# Continuous Family of Equilibria of the 3D Axisymmetric Relativistic Vlasov-
Maxwell System

Katherine Zhiyuan Zhang  Courant Institute of Mathematical Sciences, New York
University

###### Abstract.

We consider the relativistic Vlasov-Maxwell system (RVM) on a general
axisymmetric spatial domain with perfect conducting boundary which reflects
particles specularly, assuming axisymmetry in the problem. We construct
continuous global parametric solution sets for the time-independent RVM. The
solutions in these sets have arbitrarily large electromagnetic field and the
particle density functions have the form
$f^{\pm}=\mu^{\pm}(e^{\pm}(x,v),p^{\pm}(x,v))$, where $e^{\pm}$ and $p^{\pm}$
are the particle energy and angular momentum, respectively. In particular, for
a certain class of examples, we show that the spectral stability changes as
the parameter varies from $0$ to $\infty$.

##  1\. Introduction

When the temperature is high or the density is low, the effect of collisions
in a plasma becomes minor compared to the effect of the electromagnetic
forces. Such a plasma is modeled by the relativistic Vlasov-Maxwell system
(RVM). In many real world applications, the plasma is confined to a bounded
region. A typical example is the tokamak, which is one of the main foci of
research in fusion energy. A key direction of study is to construct the
stationary solutions (equilibria) of the RVM system on a bounded domain with
varied boundary conditions.

We consider the 3D relativistic Vlasov-Maxwell (RVM) system on an axisymmetric
bounded domain $\Omega\subset\mathbb{R}^{3}$. We study a plasma with two
species (ion and electron) with non-negative distribution functions
$f^{\pm}(t,x,v)$, where $t\geq 0$, $x\in\Omega\subset\mathbb{R}^{3}$,
$v\in\mathbb{R}^{3}$. The system RVM is

(1.1)
$\partial_{t}f^{\pm}+\hat{v}\cdot\nabla_{x}f^{\pm}\pm(\textbf{E}+\hat{v}\times\textbf{B})\cdot\nabla_{v}f^{\pm}=0,$
(1.2) $\nabla_{x}\cdot\textbf{E}=\rho=\int_{\mathbb{R}^{3}}(f^{+}-f^{-})dv,\
\nabla_{x}\cdot\textbf{B}=0,$ (1.3)
$\partial_{t}\textbf{E}-\nabla_{x}\times\textbf{B}=-\textbf{j}=-\int_{\mathbb{R}^{3}}\hat{v}(f^{+}-f^{-})dv,\
\partial_{t}\textbf{B}+\nabla_{x}\times\textbf{E}=0.$

In this system, $f^{\pm}(t,x,v)\geq 0$ is the density distribution of ions
($+$) and electrons ($-$). We confine the plasma in a region
$\Omega\subset\mathbb{R}^{3}$, so that $x\in\Omega$ is the particle position.
$v\in\mathbb{R}^{3}$ is the particle momentum, $\langle
v\rangle=\sqrt{1+v^{2}}$ is the particle kinetic energy, and
$\hat{v}=v/\langle v\rangle$ is the particle velocity. Also, E is the electric
field, B is the internal magnetic field. Therefore
$\pm(\textbf{E}+\hat{v}\times\textbf{B})$ is the total (internal)
electromagnetic force. Moreover, the charge density $\rho$ and the current
density j are defined as

(1.4)
$\rho=\int_{\mathbb{R}^{3}}(f^{+}-f^{-})dv,\qquad\textbf{j}=\int_{\mathbb{R}^{3}}\hat{v}(f^{+}-f^{-})dv.$

At the boundary we impose the specular condition (which means that $f^{\pm}$
is even with respect to $v_{n}=v\cdot e_{n}$ on $\partial\Omega$, with $e_{n}$
being the outward normal vector of $\partial\Omega$ at $x$):

(1.5) $f^{\pm}(t,x,v)=f^{\pm}(t,x,v-2(v\cdot e_{n}(x))e_{n}(x)),e_{n}(x)\cdot
v<0,\ \forall x\in\partial\Omega,$

as well as the perfect conductor boundary condition

(1.6) $\textbf{E}(t,x)\times n(x)=0,\ \textbf{B}(t,x)\cdot n(x)=0,\ \forall
x\in\partial\Omega.$

We assume axisymmetry in the problem. Therefore it is natural to use
cylindrical coordinates here.

We aim to construct equilibrium solutions for the system (1.1) – (1.3) with
the boundary conditions above. We consider equilibria with the form

$f^{\pm}=\mu^{\pm}(e^{\pm}(x,v),p^{\pm}(x,v))$

where $e^{\pm}$ and $p^{\pm}$ are the particle energy and angular momentum,
respectively, and use a potential formulation together with the axisymmetry
assumption of (1.1) – (1.3) so as to the equations satisfied by the equilibria
can be reformulated as a coupled elliptic system (see (2.12) – (2.14) below).
The resulting elliptic system is highly nonlinear on its right hand side.

Investigation for time-independent RVM on a domain with physical boundaries
has been carried out over the years. Rein [20] studied weak stationary
solutions of the RVM by variational methods. Batt & Fabian [2], Braasch [3],
Poupaud [18], Sinitsyn & Dulov [24] constructed stationary solutions for
boundary value problems making use of sub-solutions and super-solutions. In
particular, Batt & Fabian [2] constructed a general family of large stationary
solutions for boundary value problems when the domain is axially symmetric and
does not touch the $z$-axis and Braasch [3] constructed large stationary
solutions for problems with a specific type of axial symmetric domain. Poupaud
[18] solves for large equilibria for general domains (not necessarily axially
symmetric) with compact and connected boundary as well as general Dirichlet
boundary conditions. Moreover, the lower dimensional models have also been
considered. For example, Schaeffer [21] studied the stationary Vlasov-Maxwell
equation (VM) in the 1.5D setting, when the spatial domain being the real line
$\mathbb{R}$, see also Guo [7].

Our strategy is to consider general parametrized family of particle density
distribution function for the equilibria, and to apply the global implicit
function theorem (Theorem 2.4) so as to obtain global solution sets that
branch out from ”trivial” equilibrium solutions. The result is stated in
Theorem 3.2. Similar methods have been applied in varied PDEs, see, for
example, [4], [5], [6], [25]. For the Vlasov-Maxwell equation (VM), Sidorov &
Sinitsyn [23] considered a specific parametrized family of (single species)
particle density distribution, and gave sufficient conditions for the
existence of bifurcation points corresponding to distribution functions of a
certain form.

In this paper, we consider a general parametrized family of particle density
distribution $f^{\pm}=\mu^{\pm}(e^{\pm}(x,v),p^{\pm}(x,v))$, and construct a
_global continuum/loop_ of equilibria branching out from the trivial or
neutral steady states for the RVM on a general axisymmetric spatial domain
with perfect conducting boundary which reflects particles specularly, assuming
axisymmetry in the problem. Moreover, in the single species case, with some
additional constraints, the global continuum/loop of equilibria we constructed
contains a locally analytic curve that is _unbounded_ , see Theorem 3.3. Our
approach enables us to observe the structure of the solution set, and reveal
more information about the change of the electromagnetic potential (and
therefore the electromagnetic field) as the density distribution function
changes in different ways. In the case when the solution set is unbounded, we
make use of the maximum principles and elliptic estimates to investigate size
properties of the equilibria constructed in Theorem 3.2. In particular, we
construct a continuum of equilibrium solutions with arbitrarily large
electromagnetic field, see Proposition 4.1. It is an open question whether the
solutions obtained in this paper are the same as the ones constructed in the
historical literature [20], [2], [3], [18], [24], etc.

Our approach can also be applied in the case when an external magnetic field
is present, see Remark 2.2. This type of problem is also considered in Weber
[22], in which confined stationary solutions of the RVM in a long cylinder in
a two and one-half dimensional setting are constructed. Moreover, Lin &
Strauss [11] constructed confined small stationary solutions of the RVM in 3D
(see the appendix of [11]). Such type of solutions are related to the
mechanism of _magnetic confinement_ , see, for example, [14], [27].

On the other hand, in plasma theory, an important goal is to study the
stability properties of plasmas. In [17], the renowned Penrose’s sharp
criterion on linear stability for a spatially homogeneous equilibrium of the
Vlasov-Poisson system is derived. In [7], [8], [11], [13] and [12], the
analysis of a spatially inhomogeneous equilibrium was carried out in domains
without any spatial boundaries (i.e. whole space or periodic setting). A sharp
criterion for spectral stability was given in [12], with some families of
stable and unstable examples provided. The question of nonlinear stability is
much more difficult, see, for example, [13]. An important topic is to
understand the stability properties of a confined plasma. In [9], the
confinement of a tokamak plasma is discussed using some fluid models and the
role of different parts of the boundary are explored. For the microscopic
model RVM, there are very few rigorous studies in bounded domains. In [16] and
[26], in which the authors considered the case when the spatial domain is a
torus or an axisymmetric bounded region (for example, a tokamak), and toroidal
or axisymmetric symmetry is assumed. A sharp criterion of spectral stability
is obtained, thus reducing the problem of determining the linear stability to
the positivity of a simpler self-adjoint operator $\mathcal{L}^{0}$.

In this paper, we investigate the change of spectral stability property along
a family of solutions constructed in Theorem 3.2 using the result obtained in
[26]. Specifically, we track down the solutions along the solution set as the
parameter $K$ changes from $0$ to $\infty$, and show that the corresponding
equilibrium solutions gradually become spectrally unstable as $K$ grows large.
The details are given in Section 5, Proposition 5.2 and Theorem 5.4.

In addition, we provide explicit examples on which the results above can be
applied. These examples cover cases when the density distribution function is
parametrized as

$\mu^{K,\pm}(e,p)=\mu^{0}(e,p)+K\mu^{\pm}(e,p)$

or

$\mu^{K,\pm}(e,p)=\mu^{0}(e,p)+\mu^{\pm}(e,Kp)$

(where $K\in[0,\infty)$ is the parameter, $\mu^{0}$ and $\mu^{\pm}$ are fixed
non-negative functions satisfying certain conditions) and generate unbounded
solution sets of single-species equilibria. Solutions in these sets have
arbitrarily large electromagnetic field (as $K$ gets large). In the example in
Subsection 6.2, the equilibria turn from spectrally stable to unstable as the
parameter $K$ changes from $0$ to $\infty$.

The contents in the paper are arranged as follows. In Section 2, we set up the
problem in cylindrical coordinates, including the coordinates and the symmetry
assumptions. The description of the particle trajectories and the family of
equilibria we consider in this paper are given, and relevant function spaces
are introduced. The difficulty caused by the singularity at $r=0$ is avoided
by using the spaces $H^{k\dagger}$ and $\mathcal{X}$ (see (2.28) and (2.30)).
We also state the local and global implicit function theorems, which are the
key tools in this paper that provide the existence of the steady sates. In
Section 3 – 6, we consider the time-independent RVM with no external magnetic
field. In Section 3, we consider a family of parametrized particle density
distribution functions and construct an unbounded solution set/loop for the
time-independent problem (2.12)–(2.14) with
$\textbf{A}^{0}=A^{0}_{\varphi}e_{\varphi}$, which gives a steady solution set
for RVM. Furthermore, in the single species case, with additional constraints,
the global continuum/loop of equilibria we constructed contains a locally
analytic curve that is unbounded, yielding a global steady solution set which
contains an unbounded analytic steady solution curve for RVM. Section 4 makes
use of the maximum principles for elliptic equations and gives some properties
of the equilibria constructed in Section 3, Theorem 3.2. It is then shown in
Section 5 that by properly choosing ways of parametrization, one can create
solution sets along which the linear stability changes. After that, in Section
6, we give two examples of parametrized families of particle density
distribution functions and apply the theory in Sections 3, 4, 5. Lastly, for
the readers’ convenience, we provide the formulae for derivative operators in
the cylindrical coordinates as well as the particle energy
$e^{\pm}(x,v)=\langle v\rangle\pm\phi^{0}(r,z)$ and the particle angular
momentum $p^{\pm}(x,v)=r(v_{\varphi}\pm A^{0}_{\varphi}(r,z))$ (or
$p^{\pm}(x,v)=r(v_{\varphi}\pm A^{0}_{\varphi}(r,z)\pm A_{\varphi,ext}(r,z))$
when an external magnetic potential
$\textbf{A}_{ext}=A_{\varphi,ext}(r,z)e_{\varphi}$ is present) in Appendix A
and B, respectively.

Acknowledgement. The author would like to express her gratitude to her
advisor, Professor Walter Strauss, for bringing this research topic to her
attention, and also for the invaluable guidance, encouragement and patience,
without which this work would be impossible.

##  2\. Set up

Let us introduce the basic settings of the stationary RVM problem. We consider
the 3D relativistic Vlasov-Maxwell (RVM) system (1.1) – (1.3) on the
axisymmetric bounded domain $\Omega\subset\mathbb{R}^{3}$. We assume
axisymmetry in the problem and use cylindrical coordinates $(r,\varphi,z)$. We
consider a plasma constrained inside a closed region $\Omega$, which is a
$C^{1}$ axisymmetric (with respect to the $z$-axis) domain in
$\mathbb{R}^{3}$, i.e. rotational invariant around the $z$-axis. $\Omega$ can
be viewed as a solid of revolution, determined as follows: Consider a
counterclockwisely parametrized $C^{1}$ curve $\mathcal{C}$ in the plane
$\\{\varphi=0\\}$, where $\beta$ is the arclength parameter:

(2.1) $r=\tilde{r}(\beta),\quad
z=\tilde{z}(\beta),\quad\beta\in[0,1],\quad\tilde{r}\geq 0\ \text{for all}\
\beta\in(0,1),\quad\tilde{r},\tilde{z}\in C^{1}(0,1),$

with either

(2.2)
$\tilde{r}(0)=\tilde{r}(1)>0,\quad\tilde{z}(0)=\tilde{z}(1),\quad\tilde{z}^{\prime}(0)=\tilde{z}^{\prime}(1),$

or,

(2.3)
$\tilde{r}(0)=\tilde{r}(1)=0,\quad\tilde{z}^{\prime}/\tilde{r}^{\prime}=0\
\text{at}\ \beta=0,1.$

Rotating $\mathcal{C}$ around the $z$-axis gives the boundary for the domain
$\partial\Omega$. In all these cases, the boundary $\partial\Omega$ is $C^{1}$
smooth. We denote

$d:=\sup_{x\in\Omega}r(x).$

Let $e_{r}$, $e_{\varphi}$ and $e_{z}$ be the unit vectors in the cylindrical
coordinate system (see Appendix A), and $e_{n}$, $e_{tg}$ to be the unit
vectors in outward normal direction and tangential direction orthogonal to
$e_{\varphi}$ on $\partial\Omega$, respectively. Then on $\partial\Omega$, the
outward unit normal vector
$e_{n}(x)=(\tilde{z}^{\prime}e_{r}-\tilde{r}^{\prime}e_{z})/\sqrt{\tilde{z}^{\prime
2}+\tilde{r}^{\prime 2}}$, and
$e_{tg}(x)=(-\tilde{r}^{\prime}e_{r}-\tilde{z}^{\prime}e_{z})/\sqrt{\tilde{z}^{\prime
2}+\tilde{r}^{\prime 2}}$.

On the boundary, we assume the specular condition on the density function
$f^{\pm}$ in case the particle hits the boundary. This means that $f^{\pm}$ is
even with respect to $v_{n}=v\cdot e_{n}/|e_{n}|$ on $\partial\Omega$, i.e.

(2.4) $f^{\pm}(t,x,v)=f^{\pm}(t,x,v-2(v\cdot e_{n}(x))e_{n}(x)),\
e_{n}(x)\cdot v<0,\forall x\in\partial\Omega,\forall v\in\mathbb{R}^{3}.$

We also put down the perfect conductor boundary condition on the electric and
magnetic fields:

(2.5) $\textbf{E}(t,x)\times e_{n}(x)=0,\ \textbf{B}(t,x)\cdot
e_{n}(x)=0,\forall x\in\partial\Omega.$

We also introduce the electric potential $\phi$ and the magnetic potential A:

(2.6) $\textbf{E}=-\nabla\phi-\partial_{t}\textbf{A},\
\textbf{B}=\nabla\times\textbf{A},$

and impose the Coulomb gauge

(2.7) $\nabla\cdot\textbf{A}=0.$

######  Remark 2.1.

The choices of $\phi$ and A are not unique. Actually the choice of A can
differ by the gradient of a harmonic function.

We use the cylindrical coordinates $(r,\varphi,z)$, and make the axisymmetry
assumption:

(2.8) $\phi,\ A_{r},\ A_{z},\ A_{\varphi}\ are\ independent\ of\ \varphi.$

Therefore, $f^{\pm}$ does not depend explicitly on $\varphi$, although it
might depend on it implicitly through the components of $v$. The Maxwell
system then becomes

(2.9) $-\Delta\phi=\rho,$ (2.10)
$(\partial_{t}^{2}-\Delta+\frac{1}{r^{2}})A_{\varphi}=j_{\varphi},$ (2.11)
$(\partial_{t}^{2}-\Delta)\tilde{\textbf{A}}+\partial_{t}\nabla\phi=\tilde{\textbf{j}}.$

Here $\tilde{\textbf{A}}:=A_{r}e_{r}+A_{z}e_{z}$.

