# Normalized solutions of quasilinear Schrödinger equations with a general
nonlinearity

Ting Deng  ,  Marco Squassina  ,  Jianjun Zhang  and  Xuexiu Zhong
College of Mathematics and Statistics
Chongqing Jiaotong University
Chongqing 400074, PR China<EMAIL_ADDRESS>
Dipartimento di Matematica e Fisica
Università Cattolica del Sacro Cuore
Via dei Musei 41, Brescia, Italy<EMAIL_ADDRESS>
College of Mathematics and Statistics
Chongqing Jiaotong University
Chongqing 400074, PR China<EMAIL_ADDRESS>
South China Research Center for Applied Mathematics and Interdisciplinary
Studies
South China Normal University
Guangzhou 510631, P. R. China<EMAIL_ADDRESS>
###### Abstract.

We are concerned with solutions of the following quasilinear Schrödinger
equations

$\displaystyle-{\mathrm{div}}\left(\varphi^{2}(u)\nabla
u\right)+\varphi(u)\varphi^{\prime}(u)|\nabla u|^{2}+\lambda u=f(u),\quad
x\in\mathbb{R}^{N}$

with prescribed mass

$\int_{\mathbb{R}^{N}}u^{2}\mathrm{d}x=c,$

where $N\geq 3,c>0$, $\lambda\in\mathbb{R}$ appears as the Lagrange multiplier
and $\varphi\in C^{1}(\mathbb{R},\mathbb{R}^{+})$. The nonlinearity $f\in
C\left(\mathbb{R},\,\mathbb{R}\right)$ is allowed to be mass-subcritical,
mass-critical and mass-supercritical at origin and infinity. Via a dual
approach, the fixed point index and a global branch approach, we establish the
existence of normalized solutions to the problem above. The results extend
previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear
case.

###### Key words and phrases:

Quasilinear Schrödinger equations, Normalized solutions, Mass critical
exponent.

###### 2010 Mathematics Subject Classification:

35J62, 35B40, 35B09

Marco Squassina is supported by Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni. X.X. Zhong is partially supported by the
NSFC (No.12271184), Guangdong Basic and Applied Basic Research Foundation
(2021A1515010034), Guangzhou Basic and Applied Basic Research Foundation
(202102020225). J. J. Zhang was supported by NSFC (No.11871123).

###### Contents

  1. 1 Introduction
    1. 1.1 Background and motivation
    2. 1.2 Assumptions and main results
    3. 1.3 Strategy of this paper
  2. 2 Functional setting and preliminaries
  3. 3 Asymptotic behaviors
  4. 4 Uniqueness for $\lambda$ small or large
  5. 5 Proof of Theorem 1.3

##  1\. Introduction

###  1.1. Background and motivation

This paper is concerned with the quasilinear Schrödinger equation of the form

(1.1) $-{\mathrm{div}}\left(\varphi^{2}(u)\nabla
u\right)+\varphi(u)\varphi^{\prime}(u)|\nabla u|^{2}+\lambda u=f(u),\quad
x\in\mathbb{R}^{N},$

which is relevant to solitary wave solutions of quasilinear Schrödinger
equation

(1.2) $iz_{t}=-\Delta
z+W(x)z-h\left(|z|^{2}\right)z-\Delta\left(l\left(|z|^{2}\right)\right)l^{\prime}\left(|z|^{2}\right)z,\quad
x\in\mathbb{R}^{N},$

where $z:\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{C}$,
$W:\mathbb{R}^{N}\to\mathbb{R}$ is a given potential and $l$, $h$ are real
functions. Quasilinear equations of the form (1.2) appear naturally in
mathematical physics and have been accepted as models of several physical
phenomena corresponding to various types of nonlinear terms $l$. When
$l\left(s\right)=s$, that is, $\varphi^{2}(u)=1+2u^{2}$, Equation (1.2) arises
in plasma physics for the superfluid film equation (see [14]), while when
$l(s)=(1+s)^{1/2}$, that is,
$\varphi^{2}(u)=1+\frac{u^{2}}{2\left(1+u^{2}\right)}$, it models the self-
channeling of a high-power ultrashort laser in matter (see [16]). Such
equations were also showed in plasma physics and fluid mechanics [17], in
dissipative quantum mechanics [9], in the theory of Heisenberg ferromagnetism
and magnons [15, 25], and in condensed matter theory [23]. For more details,
we refer to [18, 19]. In the last decades, quasilinear Schrödinger equations
have received a considerable attentions from researchers on the existence of
nontrivial solutions. In [4], M. Colin and L. Jeanjean considered the
following quasilinear Schrödinger equations

(1.3) $-\Delta u-\Delta\left(u^{2}\right)u=g(x,u),\quad x\in\mathbb{R}^{N}.$

Thanks to a dual approach, they transform equation (1.3) into a semilinear
elliptic equation. By using the variational method, nontrivial solutions were
obtained. In [8], E. Gloss adopted the dual approach in [4] to study the
semiclassical states of quasilinear elliptic equations

$-\varepsilon^{2}\Delta
u-\varepsilon^{2}\Delta\left(u^{2}\right)u+V(x)u=h(u),\quad u>0\text{ in
}\mathbb{R}^{N}.$

Via the penalized argument as in [3], she established the existence and
concentration of positive solutions without some growth conditions such as
Ambrosetti-Rabinowitz. Different from [4], Y. Shen and Y. Wang [26] introduced
a new variable replacement to study the existence of nontrivial solutions for
generalized quasilinear Schrödinger equations. By using the dual approach in
[26], Y. Deng, S. Peng and S. Yan [6] considered a generalized quasilinear
Schrödinger equation with critical growth. By virtue of variational
approaches, they obtained the existence of positive solutions. In [1], C. O.
Alves, Y. Wang and Y. Shen considered the following quasilinear Schrödinger
equation with one parameter

(1.4) $-\Delta
u+V(x)u+\frac{\kappa}{2}\Delta(u^{2})u=l(u),x\in\mathbb{R}^{N},$

where $\kappa$ is allowed to be positive. When $\kappa<0$, one can use the
dual approaches in [4, 26] to deal with the existence of nontrivial solutions
for equation (1.4). The case of $\kappa>0$ becomes more complicated. By
adopting a truncation approach, the authors obtained the existence of
nontrivial classical solution for $\kappa>0$ small enough. For more related
results on quasilinear elliptic equations, we refer to [21, 22, 10] and
references therein. For a direct approach via nonsmooth critical point theory
in cases where the change of variable is not usable, see [24]. In this paper,
we focus on normalized solutions of equation (1.5) with prescribed mass, that
is to find $(\lambda,u)$ such that

(1.5) $\left\\{\begin{array}[]{ll}-\mathrm{div}\left(\varphi^{2}(u)\nabla
u\right)+\varphi(u)\varphi^{\prime}(u)|\nabla u|^{2}+\lambda u=f(u),\quad
x\in\mathbb{R}^{N},\\\
\displaystyle\int_{\mathbb{R}^{N}}\left|u\right|^{2}\mathrm{d}x=c,\end{array}\right.$

where $N\geq 3$, $f\in C\left(\mathbb{R},\,\mathbb{R}\right)$,
$\varphi:\mathbb{R}\to\mathbb{R}^{+}$ is a $C^{1}$ nondecreasing function with
respect to $\left|s\right|,c>0$ is a given mass, $\lambda\in\mathbb{R}$
appears as a Lagrange multiplier. From the view of physics, prescribed mass
represents the law of conservation of mass, so it is pretty meaningful to
investigate the existence of normalized solutions. If $\varphi\equiv 1$,
problem (1.5) is reduced to the following scalar field equation with
prescribed mass

(1.6) $\left\\{\begin{array}[]{ll}-\Delta u+\lambda u=f(u),\quad
x\in\mathbb{R}^{N},\\\
\displaystyle\int_{\mathbb{R}^{N}}\left|u\right|^{2}\mathrm{d}x=c.\end{array}\right.$

One classical argument to deal with such problems is to find critical points
$u\in H^{1}(\mathbb{R}^{N})$ of the energy functional

$\mathcal{J}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,\mathrm{d}x-\int_{\mathbb{R}^{N}}F(u)\,\mathrm{d}x,\quad\mbox{where}\quad
F(s):=\int_{0}^{s}f(t)\mathrm{d}t,$

subject to the constraint

$S_{c}:=\\{u\in H^{1}(\mathbb{R}^{N}):\|u\|_{L^{2}(\mathbb{R}^{N})}^{2}=c\\}.$

When $f(s)=|s|^{p-2}s$ with $p\in(2,2N/(N-2))$, thanks to the well-known
Gagliardo-Nirenberg inequality, it is known that the energy functional
$\mathcal{J}$ is bounded from below on $S_{a}$ for any $a>0$ if $p<2+4/N$ and
unbounded from below for any $a>0$ if $p>2+4/N$. $2+4/N$ is called the mass-
critical exponent. In [27, 28], C. A. Stuart considered problem (1.6) with a
mass-subcritical nonlinearity and obtained the existence of normalized
solutions by seeking a global minimizer of the energy functional on $S_{c}$.
In the mass supercritical case, since there exists no global minimizer of the
associated energy functional restricted on the constraint $S_{c}$, it seems
that the arguments in dealing with the mass subcritical case do not work. In
[11], L. Jeanjean imposed the following global condition on $f$.

  * ($H$)

$\exists(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}$ satisfying

$2+\frac{4}{N}<\alpha\leqslant\beta<\frac{2N}{(N-2)_{+}}$

such that

$\alpha F(s)\leqslant f(s)s\leqslant\beta F(s).$

The assumption $(H)$ is used to guarantee the mountain pass geometry of
$\tilde{\mathcal{J}}$ on $S_{c}\times\mathbb{R}$, where
$\tilde{\mathcal{J}}(u,s)=\mathcal{J}(e^{sN/2}u(e^{s}\cdot))$. Then a bounded
Palais-Smale sequence of $\mathcal{J}$ restricted on $S_{c}$ is obtained. Via
some compactness arguments, the author shows the existence of normalized
solutions to problem (1.6). More recently, L. Jeanjean, J. Zhang and X. Zhong
[13] introduced a new and non-variational approach to deal with problem (1.6)
in the mass-subcritical, mass-critical and mass-supercritical case in one
unified way. In particular, the assumption $(H)$ can be removed. For more
related results on problem (1.6), we refer to [13] and references therein.
When $\varphi^{2}(u)=1+2u^{2}$, problem (1.1) reads as the following form

(1.7) $-\Delta u-u\Delta\left(u^{2}\right)+\lambda u=f(u)\quad\text{ in
}\mathbb{R}^{N}.$

Different to the case $\varphi\equiv 1$, $4+4/N$ was proved by M. Colin, L.
Jeanjean and M. Squassina [5] to be the mass-critical exponent for problem
(1.7). In [12]. L. Jeanjean, T. Luo and Z.-Q. Wang considered problem (1.7)
with a mass-subcritical nonlinearity $f(u)=|u|^{p-2}u$, $p\in(2+4/N,4+4/N)$
and prove the existence of two solutions if the prescribed $L^{2}$-norm is
large enough. In [30], H. Ye and Y. Yu focused on the existence of normalized
solutions of problem (1.7) in the mass-critical case. For the mass-
supercritical case, we refer to [20], where H. Li and W. Zou adopted one
perturbation argument as in [21] to get the existence of the existence of
ground state normalized solutions to problem (1.7). By applying the index
theory, infinitely many normalized solutions were also obtained. Moreover,
they also established the concentration behavior of ground state solutions in
the mass-critical case. In [31], via the dual approach in [4] and genus
theory, L. Zhang, Y. Li and Z.-Q. Wang constructed multiple normalized
solutions of problem (1.7) with $f(u)=|u|^{p-2}u$ in the mass-subcritical
case. When $\varphi^{2}(u)=1+\frac{u^{2}}{2\left(1+u^{2}\right)}$, equation
(1.1) is reduced to the following quasilinear equation

(1.8) $-\Delta u+\lambda
u-\left[\Delta\left(1+u^{2}\right)^{\frac{1}{2}}\right]\frac{u}{2\left(1+u^{2}\right)^{\frac{1}{2}}}=f(u)\text{
in }\mathbb{R}^{N}.$

In [29], under the Berestycki-Lions condition, X. Yang, X. Tang and B. Cheng
show that problem (1.8) admits multiple radial and nonradial normalized
solutions in the mass-subcritical case. For more processes on normalized
solutions of quasilinear Schrödinger equations, we refer to [12, 20, 30, 31]
and the references therein.

###  1.2. Assumptions and main results

The main purpose of the present paper is to investigate the existence of
normalized solutions to problem (1.5) with a relatively general nonlinearity.
Throughout this paper, we impose the following assumptions on $\varphi$.

  * $\left(\varphi_{0}\right)$

$\varphi\in C^{1}\left(\mathbb{R},\mathbb{R}^{+}\right)$ is even,
$\varphi^{\prime}\left(t\right)\geq 0$ for all $t\geq 0$ and
$\varphi\left(0\right)=1$.

  * $\left(\varphi_{1}\right)$

There exists $a^{*}>0$ such that

$\lim_{s\to+\infty}\varphi\left(s\right)=a^{*}>0.$

  * $\left(\varphi_{2}\right)$

$\lim\limits_{t\rightarrow\infty}t\varphi^{\prime}(t)=0$.

######  Remark 1.1.

As a reference model, take
$\varphi(t)=\left(1+\frac{t^{2}}{2\left(1+t^{2}\right)}\right)^{\frac{1}{2}}$
satisfying $\left(\varphi_{0}\right)$-$\left(\varphi_{2}\right)$.

