# Generalized Chern-Simons-Schrödinger system with critical exponential
growth: the zero-mass case

Liejun Shen  and  Marco Squassina  Liejun Shen,
Department of Mathematics,
Zhejiang Normal University,
Jinhua, Zhejiang, 321004, People’s Republic of China<EMAIL_ADDRESS>Marco Squassina,
Dipartimento di Matematica e Fisica
Università Cattolica del Sacro Cuore,
Via della Garzetta 48, 25133, Brescia, Italy<EMAIL_ADDRESS>
###### Abstract.

We consider the existence of ground state solutions for a class of zero-mass
Chern-Simons-Schrödinger systems

$\left\\{\begin{array}[]{ll}\displaystyle-\Delta
u+A_{0}u+\sum\limits_{j=1}^{2}A_{j}^{2}u=f(u)-a(x)|u|^{p-2}u,\\\
\displaystyle\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2},\leavevmode\nobreak\
\partial_{1}A_{1}+\partial_{2}A_{2}=0,\\\
\displaystyle\partial_{1}A_{0}=A_{2}|u|^{2},\leavevmode\nobreak\
\partial_{2}A_{0}=-A_{1}|u|^{2},\\\ \end{array}\right.$

where $a:\mathbb{R}^{2}\to\mathbb{R}^{+}$ is an external potential,
$p\in(1,2)$ and $f\in\mathcal{C}(\mathbb{R})$ denotes the nonlinearity that
fulfills the critical exponential growth in the Trudinger-Moser sense at
infinity. By introducing an improvement of the version of Trudinger-Moser
inequality approached in [33], we are able to investigate the existence of
positive ground state solutions for the given system using variational method.

###### Key words and phrases:

Zero-mass, Chern-Simons-Schrödinger system, Trudinger-Moser inequality,
Critical exponential growth, Ground state solution, Variational method.

###### 2010 Mathematics Subject Classification:

35J20, 58E50, 35B06.

L.J. Shen is partially supported by NSFC (12201565). M. Squassina is member of
Gruppo Nazionale per l’Analisi Matematica, la Probabilita e le loro
Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

##  1\. Introduction and main results

In this article, we focus on establishing the existence of positive ground
state solutions for the following generalized Chern-Simons-Schrödinger (CSS in
short) system/equation with critical exponential growth

(1.1) $\left\\{\begin{array}[]{ll}\displaystyle-\Delta
u+A_{0}u+\sum\limits_{j=1}^{2}A_{j}^{2}u=f(u)-a(x)|u|^{p-2}u,\\\
\displaystyle\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2},\leavevmode\nobreak\
\partial_{1}A_{1}+\partial_{2}A_{2}=0,\\\
\displaystyle\partial_{1}A_{0}=A_{2}|u|^{2},\leavevmode\nobreak\
\partial_{2}A_{0}=-A_{1}|u|^{2},\end{array}\right.$

where $a:\mathbb{R}^{2}\to\mathbb{R}^{+}$ is an external potential,
$p\in(1,2)$ and $f\in\mathcal{C}(\mathbb{R})$ denotes the nonlinearity that
fulfills the critical exponential growth in the Trudinger-Moser sense at
infinity which would be specified later.

Recently, great attention has been paid to the time-dependent CSS system in
two spatial dimension

(1.2)
$\begin{cases}\text{i}D_{0}\psi+(D_{1}D_{1}+D_{2}D_{2})\psi+g(x,|\psi|^{2})\psi=0,\\\
\partial_{0}A_{1}-\partial_{1}A_{0}=-\text{Im}(\overline{\psi}D_{2}\psi),\\\
\partial_{0}A_{2}-\partial_{2}A_{0}=\text{Im}(\overline{\psi}D_{1}\psi),\\\
\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|\psi|^{2},\\\ \end{cases}$

where i stands for the imaginary unit, $\partial_{0}=\frac{\partial}{\partial
t}$, $\partial_{1}=\frac{\partial}{\partial x_{1}}$,
$\partial_{2}=\frac{\partial}{\partial x_{2}}$ for
$(t,x_{1},x_{2})\in\mathbb{R}^{1+2}$, $\psi:\mathbb{R}^{1+2}\to\mathbb{C}$
acts as the complex scalar field, $A_{j}:\mathbb{R}^{1+2}\to\mathbb{R}$
denotes the gauge field, $D_{j}=\partial_{j}+\text{i}A_{j}$ is the covariant
derivative for $j=0,1,2$ and $g$ is the nonlinearity. In real world, it is
usually exploited to describe the non-relativistic dynamics behavior of
massive number of particles in Chern-Simons gauge fields. This model plays an
important role in the study of high-temperature superconductors, Aharovnov-
Bohm scattering, and quantum Hall effect, we refer the reader to [16, 17, 18].
Moreover, there exist some further physical motivations for considering CSS
system (1.2), see [12, 14, 25, 26] for example.

For all $(t,x_{1},x_{2})\in\mathbb{R}^{1+2}$ and $j=0,1,2$, one usually
considers the situation $A_{j}(t,x)=A_{j}(x)$. If the standing wave ansatz
$\psi(t,x)=e^{\text{i}\lambda t}u(x)$ with a given $\lambda\in\mathbb{R}$ for
$u:\mathbb{R}^{2}\to\mathbb{R}$, then (1.2) reduces to

(1.3) $\left\\{\begin{array}[]{ll}\displaystyle-\Delta u+\lambda
u+A_{0}u+\sum_{j=1}^{2}A_{j}^{2}u=f(x,u),\\\
\displaystyle\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2},\\\
\displaystyle\partial_{1}A_{0}=A_{2}|u|^{2},\leavevmode\nobreak\
\partial_{2}A_{0}=-A_{1}|u|^{2},\end{array}\right.$

where $f(x,u)=g(x,|u|^{2})u$. Suppose $A_{j}$ satisfies the Coulomb gauge
condition $\sum_{j=0}^{2}\partial_{j}A_{j}=0$, then (1.3) with $\lambda\equiv
0$ becomes the original CSS equation (1.1), namely

(1.4) $\left\\{\begin{array}[]{ll}\displaystyle-\Delta
u+A_{0}u+\sum_{j=1}^{2}A_{j}^{2}u=f(x,u),\\\
\displaystyle\partial_{1}A_{0}=A_{2}|u|^{2},\leavevmode\nobreak\
\partial_{2}A_{0}=-A_{1}|u|^{2},\\\
\displaystyle\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2},\leavevmode\nobreak\
\partial_{1}A_{1}+\partial_{2}A_{2}=0.\end{array}\right.$

It follows from $\partial_{1}A_{0}=A_{2}|u|^{2}$ and
$\partial_{2}A_{0}=-A_{1}|u|^{2}$ in (1.4) that

$\Delta
A_{0}=\partial_{1}\big{(}A_{2}|u|^{2}\big{)}-\partial_{2}\big{(}A_{1}|u|^{2}\big{)},$

leading to

(1.5)
$A_{0}[u](x)=\frac{x_{1}}{2\pi|x|^{2}}\ast\big{(}A_{2}|u|^{2}\big{)}-\frac{x_{2}}{2\pi|x|^{2}}\ast\big{(}A_{1}|u|^{2}\big{)}.$

In a similar way, we depend on
$\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2}$ and
$\partial_{1}A_{1}+\partial_{2}A_{2}=0$ in (1.4) to derive

$\Delta
A_{1}=\partial_{2}\bigg{(}\frac{|u|^{2}}{2}\bigg{)}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \Delta
A_{2}=-\partial_{1}\bigg{(}\frac{|u|^{2}}{2}\bigg{)}.$

From which, the components $A_{j}$ for $j=1,2$ in (1.4) can be represented as

(1.6) $\left\\{\begin{array}[]{ll}\displaystyle
A_{1}[u](x)=\frac{x_{2}}{2\pi|x|^{2}}\ast\bigg{(}\frac{|u|^{2}}{2}\bigg{)}=-\frac{1}{4\pi}\int_{\mathbb{R}^{2}}\frac{(x_{2}-y_{2})u^{2}(y)}{|x-y|^{2}}dy,\\\
\displaystyle
A_{2}[u](x)=-\frac{x_{1}}{2\pi|x|^{2}}\ast\bigg{(}\frac{|u|^{2}}{2}\bigg{)}=\frac{1}{4\pi}\int_{\mathbb{R}^{2}}\frac{(x_{1}-y_{1})u^{2}(y)}{|x-y|^{2}}dy.\end{array}\right.$

In the sequel, we shall write $A_{j}$ in place of $A_{j}[u]$ for
$j\in\\{0,1,2\\}$ for simplicity as long as there is no misunderstanding.
There are some further properties of $A_{j}$ for $j\in\\{0,1,2\\}$ in Section
2 below.

