak TIs, with fixed $\mu_{0}$, the magnetic field can also drive the
critical fields. In the strong TI, the Weyl nodes appear when $\mu_{0}$
satisfies $\varepsilon_{0+}(\Gamma)<\mu_{0}<\varepsilon_{0-}(\Gamma)$, which
strongly depends on the carrier density; while in the weak TI, the Weyl nodes
appear when $\mu_{0}$ satisfies
$\varepsilon_{0-}(\Gamma)<\mu_{0}<\varepsilon_{0+}(\Gamma)$. The latter
condition is easily satisfied, as the magnetic field drives the zeroth LLs to
cross each other and the $0+$ LL move upwards. Therefore we suggest that the
conclusions for the 1D Weyl nodes with fixed $\mu_{0}$ are similar to those
with fixed $n_{0}$.

When the Fermi velocity $v_{z}$ is nonzero, the following consequences will be
induced: (i) the 1D Weyl nodes will become gapped, thus the behavior of the
zeroth LLs mimicks the physics of the 1D massive Weyl nodes; (ii) the upspin
and downspin will be mixed and are no longer good quantum numbers; and (iii)
in the strong TI phase, the zeroth LLs may avoid crossing each other, leading
to the additional saddle points Y.Jiang ; J.Wang . Thus in the DOS and
longitudinal conductivity $\sigma_{xx}$, more peaks will be found when the
chemical potential crosses such saddle points. The details are presented in
Appendix A. In addition, the effect of temperature on the magneotransport is
briefly discussed in Appendix B.

To summarize, our work explores the conditions for the magnetic field-driven
Weyl nodes and the dc transport property in the quantum limit of the 3D
pentatellurides. Although the quantitative results depend on the model
parameters, they are qualitatively valid and can show guiding significance for
the experiments. We hope that the 3D pentatellurides under a magnetic field
can open an avenue for studying the interactions of 1D Weyl fermions as well
as the resulting various strongly correlated electronic states.

##  VI Acknowledgment

This work was supported by the Natural Science Foundation of China (Grant No.
11804122), and the China Postdoctoral Science Foundation (Grant No.
2021M690970).

##  VII Appendix

###  VII.1 LLs with nonzero $v_{z}$

Figure 7: (Color online) The chemical potential $\mu$ (a), longitudinal
conductivity $\sigma_{xx}$ (b) and Hall conductivity $\sigma_{xy}$ (c) versus
the magnetic field $B$ for different temperatures $T$. The parameters are the
same as Fig. 1(a) and the critical fields are labeled by the dotted lines. The
legends are the same in all figures.

For a finite Fermi velocity $v_{z}$, the zeroth LLs can still be obtained
analytically, with the energies

$\displaystyle\varepsilon_{0\lambda}(k_{z})=$
$\displaystyle\lambda\Big{[}\big{(}M-\xi_{z}k_{z}^{2}-\frac{\xi}{l_{B}^{2}}+\frac{1}{2}g_{1}\mu_{B}B\big{)}^{2}+v_{z}^{2}k_{z}^{2}\Big{]}^{\frac{1}{2}}$
$\displaystyle+\frac{1}{2}g_{2}\mu_{B}B,$ (A1)

while the $n\geq 1$ LLs need to be solved numerically. Now although the 1D
Weyl nodes are gapped, the LL spectra at $k_{z}=0$ are unaffected by the
finite $v_{z}$ in both strong and weak TIs [Figs. 1(a) and 4(a)], thus the
evolution of the zeroth LLs with the magnetic field as well as the determined
critical fields remain unchanged.

In the strong TI, besides the saddle point $\Gamma$ at $k_{z}=0$, the
additional saddle points $\zeta_{n}$ may appear Y.Jiang ; J.Wang ; L.You . For
example, when $B=2$ T [Fig. 6(a)], in the zeroth LLs, the additional saddle
points $\zeta_{0}$ can be found and are located at
$k_{z}=\pm\big{(}\frac{M+g_{1}\mu_{B}/2}{\xi_{z}}-\frac{\xi}{\xi_{z}l_{B}^{2}}-\frac{v_{z}^{2}}{2\xi_{z}^{2}}\big{)}^{\frac{1}{2}}$.
With increasing $B$, $\zeta_{0}$ moves to $\Gamma$ and will finally merge with
it at the critical field $B_{0c}=\frac{M-v_{z}^{2}/2\xi_{z}}{e\xi-
g_{1}\mu_{B}/2}\simeq 6$ T. When $B>B_{0c}$, there are no additional saddle
points in the zeroth LLs [Fig. 6(b)].

###  VII.2 Effect of temperature

Here we study the effect of temperature on the magnetotransport in
pentatellurides. At a finite temperature $T$, the conductivity
$\sigma_{\alpha\beta}$ will be modified in two aspects: one is that $T$ can
shift the chemical potential $\mu$; and another is that $T$ directly enters
$\sigma_{\alpha\beta}$ via the Fermi-Dirac distribution function $f(x)$. By
multiplying the zero-temperature conductivity $\sigma_{\alpha\beta}(T=0)$ by
$\int_{-\infty}^{\infty}d\epsilon\delta(\epsilon-\mu)$, the finite-temperature
conductivity $\sigma_{\alpha\beta}(T)$ can be expressed as a weighted
integration of $\sigma_{\alpha\beta}(T=0)$ around the chemical potential $\mu$
and is written as L.Smrcka

$\displaystyle\sigma_{\alpha\beta}(T)=\int_{-\infty}^{\infty}d\epsilon\sigma_{\alpha\beta}(T=0,\epsilon)\big{[}-\frac{\partial
f(\epsilon-\mu)}{\partial\epsilon}\big{]},$ (A2)

where the derivative of the Fermi-Dirac distribution function is
$-\frac{\partial
f(\epsilon-\mu)}{\partial\epsilon}=\frac{1}{2k_{B}T\big{(}1+\text{cosh}\frac{\epsilon-\mu}{k_{B}T}\big{)}}$.

In Fig. 7, with the parameters the same as Fig. 1(a), we display the results
of the chemical potential $\mu$, longitudinal conductivity $\sigma_{xx}$, and
Hall conductivity $\sigma_{xy}$ for different temperatures $T$. In Fig. 7(a),
we see that with increasing $T$, when $B<B_{s}^{0}$ in the quantum oscillation
regime, the oscillations of $\mu$ are weakened; when $B>B_{s}^{0}$ in the
quantum limit, $\mu$ is less affected at $T\leq 20$ K. When the temperature
increases to $T=100$ K, $\mu$ even becomes negative, meaning that $\mu$ shifts
from the conduction band to the valence band. This result is consistent with
the experimental observations Y.Zhang2017a ; Y.Zhang2017b  and was believed to
be the underlying physical mechanism of the anomalous resistivity peak at a
finite $T$ in pentatellurides C.Wang . In Fig. 7(b), the peaks of
$\sigma_{xx}$ are smeared by temperature, and in Fig. 7(c), $\sigma_{xy}$ is
gradually suppressed. Note that when $T=100$ K, $\sigma_{xx}$ becomes a smooth
curve, indicating that the quantized LLs do not exist and the system enters
the semiclassical diffusive region.

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# On the absence of solutions of differential inequalities with
$\infty$-Laplacian

A. A. Kon’kov  Department of Differential Equations, Faculty of Mechanics and
Mathematics, Moscow Lomonosov State University, Vorobyovy Gory, 119992 Moscow,
Russia<EMAIL_ADDRESS>
###### Abstract.

For differential inequalities with the $\infty$-Laplacian in the principal
part, we obtain conditions for the absence of solutions in unbounded domains.
Examples are given to demonstrate the accuracy of these conditions.