As for the nonlinearity $f$, we assume that

  * $\left(F_{1}\right)$

$f\in C^{1}[0,+\infty)$, $f(s)>0$ for $s>0$.

  * $\left(F_{2}\right)$

There exist $\alpha,\beta,\mu_{1},\mu_{2}>0$ satisfying

$2<\alpha,\beta<2^{*}:=\frac{2N}{N-2}$

such that

$\lim_{t\rightarrow
0^{+}}\frac{f^{\prime}(t)}{t^{\alpha-2}}=\mu_{1}(\alpha-1)>0\text{ \quad and
}\quad\lim_{t\rightarrow+\infty}\frac{f^{\prime}(t)}{t^{\beta-2}}=\mu_{2}(\beta-1)>0.$

  * $\left(F_{3}\right)$

There exists no positive radial decreasing classical solution for
$-\mathrm{div}\left(\varphi^{2}(u)\nabla
u\right)+\varphi(u)\varphi^{\prime}(u)|\nabla u|^{2}=f(u)$ in
$\mathbb{R}^{N}$.

######  Remark 1.2.

By [13, Theorem 2.2-(ii)], $(F_{3})$ holds for $f(t)=|t|^{q-2}t$,
$q\in(2,2N/(N-2))$.

Before to state our result, denote by $U$ the unique positive solution of

(1.9) $-\Delta U+U=\mu_{1}U^{\alpha-1}\leavevmode\nobreak\
\hbox{in}\leavevmode\nobreak\
\mathbb{R}^{N},\,\,U(0)=\max_{x\in\mathbb{R}^{N}}U(x)$

and $V$ the unique positive solution of

(1.10) $-\Delta V+V=\mu_{2}V^{\beta-1}\leavevmode\nobreak\
\hbox{in}\leavevmode\nobreak\
\mathbb{R}^{N},\,\,V(0)=\max_{x\in\mathbb{R}^{N}}V(x).$

It is known that $U,V$ are non-degenerated (See [13, Proposition 2.1]). Now,
our main result reads as follows.

######  Theorem 1.3.

Let $N\geq 3$ and assume that
$\left(\varphi_{0}\right)$-$\left(\varphi_{2}\right)$,
$\left(F_{1}\right)$-$\left(F_{3}\right)$ hold, then we have the following
conclusions.

  * (i)

(mass subcritical case) If $2<\alpha$, $\beta<2+\frac{4}{N}$, then for any
given $c>0$, (2.4) possesses a positive normalized solution
$\left(\lambda,\,v_{\lambda}\right)\in(0,\,+\infty)\times
H_{\text{rad}}^{1}\left(\mathbb{R}^{N}\right)$.

  * (ii)

(exactly mass critical case) If $\alpha=\beta$ $=2+\frac{4}{N}$, denote

$c_{\ast}:=\min\\{\|U\|_{2}^{2},(a^{\ast})^{N}\|V\|_{2}^{2}\\},c^{\ast}:=\max\\{\|U\|_{2}^{2},(a^{\ast})^{N}\|V\|_{2}^{2}\\},$

then (2.4) admits at least one positive normalized solution
$(\lambda,\,v_{\lambda})\in(0,\,+\infty)\times H_{\text{rad
}}^{1}(\mathbb{R}^{N})$ provided $c\in(c_{\ast},c^{\ast})$ and no positive
normalized solution if $c>0$ small or large.

  * (iii)

(at most mass critical case)

  * (iii-1)

If $2<\alpha<\beta=2+\frac{4}{N}$, (2.4) admits at least one positive
normalized solution $\left(\lambda,\,v_{\lambda}\right)\in(0,\,+\infty)\times
H_{\text{rad }}^{1}\left(\mathbb{R}^{N}\right)$ if
$0<c<(a^{\ast})^{N}\|V\|_{2}^{2}$ and no positive normalized solution if $c>0$
large.

  * (iii-2)

If $2<\beta<\alpha=2+\frac{4}{N}$, (2.4) admits at least one positive
normalized solution $\left(\lambda,\,v_{\lambda}\right)\in(0,\,+\infty)\times
H_{\text{rad }}^{1}\left(\mathbb{R}^{N}\right)$ if $c>\|U\|_{2}^{2}$ and no
positive normalized solution if $c>0$ small.

  * (iv)

(mixed case)

  * (iv-1)

If $2<\alpha<2+\frac{4}{N}<\beta<2^{*}$, (2.4) admits at least two distinct
positive normalized solutions
$\left(\lambda_{i},\,v_{\lambda_{i}}\right)\in(0,\,+\infty)\times H_{\text{rad
}}^{1}\left(\mathbb{R}^{N}\right)$ if $c>0$ small and no positive normalized
solution provided $c>0$ large.

  * (iv-2)

If $2<\beta<2+\frac{4}{N}<\alpha<2^{*}$, (2.4) admits at least two distinct
positive normalized solutions
$\left(\lambda_{i},\,v_{\lambda_{i}}\right)\in(0,\,+\infty)\times H_{\text{rad
}}^{1}\left(\mathbb{R}^{N}\right)$ if $c>0$ large and no positive normalized
solution provided $c>0$ small.

  * (v)

(at least mass critical case)

  * (v-1)

If $2+\frac{4}{N}=\alpha<\beta<2^{*}$, (2.4) admits at least one positive
normalized solution $\left(\lambda,\,v_{\lambda}\right)\in(0,\,+\infty)\times
H_{\text{rad }}^{1}\left(\mathbb{R}^{N}\right)$ if $0<c<\|U\|_{2}^{2}$ and no
positive normalized solution provided $c>0$ large.

  * (v-2)

If $2+\frac{4}{N}=\beta<\alpha<2^{*}$, (2.4) admits at least one positive
normalized solution $\left(\lambda,\,v_{\lambda}\right)\in(0,\,+\infty)\times
H_{\text{rad }}^{1}\left(\mathbb{R}^{N}\right)$ if
$c>(a^{\ast})^{N}\|V\|_{2}^{2}$ and no positive normalized solution provided
$c>0$ small.

  * (vi)

(mass supercritical case) If $2+\frac{4}{N}<\alpha,\beta<2^{*}$, then for any
given $c>0$, (2.4) admits a normalized positive solution
$\left(\lambda,\,v_{\lambda}\right)\in(0,\,+\infty)\times H_{\text{rad
}}^{1}\left(\mathbb{R}^{N}\right)$.

###  1.3. Strategy of this paper

First, by virtue of a dual approach in [26], we transform equation (1.1) into
a semilinear elliptic equation. Second, for any fixed $\lambda>0$, we show the
semilinear elliptic equation obtained admits at least one positive and
radially symmetric solution $v_{\lambda}$. Similarly to [13], by using the
blow-up argument and a Liouville theorem, as $\lambda\rightarrow 0^{+}$ or
$\lambda\rightarrow+\infty$, the asymptotic behaviors of positive solutions
are investigated, as well as the $L^{2}$-norms of
$\phi^{-1}\left(v_{\lambda}\right)$. Finally, via the fixed point index and
the topological degree, we adopt the similar idea to [13] to establish a
global branch of positive solutions for $\lambda\in(0,\infty)$. By applying a
continuity argument, Theorem 1.3 is proved. Notations

  * •

$L^{p}\left(\mathbb{R}^{N}\right)$ denotes the Lebesgue space with norm
$\|u\|_{p}:=\big{(}\int_{\mathbb{R}^{N}}|u|^{p}\mathrm{d}x\big{)}^{1/p}$.

  * •

$H^{1}\left(\mathbb{R}^{N}\right)$ denotes the Sobolev space modeled in
$L^{2}\left(\mathbb{R}^{N}\right)$ with its usual norm
$\|u\|:=\big{(}\|u\|_{2}^{2}+\|\nabla u\|_{2}^{2}\big{)}^{1/2}$.

  * •

$C_{r,\,0}\left(\mathbb{R}^{N}\right)$ denotes the space of radial continuous
functions vanishing at infinity.

##  2\. Functional setting and preliminaries

To seek normalized solutions of (1.5), it suffices to find critical points of

(2.1)
$I\left(u\right)=\frac{1}{2}\int_{\mathbb{R}^{N}}\varphi^{2}(u)\left|\nabla
u\right|^{2}\mathrm{d}x-\int_{\mathbb{R}^{N}}F\left(u\right)\mathrm{d}x$

on the mass sphere $S_{c}$. Due to the presence of $\varphi$, some additional
difficulties arise. In this paper, we adopt a dual approach in [26] to
overcome them and make a change of variables as follows. Let $(u,\lambda)$ be
any solution of problem (1.5), that is, for any $\tilde{\phi}\in
C_{0}^{\infty}\left(\mathbb{R}^{N}\right)$, there holds

(2.2) $\int_{\mathbb{R}^{N}}\left[\varphi^{2}\left(u\right)\nabla
u\nabla\tilde{\phi}+\varphi\left(u\right)\varphi^{\prime}\left(u\right)\left|\nabla
u\right|^{2}\tilde{\phi}+\lambda
u\tilde{\phi}-f\left(u\right)\tilde{\phi}\right]\mathrm{d}x=0.$

If ones take

$v=\phi\left(u\right)=\int_{0}^{u}\varphi\left(t\right)\mathrm{d}t,$

and choose $\tilde{\phi}=\frac{1}{\varphi\left(u\right)}\psi$ in equation
(2.2) for any $\psi\in C_{0}^{\infty}\left(\mathbb{R}^{N}\right)$, then it
follows that

(2.3) $\int_{\mathbb{R}^{N}}\left[\nabla
v\nabla\psi+\lambda\frac{\phi^{-1}\left(v\right)}{\varphi\left(\phi^{-1}\left(v\right)\right)}\psi-\frac{f\left(\phi^{-1}\left(v\right)\right)}{\varphi\left(\phi^{-1}\left(v\right)\right)}\psi\right]\mathrm{d}x=0.$

Then $(v,\lambda)$ is a solution of the following semilinear elliptic problem

(2.4) $-\Delta
v+\lambda\frac{\phi^{-1}\left(v\right)}{\varphi\left(\phi^{-1}\left(v\right)\right)}-\frac{f\left(\phi^{-1}\left(v\right)\right)}{\varphi\left(\phi^{-1}\left(v\right)\right)}=0,\quad
x\in\mathbb{R}^{N}$

with the prescribed mass

$\int_{\mathbb{R}^{N}}\left|\phi^{-1}\left(v\right)\right|^{2}\mathrm{d}x=c.$

And corresponding to the energy functional $I$, the energy functional
associated with problem (2.4) is defined by

(2.5) $J\left(v\right)=\frac{1}{2}\int_{\mathbb{R}^{N}}\left|\nabla
v\right|^{2}\mathrm{d}x-\int_{\mathbb{R}^{N}}F\left(\phi^{-1}\left(v\right)\right)\mathrm{d}x.$

Since $\varphi$ is a nondecreasing positive function, we get
$\left|\phi^{-1}\left(s\right)\right|\leq\left|s\right|$ for any $s$.
Moreover, it is clear that $J$ is well defined in
$H^{1}\left(\mathbb{R}^{N}\right)$ and of $C^{1}$-class. Therefore, in order
to find normalized solutions of equation (1.5), it is sufficient to turn to
consider the existence of normalized solutions to problem (2.4).

In the following, we intend to borrow some ideas in [13] to find normalized
solutions of problem (2.4).

Let

$g_{\lambda}\left(v\right)=\frac{f\left(\phi^{-1}\left(v\right)\right)}{\varphi\left(\phi^{-1}\left(v\right)\right)}-\lambda\frac{\phi^{-1}\left(v\right)}{\varphi\left(\phi^{-1}\left(v\right)\right)}+\lambda
v,$

then equation (2.4) turns into

(2.6) $-\Delta v+\lambda v=g_{\lambda}\left(v\right),\,\quad
x\in\mathbb{R}^{N}.$

######  Lemma 2.1.

Under the assumptions $\left(F_{1}\right)$-$\left(F_{3}\right)$ and
$\left(\varphi_{0}\right)$ -$\left(\varphi_{1}\right)$, for any $\lambda>0$,
there hold that

  * (i)

For some $1<p<2^{*}-1$,

$\limsup_{s\rightarrow+\infty}\frac{g_{\lambda}(s)}{s^{p}}<\infty.$

  * (ii)

$g_{\lambda}\left(s\right)=o\left(s\right)$, $s\to 0$.

  * (iii)

There exists $T>0$ such that
$\int_{0}^{T}g_{\lambda}\left(\tau\right)\mathrm{d}\tau>\frac{\lambda}{2}T^{2}$.

  * (iv)

$\lim\limits_{s\rightarrow+\infty}\frac{g_{\lambda}\left(s\right)}{s^{\beta-1}}=\frac{\mu_{2}}{\left(a^{*}\right)^{\beta}}$.

  * (v)

$g_{\lambda}(s)\leq sg_{\lambda}^{\prime}(s)$ for $s>0$ small.

###### Proof.

  * (i)

It is easy to check $(i)$ holds for $p=\beta-1$ by the assumption $(F_{2})$.