Indeed, CSS system (1.2) can reduce to a single equation if one studies the
standing wave ansatz $\psi(t,x)=e^{\text{i}\lambda t}u(x)$ with a radially
symmetric $u$. Actually, Byeon-Huh-Seok [6] considered the standing waves of
type

(1.7) $\begin{gathered}\psi(t,x)=u(|x|)e^{\text{i}\lambda
t},\leavevmode\nobreak\ A_{0}(t,x)=k(|x|),\hfill\\\
A_{1}(t,x)=\frac{x_{2}}{|x|^{2}}h(|x|),\leavevmode\nobreak\
A_{2}(t,x)=-\frac{x_{1}}{|x|^{2}}h(|x|),\hfill\\\ \end{gathered}$

where $k$ and $h$ are real value functions depending only on $|x|$. Note that
(1.7) satisfies the Coulomb gauge condition with $\varsigma=ct+n\pi$, where
$n$ is an integer and $c$ is a real constant. To seek for solutions of CSS
system (1.2) of the type (1.7), it is enough to handle the following
semilinear elliptic equation

(1.8) $-\Delta u+\lambda
u+\bigg{(}\int_{|x|}^{\infty}\frac{h(s)}{s}u^{2}(s)ds+\frac{h^{2}(|x|)}{|x|^{2}}\bigg{)}u=f(x,u)\leavevmode\nobreak\
\text{in}\leavevmode\nobreak\ \mathbb{R}^{2},$

where $h(s)=\int_{0}^{s}\frac{r}{2}u^{2}(r)dr$. As before, we continue to
assume that $\lambda\equiv 0$ in Eq. (1.8).

At this stage, there are two kinds of CSS equations, (1.4) and (1.8), which
could be called by the so called zero-mass ones. Generally, when
$f(x,t)=\bar{f}(x,t)-V(x)t$ for all $(x,t)\in\mathbb{R}^{2}\times\mathbb{R}$
in the classic CSS equations, more and more interesting results have been
explored by many mathematicians over the past decades for various assumptions
on $\bar{f}$ and $V$. Speaking precisely, for $\bar{f}(x,t)=|t|^{p-2}t$ and
$V\equiv 1$, by exploiting the Nehari-Pohoz̆aev manifold argument, Byeon _et
al._ [6] derived the existence of positive solutions for all $p>6$.
Particularly, with the prescribed mass constraint
$\int_{\mathbb{R}^{2}}|u|^{2}dx=c^{2}$, they showed some existence results for
each $c\neq 0$ if $p\in(2,3]$ and sufficiently small $|c|$ if $p\in(3,4)$.
Afterwards, the existence, nonexistence and multiplicity of nontrivial
solutions for (1.3), or (1.8), have been considerably contemplated by a lot of
mathematicians, see [4, 8, 20, 27, 23, 29, 32, 34, 35, 30, 37, 11, 19] and the
references therein for example even if these references are far to be
exhaustive.

Next, we should turn to consider the so-called zero-mass CSS equation. Very
recently, Zhang, Tang and Chen [40] handled the following zero-mass CSS
equation

(1.9) $-\Delta
u+\bigg{(}\int_{|x|}^{\infty}\frac{h(s)}{s}u^{2}(s)ds+\frac{h^{2}(|x|)}{|x|^{2}}\bigg{)}u=f(u)-\bar{a}|u|^{p-2}u\leavevmode\nobreak\
\text{in}\leavevmode\nobreak\ \mathbb{R}^{2},$

where $\bar{a}>0$ is a constant, $p\in(3,4)$ and the nonlinearity $f$ admits
the critical exponential growth in the Trudinger-Moser sense at infinity. In
fact, we say that a function $f$ possesses the _critical exponential growth_
at infinity if there exists a constant $\alpha_{0}>0$ such that

(1.10) $\lim_{t\rightarrow+\infty}\frac{|f(t)|}{e^{\alpha
t^{2}}}=\begin{cases}0,&\forall\alpha>\alpha_{0},\\\
+\infty,&\forall\alpha<\alpha_{0}.\end{cases}$

The above definition was introduced by Adimurthi and Yadava in [1], see also
de Figueiredo, Miyagaki and Ruf [9] for example.

In [40], the authors depended on the work space below

$E\triangleq\bigg{\\{}u:u(x)\leavevmode\nobreak\ \text{is Lebesgue measurable
s.t.}\int_{\mathbb{R}^{2}}|\nabla u|^{2}dx<+\infty\leavevmode\nobreak\
\text{and}\int_{\mathbb{R}^{2}}|u|^{p}dx<+\infty\bigg{\\}}$

which is the completion of $C_{0}^{\infty}(\mathbb{R}^{2})$ under the norm

$\|u\|=\sqrt{|\nabla u|_{2}^{2}+|u|_{p}^{2}},\leavevmode\nobreak\ \forall u\in
E,$

where $|\cdot|_{q}$ denotes the usual norm corresponding to the Lebesgue space
$L^{q}(\mathbb{R}^{2})$ for every $1\leq q\leq\infty$. In order to treat the
problem variationally, proceeding as [1, 2, 7, 9, 10, 22, 39], they
established the following version of Trudinger-Moser inequality

######  Proposition 1.1.

Suppose that $3<p<4$, then $(e^{\alpha u^{2}}-1-\alpha u^{2})\in
L^{1}(\mathbb{R}^{2})$ for all $\alpha>0$ and $u\in E$. Moreover, if $u\in E$,
$|\nabla u|_{2}^{2}\leq 1$, $|u|_{p}^{p}\leq M>+\infty$ and $\alpha<4\pi$,
then there exists a constant $C(M,\alpha)>0$, which depends only on $M$ and
$\alpha$, such that

(1.11) $\int_{\mathbb{R}^{2}}(e^{\alpha u^{2}}-1-\alpha u^{2})dx\leq
C(M,\alpha).$

With the help of Proposition 1.1, they concluded the existence of mountain-
pass solutions for Eq. (1.9) with a nonlinearity $f$ involving the critical
exponential growth. Actually, to search for the nontrivial solutions, they
Actually, to look for the nontrivial solutions, they restricted themselves in
the radially symmetric subspace of $E$, namely $E_{r}=\\{u\in
E:u(x)=u((|x|)\\}$. In this situation, they immediately have the compact
imbedding $E_{r}\hookrightarrow L^{s}(\mathbb{R}^{2})$ for all $p<s<+\infty$.

Afterwards, Shen [33] generalized and improved the results in [40] to the case
that $1<p<2$ and the nonlinearity $f$ having supercritical exponential growth.
Precisely, by contemplating the work space $E$ above and introducing the the
Young function defined by

(1.12) $\Phi_{\alpha,j_{0}}(t)=e^{\alpha
t^{2}}-\sum_{j=0}^{j_{0}-1}\frac{\alpha^{j}}{j!}|t|^{2j},\leavevmode\nobreak\
\forall t\in\mathbb{R},$

where $\alpha>0$ appearing in (1.10) and
$j_{0}\triangleq\inf\\{j\in\mathbb{N}^{+}:2j\geq p^{*}\\}$ with
$p^{*}=\frac{2p}{2-p}>2$, Shen [33] firstly established the Trudinger-Moser
inequality below

######  Proposition 1.2.

Suppose that $1<p<2$, then $\Phi_{\alpha,j_{0}}(u)\in L^{1}(\mathbb{R}^{2})$
for all $\alpha>0$ and $u\in E$. Moreover

(1.13) $\mathbb{S}(\alpha)\triangleq\sup_{u\in E,\|u\|\leq
1}\int_{\mathbb{R}^{2}}\Phi_{\alpha,j_{0}}(u)dx<+\infty$

for all $0<\alpha<4\pi$. Finally, if $\alpha>4\pi$, then
$\mathbb{S}(\alpha)=+\infty$.

Then, combining the minimax procedure and elliptic regular theory, Shen
investigated the existence of a nontrivial solution with the mountain-pass
energy in [33], where the subspace $E_{r}$ was still considered.

Motivated by all of the quoted papers above, particular by [40, 33], it is
quite natural to ask some interesting questions. For example,

(I) As pointed out in [33], either (1.11) or (1.13) is a subcritical
Trudinger-Moser type inequality in the whole space $\mathbb{R}^{2}$, namely
there is no information when $\alpha$ is exactly equal to $4\pi$. Thereby, can
we given an affirmative answer that whether $\mathbb{S}(4\pi)<+\infty$ or
$\mathbb{S}(4\pi)=+\infty$.

(II) Owing to the compact imbedding $E_{r}\hookrightarrow
L^{s}(\mathbb{R}^{2})$ for each $p<s<+\infty$, although the nonlinearity $f$
possesses the (super)critical exponential growth in [40, 33], it is simple to
recover the compactness to some extent. Hence, can we find nontrivial
solutions for zero-mass CSS equation (1.9). In other words, whether the
existence results in [40, 33] remain true for Eq. (1.4) with
$f(x,t)=\bar{a}|t|^{p-2}t$ for all $x\in\mathbb{R}^{2}$ and $t\in\mathbb{R}$,
where $p\in(1,2)$, or $p\in(3,4)$.