We consider the equilibrium field, with the potential denoted by
$(\phi^{0},A_{\varphi}^{0},\tilde{\textbf{A}}^{0})$, and for which the Maxwell
equations become

(2.12) $-\Delta\phi^{0}=\rho^{0},$ (2.13)
$(-\Delta+\frac{1}{r^{2}})A_{\varphi}^{0}=j_{\varphi}^{0},$ (2.14)
$(-\Delta)\tilde{\textbf{A}}^{0}=\tilde{\textbf{j}}^{0}.$

We consider an equilibrium that satisfies

$\textbf{A}^{0}=A^{0}_{\varphi}e_{\varphi}.$

which implies

$B^{0}_{\varphi}=0.$

Under this constraint, (2.12) – (2.14) become a system with reduced dimension:

(2.15) $-\Delta\phi^{0}=\rho^{0},$ (2.16)
$(-\Delta+\frac{1}{r^{2}})A_{\varphi}^{0}=j_{\varphi}^{0}.$

According to (2.5), we assume the following boundary conditions for the
potentials, Section 4 in [26] for details:

(2.17) $\phi^{0}=0,\qquad A_{\varphi}^{0}=0,\qquad
A_{tg}^{0}=0,\qquad\tilde{z}^{\prime}\partial_{r}A_{n}^{0}-\tilde{r}^{\prime}\partial_{z}A_{n}^{0}+\frac{\tilde{z}^{\prime}}{\tilde{r}}A_{n}^{0}=0,\qquad\forall
x\in\partial\Omega.$

Then the equilibrium field is

(2.18) $\textbf{E}^{0}=-\nabla\phi^{0}=-\frac{\partial\phi^{0}}{\partial
r}e_{r}-\frac{\partial\phi^{0}}{\partial z}e_{z},$ (2.19)
$\begin{split}\textbf{B}^{0}&=-\frac{\partial A^{0}_{\varphi}}{\partial
z}e_{r}+\frac{1}{r}\frac{\partial(rA^{0}_{\varphi})}{\partial r}e_{z}.\\\
\end{split}$

The boundary condition (2.17) then becomes

(2.20) $\phi^{0}=0,\qquad A_{\varphi}^{0}=0,\qquad\forall x\in\partial\Omega.$

We define the particle trajectories as

(2.21) $\dot{X}^{\pm}=\hat{V}^{\pm},\
\dot{V}^{\pm}=\pm\textbf{E}^{0}(X^{\pm})\pm\hat{V}^{\pm}\times\textbf{B}^{0}(X^{\pm})$

with initial values $(X^{\pm}(0;x,v),V^{\pm}(0;x,v))=(x,v)$. Each particle
trajectory exists and preserves the axisymmetry up to the first time it meets
the boundary. Let $s_{0}$ be a time when the trajectory $X^{\pm}(s_{0}-;x,v)$
hits the boundary $\partial\Omega$. Recall that $v_{n}=v\cdot
e_{n}=\frac{\tilde{z}^{\prime}v_{r}-\tilde{r}^{\prime}v_{z}}{\sqrt{\tilde{z}^{\prime
2}+\tilde{r}^{\prime 2}}}$, $v_{tg}=v\cdot
e_{tg}=\frac{-\tilde{z}^{\prime}v_{z}-\tilde{r}^{\prime}v_{r}}{\sqrt{\tilde{z}^{\prime
2}+\tilde{r}^{\prime 2}}}$. For any given $(x,v)$ and $(X^{\pm},V^{\pm})$ with
$x$ and $X^{\pm}$ on $\partial\Omega$, we re-decompose $v$ and $V^{\pm}$ into
their $n$-component, $tg$-component and $\varphi$\- component:
$v=v_{n}e_{n}+v_{tg}e_{tg}+v_{\varphi}e_{\varphi}$,
$V^{\pm}=V^{\pm}_{n}e_{n}+V^{\pm}_{tg}e_{tg}+V^{\pm}_{\varphi}e_{\varphi}$,
and define

(2.22) $v_{*}=-v_{n}e_{n}+v_{tg}e_{tg}+v_{\varphi}e_{\varphi}\ ,\
V^{\pm}_{*}=-V^{\pm}_{n}e_{n}+V^{\pm}_{tg}e_{tg}+V^{\pm}_{\varphi}e_{\varphi}.$

Thus from the specular boundary condition, the trajectory can be continued by
the rule

(2.23)
$(X^{\pm}(s_{0}+;x,v),V^{\pm}(s_{0}+;x,v))=(X^{\pm}(s_{0}-;x,v),V_{*}^{\pm}(s_{0}-;x,v)).$

Furthermore, we assume the equilibrium has a particle density of the form
$f^{0,\pm}(x,v)=\mu^{\pm}(e^{\pm}(x,v),p^{\pm}(x,v))$, where

(2.24) $\begin{split}&e^{\pm}(x,v)=\langle v\rangle\pm\phi^{0}(r,z),\
p^{\pm}(x,v)=r(v_{\varphi}\pm A^{0}_{\varphi}(r,z)).\\\ \end{split}$

Here $e^{\pm}$ and $p^{\pm}$ are invariant along the particle trajectories
under our assumptions (with specular boundary condition). (The proof of the
invariance of $e^{\pm}$ and $p^{\pm}$ can be found in Appendix B.) We assume
that $\mu^{\pm}(e,p)$ are non-negative $C^{1}$ functions which satisfy

(2.25)
$\begin{split}&|\mu^{\pm}(e,p)|+|\mu^{\pm}_{p}(e,p)|+|\mu^{\pm}_{e}(e,p)|\leq\frac{C_{\mu}}{1+|e|^{\delta}},\
\delta>3.\\\ \end{split}$

######  Remark 2.2.

Also our results can be applied to the RVM with an external magnetic field
being applied. Consider an external magnetic potential

$\textbf{A}_{ext}=A_{\varphi,ext}e_{\varphi},\ A_{\varphi,ext}\text{ is
independent of }\varphi,\text{ only depends on }r,z.$

Hence the corresponding external magnetic field satisfies $B_{\varphi,ext}=0$.
Again, we assume the equilibrium has a particle density of the form
$f^{0,\pm}(x,v)=\mu^{\pm}(e^{\pm}(x,v),p^{\pm}(x,v))$, where

(2.26) $\begin{split}&e^{\pm}(x,v)=\langle v\rangle\pm\phi^{0}(r,z),\
p^{\pm}(x,v)=r(v_{\varphi}\pm A^{0}_{\varphi}(r,z)\pm
A_{\varphi,ext}(r,z)).\\\ \end{split}$

$e^{\pm}$ and $p^{\pm}$ are invariant along the particle trajectories

(2.27) $\dot{X}^{\pm}=\hat{V}^{\pm},\
\dot{V}^{\pm}=\pm\textbf{E}^{0}(X^{\pm})\pm\hat{V}^{\pm}\times\textbf{B}^{0}(X^{\pm})\pm\hat{V}^{\pm}\times\textbf{B}_{ext}(X^{\pm}).$

with all the other assumptions as mentioned in the previous case without an
external magnetic field, see Appendix B.

We are going to solve the elliptic system (2.12) – (2.14) with the boundary
conditions above as well as $\textbf{A}^{0}=A^{0}_{\varphi}e_{\varphi}$
(yielding (2.15) – (2.16)). We use the letter $\tau$ to denote the
axisymmetric constraint for functions. Denote $Y=L^{2}_{1/r^{2}}(\Omega)$,
i.e. the weighted-$L^{2}$ space with weight $1/r^{2}$. This weight gives some
singularity at $r=0$ if $\Omega$ touches the $z$-axis. For this, we define

(2.28) $H^{k\dagger}(\Omega):=\\{g\in L^{2,\tau}(\Omega)|e^{i\varphi}g\in
H^{k}(\Omega)\\}(k=1,2).$

Then $H^{2\dagger}(\Omega)\subset Y$ because the identity

(2.29) $-\Delta(ge^{i\varphi})=(-\Delta+\frac{1}{r^{2}})ge^{i\varphi}$

holds for any $\varphi$-independent function $g$. Indeed,

$\begin{split}-\Delta(ge^{i\varphi})&=-\frac{1}{r}\frac{\partial}{\partial
r}(r\frac{\partial g}{\partial r})e^{i\varphi}-\frac{\partial^{2}g}{\partial
z^{2}}e^{i\varphi}-\frac{1}{r^{2}}\frac{\partial^{2}(e^{i\varphi})}{\partial\varphi^{2}}g\\\
&=-\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial g}{\partial
r})e^{i\varphi}-\frac{\partial^{2}g}{\partial
z^{2}}e^{i\varphi}+\frac{1}{r^{2}}ge^{i\varphi}\\\
&=(-\Delta+\frac{1}{r^{2}})ge^{i\varphi}.\\\ \end{split}$

We will use the following space regularly:

(2.30) $\mathcal{X}:=\\{g\in H^{2}(\Omega):g\ \text{independent of}\ \varphi,\
\exp(i\varphi)g\in H^{2}(\Omega),\ g|_{\partial\Omega}=0\\}\subset
H^{2\dagger}(\Omega).$

In the rest of the paper, $A\lesssim B$ ($A\gtrsim B$) means $A\leq CB$
($A\geq CB$) for some constant $C>0$ independent of $K$.

Before we continue, let us state some local and global implicit function
theorems, which are key ingredients in this paper. Readers can refer to [10],
[5], [6], etc. for detailed discussions of these results.

######  Theorem 2.3.

(Local Implicit Function Theorem) Let $X$, $Y$, $Z$ be (real) Banach spaces.
Consider a mapping $F:U\times V\rightarrow Z$ with open sets $U\subset X$,
$V\subset Y$, and the equation

(2.31) $F(x,y)=0\ .$

Suppose the equation (2.31) admits a solution $(x_{0},y_{0})\in U\times V$
such that the Fréchet derivative of $F$ with respect to $x$ at $(x_{0},y_{0})$
is bijective, i.e.

(2.32) $F(x_{0},y_{0})=0\ ,$ (2.33) $D_{x}F(x_{0},y_{0}):X\rightarrow Z\ is\
bounded\ (continuous)\ with\ a\ bounded\ inverse.$

Assume also that $F$ and $D_{x}F$ are continuous:

(2.34) $F\in C(U\times V,Z)\ ,$ (2.35) $D_{x}F\in C(U\times V,L(X,Z))\ .$

Here $L(X,Z)$ denotes the Banach space of bounded linear operators from $X$ to
$Z$ endowed with the operator norm. Then there exists a neighbourhood
$U_{1}\times V_{1}$ in $U\times V$ of $(x_{0},y_{0})$ and a mapping
$f:V_{1}\rightarrow U_{1}\subset X$, such that

(2.36) $f(y_{0})=x_{0},$ (2.37) $F(f(y),y)=0\ for\ all\ y\in V_{1}.$

Furthermore, $f$ is continuous on $V_{1}$:

(2.38) $f\in C(V_{1},X).$

Finally, every solution of (2.31) in $U_{1}\times V_{1}$ is of the form
$(f(y),y)$.

######  Theorem 2.4.

(Global Implicit Function Theorem) Let $X$ and $Z$ be (real) Banach spaces and
$U$ be an open set in $X\times\mathbb{R}$. Consider a mapping $F:U\rightarrow
Z$ that is $C^{1}$ in the Fréchet sense, and the equation

(2.39) $F(x,y)=0$

with $x\in X$, $y\in\mathbb{R}$. Suppose the equation (2.39) admits a solution
$(x_{0},y_{0})\in U$ such that the Fréchet derivative of $F$ with respect to
$x$ at $(x_{0},y_{0})$ is bijective, i.e.

(2.40) $F(x_{0},y_{0})=0,$ (2.41) $D_{x}F(x_{0},y_{0}):X\rightarrow Z\ is\
bounded\ (continuous)\ with\ a\ bounded\ inverse.$

Assume also that the mapping $(x,y)\rightarrow F(x,y)-x$ is compact from $U$
to $X$. Let $\mathcal{S}$ be the closure in $X\times\mathbb{R}$ of the
solutions set $\\{(x,y):F(x,y)=0\\}$. Let $\mathcal{C}$ be the connected
component of $\mathcal{S}$ to which $(x_{0},y_{0})$ belongs. Then one of the
following three alternatives holds:
i) $\mathcal{C}$ is unbounded in $X\times\mathbb{R}$;
ii) $\mathcal{C}\setminus\\{(x_{0},y_{0})\\}$ is connected;
iii) $\mathcal{C}\cap\partial U$ is not empty.

######  Theorem 2.5.

(Analytic Global Implicit Function Theorem) Let $X$ and $Z$ be (real) Banach
spaces, $I$ be an open interval (possibly unbounded) with $0\in\overline{I}$,
and $U$ be an open set in $X$ with $0\in\partial U$. Consider a mapping
$F:U\times I\rightarrow Z$ that is real-analytic in the Fréchet sense, and the
equation

(2.42) $F(x,y)=0$

with $x\in X$, $y\in I$. Assume that for any $(x,y)\in U\times I$ with
$F(x,y)=0$, the Fréchet derivative $D_{x}F(x,y):X\rightarrow Z$ is Fredholm
with index $0$.
Suppose that there exists a continuous curve $\mathcal{C}_{loc}$ of solution
to (2.42), parametrized as

$\mathcal{C}_{loc}:=\\{(\tilde{x}(y),y):0<y<y_{*}\\}\subset F^{-1}(0)$

for some $y_{*}>0$ and continuous $\tilde{x}:(0,y_{*})\rightarrow U$. If

$\lim_{y\rightarrow 0^{+}}\tilde{x}(y)=0\in\partial U,\quad
D_{x}F(\tilde{x}(y),y):X\rightarrow Z\text{ is invertible for all }y,$

then $\mathcal{C}_{loc}$ is contained in a curve of solutions $\mathcal{C}$,
parametrized as

$\mathcal{C}:=\\{(x(s),y(s)):0<s<+\infty\\}\subset F^{-1}(0)$

for some continuous function $s\mapsto(x(s),y(s))\in U\times I$,
$s\in(0,+\infty)$, with the following properties:
1) One of the following alternatives holds:
i) As $s\rightarrow+\infty$,

$N(s):=\|x(s)\|_{X}+y(s)+\frac{1}{dist(x(s),\partial
U)}+\frac{1}{dist(y(s),\partial I)}\rightarrow+\infty.$

ii) There exists a sequence $s_{n}\rightarrow+\infty$ such that
$\sup_{n}N(s_{n})<+\infty$ but $\\{x(s_{n})\\}$ has no subsequences converging
in $X$;
2) Near each point $(x(s_{0}),y(s_{0}))\in\mathcal{C}$, we can reparametrize
$\mathcal{C}$ so that $s\mapsto(x(s),y(s))$ is real-analytic;
3) $(x(s),y(s))\notin\mathcal{C}_{loc}$ for $s$ sufficiently large.

We will also use the real value version of the following result from [19]
Volume 1:

######  Lemma 2.6.

([19] Volume 1, Theorem VI.14, analytic Fredholm theorem) Let $D$ be an open
connected subset of $\mathbb{C}$. Let $f:D\rightarrow L(X,Z)$ be an analytic
operator-valued function such that $f(z)$ is compact for each $z\in D$. Then
one of the following two alternatives holds:
i) $(I-f(z))^{-1}$ exists for no $z\in D$;
ii) $(I-f(z))^{-1}$ exists for all $z\in D\setminus S$, where $S$ is a
discrete subset of $D$. In this case, $(I-f(z))^{-1}$ is meromorphic in $D$,
analytic in $D\setminus S$, the residues at he pole are finite rank operators,
and for $z\in S$ the equation $f(z)g=g$ has a non-zero solution in $X$.

##  3\. Global continuation

In this section, we consider the system (2.15) – (2.16). Let us consider a
parametrized family of functions $\\{\mu^{K,\pm}(e,p)\\}_{K\in[0,+\infty)}$,
in which all the functions $\mu^{K,\pm}(e,p)$ are non-negative $C^{1}$
functions that satisfy

(3.1)
$\begin{split}&|\mu^{K,\pm}(e,p)|+|\mu^{K,\pm}_{p}(e,p)|+|\mu^{K,\pm}_{e}(e,p)|\leq\frac{C_{\mu}(K)}{1+|e|^{\delta}},\
\delta>3.\\\ \end{split}$

Moreover, we assume the following:

######  Assumption 3.1.

We assume the following:
1) $\mu^{K,\pm}(e,p)$ depend on $K$ in the $C^{1}$ sense, i.e.
$\frac{\partial\mu^{K,\pm}(e,p)}{\partial K}\in C^{0}$. Moreover,
$\frac{\partial\mu^{K,\pm}(e,p)}{\partial K}|_{K=0}=0$.
2) $\mu^{0,+}(e,p)=\mu^{0,-}(e,p)=M^{0}(e,p)$ for some non-negative function
$M^{0}(e,p)$ satisfying the decay assumption (3.1).
3) The matrix operator

(3.2) $\left(\begin{array}[]{ccc}\tilde{J}_{11}\ \tilde{J}_{12}\\\
\tilde{J}_{21}\ \tilde{J}_{22}\\\
\end{array}\right):\mathcal{X}\times\mathcal{X}\times\mathbb{R}\rightarrow
H^{2}(\Omega)\times H^{2}(\Omega)\times\mathbb{R}$

is bounded with a bounded inverse. Here

(3.3) $\begin{split}&\tilde{J}_{11}(\delta u)=\delta
u-(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}f_{1}(r)\delta u\big{\\}},\\\
&\tilde{J}_{12}(\delta
w)=-(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}f_{2}(r)\delta w\big{\\}},\\\
&\tilde{J}_{21}(\delta u)=\Delta^{-1}\big{\\{}f_{3}(r)\delta u\big{\\}},\\\
&\tilde{J}_{22}(\delta w)=\delta w+\gamma\Delta^{-1}\big{\\{}f_{4}(r)\delta
w\big{\\}},\\\ \end{split}$

with

$\begin{split}&f_{1}(r):=2\int_{\mathbb{R}^{3}}r\hat{v}_{\varphi}M^{0}_{p}(\langle
v\rangle,rv_{\varphi})dv,\
f_{2}(r):=2\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}M^{0}_{e}(\langle
v\rangle,rv_{\varphi})dv,\\\
&f_{3}(r):=2\int_{\mathbb{R}^{3}}rM^{0}_{p}(\langle v\rangle,rv_{\varphi})dv,\
f_{4}(r):=2\int_{\mathbb{R}^{3}}M^{0}_{e}(\langle v\rangle,rv_{\varphi})dv.\\\
\end{split}$

By (2.15) – (2.16), we aim to solve the elliptic system

(3.4)
$-\Delta\phi^{K,0}=\int_{\mathbb{R}^{3}}(\mu^{K,+}(e^{K,+},p^{K,+})-\mu^{K,-}(e^{K,-},p^{K,-}))dv,$
(3.5)
$(-\Delta+\frac{1}{r^{2}})A_{\varphi}^{K,0}=\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}(\mu^{K,+}(e^{K,+},p^{K,+})-\mu^{K,-}(e^{K,-},p^{K,-}))dv,$

where

$e^{K,\pm}=\langle v\rangle\pm\phi^{K,0}(x),\ p^{K,\pm}=r(v_{\varphi}\pm
A_{\varphi}^{K,0}(x)).$

It is obvious that $(\phi^{K,0},A_{\varphi}^{K,0},K)=(0,0,0)$ is a solution to
(3.4) – (3.5). Starting from this trivial solution, we will construct a global
solution set

$\\{(\mu^{K,\pm}(e^{K,\pm}(x,v),p^{K,\pm}(x,v)),\phi^{K,0}(x),A^{K,0}_{\varphi}(x))\\}_{K\in[0,+\infty)}$

for (3.4) - (3.5), therefore obtain a global solution curve for the time-
independent RVM.