Introduction. We study solutions of the problem

$\Delta_{\infty}u\geq f(x,u)\quad\mbox{in
}\Omega,\qquad\left.u\right|_{\partial\Omega}=0,$ (1)

where $\Omega$ is unbounded domain ${\mathbb{R}}^{n}$, $n\geq 2$,

$\Delta_{\infty}u=\frac{1}{2}\nabla|\nabla u|^{2}\nabla
u=\sum_{i,j=1}^{n}\frac{\partial^{2}u}{\partial x_{i}\partial
x_{j}}\frac{\partial u}{\partial x_{i}}\frac{\partial u}{\partial x_{j}}$

is the $\infty$-Laplacian, and $f$ is some function such that

$f(x,t)>0$ (2)

for all $t>0$ and $x\in\Omega$. Let us agree to denote by $B_{r}$ and $S_{r}$,
respectively, the open ball and sphere in ${\mathbb{R}}^{n}$ of radius $r>0$
centered at zero. We shall assume that $S_{r}\cap\Omega\neq\emptyset$ for all
$r>r_{*}$, where $r_{*}>0$ is some real number. Also let there be real numbers
$\sigma>1$ and $\theta>1$ such that

$\inf_{x\in\Omega_{r,\sigma},\,\zeta\in(t/\theta,t\theta)}f(x,|x|\zeta)\geq
q(r)g(t)$ (3)

for all $t>0$ and $r>r_{*}$, where
$\Omega_{r,\sigma}=\\{x\in\Omega:r/\sigma<|x|<r\sigma\\}$ and $q$ and $g$ are
non-negative measurable functions such that

$\inf_{K}g>0$

for any compact set $K\subset(0,\infty)$.

Solutions of the problem (1) is understood in the viscosity sense [1, 2]. If
$\partial\Omega=\emptyset$, i.e. $\Omega={\mathbb{R}}^{n}$, then the boundary
condition in (1) is assumed to be satisfied automatically.

The phenomenon of the absence of non-trivial solutions for differential
equations and inequalities (blow-up) has been studied by many authors [3–19].
In the case when the main part of the differential operator has a divergent
form, the method of nonlinear capacity has proven itself well. The main ideas
of this method are presented in [6, 18]. For inequalities (1) this method
obviously does not work. Under the assumption that $\Omega$ is a bounded
domain, the blow-up phenomenon for inequalities of the form (1) was studied in
[20–22]. These papers dealt with the so-called “boundary blow-up” or, in other
words, “large solutions”, tending to infinity when approaching
$\partial\Omega$. The dependence of the right-hand side on the spatial
variable was essentially not taken into account in the works [20–22]. In the
case of unbounded domains, the conditions for the absence of nontrivial
solutions to the problem (1) have not been known until now. Theorem 1 proved
in our paper eliminates this gap.

Definition 1. A function $u:\overline{\Omega}\to{\mathbb{R}}$ is called upper
semicontinuous at a point $x\in\overline{\Omega}$ if

$\limsup_{y\to x}u(y)\leq u(x).$

We say that $u$ is upper semicontinuous on the set
${\mathcal{G}}\subset\overline{\Omega}$ if it is upper semicontinuous at every
point of $x\in{\mathcal{G}}$.

Definition 2. The second-order superjet $J_{\Omega}^{2,+}u(x)$ of a function
$u:\Omega\to{\mathbb{R}}$ at a point $x\in\Omega$ is the set of ordered pairs
$(v,A)$, where $v\in{\mathbb{R}}^{n}$ is some vector and $A$ is a symmetric
$n\times n$ matrix such that

$u(y)-u(x)\leq\langle v,y-x\rangle+\frac{1}{2}\langle
A(y-x),y-x\rangle+\bar{\bar{o}}(|y-x|^{2})$

for all $y$ from some neighborhood of the point $x$. Here and below, angle
brackets denote the standard scalar product in ${\mathbb{R}}^{n}$.

Note that in the case of $u\in C^{2}(\Omega)$ the set $J_{\Omega}^{2,+}u(x)$
is not empty for any point $x\in\Omega$, because

$(Du(x),D^{2}u(x))\in J_{\Omega}^{2,+}u(x),$

where $Du(x)$ is the gradient vector and $D^{2}u(x)$ is the matrix of second
derivatives of the function $u$ at the point $x$.

Definition 3. A function $u\geq 0$ is called a non-negative solution of
problem (1) if $u$ is upper semicontinuous on $\overline{\Omega}$, vanishes on
$\partial\Omega$ and for any $x\in\Omega$ such that
$J_{\Omega}^{2,+}u(x)\neq\emptyset$ the relation

$\langle Av,v\rangle\geq f(x,u(x))$ (4)

is valid for all $(v,A)\in J_{\Omega}^{2,+}u(x)$.

In the general case [1], an upper semicontinuous function $u$ is called a
viscosity solution of the inequality

$F(x,u,Du,D^{2}u)\leq 0\quad\mbox{in }\Omega,$

where $F$ is some function if for any point $x\in\Omega$ such that
$J_{\Omega}^{2,+}u(x)\neq\emptyset$ the relation

$F(x,u(x),v,A)\leq 0$ (5)

is valid fro all $(v,A)\in J_{\Omega}^{2,+}u(x)$. In particular, taking

$F(x,u,Du,D^{2}u)=f(x,u)-\sum_{i,j=1}^{n}\frac{\partial^{2}u}{\partial
x_{i}\partial x_{j}}\frac{\partial u}{\partial x_{i}}\frac{\partial
u}{\partial x_{j}}$ (6)

in formula (5), we obtain (4).

In order for the classical solutions to be simultaneously viscous, the
function $F$ is subjected to the condition of degenerate ellipticity

$F(x,\zeta,v,A)\leq F(x,\zeta,v,B)$

for all $x\in\Omega$, real numbers $\zeta$, vectors $v\in{\mathbb{R}}^{n}$,
and symmetric ${n\times n}$-matrices $A\geq B$. The last inequality is
understood in the sense of quadratic forms. Namely, we say that $A\geq B$ if

$\langle Av,v\rangle\geq\langle Bv,v\rangle$

for all $v\in{\mathbb{R}}^{n}$. It is obvious that the function $F$ defined by
(6) satisfies the condition of degenerate ellipticity.

1\. Main results. The following statements are valid.

Theorem 1. Suppose that

$\int\limits_{1}^{\infty}(g(t)t)^{-1/4}dt<\infty\quad\mbox{and}\quad\int\limits_{r_{*}}^{\infty}q(r)dr=\infty.$
(7)

Then any non-negative solution of (1) is identically equal to zero.

A special case of (1) is the problem

$\Delta_{\infty}u\geq c(|x|)u^{\lambda}\quad\mbox{in
}\Omega,\qquad\left.u\right|_{\partial\Omega}=0,$ (8)

where $c$ is a positive non-increasing function.

Corollary 1. Suppose that $\lambda>3$ and

$\int\limits_{r_{*}}^{\infty}r^{\lambda}c(r)dr=\infty.$

Then any non-negative solution of (8) is identically equal to zero.

Theorem 1 and Corollary 1 will be proved in Paragraph 2. Now we give examples
demonstrating the accuracy of these results.