  * (ii)

Since

$g_{\lambda}\left(s\right)=\frac{f\left(\phi^{-1}\left(s\right)\right)}{\varphi\left(\phi^{-1}\left(s\right)\right)}-\lambda\frac{\phi^{-1}\left(s\right)}{\varphi\left(\phi^{-1}\left(s\right)\right)}+\lambda
s,$

then

$\displaystyle\lim_{s\to 0}\frac{g_{\lambda}\left(s\right)}{s}$
$\displaystyle=\lim_{s\to
0}\frac{f\left(\phi^{-1}\left(s\right)\right)}{s\varphi\left(\phi^{-1}\left(s\right)\right)}-\lambda\lim_{s\to
0}\frac{\phi^{-1}\left(s\right)}{s\varphi\left(\phi^{-1}\left(s\right)\right)}+\lambda$
$\displaystyle=\lim_{s\to 0}\frac{f\left(s\right)}{s}=0.$

  * (iii)

Since

$\displaystyle\lim_{t\to+\infty}\frac{\int_{0}^{t}g_{\lambda}\left(\tau\right)\mathrm{d}\tau}{t^{2}}$
$\displaystyle=\lim_{t\to+\infty}\frac{g_{\lambda}\left(t\right)}{2t}$
$\displaystyle=\frac{1}{2}\left[\lim_{t\to+\infty}\frac{f\left(\phi^{-1}\left(t\right)\right)}{t\varphi\left(\phi^{-1}\left(t\right)\right)}-\lambda\lim_{t\to+\infty}\frac{\phi^{-1}\left(t\right)}{t\varphi\left(\phi^{-1}\left(t\right)\right)}+\lambda\right].$

Take $s=\phi^{-1}\left(t\right)$, then we get

$t=\phi\left(s\right)=\int_{0}^{s}\varphi\left(\tau\right)\mathrm{d}\tau$

and

$\lambda\lim_{t\to+\infty}\frac{\phi^{-1}\left(t\right)}{t\varphi\left(\phi^{-1}\left(t\right)\right)}=\lambda\lim_{s\to+\infty}\frac{s}{\phi\left(s\right)\varphi\left(s\right)},$

Noting that $\lim\limits_{s\to+\infty}\varphi\left(s\right)=a^{\ast}$,

$\displaystyle\lambda\lim\limits_{s\to+\infty}\frac{s}{\phi\left(s\right)\varphi\left(s\right)}$
$\displaystyle=\frac{\lambda}{a^{\ast}}\lim\limits_{s\to+\infty}\frac{s}{\int_{0}^{s}\varphi\left(\tau\right)\mathrm{d}\tau}$
$\displaystyle=\frac{\lambda}{a^{\ast}}\lim\limits_{s\to+\infty}\frac{1}{\varphi\left(s\right)}$
$\displaystyle=\frac{\lambda}{(a^{\ast})^{2}}.$

Meanwhile, it follows from $(F_{2})$ that

$\displaystyle\lim_{t\to+\infty}\frac{f\left(\phi^{-1}\left(t\right)\right)}{t\varphi\left(\phi^{-1}\left(t\right)\right)}$
$\displaystyle=\lim_{s\to+\infty}\frac{f\left(s\right)}{\phi\left(s\right)\varphi\left(s\right)}=\frac{1}{a^{\ast}}\lim_{s\to+\infty}\frac{f\left(s\right)}{\phi\left(s\right)}$
$\displaystyle=\frac{1}{(a^{\ast})^{2}}\lim_{s\to+\infty}\frac{f\left(s\right)}{s}=+\infty.$

Thus we obtain that

$\lim_{t\to+\infty}\frac{\int_{0}^{t}g_{\lambda}\left(\tau\right)\mathrm{d}\tau}{t^{2}}=\infty,$

which implies that there exists $T>0$ such that
$\int_{0}^{T}g_{\lambda}\left(\tau\right)\mathrm{d}\tau>\frac{\lambda}{2}T^{2}$.

  * (iv)

Set $p=\beta-1$ and notice that $\phi(s)\geq s$ for any $s>0$, we have

$\displaystyle\lim\limits_{s\rightarrow+\infty}\frac{g_{\lambda}(s)}{s^{p}}$
$\displaystyle=\lim_{s\rightarrow+\infty}\frac{f\left(\phi^{-1}(s)\right.)}{s^{p}\varphi\left(\phi^{-1}(s)\right.)}+\lim_{s\rightarrow+\infty}\frac{\lambda
s-\lambda\frac{\phi^{-1}(s)}{\varphi\left(\phi^{-1}\left(s\right)\right)}}{s^{p}}$
$\displaystyle=\lim\limits_{t\rightarrow+\infty}\frac{f(t)}{(\phi(t))^{p}\varphi(t)}-\lambda\lim_{t\rightarrow+\infty}\frac{t}{(\phi(t))^{p}\varphi(t)}$
$\displaystyle=\lim\limits_{t\rightarrow+\infty}\frac{f(t)}{(\phi(t))^{p}\varphi(t)}=\lim_{t\rightarrow+\infty}\frac{f(t)}{\left(ta^{*}\right)^{p}a^{*}}$
$\displaystyle=\lim\limits_{t\rightarrow+\infty}\frac{f(t)}{t^{p}}\cdot\frac{1}{\left(a^{*}\right)^{p+1}}=\frac{\mu_{2}}{\left(a^{*}\right)^{p+1}}.$

  * (v)

It suffices to show that
$\varphi(t)g_{\lambda}(\phi(t))\leq\phi(t)\frac{\rm{d}}{\rm{d}t}[g_{\lambda}(\phi(t))]$
for $t>0$ small. Obviously,

$\displaystyle\frac{\rm{d}}{\rm{d}t}[g_{\lambda}(\phi(t))]$
$\displaystyle=\frac{f^{\prime}(t)\varphi(t)-f(t)\varphi^{\prime}(t)}{\varphi^{2}(t)}+\frac{\lambda\varphi^{3}(t)-\lambda\varphi(t)+\lambda
t\varphi^{\prime}(t)}{\varphi^{2}(t)}$

By $(\varphi_{0})$, for any $t\geq 0$, $\phi(t)\leq t\varphi(t)$ and then

$\phi(t)\varphi^{-2}(t)\left[\varphi^{3}(t)-\varphi(t)+t\varphi^{\prime}(t)\right]\geq\phi(t)\varphi(t)-t.$

Meanwhile, by $(F_{2})$,

$\lim_{t\rightarrow
0^{+}}\frac{\phi(t)\varphi(t)f^{\prime}(t)}{f(t)}=\alpha-1>1,$

which implies that for $t>0$ small,

$\phi(t)\left[f^{\prime}(t)\varphi(t)-f(t)\varphi^{\prime}(t)\right]\geq\varphi^{2}(t)f(t).$

Thus, it follows that

$\phi(t)\frac{\rm{d}}{\rm{d}t}[g_{\lambda}(\phi(t))]\geq f(t)-\lambda
t+\lambda\phi(t)\varphi(t)=\varphi(t)g_{\lambda}(\phi(t)),\,t>0\,\,\mbox{small}.$

∎

By Lemma 2.1, for any fixed $\lambda>0$, $g_{\lambda}$ satisfies the
Berestycki-Lions conditions. As a consequence of [2, Theorem 1], for any
$\lambda>0$, problem (2.6) admits a ground state solution in
$H^{1}(\mathbb{R}^{N})$, which is positive and radially symmetric. Denoting

$\displaystyle\mathcal{S}=\Biggl{\\{}$
$\displaystyle\left(\lambda,\,v_{\lambda}\right)\in\left(0,\,+\infty\right)\times
H_{rad}^{1}\left(\mathbb{R}^{N}\right):v_{\lambda}>0,\left(\lambda,\,v_{\lambda}\right)\,\text{solves}$
$\displaystyle-\Delta
v+\lambda\frac{\phi^{-1}\left(v\right)}{\varphi\left(\phi^{-1}\left(v\right)\right)}-\frac{f\left(\phi^{-1}\left(v\right)\right)}{\varphi\left(\phi^{-1}\left(v\right)\right)}=\,\,\mbox{in}\,\,\mathbb{R}^{N}\Biggr{\\}}.$

######  Remark 2.2.

As a consequence of [7, Theorem 2], for any $\lambda>0$, for any positive
$C^{2}$ solution $u$ of problem (2.6) with $u(x)\rightarrow 0$ as
$|x|\rightarrow\infty$, one can show that $u$ is radially symmetric about some
point $x_{0}\in\mathbb{R}^{N}$, that is, $u(x)=u_{0}(|x-x_{0}|)$, where
$\frac{\partial u_{0}}{\partial r}<0$ for $r=|x-x_{0}|>0$. In what follows, we
assume that $x_{0}=0$. Actually, thanks to the elliptic estimate, $u$ and
$|\nabla u|$ decays exponentially at infinity. Set
$g_{\lambda}(t)=h_{1}(t)+h_{2}(t),t\geq 0$, where

$h_{1}(t)=\frac{f\left(\phi^{-1}\left(t\right)\right)}{\varphi\left(\phi^{-1}\left(t\right)\right)},\,\,h_{2}(t)=-\lambda\frac{\phi^{-1}\left(t\right)}{\varphi\left(\phi^{-1}\left(t\right)\right)}+\lambda
t,$

then by $(F_{2})$, $h_{1}(t)=O(|t|^{\alpha-1})$ as $t\rightarrow 0$ and by
$(\varphi_{0})$,

$\displaystyle\lim_{t\rightarrow 0}\frac{h_{2}(t)}{t^{2}}$
$\displaystyle=\lambda\lim_{t\rightarrow 0}\frac{\phi(t)\varphi(t)-t}{t^{2}}$
$\displaystyle=\lambda\lim_{t\rightarrow
0}\frac{\phi(t)[\varphi(t)-1]}{t^{2}}+\lambda\lim_{t\rightarrow
0}\frac{\phi(t)-t}{t^{2}}$
$\displaystyle=\frac{3\lambda}{2}\varphi^{\prime}(0)=0.$

It follows that $g_{\lambda}(t)=O(|t|^{\min\\{\alpha-1,2\\}})$ as
$t\rightarrow 0$. Moreover,

$h_{2}^{\prime}(t)=\lambda\varphi^{-3}(\phi^{-1}(t))\left[\varphi^{3}(\phi^{-1}(t))-\varphi(\phi^{-1}(t))+\phi^{-1}(t)\varphi^{\prime}(\phi^{-1}(t))\right]\geq
0,\,\,t\geq 0,$

which implies that $h_{2}$ is nondecreasing for $t\geq 0$. As for the
condition on $h_{1}$ as follows, for some $C>0,p>1$,

$\left|h_{1}(t_{1})-h_{1}(t_{2})\right|\leqslant
C|t_{1}-t_{2}|/|\log\min(t_{1},t_{2})|^{p},\quad
t_{1},t_{2}\in[0,\max_{x\in\mathbb{R}^{N}}u(x)],$

to prove [7, Theorem 2], it is only used in the proof of [7, Lemma 6.3]. Since
$u$ decays exponentially, one can check that [7, Lemma 6.3] still holds in our
case and as well as [7, Theorem 2].

##  3\. Asymptotic behaviors

In this section, we consider the asymptotic behaviors of $v_{\lambda}$ as
$\lambda=\lambda_{n}\to 0^{+}$ or $\lambda=\lambda_{n}\to+\infty$. Similarly
to [13], we have the following result in the case of $\lambda=\lambda_{n}\to
0^{+}$.

######  Lemma 3.1.

Let $\left\\{v_{n}\right\\}_{n=1}^{\infty}\subset\mathcal{S}$ with
$\lambda=\lambda_{n}\rightarrow 0^{+}$, then the following asserts hold.

  * (i)

$\limsup\limits_{n\to+\infty}\left\|v_{n}\right\|_{\infty}<+\infty.$

  * (ii)

$\liminf_{n\rightarrow+\infty}\frac{\left\|v_{n}\right\|_{\infty}^{\alpha-2}}{\lambda_{n}}>0.$

  * (iii)

$\limsup_{n\rightarrow+\infty}\frac{\left\|v_{n}\right\|_{\infty}^{\alpha-2}}{\lambda_{n}}<+\infty.$

  * (iv)

Set

(3.1)
$w_{n}(x):=\lambda_{n}^{\frac{1}{2-\alpha}}v_{n}\left(\frac{x}{\sqrt{\lambda_{n}}}\right),$

then $w_{n}$ satisfies

$-\Delta
w_{n}+w_{n}=\frac{g_{\lambda_{n}}\left(\lambda_{n}^{\frac{1}{\alpha-2}}w_{n}\right)}{\lambda_{n}^{\frac{\alpha-1}{\alpha-2}}}\text{
in }\mathbb{R}^{N}$

and

(3.2) $w_{n}\rightarrow U\text{ in
}H^{1}\left(\mathbb{R}^{N}\right)\,\mbox{and}\,\,C_{r,0}\left(\mathbb{R}^{N}\right),$

where $U\in C_{r,0}(\mathbb{R}^{N})$ is the unique positive solution of (1.9)

###### Proof.

The proof is similar to [13] and we omit the details. ∎

In the following, we consider the case of
$\lambda=\lambda_{n}\rightarrow+\infty$.