We use the Global Implicit Function Theorem 2.4 to obtain

######  Theorem 3.2.

Let $\Omega$ be a $C^{1}$ axisymmetric bounded domain as stated in Section 2.
Let $\mu^{K,\pm}(e,p)$ satisfy (3.1) as well as Assumption 3.1. Then there
exists an unbounded continuous solution set or loop
$\mathcal{C}:=\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\subset
H^{2}(\Omega)\times H^{2}(\Omega)\times\mathbb{R}$ to the system (3.4) – (3.5)
with $e^{K,\pm}=\langle v\rangle\pm\phi^{K,0}(x),\ p^{K,\pm}=r(v_{\varphi}\pm
A_{\varphi}^{K,0}(x))$, in which the solution $(0,0,0)$ is included.

###### Proof.

We extend the definition of $\mu^{K,\pm}(e,p)$ evenly with respect to $K$ by
taking $\mu^{-K,\pm}(e,p)=\mu^{K,\pm}(e,p)$ for all $K\geq 0$. Recall

$\mathcal{X}:=\\{g\in H^{2}(\Omega):g\ \text{independent of}\ \varphi,\
\exp(i\varphi)g\in H^{2}(\Omega),\ g|_{\partial\Omega}=0\\}.$

For any $h\in L^{2}(\Omega)$, define $\Delta^{-1}h\in H^{2}(\Omega)\times
H^{1}_{0}(\Omega)$ to be the unique solution $g$ to the elliptic problem with
Dirichlet boundary condition

(3.6) $\Delta g=h,\ g|_{\partial\Omega}=0.$

Then $\Delta^{-1}$ is a compact operator from $L^{2}(\Omega)$ to
$L^{2}(\Omega)$ since the inclusion of $H^{1}(\Omega)$ into $L^{2}(\Omega)$ is
compact. Also, for any $h\in L^{2}(\Omega)$, we define
$(-\Delta+\frac{1}{r^{2}})^{-1}h\in H^{2}(\Omega)\times H^{1}_{0}(\Omega)$ to
be the unique solution $g$ to the elliptic problem with Dirichlet boundary
condition

(3.7) $(-\Delta+\frac{1}{r^{2}})g=h,\ g|_{\partial\Omega}=0.$

To do this, we first solve the equation

$-\Delta\tilde{g}=he^{i\varphi},\ \tilde{g}|_{\partial\Omega}=0,$

then take

$(-\Delta+\frac{1}{r^{2}})^{-1}h=g:=e^{-i\varphi}\tilde{g}.$

Now $(-\Delta+\frac{1}{r^{2}})^{-1}$ is a compact operator from
$L^{2}(\Omega)$ to $L^{2}(\Omega)$ since the inclusion of $H^{1}(\Omega)$ into
$L^{2}(\Omega)$ is compact.

We take $X=L^{2}_{0}(\Omega)\times L^{2}_{0}(\Omega)$,
$Z=H^{2}_{0}(\Omega)\times H^{2}_{0}(\Omega)$ and
$U=\mathcal{X}\times\mathcal{X}\times\mathbb{R}$. Let

(3.8) $\begin{split}&\begin{array}[]{ccc}G(u,w,K):=\end{array}\\\
&\left(\begin{array}[]{ccc}u-(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}(\mu^{K,+}(\langle
v\rangle+w,r(v_{\varphi}+u))-\mu^{K,-}(\langle
v\rangle-w,r(v_{\varphi}-u)))dv\big{\\}}\\\
w+\Delta^{-1}\big{\\{}\int_{\mathbb{R}^{3}}(\mu^{K,+}(\langle
v\rangle+w,r(v_{\varphi}+u))-\mu^{K,-}(\langle
v\rangle-w,r(v_{\varphi}-u)))dv\big{\\}}\end{array}\right)\begin{array}[]{ccc}.\end{array}\end{split}$

Then $G$ is a mapping from $X\times\mathbb{R}$ to $Z$. Solving the system
(3.4) – (3.5) is equivalent to solving the equation

(3.9) $G(u,w,K)=0.$

For each solution $(u,w,K)$ (if exists), $(u,w)$ is automatically in
$(H^{2}(\Omega)\cap H^{1}_{0}(\Omega))^{2}$.

Consider the point $(u_{0},w_{0},K_{0})=(0,0,0)$ in $X\times\mathbb{R}$. One
can verify that $G(u_{0},w_{0},K_{0})=0$. We have: $G$ is a $C^{1}$ mapping
form $X\times\mathbb{R}$ to $Z$, with

(3.10)
$\begin{array}[]{ccc}(D_{(u,w)}G)(u,w,K)=\end{array}\left(\begin{array}[]{ccc}J_{11}\
J_{12}\\\ J_{21}\ J_{22}\\\
\end{array}\right)\begin{array}[]{ccc},\end{array}$

where

(3.11) $\begin{split}J_{11}(\delta u)&=\delta
u-(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}\int_{\mathbb{R}^{3}}r\hat{v}_{\varphi}(\mu^{K,+}_{p}(\langle
v\rangle+w,r(v_{\varphi}+u))+\mu^{K,-}_{p}(\langle
v\rangle-w,r(v_{\varphi}-u)))\delta udv\big{\\}},\\\ \end{split}$ (3.12)
$\begin{split}J_{12}(\delta
w)&=-(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}(\mu^{K,+}_{e}(\langle
v\rangle+w,r(v_{\varphi}+u))+\mu^{K,-}_{e}(\langle
v\rangle-w,r(v_{\varphi}-u)))\delta wdv\big{\\}},\\\ \end{split}$ (3.13)
$\begin{split}J_{21}(\delta
u)&=\Delta^{-1}\big{\\{}\int_{\mathbb{R}^{3}}r(\mu^{K,+}_{p}(\langle
v\rangle+w,r(v_{\varphi}+u))+\mu^{K,-}_{p}(\langle
v\rangle-w,r(v_{\varphi}-u)))\delta udv\big{\\}},\\\ \end{split}$ (3.14)
$\begin{split}J_{22}(\delta w)&=\delta
w+\Delta^{-1}\big{\\{}\int_{\mathbb{R}^{3}}(\mu^{K,+}_{e}(\langle
v\rangle+w,r(v_{\varphi}+u))+\mu^{K,-}_{e}(\langle
v\rangle-w,r(v_{\varphi}-u)))\delta wdv\big{\\}}.\\\ \end{split}$

Hence we have

(3.15)
$\begin{array}[]{ccc}(D_{(u,w)}G)(u_{0},w_{0},K_{0})=\end{array}\left(\begin{array}[]{ccc}\tilde{J}_{11}\
\tilde{J}_{12}\\\ \tilde{J}_{21}\ \tilde{J}_{22}\\\ \end{array}\right),$

where

(3.16) $\begin{split}\tilde{J}_{11}(\delta u)&=\delta
u-2(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}\int_{\mathbb{R}^{3}}r\hat{v}_{\varphi}M^{0}_{p}(\langle
v\rangle,rv_{\varphi})\delta udv\big{\\}}\\\ &=\delta
u-(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}f_{1}(r)\delta u\big{\\}},\\\
\end{split}$ (3.17) $\begin{split}\tilde{J}_{12}(\delta
w)&=-2(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}M^{0}_{e}(\langle
v\rangle,rv_{\varphi})\delta wdv\big{\\}}\\\
&=-(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}f_{2}(r)\delta w\big{\\}}\\\
\end{split}$ (3.18) $\begin{split}\tilde{J}_{21}(\delta
u)&=2\Delta^{-1}\big{\\{}\int_{\mathbb{R}^{3}}rM^{0}_{p}(\langle
v\rangle,rv_{\varphi})\delta udv\big{\\}}\\\
&=\Delta^{-1}\big{\\{}f_{3}(r)\delta u\big{\\}},\\\ \end{split}$ (3.19)
$\begin{split}\tilde{J}_{22}(\delta w)&=\delta
w+2\Delta^{-1}\big{\\{}\int_{\mathbb{R}^{3}}M^{0}_{e}(\langle
v\rangle,rv_{\varphi})\delta wdv\big{\\}}\\\ &=\delta
w+\Delta^{-1}\big{\\{}f_{4}(r)\delta w\big{\\}}.\\\ \end{split}$

Here

$\begin{split}&f_{1}(r):=2\int_{\mathbb{R}^{3}}r\hat{v}_{\varphi}M^{0}_{p}(\langle
v\rangle,rv_{\varphi})dv,\
f_{2}(r):=2\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}M^{0}_{e}(\langle
v\rangle,rv_{\varphi})dv,\\\
&f_{3}(r):=2\int_{\mathbb{R}^{3}}rM^{0}_{p}(\langle v\rangle,rv_{\varphi})dv,\
f_{4}(r):=2\int_{\mathbb{R}^{3}}M^{0}_{e}(\langle v\rangle,rv_{\varphi})dv.\\\
\end{split}$

By Assumption 3.1, the operator
$(D_{(u,w)}G)(u_{0},w_{0},K_{0})=(D_{(u,w)}G)(0,0,0):\mathcal{X}\times\mathcal{X}\rightarrow
H^{2}(\Omega)\times H^{2}(\Omega)$ is bounded with a bounded inverse.
Moreover, we have

(3.20) $\begin{split}&\quad(u,w,K)\mapsto G(u,w,K)-(u,w)=\\\
&\left(\begin{array}[]{ccc}-(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}(\mu^{K,+}(\langle
v\rangle+w,r(v_{\varphi}+u))-\mu^{K,-}(\langle
v\rangle-w,r(v_{\varphi}-u)))dv\big{\\}}\\\
\Delta^{-1}\big{\\{}\int_{\mathbb{R}^{3}}(\mu^{K,+}(\langle
v\rangle+w,r(v_{\varphi}+u))-\mu^{K,-}(\langle
v\rangle-w,r(v_{\varphi}-u)))dv\big{\\}}\end{array}\right)\begin{array}[]{ccc}.\end{array}\end{split}$

is a compact operator from $X\times\mathbb{R}$ to $Z=X$. We can now apply the
Global Implicit Function Theorem and obtain a solution continuum $\mathcal{C}$
for the equation $G(u,w,K)=0$, which is a solution continuum for the system
(3.4) – (3.5). One of the three alternatives must hold. The third alternative
does not hold since $X\times\mathbb{R}=U$. The proof is complete.

∎

In the single species case, under additional assumptions, we use the Analytic
Global Implicit Function Theorem 2.5 to obtain:

######  Theorem 3.3.

Let $\Omega$ be a $C^{1}$ axisymmetric bounded domain as stated in Section 2.
Under the assumptions of Theorem 3.2, consider the (strict) single species
case, that is, one of the following two cases hold
1) $\mu^{0,+}=\mu^{K,-}\equiv 0$, with $\mu^{K,+}>0$ for all $K>0$;
2) $\mu^{0,-}=\mu^{K,+}\equiv 0$, with $\mu^{K,-}>0$ for all $K>0$.
Let $d=\sup_{x\in\Omega}r(x)<+\infty$, $\mu^{K,\pm}$ satisfies that
$\mu^{K,\pm}$ is real-analytic with respect to $K$, and
$\int_{\mathbb{R}^{3}}\mu^{K,\pm}(\langle v\rangle+a,r(v_{\varphi}+b))dv$,
$\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}\mu^{K,\pm}(\langle
v\rangle+a,r(v_{\varphi}+b))dv$ are real-analytic with respect to $a$ and $b$,
i.e.

(3.21)
$\begin{split}&\big{|}\int_{\mathbb{R}^{3}}\partial_{1}^{k}\partial_{2}^{l}\mu^{K,\pm}(\langle
v\rangle+a,r(v_{\varphi}+b))dv\big{|}\leq C^{k+l}\ k!\ l!,\\\
&\big{|}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}\partial_{1}^{k}\partial_{2}^{l}\mu^{K,\pm}(\langle
v\rangle+a,r(v_{\varphi}+b))dv\big{|}\leq C^{k+l}\ k!\ l!,\\\ \end{split}$

for some $C=C(a,b,K)>0$ and all $k$, $l\in\mathbb{Z}_{\geq 0}$, $a$,
$b\in\mathbb{R}$, then the solution set $\mathcal{C}$ contains a locally
analytic curve $\widetilde{\mathcal{C}}$ parametrized by $s$, with the
following holds: There exists a sequence $s_{j}\rightarrow+\infty$, such that

(3.22) $\|(A_{\varphi}^{K(s_{j}),0},\phi^{K(s_{j}),0})\|_{H^{2}\times
H^{2}}+K(s_{j})\rightarrow+\infty\ \text{as}\ j\rightarrow+\infty.$

Hence $\widetilde{\mathcal{C}}$ is an unbounded solution curve.

######  Remark 3.4.

Under the assumptions of Theorem 3.3, the result above implies that the set
$\mathcal{C}$ obtained in Theorem 3.2 is unbounded.

######  Remark 3.5.

If $d=\sup_{x\in\Omega}r(x)<+\infty$, $\mu^{K,\pm}$ satisfy

(3.23)
$\begin{split}&\partial_{e}^{k}\partial_{p}^{l}\mu^{K,\pm}(e,p)\leq\frac{C}{1+|e|^{\delta}}\
k!\ l!\\\ \end{split}$

for all $k$, $l\in\mathbb{Z}_{\geq 0}$ and some $C=C(K)>0$, then (3.21) holds.

###### Proof.

Following some argument in the proof for Theorem 3.2, we can apply the Local
Implicit Function Theorem 2.3 and obtain a continuous local solution curve
$\mathcal{C}_{loc}$.

We consider the single species case with $\mu^{K,-}\equiv 0$ and let
$I=\mathbb{R}$, $U=\\{(u,w)\in\mathcal{X}\times\mathcal{X}:w>0\text{ in
}\Omega\\}$, then $(u_{0},w_{0})=(0,0)\in\partial U$ (here we use the single
species assumption). If $d=\sup_{x\in\Omega}r(x)<+\infty$, $\mu^{K,\pm}$
satisfies (3.21) for some $C=C(a,b,K)>0$ and all $k$, $l\in\mathbb{Z}_{\geq
0}$ and $a$, $b\in\mathbb{R}$, then

(3.24)
$\begin{split}&\big{|}\partial_{a}^{k}\partial_{b}^{l}\int_{\mathbb{R}^{3}}\mu^{K,\pm}(\langle
v\rangle+a,r(v_{\varphi}+b))dv\big{|}=\big{|}r^{l}\int_{\mathbb{R}^{3}}\partial_{1}^{k}\partial_{2}^{l}\mu^{K,\pm}(\langle
v\rangle+a,r(v_{\varphi}+b))dv\big{|}\leq C^{k+l}\ k!\ l!,\\\
&\big{|}\partial_{a}^{k}\partial_{b}^{l}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}\mu^{K,\pm}(\langle
v\rangle+a,r(v_{\varphi}+b))dv\big{|}=\big{|}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}\partial_{1}^{k}\partial_{2}^{l}\mu^{K,\pm}(\langle
v\rangle+a,r(v_{\varphi}+b))dv\big{|}\leq C^{k+l}\ k!\ l!.\\\ \end{split}$

Hence $G$ is real-analytic in the Fréchet sense. Also, the Fréchet derivative
$D_{(u,w)}G(u,w,K):X\rightarrow Z$ is Fredholm with index $0$ since the
operators $(-\Delta+\frac{1}{r^{2}})^{-1}$ and $\Delta^{-1}$ are compact.
Moreover, by Lemma 2.6 we have that $D_{(u,w)}G(u,w,K):X\rightarrow Z$ is
invertible for the all the $(u,w)$’s in the local solution curve
$\mathcal{C}_{loc}$. We apply the Analytic Global Implicit Function Theorem
2.5 to obtain that the solution set $\mathcal{C}$ contains a locally analytic
curve $\widetilde{\mathcal{C}}$.