Example 1. Consider the inequality

$\Delta_{\infty}u\geq c_{0}(1+|x|)^{s}u^{\lambda}\quad\mbox{in
}{\mathbb{R}}^{n},$ (9)

where $c_{0}=const>0$. According to Corollary 1, the conditions

$\lambda>3\quad\mbox{and}\quad s\geq-\lambda-1$

guarantee that any non-negative solution of (9) is identically equal to zero.
These conditions are exact. Indeed, it is not difficult to verify that, for
$\lambda>3$ and $s<-\lambda-1$, the expression

$u(x)=\left\\{\begin{aligned}
&|x|^{-(4+s)/(\lambda-3)}-r_{0}^{-(4+s)/(\lambda-3)},&&|x|>r_{0},\\\
&0,&&|x|\leq r_{0},\end{aligned}\right.$

where $r_{0}>0$ is some real number, defines a non-negative solution of (9).
At the same time, in the case of $\lambda\leq 3$, by somewhat more complicated
reasoning, one can show that the inequality (9) has a non-trivial non-negative
solution for all $s\in{\mathbb{R}}$.

Example 2. We examine the critical exponent $s=-\lambda-1$ in the right-hand
side of (9). Let $u\geq 0$ satisfy the inequality

$\Delta_{\infty}u\geq
c_{0}(1+|x|)^{-\lambda-1}\ln^{\mu}(2+|x|)u^{\lambda}\quad\mbox{in
}{\mathbb{R}}^{n},$ (10)

where $c_{0}=const>0$. Applying Corollary 1, we obtain that, for

$\lambda>3\quad\mbox{and}\quad\mu\geq-1,$ (11)

the function $u$ is identically equal to zero. At the same time, by direct
calculations, one can verify that, for $\lambda>3$ and $\mu<-1$, the relation

$u(x)=\left\\{\begin{aligned}
&|x|\ln^{-(\mu+1)/(\lambda-3)}|x|-r_{0}\ln^{-(\mu+1)/(\lambda-3)}r_{0},&&|x|>r_{0},\\\
&0,&&|x|\leq r_{0},\end{aligned}\right.$

where $r_{0}>0$ is large enough, defines a non-negative solution (10). Thus,
the second condition in (11) is exact. The first condition in (11) is also
exact. It can be shown that, in the case of $\lambda\leq 3$, inequality (10)
has a non-trivial non-negative solution for all $\mu\in{\mathbb{R}}$.

Example 3. Now we examine investigate the critical exponent $\lambda=3$ in
(9). Namely, consider the inequality

$\Delta_{\infty}u\geq c_{0}(1+|x|)^{s}u^{3}\ln^{\nu}(2+u)\quad\mbox{in
}{\mathbb{R}}^{n},$ (12)

where $c_{0}=const>0$. According to Theorem 1, if

$\nu>4\quad\mbox{and}\quad s\geq-4,$ (13)

then any non-negative solution of (12) is identically equal to zero. In so
doing, if $\nu>4$ and $s<-4$, then

$u(x)=\left\\{\begin{aligned}
&e^{|x|^{(4+s)/(4-\nu)}}-e^{r_{0}^{(4+s)/(4-\nu)}},&&|x|>r_{0},\\\
&0,&&|x|\leq r_{0},\end{aligned}\right.$

where $r_{0}>0$ is large enough, is a non-negative solution of (12). If the
first condition in (13) is violated, i.e. $\nu\leq 4$, then (12) has a non-
trivial non-negative solution for any $s\in{\mathbb{R}}$. Thus, both
conditions in (13) are exact.

2\. Proof of Theorem 1 and Corollary 1. Below we assume that conditions (7)
are satisfied. Take a function $\omega\in C^{\infty}({\mathbb{R}})$ positive
on the interval $(-1,1)$, equal to zero on the set
${\mathbb{R}}\setminus(-1,1)$, and such that

$\int\limits_{-\infty}^{\infty}\omega(t)dt=1.$

We denote

$\omega_{\delta}(t)=\frac{1}{\delta}\omega\left(\frac{t}{\delta}\right),\quad\delta>0.$

Lemma 1. Let $\mu>1$, $\nu>1$, and $0<\alpha\leq 1$ be some real numbers and,
moreover, $\eta:(0,\infty)\to(0,\infty)$ and $H:(0,\infty)\to(0,\infty)$ be
measuable functions satisfying the inequality

$\eta(t)\leq\inf_{(t/\mu,t\mu)}H$

for all $t\in(0,\infty)$. Then

$\left(\int\limits_{M_{1}}^{M_{2}}\eta^{-\alpha}(t)t^{\alpha-1}\,dt\right)^{1/\alpha}\geq
C\int\limits_{M_{1}}^{M_{2}}\frac{dt}{H(t)}$

for all real numbers $M_{1}>0$ and $M_{2}>0$ such that $M_{2}\geq\nu M_{1}$,
where the constant $C>0$ depends only on $\alpha$, $\nu$ and $\mu$.

Lemma 2. Let $0<r_{1}<r_{2}$ and $0<\alpha\leq 1$ be some real numbers and
$\eta:(r_{1},r_{2})\to[0,\infty)$ be measurable function. Then

$\int\limits_{r_{1}}^{r_{2}}\left(\int\limits_{r_{1}}^{\rho}\eta(\xi)d\xi\right)^{\alpha}d\rho\geq
C\int\limits_{r_{1}}^{r_{2}}\left(1-\frac{\xi}{r_{2}}\right)^{\alpha}(\xi\eta(\xi))^{\alpha}d\xi,$

where the constant $C>0$ depends only on $\alpha$.

Lemma 3. Let $0<r_{1}<r_{2}$ and $0<\alpha<1$ be some real numbers and
$\eta:(r_{1},r_{2})\to[0,\infty)$ be a measurable function. Then

$\left(\int\limits_{r_{1}}^{r_{2}}\eta(\xi)d\xi\right)^{\alpha}\geq
C\int\limits_{r_{1}}^{r_{2}}\eta(\xi)\varkappa^{\alpha-1}(\xi)d\xi,$ (14)

where

$\varkappa(\xi)=\int\limits_{\xi}^{r_{2}}\eta(\zeta)d\zeta,$

and the constant $C>0$ depends only on $\alpha$.

Remark 1. If $\varkappa(\xi)=0$ for some $\xi\in(r_{1},r_{2})$, then
$\eta(\zeta)=0$ for almost all $\zeta\in(\xi,r_{2})$. In this case, we assume
that $\eta(\xi)\varkappa^{\alpha-1}(\xi)$ is also equal to zero on the right
in (14).

Lemma 4. Let $\gamma>0$, $\lambda>1$, and $r_{0}>r_{*}$ be some real numbers
and, moreover, $p:[r_{*},\infty)\to[0,\infty)$ be a locally integrable
function such that

$\frac{rp(r)}{\int\limits_{r_{*}}^{r}p(\xi)d\xi}\leq\gamma$

for all $r\geq r_{0}$. Then

$\int\limits_{r_{*}}^{\lambda r}p(\xi)d\xi\leq
C\int\limits_{r_{*}}^{r}p(\xi)d\xi$

for all $r\geq r_{0}$, where the constant $C>0$ depends only on $\gamma$ and
$\lambda$.

The proof of Lemma 1 can be found in [13, Lemma 2.3]. Lemmas 2 and 3 were
proved in [12, Lemmas 2.2 and 2.3]. Lemma 4 is proved in [14, Lemma 2.7].