######  Lemma 3.2.

Let $\left\\{v_{n}\right\\}_{n=1}^{\infty}\subset\mathcal{S}$ with
$\lambda=\lambda_{n}\rightarrow+\infty$, then up to a subsequence,

$\liminf_{n\rightarrow+\infty}\left\|v_{n}\right\|_{\infty}=+\infty.$

###### Proof.

By regularity, for any fixed $n$, $v_{n}\in C^{2}\left(\mathbb{R}^{N}\right)$
and we may suppose that $v_{n}\left(0\right)=\left\|v_{n}\right\|_{\infty}$.
Setting

$\bar{{v_{n}}}\left(x\right):=\frac{1}{v_{n}\left(0\right)}v_{n}\left(\frac{x}{\sqrt{\lambda_{n}}}\right),$

we have

(3.3) $\displaystyle 1=\bar{v_{n}}\left(0\right)\leq$
$\displaystyle-\Delta\bar{v_{n}}\left(0\right)+\bar{v_{n}}\left(0\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda_{n}v_{n}\left(0\right)}g_{\lambda_{n}}\left(v_{n}\left(0\right)\right)$
$\displaystyle=$
$\displaystyle\frac{f\left(\phi^{-1}\left(v_{n}\left(0\right)\right)\right)}{\lambda_{n}v_{n}\left(0\right)\varphi\left(\phi^{-1}\left(v_{n}\left(0\right)\right)\right)}-\frac{\phi^{-1}\left(v_{n}\left(0\right)\right)}{v_{n}\left(0\right)\varphi\left(\phi^{-1}\left(v_{n}\left(0\right)\right)\right)}+1$
$\displaystyle\leq$
$\displaystyle\frac{c\left(\left|\phi^{-1}\left(v_{n}\left(0\right)\right)\right|^{\alpha-1}+\left|\phi^{-1}\left(v_{n}\left(0\right)\right)\right|^{\beta-1}\right)}{\lambda_{n}v_{n}\left(0\right)}+1$
$\displaystyle-\frac{\phi^{-1}\left(v_{n}\left(0\right)\right)}{v_{n}\left(0\right)\varphi\left(\phi^{-1}\left(v_{n}\left(0\right)\right)\right)},$

where we used the fact that for some $c>0$, $|f(t)|\leq
c(|t|^{\alpha-1}+|t|^{\beta-1}),\,t\geq 0$.

Let $\phi^{-1}\left(v_{n}\left(0\right)\right)=a_{n}$, then
$v_{n}\left(0\right)=\phi\left(a_{n}\right)=\int_{0}^{a_{n}}\varphi\left(\tau\right)\mathrm{d}\tau$.
Next we argue by contradiction and suppose now that $\\{a_{n}\\}$ is bounded.
Then (3.3) yields that

(3.4) $\displaystyle 1$
$\displaystyle\leq\frac{c\left(a_{n}^{\alpha-1}+a_{n}^{\beta-1}\right)}{\lambda_{n}\phi\left(a_{n}\right)}+1-\frac{a_{n}}{\phi(a_{n})\varphi\left(a_{n}\right)}$
$\displaystyle\leq\frac{c\left(a_{n}^{\alpha-2}+a_{n}^{\beta-2}\right)}{\lambda_{n}}+1-\frac{a_{n}}{\phi(a_{n})\varphi\left(a_{n}\right)}.$

If $a_{n}\to 0$, then $v_{n}\left(0\right)\to 0$. Recall that $v_{n}$
satisfies

(3.5) $-\Delta
v_{n}+\lambda_{n}\frac{\phi^{-1}\left(v_{n}\right)}{\varphi\left(\phi^{-1}\left(v_{n}\right)\right)}=\frac{f\left(\phi^{-1}\left(v_{n}\right)\right)}{\varphi\left(\phi^{-1}\left(v_{n}\right)\right)}\,\,\mbox{in}\,\,\mathbb{R}^{N}.$

Thanks to $(F_{2})$, it follows that

$-\Delta
v_{n}+\frac{\left(\lambda_{n}-1\right)\phi^{-1}\left(v_{n}\right)}{\varphi\left(\phi^{-1}\left(v_{n}\right)\right)}\leq
0\,\,\mbox{in}\,\,\mathbb{R}^{N}.$

Multiplying both sides by $v_{n}$ and integrating in $\mathbb{R}^{N}$, we get

(3.6) $\int_{\mathbb{R}^{N}}\left|\nabla
v_{n}\right|^{2}\mathrm{d}x+\left(\lambda_{n}-1\right)\int_{\mathbb{R}^{N}}\frac{\phi^{-1}\left(v_{n}\right)v_{n}}{\varphi\left(\phi^{-1}\left(v_{n}\right)\right)}\mathrm{d}x\leq
0,$

which is contradiction, due to
$\phi^{-1}\left(v_{n}\left(x\right)\right)v_{n}\left(x\right)>0,x\in\mathbb{R}^{N}$.
So $\\{a_{n}\\}$ is positive and bounded away from zero. Passing to the limit
of (3.4) as $n\rightarrow\infty$, we have

$\limsup_{n\rightarrow\infty}\frac{a_{n}}{\phi(a_{n})\varphi\left(a_{n}\right)}\leq
0,$

which is a contraction. Thus, up to a subsequence,
$v_{n}\left(0\right)\to\infty$. ∎

######  Lemma 3.3.

Let $\left\\{v_{n}\right\\}_{n=1}^{\infty}\subset\mathcal{S}$ with
$\lambda=\lambda_{n}\rightarrow+\infty$, then

(3.7)
$\liminf_{n\rightarrow+\infty}\frac{\left\|v_{n}\right\|_{\infty}^{\beta-2}}{\lambda_{n}}>0.$

###### Proof.

Recall that

$-\Delta
v_{n}=\frac{f\left(\phi^{-1}\left(v_{n}\right)\right)}{\varphi\left(\phi^{-1}\left(v_{n}\right)\right)}-\lambda_{n}\frac{\phi^{-1}\left(v_{n}\right)}{\varphi\left(\phi^{-1}\left(v_{n}\right)\right)}\,\,\mbox{in}\,\,\mathbb{R}^{N}.$

Without loss of generality, we suppose that
$v_{n}\left(0\right)=\left\|v_{n}\right\|_{\infty},\,\forall n\in\mathbb{N}$.
Since $-\Delta v_{n}(0)\geq 0$,

$f\left(\phi^{-1}\left(\left\|v_{n}\right\|_{\infty}\right)\right)\geq\lambda_{n}\phi^{-1}\left(\left\|v_{n}\right\|_{\infty}\right),$

which yields that

$\liminf_{n\rightarrow+\infty}\frac{\left\|v_{n}\right\|_{\infty}^{\beta-{2}}}{\lambda_{n}}>0,$

so $(\ref{o9})$ is proved. ∎

######  Lemma 3.4.

Let $\left\\{v_{n}\right\\}_{n=1}^{\infty}\subset\mathcal{S}$ with
$\lambda=\lambda_{n}\rightarrow+\infty$, then

(3.8)
$\limsup\limits_{n\to+\infty}\frac{\left\|v_{n}\right\|_{\infty}^{\beta-2}}{\lambda_{n}}<+\infty.$

###### Proof.

We prove it by contradiction and suppose

$\lim\limits_{n\to+\infty}\frac{\left\|v_{n}\right\|_{\infty}^{\beta-2}}{\lambda_{n}}=+\infty.$

Set $k=\frac{\beta-2}{2}$ and

$\tilde{{v}_{n}}(x)=\frac{1}{\left\|v_{n}\right\|_{\infty}}v_{n}\left(\frac{x}{\left\|v_{n}\right\|_{\infty}^{k}}\right),$

then $\tilde{{v}_{n}}\left(0\right)=\left\|\tilde{{v}_{n}}\right\|_{\infty}=1$
and

$-\Delta\tilde{{v}_{n}}(x)=-\frac{1}{\left\|v_{n}\right\|_{\infty}^{1+2k}}\Delta
v_{n}\left(\frac{x}{\left\|v_{n}\right\|_{\infty}^{k}}\right),$

then

(3.9)
$-\Delta\tilde{{v}_{n}}(x)=\frac{1}{\left\|v_{n}\right\|_{\infty}^{1+2k}}\left[\frac{f\left(\phi^{-1}\left(\tilde{{v}_{n}}(x)\left\|v_{n}\right\|_{\infty}\right)\right)}{\varphi\left(\phi^{-1}\left(\tilde{{v}_{n}}(x)\left\|v_{n}\right\|_{\infty}\right)\right)}-\lambda_{n}\frac{\phi^{-1}\left(\tilde{{v}_{n}}(x)\left\|v_{n}\right\|_{\infty}\right)}{\varphi\left(\phi^{-1}\left(\tilde{{v}_{n}}(x)\left\|v_{n}\right\|_{\infty}\right)\right)}\right],$

By $(F_{2})$, the right hand side of (3.9) is in
$L^{\infty}\left(\mathbb{R}^{N}\right)$. Up to a subsequence, we may suppose
that $\tilde{v_{n}}\to\tilde{v}$ in $C_{loc}^{2}\left(\mathbb{R}^{N}\right)$
and then $\tilde{v}$ is a non-negative bounded solution to

$-\Delta\tilde{v}=\mu_{2}(a^{\ast})^{-\beta}\tilde{v}^{\beta-1}\text{ in
}\mathbb{R}^{N}.$

As a consequence of [13, Theorem 2.2-(ii)], $\tilde{v}\equiv 0$, which
contradicts $\tilde{v}\left(0\right)=1$. ∎

######  Lemma 3.5.

Let $\left\\{v_{n}\right\\}_{n=1}^{\infty}\subset\mathcal{S}$ with
$\lambda=\lambda_{n}\rightarrow+\infty$. Define

(3.10)
$w_{n}(x):=\lambda_{n}^{\frac{1}{2-\beta}}v_{n}\left(\frac{x}{\sqrt{\lambda_{n}}}\right),$

then, up to a subsequence, $w_{n}\to V^{\ast}$ in
$C_{r,0}\left(\mathbb{R}^{N}\right)$ and $H^{1}\left(\mathbb{R}^{N}\right)$ as
$n\to+\infty$, where $V^{\ast}=a^{\ast}V(\frac{\cdot}{a^{\ast}})$ and $V$ is
the unique positive solution of (1.10).

###### Proof.

By Lemma 3.3 and Lemma 3.4, we have that

$0<\liminf_{n\rightarrow+\infty}\frac{v_{n}\left(0\right)^{\beta-2}}{\lambda_{n}}\leq\limsup\limits_{n\to+\infty}\frac{v_{n}\left(0\right)^{\beta-2}}{\lambda_{n}}<+\infty,$

which implies $\left\\{w_{n}\right\\}$ is uniformly bounded in
$L^{\infty}\left(\mathbb{R}^{N}\right)$. Since $w_{n}$ satisfies

(3.11) $-\Delta
w_{n}+w_{n}=\frac{g_{\lambda_{n}}\left(\lambda_{n}^{\frac{1}{\beta-2}}w_{n}\right)}{\lambda_{n}^{\frac{\beta-1}{\beta-2}}}\text{
in }\mathbb{R}^{N},$

that is,

(3.12) $-\Delta
w_{n}=\lambda_{n}^{\frac{1-\beta}{\beta-2}}\frac{f\left(\phi^{-1}\left(\lambda_{n}^{\frac{1}{\beta^{-2}}}w_{n}\right)\right)}{\varphi\left(\phi^{-1}\left(\lambda_{n}^{\frac{1}{\beta^{-2}}}w_{n}\right)\right)}-\lambda_{n}^{\frac{1}{2-\beta}}\frac{\phi^{-1}\left(\lambda_{n}^{\frac{1}{\beta^{-2}}}w_{n}\right)}{\varphi\left(\phi^{-1}\left(\lambda_{n}^{\frac{1}{\beta-2}}w_{n}\right)\right)}.$

Thanks to $(F_{2})$, one can check that the right hand of (3.12) is in
$L^{\infty}\left(\mathbb{R}^{N}\right)$. By the elliptic regularity, up to a
subsequence, we assume that $w_{n}\rightarrow V^{\ast}$ in
$C_{loc}^{2}\left(\mathbb{R}^{N}\right)$, where $V^{\ast}$ satisfies

$-\Delta
V^{\ast}+(a^{\ast})^{-2}V^{\ast}=\mu_{2}(a^{\ast})^{-\beta}(V^{\ast})^{\beta-1}\leavevmode\nobreak\
\hbox{in}\leavevmode\nobreak\
\mathbb{R}^{N},\,\,V^{\ast}(0)=\max_{x\in\mathbb{R}^{N}}V^{\ast}(x).$

Similarly to [13], the result desired can be obtained. ∎

Next, we focus on the asymptotic behaviour of
$\left\|\phi^{-1}\left(v_{n}\right)\right\|_{2}$ as $n\rightarrow\infty$.

######  Theorem 3.6.

  * (i)

Let $\left\\{v_{n}\right\\}_{n=1}^{\infty}\subset
H^{1}\left(\mathbb{R}^{N}\right)$ be positive solutions to (2.4) with
$\lambda=\lambda_{n}\rightarrow 0^{+}$. Then

$\lim_{n\rightarrow+\infty}\left\|v_{n}\right\|_{\infty}=0,\quad\lim_{n\rightarrow+\infty}\left\|\nabla
v_{n}\right\|_{2}=0$

and

$\lim_{n\rightarrow+\infty}\left\|\phi^{-1}\left(v_{n}\right)\right\|_{2}=\begin{cases}0&\alpha<2+\frac{4}{N},\\\
\|U\|_{2}&\alpha=2+\frac{4}{N},\\\ +\infty&\alpha>2+\frac{4}{N}.\end{cases}$

  * (ii)

Let $\left\\{v_{n}\right\\}_{n=1}^{\infty}\subset
H^{1}\left(\mathbb{R}^{N}\right)$ be positive solutions to (2.4) with
$\lambda=\lambda_{n}\rightarrow+\infty$. Then

$\lim_{n\rightarrow+\infty}\left\|v_{n}\right\|_{\infty}=+\infty,\quad\lim_{n\rightarrow+\infty}\left\|\nabla
v_{n}\right\|_{2}=+\infty$

and

$\lim_{n\rightarrow+\infty}\left\|\phi^{-1}\left(v_{n}\right)\right\|_{2}=\begin{cases}+\infty&\beta<2+\frac{4}{N},\\\
(a^{\ast})^{N/2}\|V\|_{2}&\beta=2+\frac{4}{N},\\\
0&\beta>2+\frac{4}{N}.\end{cases}$

###### Proof.

Similarly to [13], one can show the asymptotic behaviour of
$\|v_{n}\|_{\infty}$ and $\|\nabla v_{n}\|_{2}$. In the following, we only
consider $\left\|\phi^{-1}\left(v_{n}\right)\right\|_{2}^{2}$.