We rule out possibility ii) in Theorem 2.5. Let $\\{s_{n}\\}$ be a sequence
such that $s_{n}\rightarrow+\infty$ such that $\sup_{n}N(s_{n})<+\infty$, then

(3.25)
$\begin{split}&\left(\begin{array}[]{ccc}\big{\\{}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}(\mu^{K,+}(\langle
v\rangle+w,r(v_{\varphi}+u))-\mu^{K,-}(\langle
v\rangle-w,r(v_{\varphi}-u)))dv\big{\\}}\\\
\big{\\{}\int_{\mathbb{R}^{3}}(\mu^{K,+}(\langle
v\rangle+w,r(v_{\varphi}+u))-\mu^{K,-}(\langle
v\rangle-w,r(v_{\varphi}-u)))dv\big{\\}}\end{array}\right)\begin{array}[]{ccc}\end{array}\end{split}$

must be bounded in $L^{2}$. Since the operators
$(-\Delta+\frac{1}{r^{2}})^{-1}$ and $\Delta^{-1}$ are compact, and
$(u(s_{n}),w(s_{n}))$ satisfies the equation $G(u,w,K)=0$ with $G$ described
in (3.8), there must exist a subsequence of $(u(s_{n}),w(s_{n}))$ that
converges in $X$. Hence possibility ii) in Theorem 2.5 cannot hold, and we
must have

$N(s)=\|(u(s),w(s))\|_{X}+K(s)+\frac{1}{dist((u(s),w(s)),\partial
U)}+\frac{1}{dist(K(s),\partial I)}\rightarrow+\infty$

as $s\rightarrow+\infty$. It is not possible that $\frac{1}{dist(K(s),\partial
I)}\rightarrow+\infty$ since $I=\mathbb{R}$. Hence there must hold

$\|(u(s),w(s))\|_{H^{2}\times H^{2}}+K(s)+\frac{1}{dist((u(s),w(s)),\partial
U)}\rightarrow+\infty$

as $s\rightarrow+\infty$.

If $\|(u(s),w(s))\|_{H^{2}\times H^{2}}+K(s)\rightarrow+\infty$ as
$s\rightarrow+\infty$, then the analytic curve $\widetilde{\mathcal{C}}$ is
unbounded. On the other hand, if $\|(u(s),w(s))\|_{H^{2}\times H^{2}}+K(s)$
stays bounded as $s\rightarrow+\infty$. Consider any sequence
$\\{s_{j}\\}_{j=1}^{\infty}$ with $s_{j}\rightarrow+\infty$ as
$j\rightarrow\infty$. By compactness, there exists a subsequence still denoted
by $\\{s_{j}\\}_{j=1}^{\infty}$, such that

(3.26) $(u(s_{j}),w(s_{j}),K(s_{j}))\rightarrow(u_{1},w_{1},K_{1})\text{ in
}H^{2}\times H^{2}\times\mathbb{R}\text{ for some }(u_{1},w_{1},K_{1}).$

Moreover, we have $w_{1}\in\partial U$, therefore there exists some
$x_{0}\in\Omega$, such that $w_{1}(x_{0})=0$, and $w_{1}\geq 0$. By strong
maximum principle, we have $w_{1}\equiv 0$. Plugging this into (3.4) – (3.5),
and noticing that $\mu^{0,+}=\mu^{K,-}\equiv 0$, $\mu^{K,+}>0$ for all $K>0$.
Hence we must have $K_{1}=0$, and therefore $u_{1}\equiv 0$. By basis analysis
fact, we have

(3.27) $\lim_{s\rightarrow+\infty}(u(s),w(s),K(s))=(0,0,0).$

By the uniqueness in the Local Implicit Function Theorem 2.3, the curve
$\widetilde{\mathcal{C}}$ must rejoin itself at some $s\in(0,+\infty)$, which
violates Theorem 2.5 3). A contradiction. Hence $\|(u(s),w(s))\|_{H^{2}\times
H^{2}}+K(s)$ cannot stay bounded when $s\rightarrow+\infty$. Combining these
arguments together, we conclude that there exists a sequence
$s_{j}\rightarrow+\infty$, such that

(3.28) $\|(A_{\varphi}^{K(s_{j}),0},\phi^{K(s_{j}),0})\|_{H^{2}\times
H^{2}}+K(s_{j})\rightarrow+\infty\ \text{as}\ j\rightarrow+\infty.$

holds.

Similar argument as above can be carried out for the other single species case
$\mu^{K,+}\equiv 0$. The proof is complete. ∎

##  4\. Properties of the solution set

In this section, we consider the case when the set $\mathcal{C}$ obtained in
Theorem 3.2 is unbounded, and show the following

######  Proposition 4.1.

Let $\Omega$ be a $C^{1}$ axisymmetric bounded domain as stated in Section 2.
Assume that the set $\mathcal{C}$ obtained in Theorem 3.2 is unbounded, and
for all $(e,p)\in\mathbb{R}^{2}$,

(4.1) $\begin{split}&\mu^{K,+}(e,p)+\mu^{K,-}(e,p)\leq
m(K)(1+|e|^{\beta})^{-1},\\\ &\int_{\mathbb{R}^{3}}[\mu^{K,+}(\langle
v\rangle+a,r(v_{\varphi}+b))-\mu^{K,-}(\langle
v\rangle-a,r(v_{\varphi}-b))]dv\geq N(a,b,K).\\\ \end{split}$

Here $m(K)$ and $N(a,b,K)$ are positive functions, and the constant $\beta>3$.
$m(K)$ satisfies $m(0)=0$, $m(K)\rightarrow+\infty$ as $K\rightarrow+\infty$.
$N(a,b,K)$ is some function satisfies that when $K$ is large enough,
$N(a,b,K)$ is positive and monotonically decreasing in $a$ and $b$. Moreover,
we assume that for each $(a,b)\in\mathbb{R}^{2}$,

(4.2) $\begin{split}&N(a,b,K)\rightarrow+\infty\text{ when
}K\rightarrow+\infty.\\\ \end{split}$

Then we have
1)

(4.3)
$\begin{split}&\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{H^{2}(\Omega)}=+\infty.\\\
\end{split}$

2) In particular, in the case that
$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}K=+\infty$ below
holds, we have:

(4.4) $\text{As $K\rightarrow\infty$,
}\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{L^{\infty}(\Omega)}\rightarrow+\infty.$

3) If $\mu^{K,-}\equiv 0$, then

(4.5) $\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}K=+\infty.$

######  Remark 4.2.

We can switch the role of ions and electrons ($+$ and $-$), and the same
results hold.

######  Remark 4.3.

In general, without the single species assumption, it is not certain if (4.5)
holds, that is, it is not clear if the parameter $K$ stays bounded in the set
$\mathcal{C}$.

Before proving the proposition, let us introduce the following tool lemmas:

######  Lemma 4.4.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$. Assume $u\in
H^{2}(\Omega)$ satisfies

(4.6) $\begin{split}&-\Delta u\geq\alpha(x)\zeta(u),\ \text{for}\
x\in\Omega,\\\ &u\geq 0,\ \text{for}\ x\in\partial\Omega,\\\ \end{split}$

where $\alpha$ is a non-negative function and
$\zeta:[0,a_{\zeta})\rightarrow[0,+\infty)$ $(0<a_{\zeta}\leq+\infty)$ is a
non-decreasing $C^{1}$ function such that $\zeta(0)>0$. Let
$F(t):=\int^{t}_{0}\frac{ds}{\zeta(s)}$,
$d_{\Omega}(x):=dist(x,\partial\Omega)$. Then

(4.7) $F(u(x))\geq\alpha_{x}(d_{\Omega}(x))\frac{d_{\Omega}(x)^{2}}{2N}\
\text{for all}\ x\in\Omega,$

and equivalently (since $F(t)$ is a non-decreasing function)

(4.8) $u(x)\geq F^{-1}(\alpha_{x}(d_{\Omega}(x))\frac{d_{\Omega}(x)^{2}}{2N})\
\text{for all}\ x\in\Omega.$

Here $\alpha_{x}(r):=\inf_{y\in B_{r}(x)}\alpha(y)$,
$B_{r}(x)=\\{y:|y-x|<r\\}$.

###### Proof.

The proof can be easily adapted from Theorem 2.1 in [1] by approximating
$H^{2}$ functions using $C^{2}$ functions. ∎

######  Lemma 4.5.

For each $x\in\Omega$, $\delta>3$, there holds

(4.9) $\int_{\mathbb{R}^{3}}\frac{1}{1+|\langle
v\rangle\pm\phi^{0}(x)|^{\delta}+|r(v_{\varphi}\pm
A^{0}_{\varphi}(x))|^{\delta}}dv\leq(2+2^{\delta}|\phi^{0}(x)|^{\delta})\int_{\mathbb{R}^{3}}\frac{1}{1+\langle
v\rangle^{\delta}}dv,$ (4.10)
$\begin{split}&\int_{\mathbb{R}^{3}}\frac{1}{1+|\langle
v\rangle\pm\phi^{0}(x)|^{\delta}+|r(v_{\varphi}\pm
A^{0}_{\varphi}(x))|^{\delta}}dv\geq\\\
&\quad\qquad\frac{1}{2^{\delta}+2^{\delta}|\phi^{0}(x)|^{\delta}+2^{\delta}r^{\delta}|A^{0}_{\varphi}(x)|^{\delta}}\int_{\mathbb{R}^{3}}\frac{1}{1+\langle
v\rangle^{\delta}+r^{\delta}|v_{\varphi}|^{\delta}}dv.\\\ \end{split}$

Here $C_{\delta}>0$ is a constant that only depends on $\delta$.

###### Proof.

Firstly, let us prove (4.9). For any $x\in\Omega$, the integral
$\int_{\mathbb{R}^{3}}\frac{1}{1+\langle v\rangle^{\delta}}dv$ is convergent
because $\delta>3$. For any $x$, it suffices to prove

(4.11) $1+\langle
v\rangle^{\delta}\leq(2+2^{\delta}|\phi^{0}(x)|^{\delta})(1+|\langle
v\rangle\pm\phi^{0}(x)|^{\delta}),$

which implies

(4.12) $1+\langle
v\rangle^{\delta}\leq(2+2^{\delta}|\phi^{0}(x)|^{\delta})(1+|\langle
v\rangle\pm\phi^{0}(x)|^{\delta}+|r(v_{\varphi}\pm
A^{0}_{\varphi}(x))|^{\delta}).$

Let $D_{1}:=\\{v\in\mathbb{R}^{3}:|\langle
v\rangle\pm\phi^{0}(x)|>\frac{1}{2}\langle v\rangle\\}$ ,
$D_{2}:=\\{v\in\mathbb{R}^{3}:|\langle
v\rangle\pm\phi^{0}(x)|\leq\frac{1}{2}\langle v\rangle\\}$. We will show that
the claim holds true on both sets. Indeed, on $D_{1}$,

(4.13) $1+\langle v\rangle^{\delta}\leq 2(1+|\langle
v\rangle\pm\phi^{0}(x)|^{\delta})\ .$

On $D_{2}$, note that $|\langle v\rangle\pm\phi^{0}(x)|\leq\frac{1}{2}\langle
v\rangle$ implies $1\leq\langle v\rangle\leq 2|\phi^{0}(x)|$. Therefore

(4.14) $1+\langle v\rangle^{\delta}\leq 1+2^{\delta}|\phi^{0}(x)|^{\delta}\ .$

(4.9) is verified.

Next, we prove (4.10). It suffices to show

(4.15) $1+|\langle v\rangle\pm\phi^{0}(x)|^{\delta}+|r(v_{\varphi}\pm
A^{0}_{\varphi}(x))|^{\delta}\leq(2^{\delta}+2^{\delta}|\phi^{0}(x)|^{\delta}+2^{\delta}r^{\delta}|A^{0}_{\varphi}(x)|^{\delta})(1+\langle
v\rangle^{\delta}+r^{\delta}|v_{\varphi}|^{\delta}).$

Indeed, the left hand side of the inequality (4.15) satisfies

$\begin{split}1+|\langle v\rangle\pm\phi^{0}(x)|^{\delta}+|r(v_{\varphi}\pm
A^{0}_{\varphi}(x))|^{\delta}&\leq 1+2^{\delta}\langle
v\rangle^{\delta}+2^{\delta}|\phi^{0}(x)|^{\delta}+2^{\delta}|rv_{\varphi}|^{\delta}+2^{\delta}r^{\delta}|A^{0}_{\varphi}(x)|^{\delta},\\\
\end{split}$

and the right hand side of (4.15) satisfies

$(2^{\delta}+2^{\delta}|\phi^{0}(x)|^{\delta}+2^{\delta}r^{\delta}|A^{0}_{\varphi}(x)|^{\delta})(1+\langle
v\rangle^{\delta}+r^{\delta}|v_{\varphi}|^{\delta})\geq 1+2^{\delta}\langle
v\rangle^{\delta}+2^{\delta}|\phi^{0}(x)|^{\delta}+2^{\delta}|rv_{\varphi}|^{\delta}+2^{\delta}r^{\delta}|A^{0}_{\varphi}(x))|^{\delta}.$

The two estimates above then gives (4.15), and hence (4.10) is proved. The
proof of the lemma is complete.

∎

Denote

$C_{int,1}:=\int_{\mathbb{R}^{3}}\frac{1}{1+\langle v\rangle^{\delta}}dv,\
C_{int,2}:=\int_{\mathbb{R}^{3}}\frac{1}{1+\langle
v\rangle^{\delta}+r^{\delta}|v_{\varphi}|^{\delta}}dv.$

The proof of Proposition 6.2 is given by the following three lemmas:

######  Lemma 4.6.

Assume that assumption (4.1) of Proposition 4.1 holds. Then we have

(4.16)
$\begin{split}&\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{H^{2}(\Omega)}=+\infty.\\\
\end{split}$

###### Proof.

Assume $\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}K=+\infty$.
We are going to show

$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{H^{2}(\Omega)}=+\infty$

by arguing by contradiction. Assume the contrary, then there must hold

$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{L^{\infty}(\Omega)}\leq
c(K)\leq C_{0}$

for some constant $C_{0}>0$ independent of $K$. By the assumption (4.1) of
Proposition 4.1, we then have

(4.17) $\begin{split}&|\int_{\mathbb{R}^{3}}[\mu^{K,+}(\langle
v\rangle+\phi^{K,0}(x),r(v_{\varphi}+A^{K,0}_{\varphi}(x)))-\mu^{K,-}(\langle
v\rangle-\phi^{K,0}(x),r(v_{\varphi}-A^{K,0}_{\varphi}(x)))]dv|\\\ &\geq
N(C_{0},C_{0},K)\\\ \end{split}$

for some positive continuous function $N(a,b,K)$, and $N(C_{0},C_{0},K)$ goes
to $+\infty$ as $K\rightarrow+\infty$ for any fixed $C_{0}$.

For large $K$, we apply Lemma 4.4 to (3.4) with Dirichlet boundary condition,
taking $\alpha(x)\equiv 1$, $\zeta\equiv N(C_{0},C_{0})$. We obtain, for all
$x\in\Omega$,

(4.18) $F(\phi^{K,0}(x))\geq\frac{d_{\Omega}(x)^{2}}{6},$

with

$\begin{split}F(\phi^{K,0}(x))&=\int^{\phi^{K,0}(x)}_{0}\frac{1}{N(C_{0},C_{0},K)}ds.\\\
\end{split}$

Hence

(4.19)
$\begin{split}&\frac{\phi^{K,0}(x)}{N(C_{0},C_{0},K)}\geq\frac{d_{\Omega}(x)^{2}}{6}.\\\
\end{split}$

Let us pick some $x$ with $d_{\Omega}(x)>0$. Letting $K\rightarrow+\infty$, we
arrive at a contradiction to the assumption
$\lim_{K\rightarrow\infty}N(C_{0},C_{0},K)=+\infty$. Therefore there must hold

(4.20) $\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}K<+\infty$

or

(4.21)
$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{H^{2}(\Omega)}=+\infty.$

In the first case, we must have

(4.22)
$\begin{split}&\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{H^{2}(\Omega)}=+\infty\\\
\end{split}$

since the set $\mathcal{C}$ is unbounded. Therefore in either case (4.16)
holds. ∎

######  Lemma 4.7.

Let all the assumptions in Proposition 4.1 and in particular the assumption in
Proposition 4.1 2) hold. We have, as $K\rightarrow\infty$,

(4.23) $\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{L^{\infty}}\rightarrow+\infty.$

###### Proof.

Denote

$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{L^{\infty}}:=c_{\infty}(K).$

Let $B_{0}$ be a maximum inscribed ball of $\Omega$ and $R$ be its radius,
then $d_{\Omega}(x)=R$. Let $B_{1}$ be the ball which has the same center as
$B_{0}$ and half the radius as it, that is, the radius of $B_{1}$ is
$\tfrac{1}{2}R$.

In the same way as in Lemma 4.6, we have

(4.24) $\begin{split}&\quad\text{The right hand side of
\eqref{system01-nonneutral-arbimass}}\geq N(c_{\infty}(K),c_{\infty}(K),K).\\\
\end{split}$

Assume that $c_{\infty}(K)$ stays bounded when $K\rightarrow+\infty$, i.e.
$c_{\infty}(K)\leq C_{\infty}$ for some constant $C_{\infty}>0$ independent of
$K$, then $N(c_{\infty}(K),c_{\infty}(K),K)\geq N_{0}$ for some $N_{0}>0$ when
$K\rightarrow\infty$.