Lemma 5. Let $\mu>1$ be some real number and $H:(0,\infty)\to(0,\infty)$ be a
measurable function such that

$\eta(t)=\inf_{(t/\mu,t\mu)}H>0$

for all $t\in(0,\infty)$ and, moreover,

$\int\limits_{1}^{\infty}(\eta(t)t)^{-1/4}dt<\infty.$ (15)

Then there exists an infinitely smooth function $h:(0,\infty)\to(0,\infty)$
satisfying the following conditions:

  1. $1)$

$h(t)\leq H(t)$ for all $t\in(0,\infty)$;

  2. $2)$

$h(t_{2})\geq h(t_{1})$ for all $t_{2}\geq t_{1}>0$;

  3. $3)$

for any real number $\alpha>0$ there is a constant $\beta>0$, depending only
on $\alpha$ and $\mu$ such that $h(\alpha t)\leq\beta h(t)$ for all $t\geq 2$;

  4. $4)$

$\int\limits_{1}^{\infty}(h(t)t)^{-1/4}dt<\infty.$ (16)

Proof. Take $t_{k}=\mu^{k/4}$, $k=0,1,2,\ldots$. Arguing by induction, we
construct a sequence of monotonically nondecreasing functions
$h_{k}:[1,t_{k}]\to(0,\infty)$, $k=1,2,\ldots$, with the following properties:

  1. i)

$h_{k}(t)\leq h_{s}(t)\leq H(t)$ for all $t\in[1,t_{s}]$, $1\leq s\leq k$;

  2. ii)

$h_{k}(\mu^{1/4}t)\leq\mu^{2}h_{k}(t)$ for all $t\in[1,t_{k-1}]$,
$k=1,2,\ldots$;

  3. iii)

$\int\limits_{1}^{t_{k}}(h_{k}(t)t)^{-1/4}dt\leq\frac{\mu^{1/2}}{1-\mu^{-1/16}}\int\limits_{1}^{t_{k}}(\eta(t)t)^{-1/4}dt$
(17)

for all $k=1,2,\ldots$.

Denote

$\gamma_{k}=\sup_{(t_{k-1},t_{k+1})}\eta,\quad k=1,2,\ldots.$

Let us put $h_{1}=\gamma_{1}$. Assume now that for some $k\geq 2$ the function
$h_{k-1}$ is already known. We take

$h_{k}(t)=\left\\{\begin{aligned}
&\min\\{\gamma_{k},\,h_{k-1}(t)\\},&&t\in[1,t_{k-1}],\\\
&\min\\{\gamma_{k},\,h_{k-1}(t_{k-1})\\},&&t\in[t_{k-1},t_{k}]\end{aligned}\right.$

if

$\mu h_{k-1}(t_{k-1})\geq\gamma_{k}$ (18)

and

$h_{k}(t)=\left\\{\begin{aligned} &h_{k-1}(t),&&t\in[1,t_{k-1}],\\\
&\frac{t_{k}-t+\mu(t-t_{k-1})}{t_{k}-t_{k-1}}h_{k-1}(t_{k-1}),&&t\in[t_{k-1},t_{k}],\end{aligned}\right.$

if condition (18) is not fulfilled. It is easy to see that $h_{k}$ satisfies
conditions i) and ii). Let us show that $h_{k}$ satisfies inequality (17). In
the case $k=1$ this inequality follows from the definitions of $h_{1}$ and
$\gamma_{1}$. Let $k\geq 2$ and

$\int\limits_{1}^{t_{s}}(h_{s}(t)t)^{-1/4}dt\leq\frac{\mu^{1/2}}{1-\mu^{-1/16}}\int\limits_{1}^{t_{s}}(\eta(t)t)^{-1/4}dt$
(19)

for all $1\leq s\leq k-1$. At first, we assume that (18) holds. Let $s\geq 0$
be the minimum integer such that $\mu h_{k}(t_{s})\geq h_{k}(t_{k})$. It
follows from the definition of the function $h_{k}$ that $h_{s}=h_{k}$ on the
interval $[1,t_{s}]$. Since

$h_{k}(t)\geq\frac{h_{k}(t_{k})}{\mu}\geq\frac{\gamma_{k}}{\mu^{2}}$

for all $t\in(t_{s},t_{k})$, we obtain

$\int\limits_{t_{s}}^{t_{k}}(h_{k}(t)t)^{-1/4}dt\leq\frac{4}{3}\mu^{1/2}\gamma_{k}^{-1/4}t_{k}^{3/4}.$

Combining the last inequality with the obvious relation

$\int\limits_{t_{k-1}}^{t_{k}}(\eta(t)t)^{-1/4}dt\geq\frac{4}{3}\gamma_{k}^{-1/4}(t_{k}^{3/4}-t_{k-1}^{3/4})=\frac{4}{3}\gamma_{k}^{-1/4}(1-\mu^{-3/16})t_{k}^{3/4},$

we have

$\int\limits_{t_{s}}^{t_{k}}(h_{k}(t)t)^{-1/4}dt\leq\frac{\mu^{1/2}}{1-\mu^{-3/16}}\int\limits_{t_{k-1}}^{t_{k}}(\eta(t)t)^{-1/4}dt.$
(20)

If $s=0$, then (20) immediately implies (17). Assume that $s\geq 1$. In view
of the fact that $s\leq k-1$, estimate (19) is valid, summing which with (20),
we derive (17).

Now assume that (18) does not hold. In this case, $h_{k}(t_{k})=\mu
h_{k}(t_{k-1})$. Let $s\geq 1$ be the minimum integer such that
$h_{k}(t_{k})=\mu^{k-s}h_{k}(t_{s})$. We have

$\int\limits_{t_{i}}^{t_{i+1}}(h_{k}(t)t)^{-1/4}dt=\mu^{-(i-s)/16}\int\limits_{t_{s}}^{t_{s+1}}(h_{k}(t)t)^{-1/4}dt$

for all $s\leq i\leq k-1$; therefore,

$\int\limits_{t_{s}}^{t_{k}}(h_{k}(t)t)^{-1/4}dt\leq\frac{1}{1-\mu^{-1/16}}\int\limits_{t_{s}}^{t_{s+1}}(h_{k}(t)t)^{-1/4}dt.$
(21)

At the same time, taking into account the monotonicity of the function $h_{k}$
and the fact that

$h_{k}(t_{s})=h_{k}(t_{s-1})\geq\frac{\gamma_{s}}{\mu},$

we have $h_{k}(t)\geq\eta(t)/\mu$ for all $t\in(t_{s},t_{s+1})$. Thus, in view
of (21), the estimate

$\int\limits_{t_{s}}^{t_{k}}(h_{k}(t)t)^{-1/4}dt\leq\frac{\mu^{1/4}}{1-\mu^{-1/16}}\int\limits_{t_{s}}^{t_{s+1}}(\eta(t)t)^{-1/4}dt$

is valid. Since $h_{s}=h_{k}$ on the interval $[1,t_{s}]$, adding this with
(19), we again arrive at (17).

By (15),

$\lim_{k\to\infty}\gamma_{k}=\infty;$

therefore, the sequence $h_{k}$, $k=1,2,\ldots$, stabilizes at $k\to\infty$ on
any compact set belonging to the set $[1,\infty)$. In particular, for all
$t\in[1,\infty)$ there exists a limit

$\tilde{h}(t)=\lim_{k\to\infty}h_{k}(t).$

Let us extend the function $\tilde{h}$ to the entire real axis, putting

$\tilde{h}(t)=\min\left\\{\tilde{h}(1),\inf_{(t,1]}H\right\\}$

for $t\in(0,1)$ and $\tilde{h}(t)=0$ for $t\in(-\infty,0]$. To complete the
proof of Lemma 5, it remasins to take

$h(t)=\int\limits_{-\infty}^{\infty}\omega_{1/2}(t-\tau)\tilde{h}\left(\tau-\frac{1}{2}\right)d\tau,\quad
t>0.$

We prove Theorem 1 in several steps. Let us agree to denote by $C$ positive,
possibly different, constants depending only on $\sigma$ and $\theta$.