(III) The reader is invited to observe that Eq. (1.9) is an autonomous one
because $\bar{a}>0$ is just a constant. Thus, can we improve this constant to
a general potential function. Moreover, if it was true, whether the obtained
nontrivial solution is indeed a ground state solution.

As a consequence, we shall try our best to introduce some new analytic tricks
and then contemplate the above Questions.

First of all, we focus on the Question (I). Let us continue to use the Young
function $\Phi_{\alpha,j_{0}}$ defined in (1.12), we shall prove the following
result.

######  Theorem 1.3.

Suppose that $1<p<2$, then $\Phi_{\alpha,j_{0}}(u)\in L^{1}(\mathbb{R}^{2})$
for all $\alpha>0$ and $u\in E$. Moreover

(1.14) $\mathbb{S}(\alpha)\triangleq\sup_{u\in E,\|u\|\leq
1}\int_{\mathbb{R}^{2}}\Phi_{\alpha,j_{0}}(u)dx<+\infty$

for all $\alpha\in(0,4\pi]$. Moreover, $\mathbb{S}(\alpha)=+\infty$ if
$\alpha>4\pi$.

######  Remark 1.4.

Due to Theorem 1.3, we can make sure that $\mathbb{S}(4\pi)<+\infty$, and so
it solves the Question (I) completely. Moreover, we do believe that the
technique for the proof of Theorem 1.3 can be also adapted to Proposition 1.1.

Next, in order to solve Questions (II) and (III), we are ready to introduce
some technical assumptions on the potential $a:\mathbb{R}^{2}\to\mathbb{R}$
and the nonlinearity $f:\mathbb{R}\to\mathbb{R}$ in Eq. (1.1) as follows.

  * $(A_{1})$

$a\in\mathcal{C}^{0}(\mathbb{R}^{2})$ with
$\inf\limits_{x\in\mathbb{R}^{2}}a(x)>0$;

  * $(A_{2})$

for almost every $x\in\mathbb{R}^{2}$,
$a(x)\leq\lim\limits_{|x|\to\infty}a(x)\triangleq a_{\infty}<+\infty$ and this
inequality is strict in a subset of positive Lebesgue measure

Concerning the nonlinearity $f$, we suppose that

  * $(f_{1})$

$f\in\mathcal{C}(\mathbb{R})$ with $f(t)\equiv 0$ for all $t\in(-\infty,0]$
and $f(t)=o(t)$ as $t\to 0^{+}$;

  * $(f_{2})$

the map $t\mapsto f(t)/t^{5}$ is strictly increasing on $t\in(0,+\infty)$;

  * $(f_{3})$

there exist constants $t_{0}>0$, $M_{0}>0$ and $\vartheta\in[0,1)$ such that

$0<t^{\vartheta}F(t)\leq M_{0}f(t),\leavevmode\nobreak\ \forall t>t_{0},$

where and in the sequel $F(t)=\int_{0}^{t}f(s)ds$ for all $t>0$;

  * $(f_{4})$

$\lim\limits_{t\to+\infty}F(t)e^{-\alpha_{0}t^{2}}\triangleq\beta_{0}>0$,
where $\alpha_{0}>0$ comes from (1.10).

We are now in a position to state the second main result in this article.

######  Theorem 1.5.

Let $1<p<2$ and suppose $(A_{1})-(A_{2})$. If $f$ satisfies (1.10) and
$(f_{1})-(f_{4})$, then Eq. (1.1) admits at least a positive ground state
solution in $E$.

######  Remark 1.6.

It is obvious that Questions (II) and (III) are uncovered by Theorem 1.5 which
in turn indicates that our results improve and replenish the counterparts in
[40, 33]. It should be mentioned here that both the assumptions on the
potential $a$ and the nonlinearity $f$ are standard. On the one hand, the
function $a$ equipping with $(A_{1})-(A_{2})$ is usually called by the well-
known Rabinowitz’s type potential introduced in [31] and it was later
exploited by Wan and Tan in [37]. On the other hand, as to the function $f$
having critical exponential growth and satisfying $(f_{1})-(f_{4})$, we prefer
to refer the reader to [30, 34, 40] and their references therein.

Finally, we shall exhibit the main idea for the proof of Theorem 1.5. The
reader is invited to see that the work space

$E_{a}\triangleq\bigg{\\{}u:u(x)\leavevmode\nobreak\ \text{is Lebesgue
measurable s.t.}\int_{\mathbb{R}^{2}}|\nabla
u|^{2}dx<+\infty\leavevmode\nobreak\
\text{and}\int_{\mathbb{R}^{2}}a(x)|u|^{p}dx<+\infty\bigg{\\}}$

endowed with the norm

$\|u\|_{a}=\left(\int_{\mathbb{R}^{2}}|\nabla
u|^{2}dx+\left(\int_{\mathbb{R}^{2}}a(x)|u|^{p}dx\right)^{\frac{2}{p}}\right)^{\frac{1}{2}},\leavevmode\nobreak\
\forall u\in E_{a},$

is equivalent to $(E,\|\cdot\|)$ because $a$ is a positive and bounded
function in $\mathbb{R}^{2}$. Thus, we will exploit the work space
$(E,\|\cdot\|)$, instead of $(E_{a},\|\cdot\|_{a})$, just for simplicity. Due
to the lack of compactness caused by the critical exponential growth and the
absence of the compact imbedding $E_{r}\hookrightarrow L^{s}(\mathbb{R}^{2})$
for every $p<s<+\infty$, the foremost point of the proof of Theorem 1.5 is to
restore the compactness. Inspired by [37], we need to investigate the
existence of ground state solutions of the associated “limit problem” of
(1.1), which is given as

(1.15) $\left\\{\begin{array}[]{ll}\displaystyle-\Delta
u+a_{\infty}|u|^{p-2}u+A_{0}u+\sum\limits_{j=1}^{2}A_{j}^{2}u=f(u),\\\
\displaystyle\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2},\leavevmode\nobreak\
\partial_{1}A_{1}+\partial_{2}A_{2}=0,\\\
\displaystyle\partial_{1}A_{0}=A_{2}|u|^{2},\leavevmode\nobreak\
\partial_{2}A_{0}=-A_{1}|u|^{2}.\end{array}\right.$

We obtain the following result.

######  Theorem 1.7.

Let $1<p<2$. If $f$ satisfies (1.10) and $(f_{1})-(f_{4})$, then Eq. (1.15)
possesses a positive ground state solution in $E$.

######  Remark 1.8.

Obviously, one realizes that Theorem 1.7 also provides a positive answer to
the Question (II) above. In the proof of Theorem 1.7, the most striking point
is that we success in establishing the Vanishing lemma corresponding to the
work space $(E,\|\cdot\|)$, see Theorem 3.8 below. Although the essential idea
originates from its classic version due to Lions, c.f. [38, Lemma 1.21], we
have to make some efforts to prove it and it may prompt some further studies
for zero-mass Schrödinger equation.

With Theorem 1.7 in hands, the solvability of Theorem 1.5 becomes available so
far, but we emphasize here that the condition $(f_{2})$ plays a crucial role
in restoring the compactness, see Lemma 3.5 and (3) for instance. As a
consequence, one naturally wonders that whether there still exists a mountain-
pass type solution for Eq. (1.15) when $(f_{2})$ is replaced with a weak type
condition below

  * $(f^{\prime}_{2})$

for all $t>0$, there holds $f(t)t-6F(t)\geq 0$.

Actually, we are going to conclude the existence result as follows.

######  Theorem 1.9.

Let $1<p<2$ and suppose $(A_{1})-(A_{2})$. If $f$ satisfies (1.10) and
$(f_{1})-(f^{\prime}_{2})$ as well as $(f_{3})-(f_{4})$, then Eq. (1.15) has a
positive mountain-pass type solution in $E$.

######  Remark 1.10.

We note that Theorem 1.9 solves Question (II) and (III) partially. Let us
point out here that we will borrow some idea adopted in [33] to reach the
proof. Nevertheless, there are some new challenges that prevent us repeating
the arguments simply. For example, as to the critical exponential case in
[33], the author strongly relied on the following condition of type

  * $(f^{\prime}_{4})$

there are $\gamma>0$ and $s>6$ such that $F(t)\geq\gamma t^{s}$ for all $t\geq
0$,

to restore the compactness, where $\gamma>0$ is sufficiently large. One would
easily deduce that the condition $(f_{4})$ in the present article is weaker
than $(f^{\prime}_{4})$ which is a global one that does never reveal the
essential feature of the critical exponential growth in (1.10).