For large $K$, we apply Lemma 4.4 to
$-\Delta\phi^{K,0}(x)\geq\alpha(x)\zeta(u)$ with Dirichlet boundary condition
and $\alpha(x)\equiv 1$, $\zeta\equiv N(c_{\infty}(K),c_{\infty}(K),K)$. We
obtain that for all $x\in\Omega$,

(4.25) $F(\phi^{K,0}(x))\geq\frac{d_{\Omega}(x)^{2}}{6}.$

Here

$\begin{split}&F(\phi^{K,0}(x))=\int^{\phi^{K,0}(x)}_{0}N_{0}^{-1}ds.\\\
\end{split}$

Hence

(4.26) $\begin{split}&\
|\phi^{K,0}(x)|N_{0}^{-1}\geq\frac{d_{\Omega}(x)^{2}}{6},\\\ \end{split}$

Let us pick some $x\in B_{1}$. Taking $K\rightarrow+\infty$ yields a
contradiction to our assumption. Hence there must hold

$c_{\infty}(K)\rightarrow+\infty\text{ as }K\rightarrow+\infty.$

∎

######  Lemma 4.8.

Let all the assumptions in Proposition 4.1 and in particular the assumption in
Proposition 4.1 3) hold. Then we have

(4.27)
$\begin{split}&\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}K=+\infty.\\\
\end{split}$

###### Proof.

We have $\mu^{K,+}\geq 0$, $\mu^{K,-}=0$, hence the solution $\phi^{K,0}$ to
the elliptic system (3.4) – (3.5) must be non-negative. Suppose
$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}|K|<+\infty$. Then
there must hold

$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{H^{2}(\Omega)}=+\infty.$

We denote
$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}K:=\kappa$.

By the standard elliptic estimates and that
$\|g\|_{L^{2}(\Omega)}\lesssim\|g\|_{L^{\infty}(\Omega)}$ (since $\Omega$ is
bounded), as well as Lemma 4.5, we have

$\begin{split}&\quad\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{H^{2}(\Omega)}\\\
&\lesssim\|\int_{\mathbb{R}^{3}}\mu^{K,+}(\langle
v\rangle+\phi^{K,0}(x),r(v_{\varphi}+A^{K,0}_{\varphi}(x)))dv\|_{L^{2}(\Omega)}\\\
&\lesssim\|\int_{\mathbb{R}^{3}}\mu^{K,+}(\langle
v\rangle+\phi^{K,0}(x),r(v_{\varphi}+A^{K,0}_{\varphi}(x)))dv\|_{L^{\infty}(\Omega)}\\\
&\lesssim m(\kappa)\big{\|}\int_{\mathbb{R}^{3}}\frac{1}{1+|\langle
v\rangle+\phi^{K,0}(x)|^{\delta}}dv\big{\|}_{L^{\infty}(\Omega)}\\\ &\leq
m(\kappa)\big{\|}\int_{\mathbb{R}^{3}}\frac{1}{1+\langle
v\rangle^{\delta}}dv\big{\|}_{L^{\infty}(\Omega)}\\\ &\lesssim m(\kappa).\\\
\end{split}$

Taking the supremum over $\mathcal{C}$ and using the assumption
$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}K<+\infty$, we
arrive at a contradiction to
$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{H^{2}(\Omega)}=+\infty$.
Hence there must hold

$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}K=+\infty.$

∎

Notice that we can switch the role of ions and electrons ($+$ and $-$), and
the same results hold. In the case of electrons ($\mu^{K,-}\geq 0$,
$\mu^{K,+}=0$), $\phi^{K,0}(x)\leq 0$, $e^{-}=\langle
v\rangle-\phi^{K,0}(x)\geq\langle v\rangle$.

##  5\. Spectral Stability in a Single Species Case

In this section, we consider a single species plasma and study the change of
spectral stability for the solutions $(\phi^{K,0},A_{\varphi}^{K,0})$ in the
set $\mathcal{C}$ under Assumption 5.3 below. Let us first state the following
theorem on spectral stability of equilibria in [26] (notice that the
$\mu^{\pm}$ in Theorem 5.1 below is different from the $\mu^{\pm}$ in all
other places in the paper):

######  Theorem 5.1.

Let $\Omega$ be a $C^{1}$ axisymmetric bounded domain as stated in Section 2.
Let $\mathcal{H}^{\pm}$ be the space of functions of $x\in\Omega$ and
$v\in\mathbb{R}^{3}$ with the norm

(5.1)
$\|g(x,v)\|_{\mathcal{H}^{\pm}}:=\big{(}\int_{\Omega}\int_{\mathbb{R}^{3}}|\mu^{\pm}_{e}(e^{\pm},p^{\pm})|g^{2}(x,v)dxdv\big{)}^{1/2}.$

Denote
$D^{\pm}=v\cdot\nabla_{x}\pm(\textbf{E}^{0}+\hat{v}\times\textbf{B}^{0})\cdot\nabla_{v}$,
and $\mathcal{P}^{\pm}$ be the orthogonal projection on the kernel of
$D^{\pm}$ in the space $\mathcal{H}^{\pm}$. Define

(5.2) $\mathcal{A}^{0}_{1}h=\Delta
h+\sum_{\pm}\int_{\mathbb{R}^{3}}\mu^{\pm}_{e}(1-\mathcal{P}^{\pm})hdv,$ (5.3)
$\mathcal{A}^{0}_{2}h=(-\Delta+\frac{1}{r^{2}})h-\sum_{\pm}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}\big{(}\mu^{\pm}_{p}rh+\mu^{\pm}_{e}\mathcal{P}^{\pm}(\hat{v}_{\varphi}h)\big{)}dv,$
(5.4)
$\mathcal{B}^{0}h=-\sum_{\pm}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}\mu^{\pm}_{e}(1-\mathcal{P}^{\pm})hdv.$

Let $(f^{0,\pm},\textbf{E}^{0},\textbf{B}^{0})$ be an equilibrium of the
relativistic Vlasov-Maxwell system satisfying
$f^{0,\pm}(x,v)=\mu^{\pm}(e^{\pm},p^{\pm})\geq 0$ and $\mu^{\pm}_{e}(e,p)<0$,
$|\mu^{\pm}_{p}|+|\mu^{\pm}_{e}|\leq\frac{C_{\mu}}{1+|e|^{\delta}}$ with
$\delta>3$, $\mu^{\pm}\in C^{1}$, $\phi^{0}\in C(\bar{\Omega})$,
$A^{0}_{\varphi}\in C(\bar{\Omega})$. Then the operator

(5.5)
$\mathcal{L}^{0}=\mathcal{A}^{0}_{2}-\mathcal{B}^{0}(\mathcal{A}^{0}_{1})^{-1}(\mathcal{B}^{0})^{*}$

on $\mathcal{X}$ is self-adjoint. Also, we have
1) If $\mathcal{L}^{0}\geq 0$, there exists no growing mode of the linearized
RVM equation.
2) Any growing mode, if it exists, must be purely growing, i.e. the exponent
$\lambda$ of instability must be a real number.
3) If $\mathcal{L}^{0}\ngeq 0$, there exists a growing mode of the linearized
Vlasov equation and the linearized Maxwell system with the boundary
conditions.
4) The operator $\mathcal{A}^{0}_{1}$ is negative definite, and hence
$\mathcal{B}^{0}(\mathcal{A}^{0}_{1})^{-1}(\mathcal{B}^{0})^{*}$ is negative
definite. Therefore spectral instability only occurs from the operator
$\mathcal{A}^{0}_{2}$. To be precise, $\mathcal{L}^{0}$ is nonnegative
definite if $\mathcal{A}^{0}_{2}$ is nonnegative definite.
Moreover, the result in the theorem holds for the single species case (either
$\mu^{+}\equiv 0$ or $\mu^{-}\equiv 0$).

We replace $\mu^{\pm}$ in the operators by $\mu^{K,\pm}$ and define

(5.6)
$D^{K,\pm}=v\cdot\nabla_{x}\pm(\textbf{E}^{K,0}+\hat{v}\times\textbf{B}^{K,0})\cdot\nabla_{v}\
.$

Here $\textbf{E}^{K,0}=-\nabla\phi^{K,0}$,
$\textbf{B}^{K,0}=\nabla\times\textbf{A}^{K,0}$,
$\textbf{A}^{K,0}=A^{K,0}_{\varphi}e_{\varphi}$. Let $\mathcal{H}^{K,\pm}$ be
the space of functions of $x\in\Omega$ and $v\in\mathbb{R}^{3}$ with the norm

(5.7)
$\|g(x,v)\|_{\mathcal{H}^{K,\pm}}:=\big{(}\int_{\Omega}\int_{\mathbb{R}^{3}}|(\mu^{K,\pm})_{e}(e^{\pm},p^{\pm})|g^{2}(x,v)dxdv\big{)}^{1/2}$

and $\mathcal{P}^{K,\pm}$ be the orthogonal projection from
$\mathcal{H}^{K,\pm}$ onto $\ker D^{K,\pm}$. In analogy with (5.2), (5.3) and
(5.4), we define

(5.8) $\mathcal{A}^{0,K}_{1}h:=\Delta
h+\sum_{\pm}\int_{\mathbb{R}^{3}}(\mu^{K,\pm})_{e}(1-\mathcal{P}^{K,\pm})hdv$
(5.9)
$\mathcal{A}^{0,K}_{2}h:=(-\Delta+\frac{1}{r^{2}})h-\sum_{\pm}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}\big{(}(\mu^{K,\pm})_{p}rh+(\mu^{K,\pm})_{e}\mathcal{P}^{K,\pm}(\hat{v}_{\varphi}h)\big{)}dv$
(5.10)
$\mathcal{B}^{0,K}h:=-\sum_{\pm}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}(\mu^{K,\pm})_{e}(1-\mathcal{P}^{K,\pm})hdv$
(5.11)
$\mathcal{L}^{0,K}:=\mathcal{A}^{0,K}_{2}-\mathcal{B}^{0,K}(\mathcal{A}^{0,K}_{1})^{-1}(\mathcal{B}^{0,K})^{*}.$

Let us first consider the case when $K$ is small. We give the following
observation:

######  Proposition 5.2.

Let $\Omega$ be a $C^{1}$ axisymmetric bounded domain as stated in Section 2.
Let $\mu^{K,\pm}$ satisfy

$(\mu^{0,\pm})_{p}\equiv 0,\quad(\mu^{0,\pm})_{e}<0.$

then for $K=K_{0}=0$, the equilibrium
$(\mu^{0,\pm},\phi^{0,0},A_{\varphi}^{0,0})=(\mu^{0,\pm},\phi^{0,0},0)$ is
spectrally stable. Hence by continuity we have, when $K$ is close to $0$, the
equilibrium $(\mu^{K,\pm},\phi^{K,0},A_{\varphi}^{K,0})$ is spectrally stable.
Moreover, the result in the theorem holds for the single species case (either
$\mu^{+}\equiv 0$ or $\mu^{-}\equiv 0$).

###### Proof.

By Theorem 5.1, it suffices for us to show that $\mathcal{A}^{0,0}_{2}$ is
nonnegative definite. For $K=K_{0}=0$, the operator $\mathcal{A}^{0,0}_{2}$
corresponding to the equilibrium
$(\mu^{0,\pm},\phi^{0,0},A_{\varphi}^{0,0})=(\mu^{0,\pm},\phi^{0,0},0)$ are
described as

(5.12)
$\mathcal{A}^{0,0}_{2}h=(-\Delta+\frac{1}{r^{2}})h-\sum_{\pm}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}(\mu^{0,\pm})_{e}\mathcal{P}^{0,\pm}(\hat{v}_{\varphi}h)dv.$

Using the Dirichlet boundary condition, we have

$\begin{split}\langle\mathcal{A}^{0,0}_{2}h,h\rangle_{L^{2}}=&\int_{\Omega}(|\nabla
h|^{2}+\frac{1}{r^{2}}|h|^{2})dx+\sum_{\pm}\|\mathcal{P}^{0,\pm}(\hat{v}_{\varphi}h)\|^{2}_{\mathcal{H}^{0,\pm}}\geq
0.\end{split}$

The proof of the proposition is complete. (The proof for the single species
case is similar. ) ∎

From now on in this section, we assume

######  Assumption 5.3.

(5.13) $\begin{split}&\mu^{K,-}\equiv 0\text{ for all }K,\\\
&\mu^{K,+}_{e}<0\text{ for all }K,\\\ &p\mu^{K,+}_{p}(e,p)\geq
C^{\prime}_{\mu}K^{1+m-\epsilon}|p|\langle p\rangle^{-\epsilon}\nu(e)\text{
when }K\text{ is large enough, }\\\ &\mu^{K,+}(e,p)\leq
K^{m}\frac{C_{\mu}}{1+|e|^{\delta}+K^{\delta}|p|^{\delta}}\text{ when }K\text{
is large enough, }\\\ &\mu^{K,+}_{e}(e,p)\leq
K^{m}\frac{C_{\mu}}{1+|e|^{\delta}+K^{\delta}|p|^{\delta}}\text{ when }K\text{
is large enough, }\\\ &\mu^{K,+}_{p}(e,p)\leq
K^{1+m}\frac{C_{\mu}}{1+|e|^{\delta}}\text{ when }K\text{ is large enough.
}\\\ \end{split}$

Here $\nu(e)$ is some positive function satisfying

(5.14) $\nu(e)\geq C_{\nu}\exp(-e)$

and the constants $\epsilon$, $\delta$, $m$ and $C_{\nu}$ satisfy
$\epsilon>0$, $\delta>4$, $m\in(-1,1)$, $C_{\nu}>0$, $\epsilon<1-|m|$ (hence
$\epsilon+m<1$, $\epsilon<1+m$).

######  Theorem 5.4.

Let $\Omega$ be a $C^{1}$ axisymmetric bounded domain as stated in Section 2,
with $\inf_{x\in\Omega}r(x)>0$. Recall $d:=\sup_{x\in\Omega}r(x)$ and assume
$d>1$. Let all the assumptions in Theorem 3.2, Theorem 3.3, Proposition 4.1,
in particular Proposition 4.1 3), hold (so we have
$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}K=+\infty$). Let
Assumption 5.3 hold. Then when $K$ is large enough, then the equilibrium
$(\mu^{K,+}(e,p),\mu^{K,-}(e,p)\equiv 0,\phi^{K,0},A_{\varphi}^{K,0})$ is
spectrally unstable. More precisely, suppose there exists some
$h\in\mathcal{X}$ (normalized in such a way that $\int_{\Omega}(|\nabla
h|^{2}+\frac{1}{r^{2}}|h|^{2})dx=1$), and $K$ is so large that

(5.15) $1-H_{1}C_{1}C^{\prime}_{\mu}K^{1+m-\epsilon}+120\cdot
2^{\delta}\pi^{2}b^{2}(H_{1}+H_{2})C_{\mu}^{2}K^{2m}+2^{\delta}H_{2}C_{2}C_{\mu}K^{m}+256\pi^{2}c_{P}C_{\mu}^{2}H_{2}K^{2m}<0\
,$

holds, where

$H_{1}=\int_{r\geq 1}r|h|^{2}dx\ ,$ $H_{2}=\|h\|^{2}_{L^{2}}\ ,$
$C_{1}=2^{-1-\epsilon/2}b^{-\epsilon}\ ,$
$C_{2}=\frac{8}{3}\pi+\frac{4}{\delta-1}\pi^{2}\ ,$

and $c_{P}$ is the square of the Poincaré constant of $\Omega$, $C_{\nu}$ is
as defined in (5.14). Then the equilibrium
$(\mu^{K,+}(e,p),\mu^{K,-}(e,p)\equiv 0,\phi^{K,0},A_{\varphi}^{K,0})$ is
spectrally unstable.

######  Remark 5.5.

The condition $p\mu^{K,+}_{p}(e,p)\geq
C^{\prime}_{\mu}K^{1+m-\epsilon}|p|\langle p\rangle^{-\epsilon}\nu(e)$
together with that $\mu^{K,+}_{p}(e,p)\leq
K^{1+m}\frac{C_{\mu}}{1+|e|^{\delta}}$ implies that $|\nu(e)|$ is controlled
by $\frac{\langle p\rangle^{\epsilon}}{1+|e|^{\delta}}$ with $\delta>4$, which
ensures the integrals in the proof are convergent. (This is where we need
$\delta>4$. Notice that $\epsilon\leq 1$.)

######  Remark 5.6.

We can switch the role of ions and electrons ($+$ and $-$), and the same
results hold.

######  Remark 5.7.

In fact, we only need $\mu^{K,-}\equiv 0$ to hold for $K$ large enough. Having
$\mu^{K,-}\equiv 0$ in Assumption 5.3 is mainly for convenience in
description. Moreover, we can extend the result for the case when
$\mu^{K,-}\equiv 0$ for $K$ large enough does not necessarily hold, with the
constraint on the constant $m\in(-1,1)$ replaced by $m\in(-1,0)$. The proof is
similar.

Now we discuss the proof of Theorem 5.4.