Step 1. Applying Lemma 5 with $\mu=\theta^{1/2}$ and

$H(t)=\sup_{(t/\mu,t\mu)}g,$

we have a non-decreasing infinitely smooth function
$h:(0,\infty)\to(0,\infty)$ such that

$h(4t)\leq Ch(t)$ (22)

for all $t\geq 2$, condition (16) is valid and, moreover,

$\inf_{x\in\Omega_{r,\sigma},\,\zeta\in(\theta^{-1/2}t,\theta^{1/2}t)}f(x,|x|\zeta)\geq
q(r)h(t)$ (23)

for all $t>0$ and $r>r_{*}$.

Step 2. We take a continuous function $p:[0,\infty)\to[0,\infty)$ and a real
number $R_{*}>r_{*}$, satisfying the following conditions:

$p(r)=0\quad\mbox{for all }0\leq r\leq R_{*};$
$p(r)=e^{-1/(r-R_{*})+1}p(R_{*}+1)>0\quad\mbox{for all }R_{*}<r\leq R_{*}+1;$
(24)
$\inf_{x\in\Omega_{r,\sigma^{1/4}},\,\zeta\in(\theta^{-1/2}t,\theta^{1/2}t)}f(x,|x|\zeta)\geq
p(r)h(t)\quad\mbox{for all }t>0,r>R_{*};$ (25) $rp(r)\leq 1\quad\mbox{for all
}r\geq 0;$ (26) $\int\limits_{r_{*}}^{\infty}p(r)dr=\infty.$ (27)

Let us show that such a function $p$ and a real number $R_{*}$ exist, and
$R_{*}$ can be made arbitrarily large. We put

$Q_{\delta}(r)=\int\limits_{-\infty}^{\infty}\omega_{\delta}(r-\rho)Q(\rho)d\rho,\quad\delta>0,$

where

$Q(\rho)=\left\\{\begin{aligned} &0,&&\rho\in(-\infty,r_{*}],\\\
&\sup_{(\sigma^{-1/4}\rho,\sigma^{1/4}\rho)\cap(r_{*},\infty)}q,&&\rho\in(r_{*},\infty).\end{aligned}\right.$

Taking into account (23), we have

$\inf_{x\in\Omega_{r,\sigma^{1/2}},\,\zeta\in(\theta^{-1/2}t,\theta^{1/2}t)}f(x,|x|\zeta)\geq
Q_{\delta}(r)h(t)$ (28)

for all $t>0$ and $r>r_{*}$, where $\delta>0$ is a sufficiently small real
number. At the same time, for any $\delta>0$ there is a real number
$r_{\delta}>r_{*}$ such that

$\inf_{(\sigma^{-1/8}r,\sigma^{1/8}r)}Q_{\delta}\geq q(r)$

for all $r\in[r_{\delta},\infty)$. Denote

$M_{\delta}(r)=\min\left\\{Q_{\delta}(r),\frac{1}{r}\right\\}.$

We show that

$\int\limits_{r_{*}}^{\infty}M_{\delta}(r)dr=\infty$ (29)

for any $\delta>0$. Indeed, let there be a real number $r>0$ in every
neighborhood of infinity such that $q(r)\geq 1/r$. Then
$M_{\delta}(\rho)\geq\sigma^{-1/8}/\rho$ for all
$\rho\in(\sigma^{-1/8}r,\sigma^{1/8}r)$; therefore,

$\int\limits_{\sigma^{-1/8}r}^{\sigma^{1/8}r}M_{\delta}(\rho)d\rho=\frac{1}{4}\sigma^{-1/8}\ln\sigma.$

According to the necessary criterion for the convergence of an improper
integral, this proves (29). Suppose now that $q(r)<1/r$ for all $r>0$ in some
neighborhood of infinity. In this case, we obtain

$M_{\delta}(r)\geq\min\left\\{q(r),\frac{1}{r}\right\\}=q(r)$

for all sufficiently large $r>0$. Hence, (29) follows from the second
condition of (7).

Thus, it remains to take

$p(r)=\left\\{\begin{aligned} &0,&&r\in[0,R_{*}],\\\
&e^{-1/(r-R_{*})+1}M_{\delta}(R_{*}+1),&&r\in(R_{*},R_{*}+1],\\\
&M_{\delta}(r),&&r\in(R_{*}+1,\infty),\end{aligned}\right.$

where the real numbers $R_{*}>r_{*}$ and $\delta>0$ are chosen so that
${M_{\delta}(R_{*}+1)}>0$ and, moreover, (25) holds. The existence of such
real numbers follows from (28) and (29). It can be seen that, in the case
where $R_{*}$ is large enough, the relation (28) implies (25). Indeed, if
$R_{*}>1/(\sigma^{1/4}-1)$, then for all $r\in(R_{*},R_{*}+1)$ the inclusion
$\Omega_{r,\sigma^{1/4}}\subset\Omega_{R_{*}+1,\sigma^{1/2}}$ is valid;
therefore,

$\inf_{x\in\Omega_{r,\sigma^{1/4}},\,\zeta\in(\theta^{-1/2}t,\theta^{1/2}t)}f(x,|x|\zeta)\geq\inf_{x\in\Omega_{R_{*}+1,\sigma^{1/2}},\,\zeta\in(\theta^{-1/2}t,\theta^{1/2}t)}f(x,|x|\zeta)$

for all $r\in(R_{*},R_{*}+1)$. Since the function $p$ does not exceed
$Q_{\delta}(R_{*}+1)$ on the interval $(R_{*},R_{*}+1)$, this allows us to
assert that (25) holds for all $r\in(R_{*},R_{*}+1)$. Finally, if
$r\in[R_{*}+1,\infty)$, then the inequality (28) implies (25), because for
these $r$ the value of the function $p$ does not exceed $Q_{\delta}$. Note
also that, in view of (29), in any neighborhood of infinity there is a point
at which the function $M_{\delta}$ is positive.

Step 3. Consider the Cauchy problem for the ordinary differential equation

$\frac{d^{2}w}{dr^{2}}\left(\frac{dw}{dr}\right)^{2}=\frac{1}{2}p(r)h\left(\frac{w}{r}\right)\quad\mbox{for
}r>0,\quad w(0)=\varepsilon,\quad\frac{dw}{dr}(0)=0,$ (30)

where $\varepsilon>0$ is some real number. It is obvious that a local solution
of this problem exists. In particular, on the interval $[0,R_{*}]$, this
solution is the function $w=\varepsilon$. Let

$R_{max}=\sup{\mathcal{R}},$

where ${\mathcal{R}}$ is the set of real numbers $R>0$ such that the solution
of problem (30) exists on the interval $[0,R)$. Reducing (30) to the integral
equation

$w(r)=\varepsilon+\int\limits_{0}^{r}\left(\frac{3}{2}\int\limits_{0}^{\rho}p(\xi)h\left(\frac{w}{\xi}\right)d\xi\right)^{1/3}d\rho,$
(31)

by standard reasoning, for example, by the contraction mapping method, one can
show that $R_{max}>R_{*}$ and on the interval $[0,R_{max})$ the solution of
problem (30) is unique. Also it is easy to show that $w\in
C^{2}([0,R_{max}))$. Indeed, $w$ is an infinitely smooth function on the set
$[0,R_{*})$, because on this set $w$ is a constant. At the same time,
differentiating (31), we have

$\frac{dw(r)}{dr}=\left(\frac{3}{2}\int\limits_{0}^{r}p(\xi)h\left(\frac{w}{\xi}\right)d\xi\right)^{1/3},\quad\frac{d^{2}w(r)}{dr^{2}}=\frac{1}{2}p(r)h\left(\frac{w}{r}\right)\left(\frac{3}{2}\int\limits_{0}^{r}p(\xi)h\left(\frac{w}{\xi}\right)d\xi\right)^{-1/3}$

for all $r\in(R_{*},R_{max})$. Since

$\int\limits_{0}^{r}p(\xi)h\left(\frac{w}{\xi}\right)d\xi>0$

for all $r\in(R_{*},R_{max})$, the function $w$ is twice continuously
differentiable on the set $(R_{*},R_{max})$. According to (24), the relation