The outline of the paper is organized as follows. In Section 2, we mainly
present some preliminary results and show the proofs of Theorem 1.3. Sections
3 and 4 are devoted to the proofs of Theorems 1.7 and 1.5, respectively. The
proof of Theorem 1.9 shall be presented in Section 5.

Notations: From now on in this paper, otherwise mentioned, we ultilize the
following notations:

  * •

$C,C_{1},C_{2},\cdots$ denote any positive constant, whose value is not
relevant and $\mathbb{R}^{+}\triangleq(0,+\infty)$.

  * •

Let $(X,\|\cdot\|_{X})$ be a Banach space with dual space
$(X^{-1},\|\cdot\|_{X^{-1}})$, and $\Psi$ be functional on $X$.

  * •

The (C) sequence at a level $c\in\mathbb{R}$ ($(C)_{c}$ sequence in short)
corresponding to $\Psi$ means that $\Psi(x_{n})\to c$ and
$(1+\|x_{n}\|_{X})\|\Psi^{\prime}(x_{n})\|_{X^{-1}}\to 0$ in $\mathbb{R}$ as
$n\to\infty$, where $\\{x_{n}\\}\subset X$.

  * •

$|\cdot|_{p}$ stands for the usual norm of the Lebesgue space
$L^{p}(\mathbb{R}^{2})$ for all $p\in[1,+\infty]$.

  * •

For any $\varrho>0$ and every $x\in\mathbb{R}^{2}$,
$B_{\varrho}(x)\triangleq\\{y\in\mathbb{R}^{2}:|y-x|<\varrho\\}$.

  * •

$o_{n}(1)$ denotes the real sequences with $o_{n}(1)\to 0$ as $n\to+\infty$.

  * •

$``\to"$ and $``\rightharpoonup"$ stand for the strong and weak convergence in
the related function spaces, respectively.

##  2\. Variational Framework and Preliminaries

In this section, we are going to exhibit some preliminary results which enable
us to treat the problems variationally.

First of all, let us recall some imbedding results which would play a foremost
role in formulating the variational structure. The following results can be
found in [33, Lemmas 2.1 and 2.2], so we shall omit the detailed proofs.

######  Lemma 2.1.

Assume $1<p<2$, then the imbedding $E\hookrightarrow L^{s}(\mathbb{R}^{2})$ is
continuous and $E\hookrightarrow L^{s}_{\text{loc}}(\mathbb{R}^{2})$ is
compact for all $p\leq s<+\infty$, respectively. Moreover, $(E,\|\cdot\|)$ is
a reflexive Banach space.

Then, we turn to contemplate the so called Chern-Simons term in Eq. (1.1). To
begin with, there exist some meaningful and significant observations.
According to the second equation and the last two equations in Eq. (1.1), for
each $u\in E$, one has

(2.1)
$\begin{gathered}\int_{\mathbb{R}^{2}}A_{0}|u|^{2}dx=2\int_{\mathbb{R}^{2}}A_{0}(\partial_{2}A_{1}-\partial_{1}A_{2})dx\hfill\\\
\ \ \ \
=2\int_{\mathbb{R}^{2}}(A_{2}\partial_{1}A_{0}-A_{1}\partial_{2}A_{0})dx=2\int_{\mathbb{R}^{2}}(A_{1}^{2}+A_{2}^{2})|u|^{2}dx.\hfill\\\
\end{gathered}$

As a by-product of the well-known Hardy-Littlewood-Sobolev inequality [24,
Theorem 4.3], we could conclude the following estimates to the gauge fields
$A_{j}$ for $j\in\\{0,1,2\\}$.

######  Lemma 2.2.

(see [15, Propositions 4.2-4.3]) Assume $1<r<2$ and
$\frac{1}{r}-\frac{1}{\widehat{r}}=\frac{1}{2}$, then

$|A_{j}|_{\widehat{r}}\leq C_{r}|u|_{2r}^{2}\leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ j=1,2,\leavevmode\nobreak\
|A_{0}|_{\widehat{r}}\leq C_{r}|u|_{2r}^{2}|u|_{4}^{2},$

where $C_{r}>0$ is a constant dependent of $r$.

Combining Lemmas 2.1 and 2.2, one can easily see that

(2.2) $|A_{j}u|_{2}\leq|A_{j}|_{\widehat{r}}|u|_{\frac{r}{r-1}}\leq
C_{r}|u|_{2r}^{2}|u|_{\frac{r}{r-1}}\leq\bar{C}_{r}\|u\|^{3},\leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ j=1,2,$

because $2r>2$ and $r/(r-1)>2$, where $\bar{C}_{r}>0$ depends only on $r>1$.
We also need the following Brézis-Lieb type lemma for the Chern-Simons term.

######  Lemma 2.3.

(see [13, Lemma 2.4]) If $u_{n}\rightharpoonup u$ in $E$ and $u_{n}\to u$ a.e.
in $\mathbb{R}^{2}$, then one has $A_{j}[u_{n}]\to A_{j}[u]$ a.e. for $j=1,2$,

(2.3)
$\left\\{\begin{array}[]{ll}\displaystyle\lim_{n\to\infty}\int_{\mathbb{R}^{2}}A_{0}[u_{n}]u_{n}\psi
dx=\int_{\mathbb{R}^{2}}A_{0}[u]u\psi dx,\leavevmode\nobreak\ \forall\psi\in
E,\\\
\displaystyle\lim_{n\to\infty}\int_{\mathbb{R}^{2}}A_{j}^{2}[u_{n}]u_{n}\psi
dx=\int_{\mathbb{R}^{2}}A_{j}^{2}[u]u\psi dx,\leavevmode\nobreak\
\forall\psi\in E\leavevmode\nobreak\ \text{\emph{with}}\leavevmode\nobreak\
j=1,2,\end{array}\right.$

and

(2.4)
$\lim_{n\to\infty}\int_{\mathbb{R}^{2}}\big{[}A_{j}^{2}[u_{n}]|u_{n}|^{2}-A_{j}^{2}[u_{n}-u]|u_{n}-u|^{2}\big{]}dx=\int_{\mathbb{R}^{2}}A_{j}^{2}[u]|u|^{2}dx,\leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ j=1,2.$

Finally, we shall focus on the nonlinearity $f$. Whereas, it possesses the
critical exponential growth at infinity in the Trudinger-Moser sense at
infinity, we have to derive the Trudinger-Moser type inequality associated
with the work space $E$ in this article.

######  _Proof of Theorem 1.3_.

Let $\alpha\in(0,4\pi]$ and $u\in E$ with $\|u\|\leq 1$. We denote by $u^{*}$
the Schwarz symmetrization of $u$, then $u^{*}$ is radial and non-increasing.
Thanks to the results in [21],

$\int_{\mathbb{R}^{2}}|\nabla u^{*}|^{2}dx\leq\int_{\mathbb{R}^{2}}|\nabla
u|^{2}dx,\leavevmode\nobreak\
\int_{\mathbb{R}^{2}}|u^{*}|^{p}dx=\int_{\mathbb{R}^{2}}|u|^{p}dx$

and

$\int_{\mathbb{R}^{2}}\Phi_{\alpha,j_{0}}(u^{*})dx=\int_{\mathbb{R}^{2}}\Phi_{\alpha,j_{0}}(u)dx.$

So, without loss of generality, we can suppose that $u\in E$ is non-
increasing. Given an $R>0$ which will be determined later, due to Lemma 2.1,
we could deduce that the function $v(x)\triangleq u(x)-u(R)$ belongs to
$H_{0}^{1}(B_{R}(0))$. Adopting the Young’s inequality, there holds

$\displaystyle u^{2}(x)$ $\displaystyle=v^{2}(x)+2v(x)u(R)+u^{2}(R)\leq
v^{2}(x)+[1+v^{2}(x)u^{2}(R)]+u^{2}(R)$
$\displaystyle=[1+u^{2}(R)]v^{2}(x)+1+u^{2}(R)\triangleq
w^{2}(x)+1+u^{2}(R),\leavevmode\nobreak\ \forall x\in B_{R}(0).$

Obviously, $w\triangleq\sqrt{1+u^{2}(R)}v\in H_{0}^{1}(B_{R}(0))$ and $\nabla
w=\sqrt{1+u^{2}(R)}\nabla v=\sqrt{1+u^{2}(R)}\nabla u$. Moreover, we recall
from [5, Lemma A.IV], because $u\in L^{p}(\mathbb{R}^{2})$ by Lemma 2.1, there
is a constant $C_{p}>0$ which is independent of $u$ such that

(2.5) $|u(x)|\leq C_{p}|x|^{-\frac{2}{p}}|u|_{p}\leavevmode\nobreak\ \text{for
all}\leavevmode\nobreak\ x\neq 0.$