In the same way as in the proofs of Lemma 8.2, 8.3 and 8.4 in [26], we can
obtain

######  Lemma 5.8.

1) We have

(5.16) $\|\mathcal{A}^{0,K}_{1}\|_{L^{2}\rightarrow L^{2}}\geq c^{-1}_{P}.$

Hence for all $\tilde{g}\in\mathcal{X}$, there holds

(5.17)
$|\langle(\mathcal{A}^{0,K}_{1})^{-1}\tilde{g},\tilde{g}\rangle_{L^{2}}|\leq
c_{P}\|\tilde{g}\|^{2}_{L^{2}}.$

Here $c_{P}$ is the square of the Poincaré constant of $\Omega$.
2) For all $K\geq 1$, $\|(\mathcal{B}^{0,K})^{*}\|_{L^{2}\rightarrow
L^{2}}\leq 8\sqrt{2}\pi C_{\mu}K^{m}$.
3) For any $h\in\mathcal{X}$, there holds

(5.18)
$\begin{split}\langle(\mathcal{A}^{0,K}_{1})^{-1}(\mathcal{B}^{0,K})^{*}h,(\mathcal{B}^{0,K})^{*}h\rangle_{L^{2}}&\leq
c_{P}\|(\mathcal{B}^{0,K})^{*}h\|^{2}_{L^{2}}\\\ &\leq 2c_{P}\cdot
128\pi^{2}C_{\mu}^{2}K^{2m}\|h\|^{2}_{L^{2}}\\\
&=256\pi^{2}c_{P}C_{\mu}^{2}H_{2}K^{2m}.\end{split}$

We will also need the following

######  Lemma 5.9.

Assume that $\inf_{x\in\Omega}r(x)>0$. A solution
$(\phi^{K,0},A_{\varphi}^{K,0})$ to the coupled elliptic system with the
settings in this section satisfies

(5.19) $\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{L^{\infty}}\lesssim K^{-1+m},$

where the constant in $\lesssim$ is independent of $K$.

###### Proof.

By standard elliptic estimates, the assumptions $\mu^{K,-}=0$ and the decay
assumption on $\mu^{K,+}$, it suffices for us to estimate

(5.20) $\int_{\mathbb{R}^{3}}K^{m}\frac{1}{1+|\langle
v\rangle+\phi^{K,0}(x)|^{\delta}+|Kr(v_{\varphi}+A^{K,0}_{\varphi}(x))|^{\delta}}dv.$

(For $A^{K,0}_{\varphi}$, we make use of the identity
$-\Delta(ge^{i\varphi})=(-\Delta+\frac{1}{r^{2}})ge^{i\varphi}$, which holds
if $g$ is cylindrically symmetric, to get rid of the term
$\frac{1}{r^{2}}A^{K,0}_{\varphi}$ in the elliptic equation.)

We make a change of variable $K(v_{\varphi}+A^{K,0}_{\varphi}(x))\mapsto
v_{\varphi}$ and compute

$\begin{split}&\quad\int_{\mathbb{R}^{3}}K^{m}\frac{1}{1+|\langle
v\rangle+\phi^{K,0}(x)|^{\delta}+|Kr(v_{\varphi}+A^{K,0}_{\varphi}(x))|^{\delta}}dv\\\
&=\int_{\mathbb{R}^{3}}K^{m}\frac{1}{1+|\langle
v\rangle+\phi^{K,0}(x)|^{\delta}+|Kr(v_{\varphi}+A^{K,0}_{\varphi}(x))|^{\delta}}d\tilde{v}dv_{\varphi}\\\
&=\int_{\mathbb{R}^{3}}K^{-1+m}\frac{1}{1+|\sqrt{1+|\tilde{v}|^{2}+|\frac{v_{\varphi}}{K}-A^{K,0}_{\varphi}(x)|^{2}}\pm\phi^{K,0}(x)|^{\delta}+|rv_{\varphi}|^{\delta}}d\tilde{v}dv_{\varphi}\\\
&\leq\int_{\mathbb{R}^{3}}K^{-1+m}\frac{1}{1+|\sqrt{1+|\tilde{v}|^{2}}+\phi^{K,0}(x)|^{\delta}+|rv_{\varphi}|^{\delta}}d\tilde{v}dv_{\varphi}.\\\
\end{split}$

By the maximum principle, $\phi^{K,0}(x)\geq 0$. Thus the last line is bounded
from above as

$\begin{split}&\quad\int_{\mathbb{R}^{3}}K^{-1+m}\frac{1}{1+|\sqrt{1+|\tilde{v}|^{2}}+\phi^{K,0}(x)|^{\delta}+|rv_{\varphi}|^{\delta}}d\tilde{v}dv_{\varphi}\\\
&\leq
K^{-1+m}\int_{\mathbb{R}^{3}}\frac{1}{1+\langle\tilde{v}\rangle^{\delta}+|rv_{\varphi}|^{\delta}}dv\\\
&\lesssim K^{-1+m}.\\\ \end{split}$

Here in the last line we have used $\inf_{x\in\Omega}r(x)>0$. (5.19) now
follows from standard elliptic estimates. ∎

Notice that we can switch the role of ions and electrons ($+$ and $-$), and
the same results hold. In the case of electrons ($\mu^{K,-}\geq 0$,
$\mu^{K,+}=0$), $\phi^{K,0}(x)\leq 0$, $e^{-}=\langle
v\rangle-\phi^{K,0}(x)\geq\langle v\rangle$.

Now we are ready to prove Proposition 5.4.

###### Proof.

By the assumptions in Proposition 4.1, in particular the assumption in
Proposition 4.1 3), we have
$\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}K=+\infty$.

For $h\in\mathcal{X}$, we first take care of the term
$\langle\mathcal{A}^{0,K}_{2}h,h\rangle_{L^{2}}$. For simplicity, we normalize
$h$ such that $\int_{\Omega}(|\nabla h|^{2}+\frac{1}{r^{2}}|h|^{2})dx=1$.
Recall that $e^{K,\pm}=\langle v\rangle\pm\phi^{K,0}(x),\
p^{K,\pm}=r(v_{\varphi}\pm A_{\varphi}^{K,0}(x))$.

We observe

(5.21)
$\begin{split}\langle\mathcal{A}^{0,K}_{2}h,h\rangle_{L^{2}}&=1-\int_{\Omega}\int_{\mathbb{R}^{3}}\langle
v\rangle^{-1}p^{K,+}\mu^{K,+}_{p}(e^{K,+},p^{K,+})|h|^{2}dvdx\\\
&+\int_{\Omega}\int_{\mathbb{R}^{3}}\frac{\mu^{K,+}_{p}(e^{K,+},p^{K,+})}{\langle
v\rangle}rA^{K,0}_{\varphi}|h|^{2}dvdx+\|\mathcal{P}^{K,+}(\hat{v}_{\varphi}h)\|^{2}_{\mathcal{H}^{K,+}}\\\
&:=1+I+II+III.\\\ \end{split}$

We are going to show that when $K\geq 1$ is large enough, the term $I$
dominates all the others and remains negative. We compute

(5.22)
$\begin{split}I&\leq-C^{\prime}_{\mu}\int_{\Omega}\int_{\mathbb{R}^{3}}\langle
v\rangle^{-1}K^{1+m-\epsilon}|p^{K,+}|\langle
p^{K,+}\rangle^{-\epsilon}\nu(e^{K,+})dv\ |h|^{2}dx\\\
&\leq-C^{\prime}_{\mu}K^{1+m-\epsilon}\int_{\Omega}\int_{|p^{K,+}(x,v)|>1}\langle
v\rangle^{-1}|p^{K,+}|^{1-\epsilon}\cdot 2^{-\epsilon/2}\nu(e^{K,+})dv\
|h|^{2}dx\\\ &\leq-C^{\prime}_{\mu}K^{1+m-\epsilon}M,\\\ \end{split}$

where

$M=2^{-\epsilon/2}\int_{r(x)\geq 1}\inf_{x\in\Omega,r(x)\geq
1}\Big{[}\int_{\\{v\in\mathbb{R}^{3},|p^{K,+}(x,v)|>1\\}}\langle
v\rangle^{-1}|v_{\varphi}+A^{K,0}_{\varphi}(r,z)|^{1-\epsilon}\nu(e^{K,+})dv\Big{]}r^{1-\epsilon}|h|^{2}dx\
.$

Noting that $\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{L^{\infty}}\lesssim
K^{-1+m}\leq 1/2$ when $K$ is large, we have

(5.23) $\begin{split}M&=2^{-\epsilon/2}\int_{r(x)\geq
1}\inf_{x\in\Omega,r(x)\geq
1}\Big{[}\int_{\\{v\in\mathbb{R}^{3},|p^{K,+}(x,v)|>1\\}}\langle
v\rangle^{-1}|v_{\varphi}+A^{K,0}_{\varphi}(r,z)|^{1-\epsilon}\nu(e^{K,+})dv\Big{]}r^{1-\epsilon}|h|^{2}dx\\\
&\geq 2^{-\epsilon/2}\int_{r(x)\geq 1}\inf_{x\in\Omega,r(x)\geq
1}\Big{[}\int_{\\{v\in\mathbb{R}^{3},|v_{\varphi}|>2+\frac{1}{r(x)}\\}}\langle
v\rangle^{-1}(1+\frac{1}{r(x)})^{1-\epsilon}\nu(\langle
v\rangle+\phi^{K,0})dv\Big{]}r^{1-\epsilon}|h|^{2}dx\\\ &\geq
2^{-\epsilon/2}\int_{r(x)\geq 1}\inf_{x\in\Omega,r(x)\geq
1}\Big{[}\int_{\\{v\in\mathbb{R}^{3},|v_{\varphi}|>3\\}}\langle
v\rangle^{-1}\nu(\langle
v\rangle+\phi^{K,0})dv\Big{]}r^{1-\epsilon}|h|^{2}dx\\\ &\geq
2^{-\epsilon/2}C_{\nu}\int_{r(x)\geq 1}\inf_{x\in\Omega,r(x)\geq
1}\Big{[}\int_{\\{v\in\mathbb{R}^{3},|v_{\varphi}|>3\\}}\langle
v\rangle^{-1}\exp(-\langle
v\rangle-\phi^{K,0})dv\Big{]}r^{1-\epsilon}|h|^{2}dx\\\ &\geq
2^{-\epsilon/2}C_{\nu}\cdot 2\exp(-\frac{1}{2})\int_{r(x)\geq
1}\int_{\\{v\in\mathbb{R}^{3},|v_{\varphi}|>3\\}}\langle
v\rangle^{-1}\exp(-\langle v\rangle)dvr^{1-\epsilon}|h|^{2}dx\\\ &\geq
2^{-\epsilon/2}C_{\nu}\cdot\frac{1}{2}\int_{r(x)\geq
1}r^{1-\epsilon}|h|^{2}dx\\\ &\geq
2^{-\epsilon/2}\cdot\frac{1}{2}d^{-\epsilon}C_{\nu}\int_{r(x)\geq
1}r|h|^{2}dx,\\\ \end{split}$

where $d=\sup_{x\in\Omega}r(x)>1$. Hence

$I\leq-H_{1}C_{1}C_{\nu}C^{\prime}_{\mu}K^{1+m-\epsilon}\ ,$

where $H_{1}=\int_{r\geq 1}r|h|^{2}dx$,
$C_{1}=2^{-1-\epsilon/2}b^{-\epsilon}$. From Remark 1, we have
$|\nu(e^{K,+})|\leq\frac{\langle
p^{K,+}\rangle^{\epsilon}}{C(1+|e^{K,+}|^{\delta})}$ with $\delta>4$, which
ensures that the integral $\int_{\mathbb{R}^{3}}\langle
v\rangle^{-1}v_{\varphi}^{2}\nu(e^{K,+})dv$ is convergent. In the end we
obtain $I=O(K^{1+m-\epsilon})$.

$II$ vanishes if $A^{K,0}_{\varphi}=0$. If $A^{K,0}_{\varphi}\neq 0$, we
estimate, using Lemma 5.9,

(5.24) $\begin{split}II&\leq 20\pi
b^{2}C_{\mu}(H_{1}+H_{2})\|A^{K,0}_{\varphi}\|_{L^{\infty}}\sup_{x}(\int_{\mathbb{R}^{3}}\langle
v\rangle^{-1}|\mu^{K,+}_{p}(e^{K,+},p^{K,+})|dv)\\\ &\leq 20\pi
b^{2}C_{\mu}(H_{1}+H_{2})C_{\mu}K^{-1+m}\cdot
K^{1+m}\sup_{x}\int_{\mathbb{R}^{3}}\frac{1}{\langle
v\rangle(1+|e^{K,+}|^{\delta})}dv\\\ &\leq 20\pi
b^{2}C_{\mu}(H_{1}+H_{2})C_{\mu}K^{2m}\sup_{x}\int_{\mathbb{R}^{3}}\frac{1}{\langle
v\rangle(1+|\langle
v\rangle+\phi^{K,0}(x)|^{\delta})}dv(1+K^{(-1+m)\delta})\\\ &\leq 40\cdot
2^{\delta}\pi
b^{2}(H_{1}+H_{2})C_{\mu}^{2}K^{2m}\int_{\mathbb{R}^{3}}\frac{1}{1+\langle
v\rangle^{\delta}}dv\\\ &\leq 40\cdot 2^{\delta}\pi
b^{2}(H_{1}+H_{2})C_{\mu}^{2}K^{2m}\cdot 3\pi\\\ &\leq 120\cdot
2^{\delta}\pi^{2}b^{2}(H_{1}+H_{2})C_{\mu}^{2}K^{2m}.\\\ \end{split}$

Here $H_{2}=\int_{\Omega}|h|^{2}dx=\|h\|^{2}_{L^{2}}$. Therefore the term $II$
is $O(K^{2m})$.

For $III=\|\mathcal{P}^{K,+}(\hat{v}_{\varphi}h)\|^{2}_{\mathcal{H}^{K,+}}$,
we estimate

(5.25)
$\begin{split}III=\|\mathcal{P}^{K,+}(\hat{v}_{\varphi}h)\|^{2}_{\mathcal{H}^{K,+}}&\leq\int_{\Omega}\sup_{v}|\mathcal{P}^{K,+}(\hat{v}_{\varphi}h)|^{2}dx\sup_{x}(\int_{\mathbb{R}^{3}}|\mu^{K,+}_{e}(e^{K,+},p^{K,+})|dv)\\\
&\leq K^{m}H_{2}\sup_{x}(\int_{\mathbb{R}^{3}}\frac{C_{\mu}}{1+|\langle
v\rangle+\phi^{K,0}(x)|^{\delta}}dv)\\\ &\leq 2^{\delta}H_{2}C_{\mu}\cdot
2K^{m}\int_{\mathbb{R}^{3}}\frac{1}{1+\langle v\rangle^{\delta}}dv\\\ &\leq
2^{\delta}H_{2}C_{2}C_{\mu}K^{m}.\\\ \end{split}$

Here $C_{2}=\frac{8}{3}\pi+\frac{4}{\delta-1}\pi^{2}$. Therefore
$III=O(K^{m})$.

Combining together all the estimates, as well as Lemma 5.8, we conclude

(5.26) $\begin{split}&\langle\mathcal{L}^{0,K}h,h\rangle_{L^{2}}\\\ &\leq
1-H_{1}C_{1}C_{\nu}C^{\prime}_{\mu}K^{1+m-\epsilon}+120\cdot
2^{\delta}\pi^{2}b^{2}(H_{1}+H_{2})C_{\mu}^{2}K^{2m}+2^{\delta}H_{2}C_{2}C_{\mu}K^{m}\\\
&+256\pi^{2}c_{P}C_{\mu}^{2}H_{2}K^{2m}.\\\ \end{split}$

Hence $(\mu^{K,\pm},\phi^{K,0},A_{\varphi}^{K,0})$ is spectrally unstable if
$C_{\mu}$, $K$ and $h$ satisfies (5.15).

∎

##  6\. Special cases

###  6.1. Special case 1

Let us take a special case of Theorem 3.2. We take some $\mu^{\pm}(e,p)$,
$\mu^{0}(e,p)$ which are non-negative $C^{1}$ functions that satisfy

(6.1)
$\begin{split}&|\mu^{0}(e,p)|+|\mu^{0}_{p}(e,p)|+|\mu^{0}_{e}(e,p)|\leq\frac{C_{\mu}}{1+|e|^{\delta}},\
\delta>3,\\\
&|\mu^{\pm}(e,p)|+|\mu^{\pm}_{p}(e,p)|+|\mu^{\pm}_{e}(e,p)|\leq\frac{C_{\mu}}{1+|e|^{\delta}},\
\delta>3,\\\ \end{split}$

and that $\mu^{0}(e,p)$ is even in $p$. Let $\gamma\geq 0$ be a temporarily
fixed parameter. For any $K\in[0,+\infty)$, consider the equilibrium
$\mu^{K,\pm}$ given by

(6.2)
$\begin{split}&\mu^{K,+}(e,p)=\gamma\mu^{0}(e,p)+a^{+}(K)\mu^{+}(e,p),\\\
&\mu^{K,-}(e,p)=\gamma\mu^{0}(e,p)+a^{-}(K)\mu^{-}(e,p).\\\ \end{split}$

Here $a^{\pm}(K)$ are nonnegative $C^{1}$ functions of $K$ which satisfies
$a^{+}(0)=a^{-}(0)=0$, and there exists no $K\neq 0$ such that
$a^{+}(K)=a^{-}(K)=0$. Moreover, we assume that
$(a^{+})^{\prime}(0)=(a^{-})^{\prime}(0)=0$. It is obvious that for each
$\gamma\in\mathbb{R}$, $(\phi^{K,0},A_{\varphi}^{K,0},K)=(0,0,0)$ is a
solution to (3.4) – (3.5).