$\int\limits_{0}^{r}p(\xi)h\left(\frac{w}{\xi}\right)d\xi=(r-R_{*})^{2}e^{-1/(r-R_{*})+1}p(R_{*}+1)h\left(\frac{\varepsilon}{R_{*}}\right)+\bar{\bar{o}}\left((r-R_{*})^{2}e^{-1/(r-R_{*})+1}\right)$

is valid as $r\to R_{*}+0$; therefore the function $w$ is twice differentiable
at the point $R_{*}$ with

$\frac{dw(R_{*})}{dr}=\frac{d^{2}w(R_{*})}{dr^{2}}=0.$

Thus, it remains to note that

$\lim_{r\to R_{*}}\frac{dw(r)}{dr}=\lim_{r\to
R_{*}}\frac{d^{2}w(r)}{dr^{2}}=0.$

Now we establish the validity of the inequality

$R_{max}<\infty.$ (32)

Assume the contrary, let the solution of the problem (30) exist on the entire
interval $[0,\infty)$. Putting

$\frac{w(r)-\varepsilon}{r}=v(r)$

in equation (31), we obtain

$v(r)=\frac{1}{r}\int\limits_{0}^{r}\left(\frac{3}{2}\int\limits_{0}^{\rho}p(\xi)h\left(v+\frac{\varepsilon}{\xi}\right)d\xi\right)^{1/3}d\rho.$
(33)

Differentiating (33), we have

$\frac{dv(r)}{dr}=\frac{1}{r^{2}}\left(r\varphi(r)-\int\limits_{0}^{r}\varphi(\rho)d\rho\right)$
(34)

for all $r>0$, where

$\varphi(\rho)=\left(\frac{3}{2}\int\limits_{0}^{\rho}p(\xi)h\left(v+\frac{\varepsilon}{\xi}\right)d\xi\right)^{1/3}.$
(35)

Note that on the set $[0,R_{*}]$ the functions $v$ and $\varphi$ are
identically equal to zero. Since $\varphi$ is non-decreasing on $(0,\infty)$,
the right-hand side of the formula (34) is non-negative; therefore, $v$ is a
non-decreasing function on $[0,\infty)$. It can also be shown that

$\lim_{r\to\infty}v(r)=\infty.$ (36)

Indeed, from (27) and the fact that the function $h$ is positive, it follows
that $v(\rho_{*})>0$ for some real number $\rho_{*}>R_{*}$. Thus,

$v(r)\geq\frac{1}{r}\int\limits_{r/2}^{r}\left(\frac{3}{2}\int\limits_{\rho_{*}}^{\rho}p(\xi)h\left(v+\frac{\varepsilon}{\xi}\right)d\xi\right)^{1/3}d\rho\geq\frac{1}{2}h^{1/3}(v(\rho_{*}))\left(\frac{3}{2}\int\limits_{\rho_{*}}^{r/2}p(\xi)d\xi\right)^{1/3}\to\infty$

as $r\to\infty$. Take a real number $r_{0}>r_{*}$ such that
$\varepsilon/r_{0}\leq 1$, $v(r_{0})\geq 4$ and

$\frac{rp(r)}{\int\limits_{r_{*}}^{r}p(\xi)d\xi}\leq 1$ (37)

for all $r\geq r_{0}$. In view of (36), (26), and (27) such a real number
$r_{0}$ exists. We put

$r_{i}=\sup\\{r\in(r_{i-1},2r_{i-1}):v(r)\leq 2v(r_{i-1})\\},\quad
i=1,2,\ldots.$

In is easy to see that $r_{i}\to\infty$ as $i\to\infty$; otherwise (32) holds.

Lemma 6. Let $v(r_{i+1})\geq 2v(r_{i-1})$ for some $i\geq 1$. Then

$\int\limits_{v(r_{i-1})}^{v(r_{i+1})}(h(t)t)^{-1/4}dt\geq\frac{C\int\limits_{r_{i-1}}^{r_{i+1}}p(\xi)d\xi}{\left(\int\limits_{r_{*}}^{r_{i+1}}p(\xi)d\xi\right)^{3/4}}.$
(38)

Proof. By (33), the inequality

$v(r_{i+1})\geq\frac{h^{1/3}(v(r_{i-1}))}{r_{i+1}}\int\limits_{r_{i-1}}^{r_{i+1}}\left(\frac{3}{2}\int\limits_{r_{i-1}}^{\rho}p(\xi)d\xi\right)^{1/3}d\rho$

iv valid. Combining this with the estimate

$\int\limits_{r_{i-1}}^{r_{i+1}}\left(\int\limits_{r_{i-1}}^{\rho}p(\xi)d\xi\right)^{1/3}d\rho\geq
C\int\limits_{r_{i-1}}^{r_{i+1}}((r_{i+1}-\xi)p(\xi))^{1/3}d\xi$

which follows from Lemma 2, we obtain

$\frac{v(r_{i+1})}{h^{1/3}(v(r_{i-1}))}\geq\frac{C}{r_{i+1}}\int\limits_{r_{i-1}}^{r_{i+1}}((r_{i+1}-\xi)p(\xi))^{1/3}d\xi.$
(39)

Applying further Lemma 3, we arrive at the relation

$\left(\int\limits_{r_{i-1}}^{r_{i+1}}((r_{i+1}-\xi)p(\xi))^{1/3}d\xi\right)^{3/4}\geq
C\int\limits_{r_{i-1}}^{r_{i+1}}((r_{i+1}-\xi)p(\xi))^{1/3}\varkappa^{-1/4}(\xi)d\xi,$
(40)

where

$\varkappa(\xi)=\int\limits_{\xi}^{r_{i+1}}((r_{i+1}-\zeta)p(\zeta))^{1/3}d\zeta.$

It is easy to see that

$\displaystyle(r_{i+1}-\xi)^{1/3}\varkappa^{-1/4}(\xi)$
$\displaystyle=\left(\frac{1}{(r_{i+1}-\xi)^{4/3}}\int\limits_{\xi}^{r_{i+1}}((r_{i+1}-\zeta)p(\zeta))^{1/3}d\zeta\right)^{-1/4}$
$\displaystyle\geq\left(\frac{1}{r_{i+1}-\xi}\int\limits_{\xi}^{r_{i+1}}p^{1/3}(\zeta)d\zeta\right)^{-1/4}$

for all $\xi\in(r_{i-1},r_{i+1})$. At the same time, in view of (37), we have

$p^{1/3}(\zeta)\leq\zeta^{-1/3}\left(\int\limits_{r_{*}}^{\zeta}p(\rho)d\rho\right)^{1/3}\leq\xi^{-1/3}\left(\int\limits_{r_{*}}^{r_{i+1}}p(\rho)d\rho\right)^{1/3}$

for all $\zeta\in(\xi,r_{i+1})$. Thus,

$(r_{i+1}-\xi)^{1/3}\varkappa^{-1/4}(\xi)\geq\xi^{1/12}\left(\int\limits_{r_{*}}^{r_{i+1}}p(\rho)d\rho\right)^{-1/12}$

for all $\xi\in(r_{i-1},r_{i+1})$ and the inequality (40) allows us to assert
that

$\left(\int\limits_{r_{i-1}}^{r_{i+1}}((r_{i+1}-\xi)p(\xi))^{1/3}d\xi\right)^{3/4}\geq\frac{C\int\limits_{r_{i-1}}^{r_{i+1}}\xi^{1/12}p^{1/3}(\xi)d\xi}{\left(\int\limits_{r_{*}}^{r_{i+1}}p(\rho)d\rho\right)^{1/12}}.$