Consequently, with the help of (2.5), we are able to see that

$\displaystyle\int_{\mathbb{R}^{2}}|\nabla w|^{2}dx$
$\displaystyle=[1+u^{2}(R)]\int_{\mathbb{R}^{2}}|\nabla
u|^{2}dx\leq[1+u^{2}(R)]\left[1-\left(\int_{\mathbb{R}^{2}}|u|^{p}dx\right)^{\frac{2}{p}}\right]$
$\displaystyle\leq
1-\left(\int_{\mathbb{R}^{2}}|u|^{p}dx\right)^{\frac{2}{p}}+u^{2}(R)\leq
1-(1-C_{p}^{2}R^{-\frac{4}{p}})\left(\int_{\mathbb{R}^{2}}|u|^{p}dx\right)^{\frac{2}{p}}.$

We then choose the constant $R$ to be sufficiently large such that
$1-C_{p}^{2}R^{-\frac{4}{p}}\geq 0$ and so $|\nabla w|_{2}^{2}\leq 1$. As a
consequence, we can apply the classic Trudinger-Moser inequality explored in
[7, 10] to arrive at

$\int_{B_{R}(0)}\Phi_{4\pi,j_{0}}(u)\mathrm{d}x\leq\int_{B_{R}(0)}e^{4\pi
u^{2}}\mathrm{d}x\leq e^{4\pi(1+u^{2}(R))}\int_{B_{R}(0)}e^{4\pi
w^{2}}\mathrm{d}x\leq C.$

From which, one concludes that

(2.6) $\sup_{u\in E:\|u\|\leq
1}\int_{B_{R}(0)}\Phi_{4\pi,j_{0}}(u)\mathrm{d}x\leq C<+\infty.$

On the other hand, we can follow the proof of [33, Theorem 1.1] to verify that
the integral on the complement of $B_{R}(0)$ is uniformly bounded. For the
sake of the reader’s convenience, we shall show the proof in detail. Using [5,
Lemma A.IV] and Lemma 2.1 again, for each $u\in L^{p^{*}}(\mathbb{R}^{2})$,
there is a constant $\bar{C}_{p}>0$ independent of $u$ such that

(2.7)
$|u(x)|\leq\bar{C}_{p}|x|^{-\frac{2-p}{p}}|u|_{p^{*}}\leavevmode\nobreak\
\text{for all}\leavevmode\nobreak\ x\neq 0.$

Since $j_{0}=\inf\\{j\in\mathbb{N}:2j\geq p^{*}\\}$, then one sees $2j\geq
p^{*}$ for all $j\geq j_{0}$ and so applying (2.7) and $R>1$,

$\begin{gathered}\int_{\mathbb{R}^{2}\backslash
B_{R}(0)}\Phi_{4\pi,j_{0}}(u)\mathrm{d}x=\sum_{j=j_{0}}^{\infty}\frac{(4\pi)^{j}}{j!}\int_{\mathbb{R}^{2}\backslash
B_{R}(0)}|u|^{2j}\mathrm{d}x=\sum_{j=j_{0}}^{\infty}\frac{(4\pi)^{j}}{j!}\int_{\mathbb{R}^{2}\backslash
B_{R}(0)}|u|^{2j-p^{*}}|u|^{p^{*}}\mathrm{d}x\hfill\\\ \ \ \ \
\leq\sum_{j=j_{0}}^{\infty}\frac{(4\pi)^{j}}{j!}\int_{\mathbb{R}^{2}\backslash
B_{R}(0)}({\bar{C}}_{p}|u|_{p^{*}})^{2j-p^{*}}|u|^{p^{*}}\mathrm{d}x=\frac{1}{{\bar{C}}_{p}^{p^{*}}|u|_{p^{*}}^{p^{*}}}\sum_{j=j_{0}}^{\infty}\frac{(4\pi)^{j}}{j!}\int_{\mathbb{R}^{2}\backslash
B_{\varrho}(0)}({\bar{C}}_{p}|u|_{p^{*}})^{2j}|u|^{p^{*}}\mathrm{d}x\hfill\\\
\ \ \ \
\leq\frac{e^{4\pi({\bar{C}}_{p}|u|_{p^{*}})^{2}}}{{\bar{C}}_{p}^{p^{*}}|u|_{p^{*}}^{p^{*}}}\int_{\mathbb{R}^{2}\backslash
B_{\varrho}(0)}|u|^{p^{*}}\mathrm{d}x\leq\frac{e^{4\pi({\bar{C}}_{p}|u|_{p^{*}})^{2}}}{{\bar{C}}_{p}^{p^{*}}}\leq\frac{e^{4\pi(\tilde{C}_{p}\|u\|)^{2}}}{\bar{C}_{p}^{p^{*}}}=\frac{e^{4\pi\tilde{C}_{p}^{2}}}{{\bar{C}}_{p}^{p^{*}}}.\hfill\\\
\end{gathered}$

From this inequality and (2.6), we obtain that

$\mathbb{S}(\alpha)\leq
C+{e^{4\pi\tilde{C}_{p}^{2}}}{{\bar{C}}_{p}^{-p^{*}}}<+\infty,\leavevmode\nobreak\
\forall\alpha\in(0,4\pi].$

The remaining parts are totally same as in [33, Theorem 1.1], so we omit them
here. ∎

Now, we are able to verify that the variational functional
$J_{a}:E\to\mathbb{R}$ defined by

${J}_{a}(u)=\frac{1}{2}\int_{\mathbb{R}^{2}}[|\nabla
u|^{2}+(A_{1}^{2}+A_{2}^{2})u^{2}]\mathrm{d}x+\frac{1}{p}\int_{\mathbb{R}^{2}}a(x)|u|^{p}\mathrm{d}x-\int_{\mathbb{R}^{2}}F(u)\mathrm{d}x,$

is well-defined and of class $\mathcal{C}^{1}(E,\mathbb{R})$. Actually, due to
(1.10) and $(f_{1})$, for all $\varepsilon>0$ and $\alpha>\alpha_{0}$, there
is a constant $C_{\varepsilon}>0$ such that

(2.8)
$|f(s)|\leq\varepsilon|s|+C_{\varepsilon}|s|^{q-1}\Phi_{\alpha,j_{0}}(s),\leavevmode\nobreak\
\forall s\in\mathbb{R},$

where $\Phi_{\alpha,j_{0}}$ is defined by (1.12) and $q>2$ can be arbitrarily
chosen later. Using $(f_{2})$, there holds

(2.9)
$|F(s)|\leq\varepsilon|s|^{2}+C_{\varepsilon}|s|^{q}\Phi_{\alpha,j_{0}}(s),\leavevmode\nobreak\
\forall s\in\mathbb{R}.$

Moreover, without mentioned any longer, let us exploit directly the following
inequality (see e.g. [39, Lemma 2.1]):

$(\Phi_{\alpha,j_{0}}(s))^{m}\leq\Phi_{m\alpha,j_{0}}(s),\leavevmode\nobreak\
\forall s\in\mathbb{R},\leavevmode\nobreak\ \alpha>0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ m>1.$

With (2.8) and (2.9) in hands, exploiting Theorem 1.3, we could proceed as the
calculations in [32, 33] to deduce that the variational functional $J_{a}$
associated with (1.1) is well-defined and belongs to
$\mathcal{C}^{1}(E,\mathbb{R})$ such that

${J}_{a}^{\prime}(u)[v]=\int_{\mathbb{R}^{2}}[\nabla u\nabla
v+(A_{1}^{2}+A_{2}^{2}+A_{0})uv]\mathrm{d}x+\int_{\mathbb{R}^{2}}a(x)|u|^{p-2}uv\mathrm{d}x-\int_{\mathbb{R}^{2}}f(u)v\mathrm{d}x,\leavevmode\nobreak\
\forall v\in E.$

In particular, it follows from (2.1) that

${J}_{a}^{\prime}(u)[u]=\int_{\mathbb{R}^{2}}[|\nabla
u|^{2}+3(A_{1}^{2}+A_{2}^{2})u^{2}]\mathrm{d}x+\int_{\mathbb{R}^{2}}a(x)|u|^{p}\mathrm{d}x-\int_{\mathbb{R}^{2}}f(u)u\mathrm{d}x.$

Hence, any (weak) solution of Eq. (1.1) corresponds to a critical point of
$J_{a}$. In order to search for the critical points of $J_{a}$, we introduce
the following results.