######  Theorem 6.1.

Let $\Omega$ be a $C^{1}$ axisymmetric bounded domain as stated in Section 2.
Let $\mu^{K,\pm}$ satisfy (6.2) with (6.1), and that the conditions above for
$a^{\pm}(K)$ hold. There exists a discrete subset $S_{\gamma}$ of
$[0,+\infty)$, such that $0\in S_{\gamma}$, and for all
$\gamma\in[0,+\infty)\setminus S_{\gamma}$, the following holds:
There exists an unbounded continuous solution set or loop
$\mathcal{C}:=\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\subset
H^{2}(\Omega)\times H^{2}(\Omega)\times\mathbb{R}$ to the system (3.4) – (3.5)
(which includes the parameter $\gamma$) with $e^{K,\pm}=\langle
v\rangle\pm\phi^{K,0}(x),\ p^{K,\pm}=r(v_{\varphi}\pm A_{\varphi}^{K,0}(x))$,
in which the solution $(0,0,0)$ is included.
Moreover, consider the (strict) single species case (either
$\mu^{0,+}=\mu^{K,-}\equiv 0$, with $\mu^{K,+}>0$ for all $K>0$, or
$\mu^{0,-}=\mu^{K,+}\equiv 0$, with $\mu^{K,-}>0$ for all $K>0$), and let
$d=\sup_{x\in\Omega}r(x)<+\infty$, $a^{\pm}(K)$ are analytic functions,
$\mu^{0}$ and $\mu^{\pm}$ satisfy

(6.3)
$\begin{split}&\big{|}\partial_{e}^{k}\partial_{p}^{l}\mu^{0}(e,p)\big{|}\leq\frac{C}{1+|e|^{\delta}}\
k!\ l!,\\\
&\big{|}\partial_{e}^{k}\partial_{p}^{l}\mu^{\pm}(e,p)\big{|}\leq\frac{C}{1+|e|^{\delta}}\
k!\ l!\\\ \end{split}$

for all $k$, $l\in\mathbb{Z}_{\geq 0}$ and some $C>0$, $\delta>3$, then (3.21)
holds, and hence the solution set $\mathcal{C}$ contains a locally analytic
curve $\widetilde{\mathcal{C}}$ parametrized by $s$. Moreover,
$\widetilde{\mathcal{C}}$ is unbounded in the sense that there exists a
sequence $s_{j}\rightarrow+\infty$, such that (3.22) holds.

###### Proof.

By Theorem 3.2, we only need to show that there exists a discrete subset
$S_{\gamma}$ of $[0,+\infty)$, such that for all
$\gamma\in[0,+\infty)\setminus S_{\gamma}$, $\mu^{K,\pm}$ satisfy Assumption
3.1. It suffices to verify 4) in Assumption 3.1, since all others are
straightforward (in particular, $(a^{+})^{\prime}(0)=(a^{-})^{\prime}(0)=0$
implies $\frac{\partial\mu^{K,\pm}(e,p)}{\partial K}|_{K=0}=0$). For this
purpose, we use the real value version of Lemma 2.6 to analyize the matrix
operator in 4) in Assumption 3.1.

Notice that the matrix operator in 4) in Assumption 3.1 now can be written as

(6.4) $\left(\begin{array}[]{ccc}\tilde{J}_{11}\ \tilde{J}_{12}\\\
\tilde{J}_{21}\ \tilde{J}_{22}\\\
\end{array}\right):\mathcal{X}\times\mathcal{X}\times\mathbb{R}\rightarrow
H^{2}(\Omega)\times H^{2}(\Omega)\times\mathbb{R}$

where

(6.5) $\begin{split}&\tilde{J}_{11}(\delta u)=\delta
u-\gamma(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}f_{1}(r)\delta u\big{\\}},\\\
&\tilde{J}_{12}(\delta
w)=-\gamma(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}f_{2}(r)\delta w\big{\\}},\\\
&\tilde{J}_{21}(\delta u)=\gamma\Delta^{-1}\big{\\{}f_{3}(r)\delta
u\big{\\}},\\\ &\tilde{J}_{22}(\delta w)=\delta
w+\gamma\Delta^{-1}\big{\\{}f_{4}(r)\delta w\big{\\}},\\\ \end{split}$

and

$\begin{split}&f_{1}(r):=2\int_{\mathbb{R}^{3}}r\hat{v}_{\varphi}\mu^{0}_{p}(\langle
v\rangle,rv_{\varphi})dv,\
f_{2}(r):=2\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}\mu^{0}_{e}(\langle
v\rangle,rv_{\varphi})dv,\\\
&f_{3}(r):=2\int_{\mathbb{R}^{3}}r\mu^{0}_{p}(\langle
v\rangle,rv_{\varphi})dv,\ f_{4}(r):=2\int_{\mathbb{R}^{3}}\mu^{0}_{e}(\langle
v\rangle,rv_{\varphi})dv.\\\ \end{split}$

This matrix operator is bounded with a bounded inverse when $\gamma=0$. We
apply Lemma 2.6 to this matrix operator with the varying parameter $\gamma$
playing the role of $z$ in the lemma. We obtain that there exists a discrete
set $S_{\gamma}\subset[0,+\infty)$ such that for all
$\gamma\in[0,+\infty)\setminus S_{\gamma}$, this matrix operator is bounded
with a bounded inverse. This completes the proof of the first part of the
theorem.

Moreover, if $d=\sup_{x\in\Omega}r(x)<+\infty$, $a^{\pm}(K)$ are analytic
functions, $\mu^{0}$ and $\mu^{\pm}$ satisfy (6.3) for all $k$,
$l\in\mathbb{Z}_{\geq 0}$ and some $C>0$, then

(6.6)
$\begin{split}&\big{|}\int_{\mathbb{R}^{3}}\partial_{1}^{k}\partial_{2}^{l}\mu^{0}(\langle
v\rangle+a,r(v_{\varphi}+b))dv\big{|}\leq\int_{\mathbb{R}^{3}}\frac{C}{1+|\langle
v\rangle+a|^{\delta}+|r(v_{\varphi}+b)|^{\delta}}dv\leq C_{1}^{k+l}\ k!\
l!,\\\
&\big{|}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}\partial_{1}^{k}\partial_{2}^{l}\mu^{0}(\langle
v\rangle+a,r(v_{\varphi}+b))dv\big{|}\leq\int_{\mathbb{R}^{3}}\frac{C}{1+|\langle
v\rangle+a|^{\delta}+|r(v_{\varphi}+b)|^{\delta}}dv\leq C_{1}^{k+l}\ k!\
l!,\\\
&\big{|}\int_{\mathbb{R}^{3}}\partial_{1}^{k}\partial_{2}^{l}\mu^{\pm}(\langle
v\rangle+a,r(v_{\varphi}+b))dv\big{|}\leq\int_{\mathbb{R}^{3}}\frac{C}{1+|\langle
v\rangle+a|^{\delta}+|r(v_{\varphi}+b)|^{\delta}}dv\leq C_{2}^{k+l}\ k!\
l!,\\\
&\big{|}\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}\partial_{1}^{k}\partial_{2}^{l}\mu^{\pm}(\langle
v\rangle+a,r(v_{\varphi}+b))dv\big{|}\leq\int_{\mathbb{R}^{3}}\frac{C}{1+|\langle
v\rangle+a|^{\delta}+|r(v_{\varphi}+b)|^{\delta}}dv\leq C_{2}^{k+l}\ k!\ l!\\\
\end{split}$

hold for all $k$, $l\in\mathbb{Z}_{\geq 0}$, $a$, $b\in\mathbb{R}$,
$C_{1}(a,b)>0$, $C_{2}(a,b)>0$. Taking $C(a,b,K)=\gamma
C_{1}(a,b)+(a_{+}(K)+a_{-}(K))C_{2}(a,b)$, then (3.21) holds. Therefore the
solution set $\mathcal{C}$ contains a locally analytic curve by Theorem 3.3.
The rest of the proof is the same as the argument in the proof of Theorem 3.3.

∎

Let us consider the single species case and assume furthermore

(6.7) $\begin{split}&a^{+}(K)\rightarrow+\infty\text{ as }K\rightarrow\infty,\
a^{-}(K)\equiv 0,\\\ \end{split}$

and

(6.8) $\begin{split}&\mu^{+}(e,p)\geq
C_{\mu^{+}}(1+|e|^{\alpha}+|p|^{\alpha})^{-1},\ \mu^{0}(e,p)\equiv 0\\\
\end{split}$

for all $(e,p)\in\mathbb{R}^{2}$ and some constants $\alpha>3$,
$C_{\mu^{+}}>0$.

Using Proposition 4.1, we derive

######  Proposition 6.2.

Let $\Omega$ be a $C^{1}$ axisymmetric bounded domain as stated in Section 2.
Then with the assumptions (6.7) and (6.8), the set $\mathcal{C}$ obtained in
Theorem 6.1 is unbounded. Moreover, there holds:

(6.9) $\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}K=+\infty,$

and

(6.10) $\text{As $K\rightarrow\infty$,
}\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{L^{\infty}}\rightarrow+\infty.$

###### Proof.

Applying Theorem 6.1, we conclude that the set $\mathcal{C}$ obtained in
Theorem 6.1 is unbounded. It then suffices to verify the conditions in
Proposition 4.1, and the only non-trivial one is (4.1). Indeed, by Lemma 4.5,
we have

(6.11) $\begin{split}&\quad\int_{\mathbb{R}^{3}}[\mu^{K,+}(\langle
v\rangle+a,r(v_{\varphi}+b))-\mu^{K,-}(\langle
v\rangle-a,r(v_{\varphi}-b))]dv\\\
&=\int_{\mathbb{R}^{3}}(a^{+}(K)\mu^{+})(\langle
v\rangle+a,r(v_{\varphi}+b))\\\ &\geq
2^{-\delta}a^{+}(K)C_{int,2}C_{\mu^{+}}(1+|a|^{\alpha}+d^{\alpha}|b|^{\alpha})^{-1}\\\
&=:N(a,b,K).\\\ \end{split}$

Then when $K$ is large enough, $N(a,b,K)$ is positive and monotonically
decreasing in $a$ and $b$. Moreover, for each $(e,p)\in\mathbb{R}^{2}$,

(6.12) $\begin{split}&N(e,p,K)\rightarrow+\infty\text{ when
}K\rightarrow+\infty.\\\ \end{split}$

(4.1) is verified. Therefore we can apply Proposition 4.1 and obtain the
desired results. ∎

######  Remark 6.3.

We can switch the role of ions and electrons ($+$ and $-$), and the same
results in this subsection hold.

###  6.2. Special case 2

Let us consider another another special case of Theorem 3.2 other than the one
in Section 6.1 (Theorem 6.1). We take some $\mu^{\pm}(e,p)$,
$\mu^{0,\pm}(e,p)$ which are non-negative $C^{1}$ functions that satisfy
(6.1), and $\mu^{0}(e,p)$ is even in $p$. Let $\gamma\geq 0$ be a temporarily
fixed parameter. For any $K\in[0,+\infty)$, consider the equilibrium
$\mu^{K,\pm}$ given by

(6.13)
$\begin{split}&\mu^{K,+}(e,p)=\gamma\mu^{0}(e,Kp)+a^{+}(K)\mu^{+}(e,Kp),\\\
&\mu^{K,-}(e,p)=\gamma\mu^{0}(e,Kp)+a^{-}(K)\mu^{-}(e,Kp).\\\ \end{split}$

Here $a^{\pm}(K)$ are nonnegative functions of $K$ which satisfies
$a^{+}(0)=a^{-}(0)=0$, and there exists no $K\neq 0$ such that
$a^{+}(K)=a^{-}(K)=0$. Moreover, we assume that
$(a^{+})^{\prime}(0)\mu^{+}(e,0)+a^{+}(0)p\mu^{+}_{p}(e,0)=(a^{-})^{\prime}(0)\mu^{-}(e,0)+a^{-}(0)p\mu^{-}_{p}(e,0)=0$.
It is obvious that for each $\gamma\in\mathbb{R}$,
$(\phi^{K,0},A_{\varphi}^{K,0},K)=(0,0,0)$ is a solution to (3.4) – (3.5).

######  Theorem 6.4.

Let $\Omega$ be a $C^{1}$ axisymmetric bounded domain as stated in Section 2.
Let $\mu^{K,\pm}$ satisfy (6.13) with (6.1), and that the conditions above for
$a^{\pm}(K)$ hold. There exists a discrete subset $S_{\gamma}$ of
$[0,+\infty)$, such that $0\in S_{\gamma}$, and for all
$\gamma\in[0,+\infty)\setminus S_{\gamma}$, the following holds:
There exists an unbounded continuous solution set or loop
$\mathcal{C}:=\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\subset
H^{2}(\Omega)\times H^{2}(\Omega)\times\mathbb{R}$ to the system (3.4) – (3.5)
(which includes the parameter $\gamma$) with $e^{K,\pm}=\langle
v\rangle\pm\phi^{K,0}(x),\ p^{K,\pm}=r(v_{\varphi}\pm A_{\varphi}^{K,0}(x))$,
in which the solution $(0,0,0)$ is included.
Moreover, consider the single species case ($\gamma\mu^{0}(e,p)\equiv 0$,
together with $a^{-}(K)\mu^{-}(e,p)\equiv 0$ or $a^{+}(K)\mu^{+}(e,p)\equiv
0$) and let $d=\sup_{x\in\Omega}r(x)<+\infty$, $a^{\pm}(K)$ are analytic
functions, $\mu^{0}$ and $\mu^{\pm}$ satisfy (6.3) for all $k$,
$l\in\mathbb{Z}_{\geq 0}$ and some $C>0$, then (3.21) holds, and hence the
solution set $\mathcal{C}$ contains a locally analytic curve
$\widetilde{\mathcal{C}}$. Moreover, $\widetilde{\mathcal{C}}$ is unbounded in
the sense that (3.22) holds.

###### Proof.

By Theorem 3.2, we only need to show that there exists a discrete subset
$S_{\gamma}$ of $[0,+\infty)$, such that for all
$\gamma\in[0,+\infty)\setminus S_{\gamma}$, $\mu^{K,\pm}$ satisfy Assumption
3.1. It suffices to verify 4) in Assumption 3.1, since all others are
straightforward (in particular,
$(a^{+})^{\prime}(0)\mu^{+}(e,0)+a^{+}(0)p\mu^{+}_{p}(e,0)=(a^{-})^{\prime}(0)\mu^{-}(e,0)+a^{-}(0)p\mu^{-}_{p}(e,0)=0$
implies $\frac{\partial\mu^{K,\pm}(e,p)}{\partial K}|_{K=0}=0$).

Notice that the matrix operator in 4) in Assumption 3.1 now can be written as

(6.14) $\left(\begin{array}[]{ccc}\tilde{J}_{11}\ \tilde{J}_{12}\\\
\tilde{J}_{21}\ \tilde{J}_{22}\\\
\end{array}\right):\mathcal{X}\times\mathcal{X}\times\mathbb{R}\rightarrow
H^{2}(\Omega)\times H^{2}(\Omega)\times\mathbb{R}$

where

(6.15) $\begin{split}&\tilde{J}_{11}(\delta u)=\delta u,\\\
&\tilde{J}_{12}(\delta
w)=-\gamma(-\Delta+\frac{1}{r^{2}})^{-1}\big{\\{}f_{2}(r)\delta w\big{\\}},\\\
&\tilde{J}_{21}(\delta u)=0,\\\ &\tilde{J}_{22}(\delta w)=\delta
w+\gamma\Delta^{-1}\big{\\{}f_{4}(r)\delta w\big{\\}},\\\ \end{split}$

and

$\begin{split}\\\
&f_{2}(r):=2\int_{\mathbb{R}^{3}}\hat{v}_{\varphi}\mu^{0}_{e}(\langle
v\rangle,rv_{\varphi})dv,\\\
&f_{4}(r):=2\int_{\mathbb{R}^{3}}\mu^{0}_{e}(\langle v\rangle,0)dv.\\\
\end{split}$

This matrix operator is bounded with a bounded inverse when $\gamma=0$. We
apply Lemma 2.6 to this matrix operator with the varying parameter $\gamma$
playing the role of $z$ in the lemma. We obtain that there exists a discrete
set $S_{\gamma}\subset[0,+\infty)$ such that for all
$\gamma\in[0,+\infty)\setminus S_{\gamma}$, this matrix operator is bounded
with a bounded inverse. This completes the proof of the first part of the
theorem. The analytic part can be shown in the same way as in Theorem 6.1. ∎

Let us consider the single species case and assume furthermore

(6.16) $\begin{split}&a^{+}(K)\rightarrow+\infty\text{ as
}K\rightarrow\infty,\ a^{-}(K)\equiv 0,\\\ \end{split}$

and

(6.17) $\begin{split}&\mu^{+}(e,p)\geq
C_{\mu^{+}}(1+|e|^{\alpha}+|p|^{\alpha})^{-1},\ \mu^{0}(e,p)\equiv 0\\\
\end{split}$

for all $(e,p)\in\mathbb{R}^{2}$ and some constants $\alpha>3$,
$C_{\mu^{+}}>0$.