  * (i)

Let $w_{n}$ be defined by (3.1) and by Lemma 3.1, we have

$\displaystyle\left\|\phi^{-1}\left(v_{n}\right)\right\|_{2}^{2}$
$\displaystyle=\int_{\mathbb{R}^{N}}\left|\phi^{-1}\left(v_{n}\left(x\right)\right)\right|^{2}\mathrm{d}x$
$\displaystyle=\lambda_{n}^{-\frac{N}{2}}\int_{\mathbb{R}^{N}}\left|\phi^{-1}\left(v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)\right)\right|^{2}\mathrm{d}y$
$\displaystyle=\lambda_{n}^{-\frac{N}{2}}\int_{\mathbb{R}^{N}}\left(\frac{\phi^{-1}\left(v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)\right)}{v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)}\right)^{2}\left(v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)\right)^{2}\mathrm{d}y$
$\displaystyle=\lambda_{n}^{-\frac{N}{2}}\int_{\mathbb{R}^{N}}\left(\frac{\phi^{-1}\left(v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)\right)}{v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)}\right)^{2}\left(\frac{w_{n}\left(y\right)}{\lambda_{n}^{\frac{1}{2-\alpha}}}\right)^{2}\mathrm{d}y$
$\displaystyle=\lambda_{n}^{-\frac{N}{2}+\frac{2}{\alpha-2}}\int_{\mathbb{R}^{N}}\left(\frac{\phi^{-1}\left(v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)\right)}{v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)}\right)^{2}w_{n}^{2}\left(y\right)\mathrm{d}y$
$\displaystyle=\lambda_{n}^{-\frac{N}{2}+\frac{2}{\alpha-2}}(\left\|U\right\|_{2}^{2}+o_{n}(1)),$

which yields the result desired.

  * (ii)

For the case of $\lambda=\lambda_{n}\to+\infty$, let $w_{n}$ be defined by
(3.10). Since

$\lim_{s\to+\infty}\frac{\phi^{-1}\left(s\right)}{s}=\lim_{t\to+\infty}\frac{t}{\phi\left(t\right)}=\frac{1}{a^{\ast}},$

by Lemma 3.5,

$\displaystyle\left\|\phi^{-1}\left(v_{n}\right)\right\|_{2}^{2}=$
$\displaystyle\lambda_{n}^{-\frac{N}{2}}\int_{\mathbb{R}^{N}}\left(\frac{\phi^{-1}\left(v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)\right)}{v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)}\right)^{2}\left(v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)\right)^{2}\mathrm{d}y$
$\displaystyle=\lambda_{n}^{-\frac{N}{2}+\frac{2}{\beta-2}}\int_{\mathbb{R}^{N}}\left(\frac{\phi^{-1}\left(v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)\right)}{v_{n}\left(\frac{y}{\sqrt{\lambda_{n}}}\right)}\right)^{2}w_{n}^{2}\left(y\right)\mathrm{d}y$
$\displaystyle=\lambda_{n}^{-\frac{N}{2}+\frac{2}{\beta-2}}((a^{\ast})^{-2}\left\|V^{\ast}\right\|_{2}^{2}+o_{n}(1)),$

which yields the result desired.

∎

##  4\. Uniqueness for $\lambda$ small or large

In this section, we prove the uniqueness of positive solutions to (2.4)
provided $\lambda>0$ small or large enough.

######  Theorem 4.1.

Let $N\geqslant 3$ and assume that
$\left(\varphi_{0}\right)$-$\left(\varphi_{1}\right)$,
$\left(F_{1}\right)$-$\left(F_{2}\right)$ hold. Then (2.4) has at most one
positive solution if $\lambda>0$ large and in addition
$\left(\varphi_{2}\right)$ holds or if $\lambda>0$ small and in addition
$\left(F_{3}\right)$ holds.

###### Proof.

We adopt the idea in [13] to give the proof.

If $\lambda>0$ small and $\left(F_{1}\right)$-$\left(F_{3}\right)$ hold,
assume that problem (2.4) admits two families of positive solutions
$u_{\lambda}^{(1)}$ and $u_{\lambda}^{(2)}$ with $\lambda\rightarrow 0^{+}$.
Let

$v_{\lambda}^{(i)}(\cdot):=\lambda^{-\frac{1}{\alpha-2}}u_{\lambda}^{(i)}\left(\cdot/\sqrt{\lambda}\right),\quad
i=1,2.$

Then by Lemma 3.1, $v_{\lambda}^{(1)},v_{\lambda}^{(2)}\in
H_{rad}^{1}(\mathbb{R}^{N})$ satisfy

$-\Delta
v+v=\frac{g_{\lambda}(\lambda^{\frac{1}{\alpha-2}}v)}{\lambda^{\frac{\alpha-1}{\alpha-2}}}\,\mbox{in}\,\,\mathbb{R}^{N}$

and as $\lambda\rightarrow 0^{+}$, $v_{\lambda}^{(i)}\rightarrow U\;\hbox{
both in}\;C_{r,0}(\mathbb{R}^{N})\leavevmode\nobreak\ \hbox{and
in}\leavevmode\nobreak\ H^{1}(\mathbb{R}^{N}),\;i=1,2.$ Setting

$\xi_{\lambda}:=\frac{v_{\lambda}^{(1)}-v_{\lambda}^{(2)}}{\left\|v_{\lambda}^{(1)}-v_{\lambda}^{(2)}\right\|_{\infty}},$

for any $x\in\mathbb{R}^{N}$, there exists some $\theta(x)\in[0,1]$ such that

$-\Delta\xi_{\lambda}+\xi_{\lambda}=\lambda^{-1}g_{\lambda}^{\prime}\left(\lambda^{\frac{1}{\alpha-2}}[\theta(x)v_{\lambda}^{(1)}(x)+(1-\theta(x))v_{\lambda}^{(2)}(x)]\right)\xi_{\lambda}\,\,\,\mbox{in}\,\,\mathbb{R}^{N},$

where for any $t\in\mathbb{R}$,

$\displaystyle\lambda^{-1}g_{\lambda}^{\prime}(t)=$
$\displaystyle\frac{f^{\prime}(\phi^{-1}(t))\varphi(\phi^{-1}(t))-f(\phi^{-1}(t))\varphi^{\prime}(\phi^{-1}(t))}{\lambda[\varphi(\phi^{-1}(t))]^{3}}$
$\displaystyle-\frac{\varphi(\phi^{-1}(t))-\phi^{-1}(t)\varphi^{\prime}(\phi^{-1}(t))}{[\varphi(\phi^{-1}(t))]^{3}}+1.$

Thanks to $v_{\lambda}^{(i)}\rightarrow U$ in $C_{r,0}(\mathbb{R}^{N})$, one
can see that

$\lim_{\lambda\rightarrow
0^{+}}\lambda^{\frac{1}{\alpha-2}}[\theta(x)v_{\lambda}^{(1)}(x)+(1-\theta(x))v_{\lambda}^{(2)}(x)]=0\,\,\,\mbox{uniformly
for}\,\,x\in\mathbb{R}^{N},$

which implies that for any $x\in\mathbb{R}^{N}$,

$\lim_{\lambda\rightarrow
0^{+}}\lambda^{-1}g_{\lambda}^{\prime}\left(\lambda^{\frac{1}{\alpha-2}}[\theta(x)v_{\lambda}^{(1)}(x)+(1-\theta(x))v_{\lambda}^{(2)}(x)]\right)=\mu_{1}(\alpha-1)U^{\alpha-2}.$

Hence, up to a subsequence, $\xi_{\lambda}\rightarrow\xi$ in
$C_{loc}^{2}(\mathbb{R}^{N})$ and $\xi$ is a radial bounded solution of

$-\Delta\xi+\xi=(\alpha-1)\mu_{1}U^{\alpha-2}\xi.$

Similarly to [13], we can get one contradiction. Now, we consider the case of
$\lambda>0$ large and assume $\left(\varphi_{2}\right)$ holds. Assume problem
(2.4) admits two families of positive solutions $u_{\lambda}^{(1)}$ and
$u_{\lambda}^{(2)}$ with $\lambda\rightarrow+\infty$. Let

$v_{\lambda}^{(i)}(\cdot):=\lambda^{-\frac{1}{\beta-2}}u_{\lambda}^{(i)}\left(\cdot/\sqrt{\lambda}\right),\quad
i=1,2,$

and

$\xi_{\lambda}:=\frac{v_{\lambda}^{(1)}-v_{\lambda}^{(2)}}{\left\|v_{\lambda}^{(1)}-v_{\lambda}^{(2)}\right\|_{\infty}}.$

Similarly as above, there exists some $\theta(x)\in[0,1]$ such that

$-\Delta\xi_{\lambda}+\xi_{\lambda}=\lambda^{-1}g^{\prime}\left(\lambda^{\frac{1}{\beta-2}}[\theta(x)v_{\lambda}^{(1)}(x)+(1-\theta(x))v_{\lambda}^{(2)}(x)]\right)\xi_{\lambda}\leavevmode\nobreak\
\hbox{in}\leavevmode\nobreak\ \mathbb{R}^{N}.$

By Lemma 3.5, $v_{\lambda}^{(i)}\rightarrow V$ in $C_{r,0}(\mathbb{R}^{N})$ as
$\lambda\rightarrow+\infty$ for $i=1,2$. It is easy to know that, for any
$x\in\mathbb{R}^{N}$,

$\lim_{\lambda\rightarrow+\infty}\lambda^{\frac{1}{\beta-2}}[\theta(x)v_{\lambda}^{(1)}(x)+(1-\theta(x))v_{\lambda}^{(2)}(x)]=+\infty.$

By $\left(\varphi_{2}\right)$, for any $x\in\mathbb{R}^{N}$,

$\lim_{\lambda\rightarrow+\infty}\left.\frac{\varphi(\phi^{-1}(t))-\phi^{-1}(t)\varphi^{\prime}(\phi^{-1}(t))}{[\varphi(\phi^{-1}(t))]^{3}}\right|_{t=\lambda^{\frac{1}{\beta-2}}[\theta(x)v_{\lambda}^{(1)}(x)+(1-\theta(x))v_{\lambda}^{(2)}(x)]}=(a^{\ast})^{-2}$

and

$\lim_{\lambda\rightarrow+\infty}\left.\frac{f^{\prime}(\phi^{-1}(t))\varphi(\phi^{-1}(t))-f(\phi^{-1}(t))\varphi^{\prime}(\phi^{-1}(t))}{\lambda[\varphi(\phi^{-1}(t))]^{3}}\right|_{t=\lambda^{\frac{1}{\beta-2}}[\theta(x)v_{\lambda}^{(1)}(x)+(1-\theta(x))v_{\lambda}^{(2)}(x)]}=\mu_{2}(a^{\ast})^{-\beta}V^{\beta-2}.$

Hence, up to a subsequence, $\xi_{\lambda}\rightarrow\xi$ in
$C_{loc}^{2}(\mathbb{R}^{N})$ and $\xi$ is a radial bounded solution of

$-\Delta\xi+(a^{\ast})^{-2}\xi=\mu_{2}(a^{\ast})^{-\beta}V^{\beta-2}\xi.$

Similarly, we get a contradiction. ∎

##  5\. Proof of Theorem 1.3

We give the proof of Theorem 1.3. Being similar to that of [13], we only
sketch it.

$\bullet$ Step 1. For any $\lambda>0$, problem (2.4) admits a positive and
radially symmetric solution $u_{\lambda}$. By Theorem 4.1, there exists
$\lambda_{0}>0$ such that, up to translation, $u_{\lambda}$ is the unique
positive solution of problem (2.4) and then is of mountain-pass-type (see [13,
Definition 2.1]). Furthermore, the map $\lambda\mapsto
u_{\lambda},\lambda\in(0,\lambda_{0})$ is continuous. That is,
$\left\\{(\lambda,u_{\lambda}):\lambda\in(0,\lambda_{0})\right\\}$ is a curve
in $\mathbb{R}\times H_{rad}^{1}(\mathbb{R}^{N})$. $\bullet$ Step 2. Let
$\widetilde{\mathcal{S}}\subset\mathcal{S}$ be the connected component of
$\mathcal{S}$ containing the curve
$\left\\{(\lambda,u_{\lambda}):\lambda\in(0,\lambda_{0})\right\\}$, where
$\mathcal{S}$ is given in Section 2. Denoting by $P_{1}:(0,+\infty)\times
H_{\text{rad }}^{1}\left(\mathbb{R}^{N}\right)\rightarrow(0,+\infty)$ the
projection onto the $\lambda$-component, we show that
$P_{1}(\widetilde{\mathcal{S}})=(0,+\infty)$.