######  Lemma 2.4.

Let $1<p<2$ and $f$ satisfies (1.10) and $(f_{1})-(f_{4})$. Suppose there
exists a sequence $\\{u_{n}\\}\subset E$ such that $u_{n}\rightharpoonup u$ in
$E$ and $u_{n}\to u$ a.e. in $\mathbb{R}^{2}$. If in addition, we assume that

(2.10) $\sup_{n\in\mathbb{N}}\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\mathrm{d}x\leq
K_{0}$

for some $K_{0}\in(0,+\infty)$ independent of $n\in\mathbb{N}$, then, going to
a subsequence if necessary,

(2.11)
$\lim_{n\to\infty}\int_{\Omega}F(u_{n})\mathrm{d}x=\int_{\Omega}F(u)\mathrm{d}x\leavevmode\nobreak\
\text{\emph{for any compact set}}\leavevmode\nobreak\
\Omega\subset\mathbb{R}^{2}.$

Moreover, passing to a subsequence if necessary, there holds

(2.12)
$\lim_{n\to\infty}\int_{\mathbb{R}^{2}}f(u_{n})\psi\mathrm{d}x=\int_{\mathbb{R}^{2}}f(u)\psi\mathrm{d}x\leavevmode\nobreak\
\text{\emph{for all}}\leavevmode\nobreak\ \psi\in
C_{0}^{\infty}(\mathbb{R}^{2}).$

###### Proof.

We can follow the essential ideas adopted in [9, Lemma 2.1] and the details
will be omitted. ∎

##  3\. The limit problem (1.15)

The main objective of this section is to investigate the existence of positive
ground state solutions for the CSS equation (1.15) which acts as the “limit
problem” of Eq. (1.1).

In order to solve Eq. (1.15), we are going to look for the critical points of
its corresponding variational functional $J_{\infty}:E\to\mathbb{R}$ below

(3.1) ${J}_{\infty}(u)=\frac{1}{2}\int_{\mathbb{R}^{2}}[|\nabla
u|^{2}+(A_{1}^{2}+A_{2}^{2})u^{2}]\mathrm{d}x+\frac{a_{\infty}}{p}\int_{\mathbb{R}^{2}}|u|^{p}\mathrm{d}x-\int_{\mathbb{R}^{2}}F(u)\mathrm{d}x,\leavevmode\nobreak\
\forall u\in E.$

Arguing as before, it is simple to show that $J_{\infty}$ is well-defined and
it is of class $\mathcal{C}^{1}(E,\mathbb{R})$ satisfying

${J}_{\infty}^{\prime}(u)[v]=\int_{\mathbb{R}^{2}}[\nabla u\nabla
v+(A_{1}^{2}+A_{2}^{2}+A_{0})uv]\mathrm{d}x+\frac{a_{\infty}}{p}\int_{\mathbb{R}^{2}}|u|^{p-2}uv\mathrm{d}x-\int_{\mathbb{R}^{2}}f(u)v\mathrm{d}x,\leavevmode\nobreak\
\forall v\in E.$

Moreover, we are derived from (2.1) that

${J}_{\infty}^{\prime}(u)[u]=\int_{\mathbb{R}^{2}}[|\nabla
u|^{2}+3(A_{1}^{2}+A_{2}^{2})u^{2}]\mathrm{d}x+a_{\infty}\int_{\mathbb{R}^{2}}|u|^{p}\mathrm{d}x-\int_{\mathbb{R}^{2}}f(u)u\mathrm{d}x.$

In what follows, we shall denote the Nehari manifold associated with
$J_{\infty}$ by

$\mathcal{N}_{\infty}\triangleq\\{u\in
E\backslash\\{0\\}:J^{\prime}_{\infty}(u)[u]=0\\}$

and the corresponding ground state energy level on $\mathcal{N}_{\infty}$ is
defined by

(3.2) $m_{\infty}\triangleq\min_{u\in\mathcal{N}_{\infty}}J_{\infty}(u).$

Our main result concerning the autonomous CSS equation (1.15) is the
following:

######  Theorem 3.1.

Let $1<p<2$ and suppose that $f$ satisfies (1.10) and $(f_{1})-(f_{5})$, then
Eq. (1.15) admits a positive ground state solution $u_{\infty}\in E$ such that

$J_{\infty}(u_{\infty})=m_{\infty}=\inf_{u\in E\backslash\\{0\\}}\max_{t\geq
0}J_{\infty}(tu).$

The proof of the above theorem will be divided into several lemmas. For
simplicity, we shall always suppose that the nonlinearity $f$ satisfies (1.10)
and $(f_{1})-(f_{4})$ and do not mention them unless needed.

First of all, let us give some key observations on the shape of the functional
$J_{\infty}$.

######  Lemma 3.2.

Let $1<p<2$, then there exists a constant $\zeta>0$ such that

(3.3) $m_{\rho}\triangleq\inf\big{\\{}J_{\infty}(u):u\in
E,\|u\|=\rho\big{\\}}>0,\leavevmode\nobreak\ \forall\rho\in(0,\zeta],$

and

(3.4) $n_{\rho}\triangleq\inf\big{\\{}J^{\prime}_{\infty}(u)[u]:u\in
E,\|u\|=\rho\big{\\}}>0,\leavevmode\nobreak\ \forall\rho\in(0,\zeta].$

###### Proof.

It follows from (2.9) that

$J_{\infty}(u)\geq\frac{1}{2}\int_{\mathbb{R}^{2}}(|\nabla
u|^{2}+a_{\infty}|u|^{p})\mathrm{d}x-\varepsilon\int_{\mathbb{R}^{2}}|u|^{2}dx-C(\varepsilon,q,\alpha)\int_{\mathbb{R}^{2}}|u|^{q}\Phi_{\alpha,j_{0}}(u)dx.$

Using Lemma 2.1 and letting $\varepsilon>0$ be suitably small, with the help
of (1.14), there exists a constant $\overline{\zeta}\in(0,1)$ such that

$J_{\infty}(u)\geq\frac{1}{4}\|u\|^{2}-C\|u\|^{q}\leavevmode\nobreak\
\text{when}\leavevmode\nobreak\ \|u\|\leq\overline{\zeta}.$

In light of $q>2$, we can determine a constant $\zeta\in(0,\overline{\zeta})$
such that (3.3) holds true. According to the definition of
$J^{\prime}_{\infty}$, it is easy to reach (3.4) as before. The proof is
completed. ∎

######  Lemma 3.3.

Let $1<p<2$ and suppose that $u\in E\backslash\\{0\\}$, then for all $t>0$,
there holds

$J_{\infty}(tu)\to-\infty\leavevmode\nobreak\ \text{as}\leavevmode\nobreak\
t\to+\infty.$

In particular, the functional $J_{\infty}$ is not bounded from below.

###### Proof.

For any fixed positive function $u\in E\backslash\\{0\\}$ and $t>1$, we have
that

$\frac{J_{\infty}(tu)}{t^{6}}\leq\frac{1}{p}\int_{\mathbb{R}^{2}}[|\nabla
u|^{2}+(A_{1}^{2}+A_{2}^{2})u^{2}+a_{\infty}|u|^{p}]\mathrm{d}x-\frac{1}{t^{6}}\int_{\mathbb{R}^{2}}F(tu)dx.$

Due to $(f_{4})$, one sees that $F(t)t^{-6}\to+\infty$ as $t\to+\infty$.
Thereby, using the Fatou’s lemma, we arrive at
$J_{\infty}(tu)/t^{6}\to-\infty$ as $t\to+\infty$, and the claim follows. ∎

Relying on Lemmas 3.2 and 3.3, we shall exploit the following critical point
theorem without the $(C)$ condition introduced in [28] to find a $(C)$
sequence for $J_{\infty}$.

######  Proposition 3.4.

Let $X$ be a Banach space and $\varphi\in\mathcal{C}^{1}(X,\mathbb{R})$
Gateaux differentiable for all $v\in X$, with G-derivative
$\varphi^{\prime}(v)\in X^{-1}$ continuous from the norm topology of $X$ to
the weak $*$ topology of $X^{-1}$ and $\varphi(0)=0$. Let $S$ be a closed
subset of $X$ which disconnects (archwise) $X$. Let $v_{0}=0$ and $v_{1}\in X$
be points belonging to distinct connected components of $\bar{X}\backslash X$.
Suppose that

$\inf_{S}\varphi\geq\varrho>0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \varphi(v_{1})\leq 0$

and let $\Gamma=\\{\gamma\in C([0,1],X):\gamma(0)\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \gamma(1)=v_{1}\\}$. Then

$c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}\varphi(\gamma(t))\geq\varrho>0$

and there is a $(C)_{c}$ sequence for $\varphi$.

Combining Lemmas 3.2 and 3.3 as well as Proposition 3.4, there is a sequence
$\\{u_{n}\\}\subset E$ such that

(3.5) $J_{\infty}(u_{n})\to c_{\infty}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\
(1+\|u_{n}\|)\|J_{\infty}^{\prime}(u_{n})\|_{E^{-1}}\to 0,$

where

(3.6)
$c_{\infty}\triangleq\inf_{\gamma\in\Gamma_{\infty}}\max_{t\in[0,1]}J_{\infty}(\gamma(t))>0$

with
$\Gamma_{\infty}=\\{\gamma\in\mathcal{C}([0,1],E):\gamma(0)=0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ J_{\infty}(\gamma(1))<0\\}$.