Combining the last expression with formula (39), we obtain

$\frac{v^{3/4}(r_{i+1})}{h^{1/4}(v(r_{i-1}))}\geq\frac{C\int\limits_{r_{i-1}}^{r_{i+1}}\xi^{-2/3}p^{1/3}(\xi)d\xi}{\left(\int\limits_{r_{*}}^{r_{i+1}}p(\rho)d\rho\right)^{1/12}}.$
(41)

According to (37),

$\xi^{-2/3}p^{1/3}(\xi)=(p(\xi)\xi)^{-2/3}p(\xi)\geq\frac{p(\xi)}{\left(\int\limits_{r_{*}}^{\xi}p(\rho)d\rho\right)^{2/3}}\geq\frac{p(\xi)}{\left(\int\limits_{r_{*}}^{r_{i+1}}p(\rho)d\rho\right)^{2/3}},$

for all $\xi\in(r_{i-1},r_{i+1})$; therefore, (41) leads to the estimate

$\frac{v^{3/4}(r_{i+1})}{h^{1/4}(v(r_{i-1}))}\geq\frac{C\int\limits_{r_{i-1}}^{r_{i+1}}p(\xi)d\xi}{\left(\int\limits_{r_{*}}^{r_{i+1}}p(\rho)d\rho\right)^{3/4}}.$
(42)

Considering (22) and the fact that $v(r_{i+1})-v(r_{i-1})\geq v(r_{i+1})/2,$
we have

$\int\limits_{v(r_{i-1})}^{v(r_{i+1})}(h(t)t)^{-1/4}dt\geq\frac{v(r_{i+1})-v(r_{i-1})}{(h(v(r_{i+1}))v(r_{i+1}))^{1/4}}\geq\frac{v^{3/4}(r_{i+1})}{2h^{1/4}(v(r_{i+1}))}\geq\frac{Cv^{3/4}(r_{i+1})}{h^{1/4}(v(r_{i-1}))}.$

Thus, (42) implies (38). Lemma 6 is proved.

Lemma 7. Let $v(r_{i+1})<2v(r_{i-1})$ for some $i\geq 1$. Then

$\int\limits_{v(r_{i-1})}^{v(r_{i+1})}\frac{dt}{h^{1/3}(t)}\geq\frac{C\int\limits_{r_{i-1}}^{r_{i}}p(\xi)d\xi}{\left(\int\limits_{r_{*}}^{r_{i}}p(\rho)d\rho\right)^{2/3}}.$
(43)

Proof. It follows from the conditions of the lemma that $r_{i}=2r_{i-1}$ and
$r_{i+1}=2r_{i}$. By (34), we get

$v(r_{i+1})-v(r_{i-1})=\int\limits_{r_{i-1}}^{r_{i+1}}v^{\prime}(r)dr\geq\frac{1}{r_{i+1}^{2}}\int\limits_{r_{i-1}}^{r_{i+1}}\psi(r)dr,$
(44)

where

$\psi(r)=r\varphi(r)-\int\limits_{0}^{r}\varphi(\rho)d\rho.$

By direct differentiation, one can verify that
$\psi^{\prime}(r)=r\varphi^{\prime}(r)\geq 0$ for all $r>0$. According to the
Newton-Leibniz formula

$\psi(r)=\int\limits_{0}^{r}\xi\varphi^{\prime}(\xi)d\xi\geq\int\limits_{r_{i-1}}^{r}\xi\varphi^{\prime}(\xi)d\xi$

for all $r\in(r_{i-1},r_{i+1})$; therefore, (44) implies the estimate

$v(r_{i+1})-v(r_{i-1})\geq\frac{1}{r_{i+1}^{2}}\int\limits_{r_{i-1}}^{r_{i+1}}\int\limits_{r_{i-1}}^{r}\xi\varphi^{\prime}(\xi)d\xi
dr\geq\frac{C}{r_{i+1}}\int\limits_{r_{i-1}}^{r_{i+1}}\int\limits_{r_{i-1}}^{r}\varphi^{\prime}(\xi)d\xi
dr,$

combining which with the elementary inequality

$\int\limits_{r_{i-1}}^{r_{i+1}}\int\limits_{r_{i-1}}^{r}\varphi^{\prime}(\xi)d\xi
dr\geq\int\limits_{r_{i}}^{r_{i+1}}\int\limits_{r_{i-1}}^{r}\varphi^{\prime}(\xi)d\xi
dr\geq(r_{i+1}-r_{i})\int\limits_{r_{i-1}}^{r_{i}}\varphi^{\prime}(\xi)d\xi,$

we have

$v(r_{i+1})-v(r_{i-1})\geq
C\int\limits_{r_{i-1}}^{r_{i}}\varphi^{\prime}(\xi)d\xi.$ (45)

At the same time, differentiating (35), we obtain

$\varphi^{\prime}(\xi)=\frac{2^{-1/3}3^{-2/3}p(\xi)h\left(v(\xi)+\varepsilon/\xi\right)}{\left(\int\limits_{r_{*}}^{\xi}p(\rho)h\left(v+\varepsilon/\rho\right)d\rho\right)^{2/3}}\geq\frac{2^{-1/3}3^{-2/3}p(\xi)h(v(r_{i-1}))}{\left(\int\limits_{r_{*}}^{r_{i}}p(\rho)d\rho\right)^{2/3}h^{2/3}(2v(r_{i}))}$

for all $\xi\in(r_{i-1},r_{i})$. Since $h(2v(r_{i}))\leq Ch(v(r_{i-1}))$ in
view of the condition (22), this allows us to assert that

$\varphi^{\prime}(\xi)\geq\frac{Cp(\xi)h^{1/3}(v(r_{i-1}))}{\left(\int\limits_{r_{*}}^{r_{i}}p(\rho)d\rho\right)^{2/3}}$

for all $\xi\in(r_{i-1},r_{i})$. Thus, the formula (45) implies the estimate

$\frac{v(r_{i+1})-v(r_{i-1})}{h^{1/3}(v(r_{i-1}))}\geq\frac{C\int\limits_{r_{i-1}}^{r_{i}}p(\xi)d\xi}{\left(\int\limits_{r_{*}}^{r_{i}}p(\rho)d\rho\right)^{2/3}},$

combining which with the inequality

$\int\limits_{v(r_{i-1})}^{v(r_{i+1})}\frac{dt}{h^{1/3}(t)}\geq\frac{C(v(r_{i+1})-v(r_{i-1}))}{h^{1/3}(v(r_{i-1}))},$

we arrive at (43). Lemma 7 is proved.