In the same way as in the proof of Proposition 6.2, we obtain, using
Proposition 4.1:

######  Proposition 6.5.

Let $\Omega$ be a $C^{1}$ axisymmetric bounded domain as stated in Section 2.
Then with the assumptions (6.16) and (6.17), the set $\mathcal{C}$ obtained in
Theorem 6.4 is unbounded. Moreover, there holds:

(6.18) $\sup_{\\{(A_{\varphi}^{K,0},\phi^{K,0},K)\\}\in\mathcal{C}}K=+\infty,$

and

(6.19) $\text{As $K\rightarrow\infty$,
}\|(\phi^{K,0},A^{K,0}_{\varphi})\|_{L^{\infty}}\rightarrow+\infty.$

###### Proof.

The proof is very similar to the one for Proposition 6.2 so we omit it. ∎

Next, we consider the spectral stability of the solutions
$(\phi^{K,0},A_{\varphi}^{K,0})$. We first have the following observation for
small $K$:

######  Proposition 6.6.

Let $\Omega$ be a $C^{1}$ axisymmetric bounded domain as stated in Section 2.
Let $\mu^{0}_{p}\equiv 0,\quad\mu^{0}_{e}<0$, then for $K=K_{0}=0$, the
equilibrium
$(\mu^{0,\pm},\phi^{0,0},A_{\varphi}^{0,0})=(\mu^{0,\pm},\phi^{0,0},0)$ is
spectrally stable. Hence by continuity we have, when $K$ is close to $0$
(depends on $\gamma$, $a^{\pm}$, $\mu^{0}$, $\mu^{\pm}$), the equilibrium
$(\mu^{K,\pm},\phi^{K,0},A_{\varphi}^{K,0})$ is spectrally stable.

###### Proof.

The conclusion directly follows from Proposition 5.2. ∎

Now, let us assume furthermore the following conditions and discuss the
spectral stability of the solutions $(\phi^{K,0},A_{\varphi}^{K,0})$ for large
$K$:

######  Assumption 6.7.

$\mu^{+}_{e}<0$

and

$a^{+}(K)\sim K^{m}$

for some $m\in(-1,1)$ when $K\rightarrow+\infty$.

Hence we have only ions in the system.

######  Theorem 6.8.

Let $\Omega$ be a $C^{1}$ axisymmetric bounded domain as stated in Section 2,
with $\inf_{x\in\Omega}r(x)>0$. Let $\mu^{K,+}$ satisfy (6.13) and the decay
assumption (6.1), with $\delta>3$ replaced by $\delta>4$. Assume that (6.16),
(6.17) and Assumption 6.7 hold. Recall $d:=\sup_{x\in\Omega}r(x)$ and assume
$d>1$. Let $\epsilon$ be a positive constant such that $\epsilon<1-|m|$, so
$\epsilon+m<1$, $\epsilon<1+m$. Let $\mu^{+}$ be such that

(6.20) $p\mu^{+}_{p}(e,p)\geq C^{\prime}_{\mu}|p|\langle
p\rangle^{-\epsilon}\nu(e)$

for some positive function $\nu(e)$, with

(6.21) $\nu(e)\geq C_{\nu}\exp(-e)$

for some constant $C_{\nu}>0$. Then if $K$ is large enough, then the
equilibrium $(\mu^{K,+}(e,p),\mu^{K,-}(e,p)\equiv
0,\phi^{K,0},A_{\varphi}^{K,0})$ is spectrally unstable.

######  Remark 6.9.

The condition $p\mu^{+}_{p}\geq C^{\prime}_{\mu}|p|\langle
p\rangle^{-\epsilon}\nu(e)$ together with that
$|\mu^{+}_{p}(e,p)|\leq\frac{C_{\mu}}{1+|e|^{\gamma}}$ (assumed throughout all
discussion) implies that $|\nu(e)|\leq\frac{\langle
p\rangle^{\epsilon}}{1+|e|^{\gamma}}$ with $\gamma>4$, which ensures the
integrals in the proof are convergent. (This is where we need $\delta>4$.
Notice that $\epsilon\leq 1$.)

###### Proof.

It is easy to verify that the assumptions in Theorem 5.4, that is, Assumption
5.3, hold. In particular, $p\mu^{+}_{p}(e,p)\geq C^{\prime}_{\mu}|p|\langle
p\rangle^{-\epsilon}\nu(e)$ together with that
$\mu^{K,+}(e^{K,+},p^{K,+})=\gamma\mu^{0}(e^{K,+},Kp^{K,+})+a^{+}(K)\mu^{+}(e^{K,+},Kp^{K,+})\sim
K^{m}\mu^{+}(e^{K,+},Kp^{K,+})$ for large $K$ imply $p\mu^{K,+}_{p}(e,p)\geq
C^{\prime}_{\mu}K^{1+m-\epsilon}|p|\langle p\rangle^{-\epsilon}\nu(e)$ for
large $K$. Therefore we can directly apply Theorem 5.4 and obtain Theorem 6.8.
∎

######  Remark 6.10.

We can switch the role of ions and electrons ($+$ and $-$), and the same
results in this subsection hold.

##  Appendix A Derivative Operators in the Cylindrical Coordinates

For the readers’ convenience, we provide the information of the derivatives
under the cylindrical coordinates. Using $(x_{1},x_{2},x_{3})$ to denote the
Cartesian coordinates and $(r,\varphi,z)$ to be the cylindrical coordinates,
we have

$x_{1}=r\cos\varphi,\qquad x_{2}=r\sin\varphi,\qquad x_{3}=z$

and therefore

$e_{r}=(\cos\varphi,\sin\varphi,0),\qquad
e_{\varphi}=(-\sin\varphi,\cos\varphi,0),\qquad e_{z}=(0,0,1).$

Hence for any function $f(r,\varphi,z)$ and any vector field
$\textbf{A}(r,\varphi,z)$, we have

$\nabla_{x}f=\frac{\partial f}{\partial r}e_{r}+\frac{1}{r}\frac{\partial
f}{\partial\varphi}e_{\varphi}+\frac{\partial f}{\partial z}e_{z},$
$\Delta_{x}f=\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial
f}{\partial
r})+\frac{1}{r^{2}}\frac{\partial^{2}f}{\partial\varphi^{2}}+\frac{\partial^{2}f}{\partial
z^{2}},$
$\nabla_{x}\cdot\textbf{A}=\frac{1}{r}\frac{\partial(rA_{r})}{\partial
r}+\frac{1}{r}\frac{\partial A_{\varphi}}{\partial\varphi}+\frac{\partial
A_{z}}{\partial z},$ $\nabla_{x}\times\textbf{A}=(\frac{1}{r}\frac{\partial
A_{z}}{\partial\varphi}-\frac{\partial A_{\varphi}}{\partial
z})e_{r}+(\frac{\partial A_{r}}{\partial z}-\frac{\partial A_{z}}{\partial
r})e_{\varphi}+\frac{1}{r}(\frac{\partial(rA_{\varphi})}{\partial
r}-\frac{\partial A_{r}}{\partial\varphi})e_{z}.$

##  Appendix B Key Invariants

In this section, we prove the invariance of $e^{\pm}(x,v)=\langle
v\rangle\pm\phi^{0}(r,z)$ and $p^{\pm}(x,v)=r(v_{\varphi}\pm
A^{0}_{\varphi}(r,z))$ (or $p^{\pm}(x,v)=r(v_{\varphi}\pm
A^{0}_{\varphi}(r,z)\pm A_{\varphi,ext}(r,z))$ when an external magnetic
potential $\textbf{A}_{ext}=A_{\varphi,ext}(r,z)e_{\varphi}$ is present) along
the particle trajectories given in Section 2:

(B.1) $\begin{split}&\dot{X}^{\pm}=\hat{V}^{\pm},\
\dot{V}^{\pm}=\pm\textbf{E}^{0}(X^{\pm})\pm\hat{V}^{\pm}\times\textbf{B}^{0}(X^{\pm}).\\\
\end{split}$

(Or

(B.2) $\dot{X}^{\pm}=\hat{V}^{\pm},\
\dot{V}^{\pm}=\pm\textbf{E}^{0}(X^{\pm})\pm\hat{V}^{\pm}\times\textbf{B}^{0}(X^{\pm})\pm\hat{V}^{\pm}\times\textbf{B}_{ext}(X^{\pm})$

when the external magnetic potential is present.)

We only compute the $+$ case for simplicity. The $-$ case is similar. In fact,
along the particle trajectories, we have

(B.3) $\begin{split}\dot{e}&=\hat{V}\cdot\dot{V}+\dot{X}\cdot\nabla\phi^{0}\\\
&=\hat{V}\cdot(\textbf{E}^{0}+\hat{V}\times\textbf{B}^{0})+\hat{V}\cdot\nabla\phi^{0}\\\
&=\hat{V}\cdot(-\nabla\phi^{0}+\hat{V}\times\textbf{B}^{0})+\hat{V}\cdot\nabla\phi^{0}\\\
&=0,\end{split}$ (B.4)
$\begin{split}\dot{p}&=\dot{X}\cdot\nabla_{X}p+\dot{V}\cdot\nabla_{V}p\\\
&=\dot{X}\cdot\nabla_{X}(r(v_{\varphi}+A^{0}_{\varphi}(r,z)))+\dot{V}\cdot\nabla_{V}(r(v_{\varphi}+A^{0}_{\varphi}(r,z)))\\\
&=\dot{X}\cdot\nabla_{X}(rv_{\varphi})+\dot{X}\cdot\nabla_{X}(rA^{0}_{\varphi})+\dot{V}\cdot
r\nabla_{V}v_{\varphi}\\\ &=\dot{X}\cdot\nabla_{X}(rv_{\varphi})+\dot{X}\cdot
A^{0}_{\varphi}e_{r}+\dot{X}\cdot r\frac{\partial A^{0}_{\varphi}}{\partial
r}e_{r}+\dot{X}\cdot r\frac{\partial A^{0}_{\varphi}}{\partial
z}e_{z}+r(\textbf{E}^{0}+\hat{V}\times\textbf{B}^{0})\cdot e_{\varphi}\\\
&=\dot{X}\cdot\nabla_{X}(rv_{\varphi})+\dot{X}\cdot
A^{0}_{\varphi}e_{r}+\dot{X}\cdot r\frac{\partial A^{0}_{\varphi}}{\partial
r}e_{r}+\dot{X}\cdot r\frac{\partial A^{0}_{\varphi}}{\partial z}e_{z}\\\
&+r(-\nabla\phi^{0}+\dot{X}\times\frac{\partial A^{0}_{\varphi}}{\partial
z}e_{r}+\dot{X}\times\frac{1}{r}A^{0}_{\varphi}e_{z}+\dot{X}\times\frac{\partial
A^{0}_{\varphi}}{\partial r}e_{z})\cdot e_{\varphi}\\\
&=\dot{X}\cdot\nabla_{X}(rv_{\varphi})+\dot{X}\cdot
A^{0}_{\varphi}e_{r}+\dot{X}\cdot r\frac{\partial A^{0}_{\varphi}}{\partial
r}e_{r}+\dot{X}\cdot r\frac{\partial A^{0}_{\varphi}}{\partial
z}e_{z}-r\frac{\partial A^{0}_{\varphi}}{\partial z}\dot{X}\cdot
e_{z}-A^{0}_{\varphi}\dot{X}\cdot e_{r}-r\frac{\partial
A^{0}_{\varphi}}{\partial r}\dot{X}\cdot e_{r}\\\
&=\dot{X}\cdot\nabla_{X}(rv_{\varphi}).\\\ \end{split}$

(Here $r$, $\varphi$, $z$, $v_{r}$, $v_{\varphi}$, $v_{z}$ are components of
$X$ and $V$.) In the computation above, we used that $\nabla\phi^{0}$ does not
have $e_{\varphi}$-component. We evaluate
$\dot{X}\cdot\nabla_{X}(rv_{\varphi})$ using the Cartesian coordinates:

(B.5)
$\begin{split}\dot{X}\cdot\nabla_{X}(rv_{\varphi})&=\dot{X}\cdot\nabla_{X}(rV\cdot(-\frac{X_{2}}{r},\frac{X_{1}}{r},0))\\\
&=\dot{X}\cdot\nabla_{X}(-X_{2}V_{1}+X_{1}V_{2})\\\ &=\frac{1}{\langle
V\rangle}(V_{1},V_{2},V_{3})\cdot(V_{2},-V_{1},0).\\\ &=0\end{split}$

Hence $\dot{p}=0$. Moreover, one can show in a similar way that
$p(x,v)=r(v_{\varphi}+A^{0}_{\varphi}(r,z)+A_{\varphi,ext}(r,z))$ is invariant
along the particle trajectories satisfying the ODE $\dot{X}=\hat{V}$,
$\dot{V}=\textbf{E}^{0}(X)+\hat{V}\times\textbf{B}^{0}(X)+\hat{V}\times\textbf{B}_{ext}(X)$.
The invariance along the particle trajectories is now verified.

## References

  * [1] A. Aghajani and A. M. Tehrani, _Pointwise bounds for positive supersolutions of nonlinear elliptic problems involving the $p$-Laplacian and application_, arXiv preprint arXiv:1609.05437 (2016)
  * [2] J. Batt and K. Fabian, _Stationary Solutions of the Relativistic Vlasov-Maxwell System of Plasma Physics_ , Chin. Ann. Math. Ser. B, 14 (1993), 253–278
  * [3] P. Braasch, _On Stationary Solutions of the Relativistic Vlasov–Maxwell System_ , Mathematical Methods in the Applied Sciences (20), no. 8 (1997): 667–677
  * [4] A. Constantin, W. A. Strauss, E. Vărvărucă, _Global bifurcation of steady gravity water waves with critical layers_ , Acta Math. (2016), 217(2): 195–262
  * [5] R. M. Chen, S. Walsh, M. H. Wheeler, _Existence and qualitative theory for stratified solitary water waves_ , In Annales de l’Institut Henri Poincaré C, Analyse non linéaire, vol. 35, no. 2, pp. 517–576. Elsevier Masson, 2018
  * [6] R. M. Chen, S. Walsh, M. H. Wheeler, _Global bifurcation for monotone fronts of elliptic equations_ , preprint arXiv:2005.00651
  * [7] Y. Guo, _Stable magnetic equilibria in collisionless plasmas_ , Commun. Pure Appl. Math. (1997) 50 (9): 891–933
  * [8] Y. Guo and W. A. Strauss, _Magnetically created instability in a collisionless plasma_ J. Math. Pure Appl. (2000) (9) 79 (10): 975–1009
  * [9] D. Han-Kwan, _On the confinement of a tokamak plasma_ , SIAM J. Math. Anal. (2010) 42 (6): 2337–2367
  * [10] H. Kielhöfer, _Bifurcation Theory: An Introduction with Applications to Partial Differential Equations_ , Springer-Verlag (2012), New York
  * [11] Z. Lin and W. A. Strauss, _Linear stability and instability of relativistic Vlasov-Maxwell systems_ , Invent. Math. (2007) 60 (5): 724–787
  * [12] Z. Lin and W. A. Strauss, _A sharp stability criterion for the Vlasov-Maxwell system_ , Commun. Pure. Appl. Math. (2008) 173 (3): 497–546
  * [13] Z. Lin and W. Strauss, _Nonlinear stability and instability of relativistic Vlasov-Maxwell systems_ , Commun. Pure. Appl. Math. (2007) 60: 789–837
  * [14] T. T. Nguyen, T. V. Nguyen and W. A. Strauss, _Global magnetic confinement for the 1.5D Vlasov-Maxwell System_ , Kinetic and Related Models (2015) 8 (1): 153-168
  * [15] T. T. Nguyen, T. V. Nguyen and W. A. Strauss, _Erratum to: Global magnetic confinement for the 1.5D Vlasov-Maxwell System_ , Kinetic and Related Models (2015) 8 (3): 615-616
  * [16] T. T. Nguyen and W. A. Strauss, _Linear stability analysis of a hot plasma in a solid torus_ , Arch. Rational. Mech. Anal. (2014) 211: 619–672
  * [17] O. Penrose, _Electrostatic instability of a non-Maxwellian plasma_ , Phys. Fluids (1960) 3: 258–265
  * [18] F. Poupaud, _Boundary Value Problems for the Stationary Vlasov-Maxwell System_ , Forum Math., 4 (1992), 499–527
  * [19] M. Reed and B. Simon, _Methods of Modern Mathematical Physics_ , Volume 1, Academic Press, Inc. 1980
  * [20] G. Rein, _Existence of stationary, collisionless plasmas in bounded domains_ , Math. Meth. in Appl. Sci. , 15 (1992), 365–374
  * [21] J. Schaeffer, _Steady states of the Vlasov-Maxwell system_ , Quarterly of Applied Mathematics (63), no. 4 (2005): 619–643
  * [22] J. Weber, _Confined Steady States of a Relativistic Vlasov-Maxwell Plasma in a Long Cylinder_ , Kinetic and Related Models, 2020, 13 (6): 1135–1161
  * [23] N. A. Sidorov and A. V. Sinitsyn, _Analysis of bifurcation points and nontrivial branches of solutions to the stationary Vlasov-Maxwell system_ , Mathematical Notes (62), no. 2 (1997): 223–243
  * [24] A. V. Sinitsyn and E. V. Dulov, _Boundary value problems for the Vlasov-Maxwell system_ , Revista Colombiana de Matemásticas (41) (2007): 151–187
  * [25] W. A. Strauss and Y. Wu, _Steady States of Rotating Stars and Galaxies_ , SIAM J. Math. Anal., 49(6), 4865–4914
  * [26] K. Z. Zhang, _Linear stability analysis of the Relativistic Vlasov-Maxwell System in an axisymmetric domain_ , SIAM Journal on Mathematical Analysis, 51(6), 4683–4723
  * [27] K. Z. Zhang, _Magnetic confinement at boundary approximates specular reflecting wall_ , Mathematical Methods in the Applied Sciences 44, no. 7 (2021): 5705–5738