######  Lemma 3.5.

Let $1<p<2$, then for every $u\in E\backslash\\{0\\}$, there exists a unique
constant $t_{u}>0$ such that $J_{\infty}(t_{u}u)=\max\limits_{t\geq
0}J_{\infty}(tu)$ and $t_{u}u\in\mathcal{N}_{\infty}$. In particular, we could
conclude that $c_{\infty}=m_{\infty}=d_{\infty}$, where

$d_{\infty}\triangleq\inf_{u\in E\backslash\\{0\\}}\max_{t\geq
0}J_{\infty}(tu).$

###### Proof.

For any $u\in E\backslash\\{0\\}$ and $t>0$, we define $\xi(t)=J_{\infty}(tu)$
and so

$\displaystyle\xi^{\prime}(t)=0$
$\displaystyle\Longleftrightarrow\int_{\mathbb{R}^{2}}[t|\nabla
u|^{2}+3t^{5}(A_{1}^{2}+A_{2}^{2})u^{2}]\mathrm{d}x+a_{\infty}t^{p-1}\int_{\mathbb{R}^{2}}|u|^{p}\mathrm{d}x-\int_{\mathbb{R}^{2}}f(tu)u\mathrm{d}x=0$
$\displaystyle\Longleftrightarrow
J^{\prime}_{\infty}(tu)[tu]/t=0\Longleftrightarrow
J^{\prime}_{\infty}(tu)[tu]=0\Longleftrightarrow tu\in\mathcal{N}_{\infty}.$

Proceeding as in the proofs of Lemmas 3.2 and 3.3, $\xi(t)$ possesses a
critical point which corresponds to its maximum, that is, there exists a
constant $t_{u}>0$ such that $\xi^{\prime}(t_{u})=0$. Next, we verify that
$t_{u}$ is unique. Arguing it indirectly, we would assume that there exist two
constants $t_{1},t_{2}>0$ with $t_{1}\neq t_{2}$ such that
$u_{t_{i}}\in\mathcal{N}_{\infty}$ for $i\in\\{1,2\\}$. It follows from some
elementary computations that

$\begin{gathered}{J}_{\infty}(t_{1}u)-{J}_{\infty}(t_{2}u)-\frac{t_{1}^{6}-t_{2}^{6}}{6t_{1}^{6}}{J}^{\prime}_{\infty}(t_{1}u)[t_{1}u]\hfill\\\
\ \ \ \
=\frac{t_{1}^{2}}{6}\left[2-3\left(\frac{t_{2}}{t_{1}}\right)^{2}+\left(\frac{t_{2}}{t_{1}}\right)^{6}\right]\int_{\mathbb{R}^{2}}|\nabla
u|^{2}\mathrm{d}x\hfill\\\ \ \ \ \ \ \
+\frac{t_{1}^{p}}{6p}\left[(6-p)-6\left(\frac{t_{2}}{t_{1}}\right)^{p}+p\left(\frac{t_{2}}{t_{1}}\right)^{6}\right]a_{\infty}\int_{\mathbb{R}^{2}}|u|^{p}\mathrm{d}x\hfill\\\
\ \ \ \ \ \
+\int_{\mathbb{R}^{2}}\left[\frac{1-(t_{1}^{-1}t_{2})^{6}}{6}f(t_{1}u)t_{1}u-F(t_{1}u)+F((t_{1}^{-1}t_{2})t_{1}u)\right]\mathrm{d}x\hfill\\\
\end{gathered}$

and

$\begin{gathered}{J}_{\infty}(t_{2}u)-{J}_{\infty}(t_{1}u)-\frac{t_{2}^{6}-t_{1}^{6}}{6t_{2}^{6}}{J}^{\prime}_{\infty}(t_{2}u)[t_{2}u]\hfill\\\
\ \ \ \
=\frac{t_{2}^{2}}{6}\left[2-3\left(\frac{t_{1}}{t_{2}}\right)^{2}+\left(\frac{t_{1}}{t_{2}}\right)^{6}\right]\int_{\mathbb{R}^{2}}|\nabla
u|^{2}\mathrm{d}x\hfill\\\ \ \ \ \ \ \
+\frac{t_{2}^{p}}{6p}\left[(6-p)-6\left(\frac{t_{1}}{t_{2}}\right)^{p}+p\left(\frac{t_{1}}{t_{2}}\right)^{6}\right]a_{\infty}\int_{\mathbb{R}^{2}}|u|^{p}\mathrm{d}x\hfill\\\
\ \ \ \ \ \
+\int_{\mathbb{R}^{2}}\left[\frac{1-(t_{2}^{-1}t_{1})^{6}}{6}f(t_{2}u)t_{2}u-F(t_{2}u)+F((t_{2}^{-1}t_{1})t_{2}u)\right]\mathrm{d}x.\hfill\\\
\end{gathered}$

Combining the above two formulas and ${J}^{\prime}_{\infty}(t_{i}u)[t_{i}u]=0$
for $i\in\\{1,2\\}$, we arrive at a contradiction if $t_{1}\neq t_{2}$. Next,
to coincide the three numbers with each other, we shall firstly conclude that
$c_{\infty}\leq d_{\infty}$, then $d_{\infty}\leq m_{\infty}$ and finally
$m_{\infty}\leq c_{\infty}$ step by step.

(1). $c_{\infty}\leq d_{\infty}$. In view of Lemma 3.3, there is a
sufficiently large $t_{0}>0$ such that $J_{\infty}(t_{0}u)<0$ for every $u\in
E\backslash\\{0\\}$. Define $\gamma_{0}(t)=tt_{0}u\in\Gamma_{\infty}$, then
$c_{\infty}\leq\max_{t\in[0,1]}J_{\infty}(tt_{0}u)\leq\max_{t\geq
0}J_{\infty}(tu)$ which implies that $c_{\infty}\leq d_{\infty}$.

(2). $d_{\infty}\leq m_{\infty}$. We claim that, for all $u\in E$ and $t>0$,
there holds

(3.7)
$\begin{gathered}{J}_{\infty}(u)-{J}_{\infty}(tu)-\frac{1-t^{6}}{6}{J}^{\prime}_{\infty}(u)[u]=\frac{1}{6}\left(2-3t^{2}+t^{6}\right)\int_{\mathbb{R}^{2}}|\nabla
u|^{2}\mathrm{d}x\hfill\\\ \ \ \ \ \ \
+\frac{a_{\infty}}{6p}\left[(6-p)-6t^{p}+pt^{6}\right]\int_{\mathbb{R}^{2}}|u|^{p}\mathrm{d}x+\int_{\mathbb{R}^{2}}\left[\frac{1-t^{6}}{6}f(u)u-F(u)+F(tu)\right]\mathrm{d}x.\hfill\\\
\end{gathered}$

Due to $(f_{2})$, it suffices to verify that

$\tau(s,t)\triangleq\frac{1-t^{6}}{6}f(s)s+F(st)-F(s)\geq 0\quad\text{for
all}\leavevmode\nobreak\ s\geq 0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ t>0.$

Indeed, it is simple to calculate that

$\displaystyle\frac{\partial}{\partial t}\tau(s,t)$
$\displaystyle=f(st)s-t^{5}f(s)s=t^{5}s^{6}\left[\frac{f(st)}{(st)^{5}}-\frac{f(s)}{s^{5}}\right]$
$\displaystyle\ \ \left\\{\begin{array}[]{ll}\geq
0,&\text{if}\leavevmode\nobreak\ t\in[1,+\infty),\\\ \leq
0,&\text{if}\leavevmode\nobreak\ t\in(0,1],\end{array}\right.$

where we have used $(f_{2})$ in the last inequalities. Therefore, we obtain
that $t\mapsto\tau(s,t)$ is decreasing in $(0,1)$ and increasing in
$(1,+\infty)$ for all $s\geq 0$, respectively. It has that
$\tau(s,t)\geq\min_{t\in(0,+\infty)}\tau(s,t)=\tau(s,1)=0$ for every $s\geq 0$
and the claim concludes. Owing to the claim, for all
$u\in\mathcal{N}_{\infty}$, one sees that
${J}_{\infty}(u)\geq{J}_{\infty}(tu)$ for all $t>0$ yielding that
$d_{\infty}\leq m_{\infty}$.