For any $v\in H_{\text{rad }}^{1}(\mathbb{R}^{N})$, define

$\mathbb{T}_{\lambda}(v)=:(-\Delta+\lambda)^{-1}g_{\lambda}(v).$

Then $\mathbb{T}_{\lambda}:H_{rad}^{1}(\mathbb{R}^{N})\rightarrow
H_{rad}^{1}(\mathbb{R}^{N})$ is completely continuous and $u$ is a radial
solution of problem (2.4) if and only if $u$ is a fixed point of
$\mathbb{T}_{\lambda}$ in $H_{\text{rad }}^{1}(\mathbb{R}^{N})$. For
$\lambda\in(0,\lambda_{0})$, thanks to Lemma 2.1-$(v)$, similarly to [13,
Lemma 7.2], one can get the local fixed point index

${\rm ind}(\mathbb{T}_{\lambda},u_{\lambda})={\rm deg}_{LS}\left({\rm
Id}-\mathbb{T}_{\lambda},N_{\varepsilon}(u_{\lambda}),0\right)=-1,$

where $\varepsilon>0$ is small, $N_{\varepsilon}$ denotes the
$\varepsilon$-neighborhood in $H^{1}_{rad}(\mathbb{R}^{N})$.

$\bullet$ Step 3. For any fixed $0<a<b$, define

$\displaystyle\mathcal{S}(a,b)=\Biggl{\\{}$
$\displaystyle\left(\lambda,\,v_{\lambda}\right)\in\left[a,\,b\right]\times
H_{rad}^{1}\left(\mathbb{R}^{N}\right):v_{\lambda}>0,\left(\lambda,\,v_{\lambda}\right)\,\text{solves}$
$\displaystyle-\Delta
v+\lambda\frac{\phi^{-1}\left(v\right)}{\varphi\left(\phi^{-1}\left(v\right)\right)}-\frac{f\left(\phi^{-1}\left(v\right)\right)}{\varphi\left(\phi^{-1}\left(v\right)\right)}=\,\,\mbox{in}\,\,\mathbb{R}^{N}\Biggr{\\}}.$

By using the similar blow-up technique and ODE approach, one can show that the
set $\mathcal{S}(a,b)$ is compact in $C_{r,0}(\mathbb{R}^{N})$ and
$H_{rad}^{1}(\mathbb{R}^{N})$. Via the topological degree theory, thanks to
the compactness of $\mathcal{S}(a,b)$ and ${\rm
ind}(\mathbb{T}_{\lambda},u_{\lambda})=-1$ for $\lambda\in(0,\lambda_{0})$,
there holds that $P_{1}(\widetilde{\mathcal{S}})=(0,\infty)$.

$\bullet$ Step 4. Similarly to [13], define

$\tilde{\rho}:\tilde{\mathcal{S}}\rightarrow\mathbb{R}^{+},\quad(\lambda,v)\mapsto\|\phi^{-1}\left(v\right)\|_{2}^{2}.$

We just consider the case $(i)$. By
$P_{1}(\widetilde{\mathcal{S}})=(0,+\infty)$ and Theorem 3.6, if
$2<\alpha<2+\frac{4}{N}$, then there exists
$\left(\lambda_{n},\,v_{n}\right)\subset\widetilde{\mathcal{S}}$ with
$\lambda_{n}\rightarrow 0^{+}$ and
$\left\|\phi^{-1}\left(v_{n}\right)\right\|_{2}^{2}\rightarrow 0$. Similarly,
if $2<\beta<2+\frac{4}{N}$, there exists
$\left(\lambda_{n}^{\prime},\,v_{n}^{\prime}\right)\subset\widetilde{\mathcal{S}}$
with $\lambda_{n}^{\prime}\rightarrow+\infty$ and
$\|\phi^{-1}\left(v_{n}^{\prime}\right)\|_{2}^{2}\rightarrow+\infty$. Since
$\widetilde{\mathcal{S}}$ is connected, for any given $c>0$, there exists
$(\lambda_{c},u_{c})\in\widetilde{\mathcal{S}}$ such that
$\tilde{\rho}(u_{c})=c$, that is, (2.4) possesses a positive normalized
solution. Since the proof of the other cases can be done similarly as in [13],
we omit the details.

Conflict of interest. The authors have no competing interests to declare that
are relevant to the content of this article

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# Outer forms of type $A_{2}$ with infinite genus

Sergey V. Tikhonov  Belarusian State University, Nezavisimosti Ave., 4,
220030, Minsk, Belarus<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be an absolutely almost simple algebraic group over a field $K$. The
genus ${\bf gen}_{K}(G)$ of $G$ is the set of $K$-isomorphism classes of
$K$-forms $G^{\prime}$ of $G$ that have the same $K$-isomorphism classes of
maximal $K$-tori as $G$. We construct an example of outer forms of type
$A_{2}$ with infinite genus.

Let $K$ be a field and $K^{sep}$ its separable closure. Two absolutely almost
simple algebraic $K$-groups $G_{1}$ and $G_{2}$ are said to have the same
$K$-isomorphism classes of maximal $K$-tori if every maximal $K$-torus of
$G_{1}$ is $K$-isomorphic to some maximal $K$-torus of $G_{2}$, and vice
versa. An algebraic $K$-group $G^{\prime}$ is called a $K$-form of an
algebraic $K$-group $G$ if $G$ and $G^{\prime}$ are isomorphic over $K^{sep}$.

######  Definition 1.

[3, Def. 6.1] Let $G$ be an absolutely almost simple algebraic group over a
field $K$. The genus ${\bf gen}_{K}(G)$ of $G$ is the set of $K$-isomorphism
classes of $K$-forms $G^{\prime}$ of $G$ that have the same $K$-isomorphism
classes of maximal $K$-tori as $G$.

The genus is trivial in some special cases and it is conjectured to be finite
whenever the field $K$ is finitely generated of ”good” characteristic (see
details in [7, §8]).

In a similar way one can define the genus of a division algebra.

######  Definition 2.

The genus ${\bf gen}({\mathcal{D}})$ of a finite-dimensional central division
algebra ${\mathcal{D}}$ over a field $K$ is defined as the set of classes
$[{\mathcal{D}}^{\prime}]\in\text{\rm Br}(K)$, where ${\mathcal{D}}^{\prime}$
is a central division $K$-algebra having the same maximal subfields as
${\mathcal{D}}$.

If ${\mathcal{D}}$ is a finite-dimensional central division $K$-algebra, then
it is well-known that any maximal $K$-torus of the corresponding algebraic
group $G={\rm SL}_{1,{\mathcal{D}}}$ is of the form $\text{\rm
R}_{E/K}(\mathbb{G}_{m})\cap G$ (where $\text{\rm R}_{E/K}(\mathbb{G}_{m})$ is
the Weil restriction of the 1-dimensional split torus $\mathbb{G}_{m}$) for
some maximal separable subfield $E$ of ${\mathcal{D}}$. Thus the results on
genus of division algebras from [5] and [10] rephrased in the language of
algebraic groups say that for any prime $p$, there exist fields (with infinite
transcendence degree over the prime subfield) over which there are inner forms
of type $A_{p-1}$ with infinite genus. An example of groups of type $G_{2}$
with infinite genus is obtained in [2, Rem. 3.6(b)]. In the present paper, we
construct such an example for outer forms of type $A_{2}$.

Let $F/K$ be a quadratic separable field extension and $\sigma$ the non-
trivial $K$-automorphism of $F$. An involution on an $F$-algebra
${\mathcal{R}}$ is called an $F/K$-involution if its restriction to $F$ is
$\sigma$. An isomorphism of $F$-algebras with involution
$f:({\mathcal{R}},\tau)\longrightarrow({\mathcal{R}}^{\prime},\tau^{\prime})$
is an $F$-algebra isomorphism
$f:{\mathcal{R}}\longrightarrow{\mathcal{R}}^{\prime}$ such that
$\tau^{\prime}\circ f=f\circ\tau$. Let also $E/F$ be a field extension such
that $E$ has an automorphism $\psi$ of order 2 such that $\psi|F=\sigma$
(i.e., $E$ has an $F/K$-involution). Then $\tau_{E}$ denotes the involution on
${\mathcal{R}}\otimes_{F}E$ defined by the formula ${\tau_{E}}(r\otimes
e):=\tau(r)\otimes\psi(e)$, where $r\in{\mathcal{R}}$, $e\in E$. In
particular, if $L/F$ is a field extension linearly disjoint to $E$ over $F$
with an automorphism $\phi$ of order two extending $\sigma$, then $\phi_{E}$
is the automorphism of order two of the field $EL=L\otimes_{F}E$ which extends
$\phi$.

Let ${\mathcal{A}}$ be a central division $F$-algebra of degree $n$ with an
$F/K$-involution $\tau$. Let $L/F$ be a separable field extension of degree
$n$, and let $\phi:L\longrightarrow L$ be an automorphism of order two such
that $\phi|F=\sigma$. An embedding of algebras with involution
$(L,\phi)\hookrightarrow({\mathcal{A}},\tau)$ is by definition an injective
$F$-homomorphism $f:L\longrightarrow{\mathcal{A}}$ such that $\tau\circ
f=f\circ\phi$. It is known that embeddings of maximal tori into the special
unitary group ${\rm SU}({\mathcal{A}},\tau)$ can be described in terms of
embeddings of fields with involution into the central simple algebra with
involution $({\mathcal{A}},\tau)$ ([6, Prop. 2.3]).

In this paper, we construct a field $E$ and a subfield $T\subset E$ such that
$[E:T]=2$ and there is an infinite set of (pairwise non-isomorphic) division
$E$-algebras of degree 3 with $E/T$-involution such that a field extension
$L/E$ of degree 3 can be embedded as an algebra with involution into one
algebra of this set if and only if it can be embedded as an algebra with
involution into all other algebras from this set. Passing to the corresponding
special unitary groups, we obtain an example of outer forms of type $A_{2}$
with infinite genus.

Below we use the following notation: $Alg_{3}(F/K)$ is the set of isomorphism
classes of central division $F$-algebras of degree $3$ with $F/K$ involution;
$Ext_{3}(F/K)$ is the set of isomorphism classes of field extensions of $F$ of
degree $3$ with $F/K$-involution. The $3$-torsion of the Brauer group
$\text{\rm Br}(F)$ is denoted by ${}_{3}\text{\rm Br}(F)$. For a field
extension $E/F$ and a central simple $F$-algebra ${\mathcal{A}}$,
${\mathcal{A}}_{E}$ denotes the tensor product ${\mathcal{A}}\otimes_{F}E$ and
$\text{\rm res}_{E/F}:\text{\rm Br}(F)\longrightarrow\text{\rm Br}(E)$ denotes
the restriction homomorphism. The restriction of $\text{\rm res}_{E/F}$ to the
subgroup ${}_{3}\text{\rm Br}(F)$ will also be denoted by $\text{\rm
res}_{E/F}$. For a central simple $F$-algebra ${\mathcal{A}}$,
${\mathcal{A}}^{op}$ denotes the opposite algebra and ${\mathcal{A}}^{m}$
denotes ${\mathcal{A}}\otimes_{F}\dots\otimes_{F}{\mathcal{A}}$ ($m$ times).
For a quadratic form $q$ over $K$ and a field extension $E/K$, $q_{E}$ denotes
the quadratic form obtained by extension of scalars from $K$ to $E$. Recall
that a field extension $E/F$ is called regular if $E/F$ is separable and $F$
is algebraically closed in $E$.

We start with the following

######  Lemma 3.

Let $n$ be a positive integer, $F$ a field of characteristic not dividing
$2n$, $F/K$ a quadratic field extension, $\sigma$ the non-trivial
$K$-automorphism of $F$, ${\mathcal{A}}$ a central simple $F$-algebra of
degree $n$, and $L/F$ a cyclic field extension of degree $n$. Then there
exists a regular field extension $M/F$ and a subfield $T\subset M$ such that
$[M:T]=2$ and

(1) $M=TF$ and the non-trivial $T$-automorphism of $M$ extends $\sigma$;

(2) the composite $ML$ splits ${\mathcal{A}}_{M}$;

(3) the homomorphism $\text{\rm res}_{M/F}:\text{\rm
Br}(F)\longrightarrow\text{\rm Br}(M)$ is injective.

Proof. Let $F(x)$ be a purely transcendental extension of $F$ of transcendence
degree 1. Let also $\phi$ be a generator of the Galois group $\text{\rm
Gal}(L(x)/F(x))$ and

${\mathcal{C}}:={\mathcal{A}}_{F(x)}^{op}\otimes_{F(x)}(L(x)/F(x),\phi,x),$

where $(L(x)/F(x),\phi,x)$ is a cyclic $F(x)$-algebra of degree $n$. Let also
$E$ be the function field of the Severi-Brauer variety of ${\mathcal{C}}$.
Note that the kernel of the restriction homomorphism $\text{\rm
res}_{E/F(x)}:\text{\rm Br}(F(x))\longrightarrow\text{\rm Br}(E)$ is generated
by $[{\mathcal{C}}]$ (see, e.g., [9, Cor. 13.16]).

Let ${\mathcal{B}}$ be a central simple $F$-algebra of exponent bigger than 1.
Assume that ${\mathcal{B}}$ is split by $E$, then
$[{\mathcal{B}}_{F(x)}]=[{\mathcal{C}}^{i}]$ for some $1\leq i\leq n$. If
$i<n$, then the $F(x)$-algebra ${\mathcal{C}}^{i}$ ramifies at the discrete
valuation (trivial on $F$) of $F(x)$ defined by the polynomial $x$, but
${\mathcal{B}}_{F(x)}$ is unramified at this valuation, hence
$[{\mathcal{B}}_{F(x)}]\neq[{\mathcal{C}}^{i}]$. Since the exponent of
${\mathcal{B}}_{F(x)}$ is bigger than 1, then
$[{\mathcal{B}}_{F(x)}]\neq[{\mathcal{C}}^{n}]=[F(x)]$. Thus
${\mathcal{B}}_{F(x)}$ is not split by $E$, i.e., the homomorphism $\text{\rm
res}_{E/F}:{\text{\rm Br}(F)}\longrightarrow{\text{\rm Br}(E)}$ is injective.