For a natural number $m$, we denote by $\Xi_{1,m}$ the set of integers ${1\leq
i\leq m}$ satisfying the condition $v(r_{i+1})\geq 2v(r_{i-1})$. Also let
$\Xi_{2,m}=\\{1,\ldots,m\\}\setminus\Xi_{1,m}.$

In view of (36) and (27), there is a natural number $l$ such that
$v(r_{m})\geq 2v(r_{0})$ and

$\int\limits_{r_{0}}^{r_{m}}p(\xi)d\xi\geq\frac{1}{2}\int\limits_{r_{*}}^{r_{m}}p(\xi)d\xi$
(46)

for all $m\geq l$. Let us show that

$\int\limits_{v(r_{0})}^{v(r_{m+1})}(h(t)t)^{-1/4}dt\geq
C\left(\int\limits_{r_{*}}^{r_{m+1}}p(\xi)d\xi\right)^{1/4}$ (47)

for all $m\geq l$. At first, we assume that

$\sum_{i\in\Xi_{1,m}}\,\int\limits_{r_{i-1}}^{r_{i}}p(\xi)d\xi\geq\frac{1}{2}\int\limits_{r_{0}}^{r_{m}}p(\xi)d\xi.$
(48)

Summing estimate (38) of Lemma 6 over all $i\in\Xi_{1,m}$, we have

$\int\limits_{v(r_{0})}^{v(r_{m+1})}(h(t)t)^{-1/4}dt\geq\frac{C\int\limits_{r_{0}}^{r_{m}}p(\xi)d\xi}{\left(\int\limits_{r_{*}}^{r_{m+1}}p(\xi)d\xi\right)^{3/4}},$

whence in view of (46) and the inequality

$\int\limits_{r_{*}}^{r_{m}}p(\xi)d\xi\geq
C\int\limits_{r_{*}}^{r_{m+1}}p(\xi)d\xi,$ (49)

which follows from Lemma 4, we immediately arrive at (47). Now assume that
(48) is not valid. In this case, it obvious that

$\sum_{i\in\Xi_{2,m}}\,\int\limits_{r_{i-1}}^{r_{i}}p(\xi)d\xi\geq\frac{1}{2}\int\limits_{r_{0}}^{r_{m}}p(\xi)d\xi.$

Thus, summing estimate (43) of Lemma 7 over all $i\in\Xi_{2,m}$, we arrive at
the relation

$\int\limits_{v(r_{0})}^{v(r_{m+1})}\frac{dt}{h^{1/3}(t)}\geq\frac{C\int\limits_{r_{0}}^{r_{m}}p(\xi)d\xi}{\left(\int\limits_{r_{*}}^{r_{m}}p(\rho)d\rho\right)^{2/3}},$

combining which with (46) and (49), we obtain

$\int\limits_{v(r_{0})}^{v(r_{m+1})}\frac{dt}{h^{1/3}(t)}\geq
C\left(\int\limits_{r_{*}}^{r_{m+1}}p(\rho)d\rho\right)^{1/3}.$ (50)

Since $h$ is a non-decreasing function satisfying condition (22), Lemma 1
yields

$\left(\int\limits_{v(r_{0})}^{v(r_{m+1})}(h(t)t)^{-1/4}dt\right)^{4/3}\geq
C\int\limits_{v(r_{0})}^{v(r_{m+1})}\frac{dt}{h^{1/3}(t)}.$

Combining the last estimate with (50), we again obtain (47).

Passing in (47) to the limit as $m\to\infty$, we obtain a contradiction with
the conditions (16) and (27). This proves the validity of (32). Note that, in
view of (32), the relation

$\lim_{r\to R_{\max}-0}w(r)=\infty;$ (51)

otherwise the solution of the problem (30) exists on the interval $(0,R)$ for
some $R>R_{max}$.

Step 4. Below we will assume that $u$ is a non-negative solution (1). We need
the comparison principle in the following simple form.

Lemma 8. Let $U\in C^{2}(B_{R}\cap\Omega)\cap C(\overline{B_{R}\cap\Omega})$
be a classical solution of the inequality

$\Delta_{\infty}U\leq{\mathcal{F}}(x,U)\quad\mbox{in }B_{R}\cap\Omega$

non-negative on the set $B_{R}\cap\Omega$ and such that

$\left.U\right|_{\partial(B_{R}\cap\Omega)}\geq\left.u\right|_{\partial(B_{R}\cap\Omega)},$
(52)

where $R>0$ is a real number and ${\mathcal{F}}$ is some function non-
decreasing with respect to the last argument with

${\mathcal{F}}(x,t)<f(x,t)$ (53)

for all $x\in B_{R}\cap\Omega$ and $t>0$. Then

$U\geq u\quad\mbox{in }\overline{B_{R}\cap\Omega}.$ (54)

Proof. Assume the contrary. Let

$\sup_{\overline{B_{R}\cap\Omega}}(u-U)>0.$

The function $u-U$ is upper semicontinuous on the set
$\overline{B_{R}\cap\Omega}$; therefore, there is a point
$x\in\overline{B_{R}\cap\Omega}$ such that

$u(x)-U(x)=\sup_{\overline{B_{R}\cap\Omega}}(u-U).$

In view of (52), one can also assert that $x\in B_{R}\cap\Omega$. Hence,

$u(y)-u(x)\leq U(y)-U(x)=\langle DU(x),y-x\rangle+\frac{1}{2}\langle
D^{2}U(x)(y-x),y-x\rangle+\bar{\bar{o}}(|y-x|^{2})$

for all $y$ from some neighborhood of the point $x$, where $DU(x)$ is the
gradient and $D^{2}U(x)$ is the matrix of the second derivatives of the
function $U$ at the point $x$ , which immediately implies that

$(DU(x),D^{2}U(x))\in J_{\Omega}^{2,+}u(x).$

Substituting $v=DU(x)$ and $A=D^{2}U(x)$ in (4), we have

$\sum_{i,j=1}^{n}\frac{\partial^{2}U(x)}{\partial x_{i}\partial
x_{j}}\frac{\partial U(x)}{\partial x_{i}}\frac{\partial U(x)}{\partial
x_{j}}\geq f(x,u(x))>{\mathcal{F}}(x,u(x)).$

Thus, taking into account the inequalities

${\mathcal{F}}(x,u(x))\geq{\mathcal{F}}(x,U(x))\geq\sum_{i,j=1}^{n}\frac{\partial^{2}U(x)}{\partial
x_{i}\partial x_{j}}\frac{\partial U(x)}{\partial x_{i}}\frac{\partial
U(x)}{\partial x_{j}},$

we conclude that

$\sum_{i,j=1}^{n}\frac{\partial^{2}U(x)}{\partial x_{i}\partial
x_{j}}\frac{\partial U(x)}{\partial x_{i}}\frac{\partial U(x)}{\partial
x_{j}}>\sum_{i,j=1}^{n}\frac{\partial^{2}U(x)}{\partial x_{i}\partial
x_{j}}\frac{\partial U(x)}{\partial x_{i}}\frac{\partial U(x)}{\partial
x_{j}}.$

This contradiction proves the lemma.

Proof of Theorem 1. According to (30), the function $U(x)=w(|x|)$ is a
classical solution of the equation

$\Delta_{\infty}U={\mathcal{F}}(x,U)\quad\mbox{in }B_{R_{max}}$

positive in the ball $B_{R_{max}}$, where

${\mathcal{F}}(x,t)=\left\\{\begin{aligned}
&\frac{1}{2}p(|x|)h\left(\frac{t}{|x|}\right),&&R_{*}<|x|<R_{max},\,t>0,\\\
&0,&&|x|\leq R_{*},\,t>0.\end{aligned}\right.$

Since a solution of (1) is upper semicontinuous on $\overline{\Omega}$, the
function $u$ is bounded on the closure of $B_{R_{max}}\cap\Omega$; therefore,
in accordance with (51) there is a real number $R\in(R_{*},R_{max})$ for which
(52) holds. Taking into account (2) and (25), one can also claim that (53)
holds. Thus, by Lemma 8, relation (54) is valid. In its turn, this implies
that

$\left.u\right|_{B_{R_{*}}\cap\Omega}\leq\varepsilon.$

Since $\varepsilon>0$ can be taken arbitrarily small and $R_{*}>r_{*}$ can be
taken arbitrarily large, we obviously obtain $u(x)=0$ for all $x\in\Omega$.

Proof of Corallary 1. We put $f(x,t)=c(|x|)t^{\lambda}$, $g(t)=t^{\lambda}$,
$q(r)=4^{-\lambda}r^{\lambda}c(r/2)$, and $\sigma=\theta=2$ in relation (3)
and use Theorem 1.

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