Since $E$ splits ${\mathcal{C}}$, then

$[{\mathcal{A}}_{E}]=[(L(x)/F(x),\phi,x)_{E}]=[(EL/E,\phi^{\prime},x)],$

where $\phi^{\prime}$ is the generator of $\text{\rm Gal}(EL/E)$. Thus $EL$
splits ${\mathcal{A}}_{E}$.

Note that $E/F$ is a regular extension of $F(x)$. For the following
construction of the transfer of a regular field extension, we refer to [8, p.
220]).

Let $\sigma$ also denotes the $K(x)$-automorphism of $F(x)$ extending the
automorphism $\sigma$ of $F$. The automorphism $\sigma$ of $F(x)$ can be
extended to an isomorphism (which we also denote by $\sigma$) of $E$ and
another regular extension of $F(x)$ denoted by $E_{\sigma}$. Thus the
following diagram commutes:

$None$
$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{F(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\sigma}\hbox
to0.0pt{\ .\hss}}$

Let $M=EE_{\sigma}$ be the free composite over $F$ of $E$ and $E_{\sigma}$.
This free composite is $F$-isomorphic to the function field of the Severi-
Brauer varity of the $E_{\sigma}(y)$-algebra
${\mathcal{A}}_{E_{\sigma}(y)}^{op}\otimes_{E_{\sigma}(y)}(E_{\sigma}L(y)/E_{\sigma}(y),\psi^{\prime},y)$,
where $y$ is transcendental over $E_{\sigma}$ (we replace $x$ by $y$ since the
composite is free) and $\psi^{\prime}$ is the generator of the Galois group
$\text{\rm Gal}(E_{\sigma}L(y)/E_{\sigma}(y))$. The field $M$ is a regular
extension of $F$. The isomorphisms $\sigma:E\longrightarrow E_{\sigma}$ and
$\sigma^{-1}:E_{\sigma}\longrightarrow E$ have a unique extension to an
automorphism $\bar{\sigma}$ of $M$ of order two. Let $T:=T_{F/K}(E)$ be the
transfer of $E$ with respect to the ground field descent $F\supset K$, i.e.,
$T$ is the subfield of $M$ of elements fixed under the action of
$\bar{\sigma}$. Note that the composite $TF$ coincides with $M$, $[M:T]=2$,
and $\bar{\sigma}$ extends $\sigma$.

The algebra ${\mathcal{A}}_{M}$ is split by $ML$ since ${\mathcal{A}}_{E}$ is
split by $EL$.

Finally, the diagram (1) induces the following commutative diagram for the
corresponding Brauer groups:

$\textstyle{\text{\rm
Br}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$res$\scriptstyle{\cong}$$\textstyle{\text{\rm
Br}(F(x))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$res$\scriptstyle{\cong}$$\textstyle{\text{\rm
Br}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\text{\rm
Br}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$res$\textstyle{\text{\rm
Br}(F(x))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$res$\textstyle{\text{\rm
Br}(E_{\sigma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$res$\textstyle{\text{\rm
Br}(M)\hbox to0.0pt{\ .\hss}}$

The injectivity of $\text{\rm res}_{E/F}$ implies the injectivity of
$\text{\rm res}_{E_{\sigma}/F}$. Moreover, $\text{\rm res}_{M/E_{\sigma}}$ is
injective by the same arguments as for $\text{\rm res}_{E/F}$, we just replace
the ground field $F$ by $E_{\sigma}$. Hence the homomorphism $\text{\rm
res}_{M/F}$ is also injective.

∎

We also need the following

######  Lemma 4.

Let $F$ be a field of characteristic $\neq 2,3$; $F/K$ a quadratic field
extension, $\sigma$ the non-trivial $K$-automorphism of $F$, and $L/F$ a field
extension of degree $3$ with an automorphism $\phi$ of order two such that
$\phi|F=\sigma$. Then there exists a field extension $F(L)/F$ and a subfield
$K(L)\subset F(L)$ such that $[F(L):K(L)]=2$ and

(1) $F(L)=K(L)F$ and the non-trivial $K(L)$-automorphism of $F(L)$, denoted by
$\sigma_{F(L)}$, extends $\sigma$;

(2) $[F(L):F]\leq 2$;

(3) the composite $F(L)L$ is a cyclic extension of $F(L)$ of degree 3;

(4) the homomorphism $\text{\rm res}_{F(L)/F}:{{}_{3}\text{\rm
Br}(F)}\longrightarrow{{}_{3}\text{\rm Br}(F(L))}$ is injective.

Proof. If the extension $L/F$ is cyclic, then one can take $F(L):=F$,
$K(L):=K$, and $\sigma_{F(L)}:=\sigma$.

Assume that the extension $L/F$ is not cyclic. Let $N$ be the normal closure
of the extension $L^{\phi}/K$, where $L^{\phi}\subset L$ is the subfield of
elements fixed by $\phi$. Then $F\not\subset N$ and $NF$ is the normal closure
of the extension $L/F$. Thus we have the following diagram of field
extensions:

$\textstyle{N}$$\textstyle{L}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
L^{\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2}$$\scriptstyle{2}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{3}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{3}$$\scriptstyle{2}$

Let $H$ be the Sylow 3-subgroup of the Galois group $\text{\rm Gal}(N/K)$.
Then $N^{H}$, the fixed field of $H$, is an extension of $K$ of degree 2 and
$N/N^{H}$ is a cyclic extension of degree 3. Hence $NF$ is a cyclic extension
of $N^{H}F$ of degree 3 and $[N^{H}F:F]=2$. Let $K(L):=N^{H}$ and
$F(L):=K(L)F$. Since $F\not\subset N$, then $[F(L):K(L)]=2$ and the field
$F(L)$ has a $K(L)$-automorphism of order two extending $\sigma$. Note that
$F(L)L=NF$, hence $F(L)L/F(L)$ is a cyclic extension of degree 3. Finally,
since $[F(L):F]=2$, then the homomorphism $\text{\rm
res}_{F(L)/F}:{{}_{3}\text{\rm Br}(F)}\longrightarrow{{}_{3}\text{\rm
Br}(F(L))}$ is injective. ∎

######  Remark 5.

In the notations of Lemma 4, for any field extension $L^{\prime}$ of $F$ of
degree 3, $F(L)$ and $L^{\prime}$ are linearly disjoint over $F$. Moreover, if
$L^{\prime}$ has an automorphism $\phi^{\prime}$ of order 2 extending
$\sigma$, then the composite $F(L)L^{\prime}$ has the automorphism
$\phi^{\prime}_{F(L)}$ which extends the automorphisms $\phi^{\prime}$ and
$\sigma_{F(L)}$.

######  Lemma 6.

Let $F$ be a field of characteristic $\neq 2,3$; $F/K$ a quadratic field
extension, $\sigma$ the non-trivial $K$-automorphism of $F$, ${\mathcal{A}}$ a
central simple $F$-algebra of degree $3$ with an $F/K$-involution $\tau$, and
$L/F$ a field extension of degree $3$ with an automorphism $\phi$ of order two
such that $\phi|F=\sigma$. Then there exists a field extension
$F_{(K,L,{\mathcal{A}})}/F$ and a subfield $K_{(L,{\mathcal{A}})}\subset
F_{(K,L,{\mathcal{A}})}$ such that
$[F_{(K,L,{\mathcal{A}})}:K_{(L,{\mathcal{A}})}]=2$ and

(1) $F_{(K,L,{\mathcal{A}})}=K_{(L,{\mathcal{A}})}F$ and the non-trivial
$K_{(L,{\mathcal{A}})}$-automorphism of $F_{(K,L,{\mathcal{A}})}$, denoted by
$\sigma_{F_{(K,L,{\mathcal{A}})}}$, extends $\sigma$;

(2) the homomorphism $\text{\rm
res}_{F_{(K,L,{\mathcal{A}})}/F}:{{}_{3}\text{\rm
Br}(F)}\longrightarrow{{}_{3}\text{\rm Br}(F_{(K,L,{\mathcal{A}})})}$ is
injective;

(3) for any field extension $L^{\prime}$ of $F$ of degree 3,
$F_{(K,L,{\mathcal{A}})}$ and $L^{\prime}$ are linearly disjoint over $F$;

(4) there is an embedding
$(F_{(K,L,{\mathcal{A}})}L,\phi_{F_{(K,L,{\mathcal{A}})}})\hookrightarrow({\mathcal{A}}_{F_{(K,L,{\mathcal{A}})}},\tau_{{F_{(K,L,{\mathcal{A}})}}})$
of algebras with involution.

Proof. Let $F(L)$, $K(L)$ and $\sigma_{F(L)}$ be as in Lemma 4. Let also $M$
and $T$ be fields obtained by applying Lemma 3 for the quadratic field
extension $F(L)/K(L)$, the $F(L)$-algebra ${\mathcal{A}}_{F(L)}$, the cyclic
field extension $F(L)L/F(L)$ of degree $3$. Then by Lemmas 3 and 4, the
homomorphism $\text{\rm res}_{M/F}:{{}_{3}\text{\rm
Br}(F)}\longrightarrow{{}_{3}\text{\rm Br}(M)}$ is injective; for any field
extension $L^{\prime}$ of $F$ of degree 3, $M$ and $L^{\prime}$ are linearly
disjoint over $F$ and the composite $ML$ splits ${\mathcal{A}}_{M}$. Thus
there is an $M$-embedding $\varepsilon:ML\hookrightarrow{\mathcal{A}}_{M}$ of
$M$-algebras.

Note that ${\mathcal{A}}_{M}$ has the $M/T$-involution $\tau_{M}$ which
extends $\tau$ and $ML$ has the automorphism $\phi_{M}$ of order two extending
$\phi$. By [6, Proposition 3.1], there exists an $M/T$-involution $\delta$ on
${\mathcal{A}}_{M}$ such that
$\varepsilon:(ML,\phi_{M})\hookrightarrow({\mathcal{A}}_{M},\delta)$ is an
embedding of algebras with involution.

Let $\pi(\delta)$ and $\pi(\tau_{M})$ be the 3-fold Pfister forms of
involutions $\delta$ and $\tau_{M}$ respectively (see [4, §19.B]). Let
$T(\pi(\delta))$ and $T(\pi(\tau_{M}))$ be the function fields of
$\pi(\delta)$ and $\pi(\tau_{M})$ respectively. Then the quadratic forms
$\pi(\delta)_{T(\pi(\delta))}$ and $\pi(\tau_{M})_{T(\pi(\tau_{M}))}$ are
isotropic and hence hyperbolic since they are Pfister forms.

Let $K_{(L,{\mathcal{A}})}$ be the free composite over $T$ of the fields
$T(\pi(\delta))$ and $T(\pi(\tau_{M}))$. Let also
$F_{(K,L,{\mathcal{A}})}:=K_{(L,{\mathcal{A}})}F$. Since $F\not\subset
K_{(L,{\mathcal{A}})}$, then
$[F_{(K,L,{\mathcal{A}})}:K_{(L,{\mathcal{A}})}]=2$ and
$F_{(K,L,{\mathcal{A}})}$ has a $K_{(L,{\mathcal{A}})}$-automorphism
$\sigma_{F_{(K,L,{\mathcal{A}})}}$ of order 2 extending $\sigma$. Note that
the algebraic closure of $F$ in $F_{(K,L,{\mathcal{A}})}$ is $F(L)$ and
$[F(L):F]$ is either 1 or 2. Therefore, $F_{(K,L,{\mathcal{A}})}$ and
$L^{\prime}$ are linearly disjoint over $F$ for any field extension
$L^{\prime}$ of $F$ of degree 3.

The extensions $T(\pi(\delta))/T$ and $T(\pi(\tau_{M}))/T$ are composition of
a purely transcendental extension with a quadratic extension, hence the
homomorphism $\text{\rm res}_{F_{(K,L,{\mathcal{A}})}/M}:{{}_{3}\text{\rm
Br}(M)}\longrightarrow{{}_{3}\text{\rm Br}(F_{(K,L,{\mathcal{A}})})}$ is
injective. Hence $\text{\rm res}_{F_{(K,L,{\mathcal{A}})}/F}:{{}_{3}\text{\rm
Br}(F)}\longrightarrow{{}_{3}\text{\rm Br}(F_{(K,L,{\mathcal{A}})})}$ is also
injective.

The quadratic forms $\pi(\delta)_{K_{(L,{\mathcal{A}})}}$ and
$\pi(\tau_{M})_{K_{(L,{\mathcal{A}})}}$ are hyperbolic. Then by [4, Theorem
19.6]), the involutions $\delta_{F_{(K,L,{\mathcal{A}})}}$ and
$\tau_{F_{(K,L,{\mathcal{A}})}}$ on ${\mathcal{A}}_{F_{(K,L,{\mathcal{A}})}}$
are conjugate. This means that there is an isomorphism
$\xi:({\mathcal{A}}_{F_{(K,L,{\mathcal{A}})}},\delta_{F_{(K,L,{\mathcal{A}})}})\longrightarrow({\mathcal{A}}_{F_{(K,L,{\mathcal{A}})}},\tau_{{F_{(K,L,{\mathcal{A}})}}})$
of algebras with involution.

Moreover, the embedding
$\varepsilon:(ML,\phi_{M})\hookrightarrow({\mathcal{A}}_{M},\delta)$ of
algebras with involution induces an embedding
$(F_{(K,L,{\mathcal{A}})}L,\phi_{F_{(K,L,{\mathcal{A}})}})\hookrightarrow({\mathcal{A}}_{F_{(K,L,{\mathcal{A}})}},\delta_{F_{(K,L,{\mathcal{A}})}})$
of algebras with involution. Indeed,
$F_{(K,L,{\mathcal{A}})}L=ML\otimes_{M}F_{(K,L,{\mathcal{A}})}$. Let

$\varepsilon_{F_{(K,L,{\mathcal{A}})}}:ML\otimes_{M}F_{(K,L,{\mathcal{A}})}\longrightarrow{\mathcal{A}}_{F_{(K,L,{\mathcal{A}})}}$

be an $F_{(K,L,{\mathcal{A}})}$-embedding defined by the formula
$\varepsilon_{F_{(K,L,{\mathcal{A}})}}(m\otimes a):=\varepsilon(m)\otimes a$,
where $m\in ML$, $a\in F_{(K,L,{\mathcal{A}})}$. Then

$\varepsilon_{F_{(K,L,{\mathcal{A}})}}(\phi_{F_{(K,L,{\mathcal{A}})}}(m\otimes
a))=\varepsilon_{F_{(K,L,{\mathcal{A}})}}(\phi_{M}(m)\otimes\sigma_{F_{(K,L,{\mathcal{A}})}}(a))=\varepsilon(\phi_{M}(m))\otimes\sigma_{F_{(K,L,{\mathcal{A}})}}(a)=$
$\delta(\varepsilon(m))\otimes\sigma_{F_{(K,L,{\mathcal{A}})}}(a)=\delta_{F_{(K,L,{\mathcal{A}})}}(\varepsilon(m)\otimes
a)=\delta_{F_{(K,L,{\mathcal{A}})}}(\varepsilon_{F_{(K,L,{\mathcal{A}})}}(m\otimes
a)).$

Thus $\varepsilon_{F_{(K,L,{\mathcal{A}})}}$ is an embedding of algebras with
involutions. Then $\xi\circ\varepsilon_{F_{(K,L,{\mathcal{A}})}}$ is an
embedding
$(F_{(K,L,{\mathcal{A}})}L,\phi_{F_{(K,L,{\mathcal{A}})}})\hookrightarrow({\mathcal{A}}_{F_{(K,L,{\mathcal{A}})}},\tau_{{F_{(K,L,{\mathcal{A}})}}})$
of algebras with involution.

∎

The following construction of the field $F_{(K,S,A)}$ is an adaptation of the
construction from [10] for algebras with involutions. We give the details
below for the reader’s convenience.

######  Proposition 7.

Let $F$ be a field of characteristic $\neq 2,3$; $F/K$ a quadratic field
extension, $\sigma$ the non-trivial $K$-automorphism of $F$, $A\subset
Alg_{3}(F/K)$ and $S\subset Ext_{3}(F/K)$. Then there exists a field extension
$F_{(K,S,A)}/F$ and a subfield $K_{(S,{\mathcal{A}})}\subset F_{(K,S,A)}$ such
that $[F_{(K,S,A)}:K_{(S,A)}]=2$ and

(1) $F_{(K,S,A)}=K_{(S,A)}F$ and the non-trivial $K_{(S,A)}$-automorphism,
denoted by $\sigma_{F_{(K,S,A)}}$, of $F_{(K,S,A)}$ extends $\sigma$;

(2) the homomorphism $\text{\rm res}_{F_{(K,S,A)}/F}:{{}_{3}\text{\rm
Br}(F)}\longrightarrow{{}_{3}\text{\rm Br}(F_{(K,S,A)})}$ is injective;

(3) for any field extension $L^{\prime}$ of $F$ of degree 3,
$F_{(K,L,{\mathcal{A}})}$ and $L^{\prime}$ are linearly disjoint over $F$;

(4) for any ${\mathcal{A}}\in A$ with an $F/K$-involution $\tau$ and $L\in S$
with a $K$-automorphism $\phi$ of order 2 extending $\sigma$, there is an
embedding
$(F_{(K,S,A)}L,\phi_{F_{(K,S,A)}})\hookrightarrow({\mathcal{A}}_{F_{(K,S,A)}},\tau_{{F_{(K,S,A)}}})$
of algebras with involution.

Proof. Let ${\mathcal{P}}:=\\{(L,{\mathcal{D}})|L\in S\mbox{ and
}{\mathcal{D}}\in A\\}$ be the set of pairs. Let also $<$ be a well-ordering
on $\mathcal{P}$ and let $t_{0}=(L_{0},{\mathcal{D}}_{0})$ denote its least
element. Set $E_{t_{0}}:=F_{(K,L_{0},{\mathcal{D}}_{0})}$ and
$T_{0}:=K_{(L_{0},{\mathcal{D}}_{0})}$, where the fields
$F_{(K,L_{0},{\mathcal{D}}_{0})}$ and $K_{(L_{0},{\mathcal{D}}_{0})}$ are
constructed in Lemma 6. For $t=(L,{\mathcal{D}})\in{\mathcal{P}}$, set

$E^{<t}:=\bigcup_{t^{\prime}<t}E_{t^{\prime}},T^{<t}:=\bigcup_{t^{\prime}<t}T_{t^{\prime}},T_{t}:={T^{<t}}_{(E^{<t}L,{\mathcal{D}}_{E^{<t}})},\mbox{
and }E_{t}:={E^{<t}}_{(T^{<t},E^{<t}L,{\mathcal{D}}_{E^{<t}})},$

where the fields $E_{t}$ and $T_{t}$ are obtained by applying Lemma 6 to the
quadratic field extension $E^{<t}/T^{<t}$, the field extension
$E^{<t}L/E^{<t}$ of degree 3, the automorphism $\phi_{E^{<t}}$ of $E^{<t}L$
extending the automorphism $\phi$ of $L$ and the $E^{<t}$-algebra
${\mathcal{D}}_{E^{<t}}$. We also define
$F_{(K,S,A)}:=\bigcup_{t\in\mathcal{P}}E_{t}$ and
$K_{(S,A)}:=\bigcup_{t\in\mathcal{P}}T_{t}$.

By Lemma 6, $E_{t}=T_{t}F$ and $[E_{t}:T_{t}]=2$ for any $t\in\mathcal{P}$.
Then $F_{(K,S,A)}=K_{(S,A)}F$, $[F_{(K,S,A)}:K_{(S,A)}]=2$ and the non-trivial
$K_{(S,A)}$-automorphism of $F_{(K,S,A)}$ extends $\sigma$.

By Lemma 6 and transfinite induction, the homomorphism $\text{\rm
res}_{F_{(K,S,A)}/F}:{{}_{3}\text{\rm Br}(F)}\longrightarrow{{}_{3}\text{\rm
Br}(F_{(K,S,A)})}$ is injective and for any field extension $L^{\prime}$ of
$F$ of degree 3, $F_{(K,S,A)}$ and $L^{\prime}$ are linearly disjoint over
$F$.

Finally, let ${\mathcal{A}}\in A$ with an $F/K$-involution $\tau$, $L\in S$
with an automorphism $\phi$ of order 2 extending $\sigma$ and
$t=(L,{\mathcal{A}})$. By Lemma 6, there is an embedding
$(E_{t}L,\phi_{E_{t}})\hookrightarrow({\mathcal{A}}_{E_{t}},\tau_{E_{t}})$ of
algebras with involution. Moreover, as in the proof of Lemma 6, this embedding
induces the embedding

$(F_{(K,S,A)}L,\phi_{F_{(K,S,A)}})\hookrightarrow({\mathcal{A}}_{F_{(K,S,A)}},\tau_{{F_{(K,S,A)}}})$

of algebras with involution. ∎

######  Theorem 8.

Let $F$ be a field of characteristic $\neq 2,3$, $F/K$ a quadratic field
extension, $\sigma$ the non-trivial $K$-automorphism of $F$, $A\subset
Alg_{3}(F/K)$. Then there exists a field extension $F_{A}/F$ and a subfield
$K_{A}\subset F_{A}$ such that $[F_{A}:K_{A}]=2$ and

(1) $F_{A}=K_{A}F$ and the non-trivial $K_{A}$-automorphism, denoted by
$\sigma_{F_{A}}$, of $F_{A}$ extends $\sigma$;

(2) the homomorphism $\text{\rm res}_{F_{A}/F}:{{}_{3}\text{\rm
Br}(F)}\longrightarrow{{}_{3}\text{\rm Br}(F_{A})}$ is injective;

(3) for any central simple $F$-algebra ${\mathcal{B}}$ of degree $3$ with an
$F/K$-involution $\theta$, the algebra ${\mathcal{B}}_{F_{A}}$ has an
$F_{A}/K_{A}$-involution $\theta_{F_{A}}$ extending $\theta$;

(4) if $L\in Ext_{3}(F_{A}/K_{A})$ with a $K_{A}$-automorphism $\phi$ of order
2 extending $\sigma_{F_{A}}$, then there is an embedding
$(L,\phi)\hookrightarrow({\mathcal{A}}_{F_{A}},\tau_{F_{A}})$ of algebras with
involution for any ${\mathcal{A}}\in A$ with an $F/K$-involution $\tau$.

Proof. Let $F_{0}:=F$ and $K_{0}:=K$. We recursively define $F_{i}$ and
$K_{i}$, $i\in{\mathbb{Z}}_{>0}$, to be the fields
${F_{i-1}}_{(K_{i-1},Ext_{3}(F_{i-1}/K_{i-1}),\text{\rm res}_{F_{i-1}/F}(A))}$
and ${K_{i-1}}_{(Ext_{3}(F_{i-1}/K_{i-1}),\text{\rm res}_{F_{i-1}/F}(A))}$
constructed by applying Proposition 7 to the quadratic field extension
$F_{i-1}/K_{i-1}$, the set $\text{\rm res}_{F_{i-1}/F}(A)\subset
Alg_{3}(F_{i-1}/K_{i-1})$ and the set $Ext_{3}(F_{i-1}/K_{i-1})$.

Let $F_{A}:=\bigcup_{i\geq 0}F_{i}$ and $K_{A}:=\bigcup_{i\geq 0}K_{i}$. Hence
$F_{A}=K_{A}F$ and the non-trivial $K_{A}$-automorphism $\sigma_{F_{A}}$ of
$F_{A}$ extends $\sigma$. Therefore, for any central simple $F$-algebra
${\mathcal{B}}$ of degree $3$ with an $F/K$-involution $\theta$, the
$F_{A}/K_{A}$-involution $\theta_{F_{A}}$ extends $\theta$.

By induction and Proposition 7, $\text{\rm res}_{F_{A}/F}:{{}_{3}\text{\rm
Br}(F)}\longrightarrow{{}_{3}\text{\rm Br}(F_{A})}$ is injective.

Assume that ${\mathcal{A}}\in A$ with an $F/K$-involution $\tau$ and $L\in
Ext_{3}(F_{A}/K_{A})$ with an automorphism $\phi$ of order two extending
$\sigma_{F_{A}}$. Then there exists $i\geq 0$ and a field extension
$L^{\prime}$ of $F_{i}$ of degree 3 such that $L=F_{A}L^{\prime}$ and
$\phi_{i}:=\phi_{|L^{\prime}}$ is a $K_{i}$-automorphism of order two. This
means that $L^{\prime}\in Ext_{3}(F_{i}/K_{i})$. By Proposition 7, there is an
embedding

$(F_{i+1}L^{\prime},{\phi_{i}}_{F_{i+1}})\hookrightarrow({\mathcal{A}}_{{F_{i+1}}},\tau_{F_{i+1}})$

of algebras with involution. As in the proof of Lemma 6, this embedding can be
extended to an embedding
$(L,\phi)\hookrightarrow({\mathcal{A}}_{F_{A}},\tau_{F_{A}})$ of algebras with
involution.

∎

As a corollary to Theorem 8, we obtain the following

######  Corollary 9.

There exists a field $E$ and a subfield $T\subset E$ with $[E:T]=2$ such that
there is an infinite set $B$ of pairwise non-isomorphic division $E$-algebras
of degree 3 with $E/T$-involution and such that for any field extension $L/E$
of degree 3 with an automorphism $\phi$ of order 2 extending the non-trivial
$T$-automorphism of $E$, there is an embedding
$(L,\phi)\hookrightarrow({\mathcal{A}},\tau)$ of algebras with involution for
any ${\mathcal{A}}\in B$ with an $E/T$-involution $\tau$.

Proof. Let $\xi_{3}$ be a primitive 3th root of unity,
$K=\mathbb{Q}(\xi_{3})(x,y,z)$, the purely transcendental extension of the
field $\mathbb{Q}(\xi_{3})$ and $F=K(\sqrt{2})$. Then for $i>0$, the symbol
$F$-algebras $(z,\frac{x+\sqrt{2}y^{i}}{x-\sqrt{2}y^{i}})_{3}$ of degree 3 are
pairwise non-isomorphic (since they have different ramification) and have
$F/K$-involutions (since the corestriction to $K$ of these algebras is
trivial). Now we apply Theorem 8 for the infinite set $A\subset Alg_{3}(F/K)$
consisting of these algebras and set $E:=F_{A}$, $T:=K_{A}$, $B:=\text{\rm
res}_{F_{A}/F}(A)$.

∎

Rephrasing the previous corollary in the language of algebraic groups, we
obtain the following

######  Corollary 10.

There exists a field $T$ such that there are infinitely many (pairwise non-
isomorphic) outer forms of type $A_{2}$ over $T$ having the same infinite
genus.

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