(3). $m_{\infty}\leq c_{\infty}$. We follow [38, Theorem 4.2] and present the
details for the sake of completeness. The manifold $\mathcal{N}_{\infty}$
clearly separates $E$ into two components, we say them by $\\{J_{\infty}>0\\}$
and $\\{J_{\infty}<0\\}$, respectively. According to Lemma 3.2, one can
conclude that $\\{J_{\infty}>0\\}$ contains the origin and a small ball around
the origin. Moreover, adopting $(f_{2})$, $J_{\infty}(u)\geq
J_{\infty}(u)-\frac{1}{6}J^{\prime}_{\infty}(u)[u]\geq 0$ for all
$u\in\\{J_{\infty}>0\\}$, so one must have that
$\gamma(1)\in\\{J_{\infty}<0\\}$ for all $\gamma\in\Gamma_{\infty}$. Because
$\gamma\in\mathcal{C}^{0}$, there exists a $t_{0}\in(0,1)$ such that
$\gamma(t_{0})\in\mathcal{N}_{\infty}$ showing that $m_{\infty}\leq
c_{\infty}$ by the arbitrariness of $\gamma$. The proof is completed. ∎

Since the nonlinearity $f$ fulfills the critical exponential growth at
infinity which leads to the lack of compactness, to recover it, we have to
pull the mountain-pass level $c_{\infty}$ down below a critical value. Have
this aim in mind, inspired by [1, 2, 7, 9, 10, 22, 39], we will consider the
Moser sequence functions defined by

$\bar{w}_{n}(x)\triangleq\frac{1}{\sqrt{2\pi}}\left\\{\begin{array}[]{ll}\sqrt{\log
n},&\text{if}\leavevmode\nobreak\ 0\leq|x|\leq\frac{1}{n},\vspace{2mm}\\\
\frac{\log(\frac{1}{|x|})}{\sqrt{\log n}},&\text{if}\leavevmode\nobreak\
\frac{1}{n}<|x|\leq 1,\vspace{2mm}\\\ 0,&\text{if}\leavevmode\nobreak\
|x|>1,\end{array}\right.$

######  Lemma 3.6.

Let $1<p<2$, then $0<c_{\infty}<\frac{2\pi}{\alpha_{0}}$.

###### Proof.

We have $c_{\infty}\geq m_{\zeta}>0$ by Lemma 3.2. Taking advantage of Lemma
3.3, it is not difficult to observe that
$c_{\infty}=\inf_{\gamma\in\Gamma}\max_{t\in(0,1]}J_{\infty}(\gamma(t))\leq\inf_{u\in
E\backslash\\{0\\}}\max_{t>0}J_{\infty}(tu)$. As a consequence, it suffices to
conclude that there exists a function $w\in E\backslash\\{0\\}$ such that
$\max_{t>0}J_{\infty}(tw)<\frac{2\pi}{\alpha_{0}}$. It follows from some
elementary computations that $\overline{w}_{n}\in
C_{0}^{\infty}(B_{1}(0))\subset E$ and it satisfies

$\int_{\mathbb{R}^{2}}|\nabla\bar{w}_{n}|^{2}dx=\frac{1}{2\pi\log
n}\int_{B_{1}(0)\backslash
B_{\frac{1}{n}}(0)}\frac{1}{|x|^{2}}dx=\frac{1}{\log
n}\int^{1}_{\frac{1}{n}}\frac{1}{\rho}d\rho=1,$

and

$\int_{\mathbb{R}^{2}}|\bar{w}_{n}|dx=\sqrt{\frac{\log
n}{2\pi}}\frac{\pi}{n^{2}}+\sqrt{\frac{2\pi}{\log
n}}\left(\frac{1}{4}-\frac{\log n}{2n^{2}}-\frac{1}{4n^{2}}\right)=o_{n}(1).$

Thanks to the interpolation inequality, there holds
$|\bar{w}_{n}|_{s}^{s}=o_{n}(1)$ for all $1\leq s<+\infty$.

Denoting $|\bar{w}_{n}|_{p}^{p}=\delta_{n}$ with $\delta_{n}\to 0$ and we then
define $w_{n}=\bar{w}_{n}/(1+\sqrt[p]{\delta_{n}})$ for all $n\in\mathbb{N}$.
Obviously, it has that $\|w_{n}\|\equiv 1$ which together with $\delta_{n}\to
0$ indicates that

(3.8) $|\nabla w_{n}|_{2}^{2}\to 1\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ |w_{n}|_{p}^{p}\to 0.$

With the help of (2.2) and (3.8), we follow [35, Lemma 3.10] to show that

(3.9)
$c(w_{n})\triangleq\int_{\mathbb{R}^{2}}\left(A_{1}^{2}[w_{n}]+A_{2}^{2}[w_{n}]\right)w_{n}^{2}dx\to
0.$

We now claim that there is a $n\in\mathbb{N}^{+}$ such that

(3.10) $\max_{t>0}J_{\infty}(tw_{n})<\frac{2\pi}{\alpha_{0}}.$

Otherwise, for all $n\in\mathbb{N}^{+}$, there exists a $t_{n}>0$
corresponding to the maximum point of $\max_{t>0}J(tw_{n})$

(3.11) $J^{\prime}_{\infty}(t_{n}w_{n})[t_{n}w_{n}]=0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\
J_{\infty}(t_{n}w_{n})=\max_{t>0}J_{\infty}(tw_{n})\geq\frac{2\pi}{\alpha_{0}}.$

From $(f_{3})-(f_{4})$, for all $\epsilon\in(0,\beta_{0})$, there exists a
constant $R_{\epsilon}=R(\epsilon)>0$ such that

$f(s)s\geq
M^{-1}_{0}(\beta_{0}-\epsilon)s^{\vartheta+1}e^{\alpha_{0}|s|^{2}},\leavevmode\nobreak\
\forall|s|\geq R_{\epsilon}.$

According to the second formula in (3.11), $\\{t_{n}\\}$ is bounded below by
some positive constant. For some sufficiently large $n\in\mathbb{N}$, one
knows that $t_{n}w_{n}\geq R_{\epsilon}$ on $B_{1/n}(0)$. Using (3.11) again,

$\begin{gathered}t_{n}^{2}|\nabla
w_{n}|_{2}^{2}+a_{\infty}t_{n}^{p}|w_{n}|_{p}^{p}+3t_{n}^{6}c(w_{n})=\int_{\mathbb{R}^{2}}f(t_{n}w_{n})t_{n}w_{n}dx\geq\int_{B_{1/n}(0)}f(t_{n}w_{n})t_{n}w_{n}dx\hfill\\\
\ \ \ \ \geq\pi
M^{-1}_{0}(\beta_{0}-\epsilon)(t_{n}w_{n})^{\vartheta+1}e^{\alpha_{0}|t_{n}w_{n}|^{2}}n^{-2}\hfill\\\
\ \ \ \ =\pi M^{-1}_{0}(\beta_{0}-\epsilon)\bigg{[}\frac{t_{n}\sqrt{\log
n}}{(1+\sqrt[p]{\delta_{n}})\sqrt{2\pi}}\bigg{]}^{\vartheta+1}\exp\bigg{[}\alpha_{0}t_{n}^{2}\frac{\log
n}{2\pi(1+\sqrt[p]{\delta_{n}})^{2}}-2\log n\bigg{]}\hfill\\\ \end{gathered}$

indicating that $\\{t_{n}\\}$ is uniformly bounded in $n\in\mathbb{N}$. Up to
a subsequence if necessary, there exists a constant $t_{0}\in(0,+\infty)$ such
that $t_{n}\to t_{0}$. Since $F(s)\geq 0$ for all $s\in\mathbb{R}$, we invoke
from (3.8) and the second formula in (3.11) that

(3.12) $t_{0}^{2}\geq\frac{4\pi}{\alpha_{0}}.$

Choosing $\epsilon=\beta_{0}/2$, we apply (3.8)-(3.9) and $t_{n}\to t_{0}$ to
get

$(1-\vartheta)\log t_{0}\geq C\bigg{[}1+\frac{\vartheta+1}{2}\log(\log
n)+\big{(}\alpha_{0}t_{0}^{2}(2\pi)^{-1}-2\big{)}\log n\bigg{]}+o_{n}(1),$

where $C>0$ is independent of $n\in\mathbb{N}$. Recalling (3.12), we would
arrive at a contradiction by tending $n\to\infty$ and so (3.10) holds true.
The proof is completed. ∎

######  Lemma 3.7.

Let $1<p<2$, then each sequence $\\{u_{n}\\}\subset E$ satisfying (3.5) is
uniformly bounded in $E$. Moreover, there is a constant $K_{0}>0$ independent
of $n\in\mathbb{N}$ such that (2.10) holds true.

###### Proof.

Given a sequence $\\{u_{n}\\}\subset E$ satisfying (3.5), it follows from
$(f_{2})$ that

$\displaystyle c_{\infty}+o_{n}(1)$
$\displaystyle=J_{\infty}(u_{n})-\frac{1}{6}J^{\prime}_{\infty}(u_{n})[u_{n}]$
$\displaystyle\geq\frac{1}{3}\int_{\mathbb{R}^{2}}|\nabla
u_{n}|^{2}\mathrm{d}x+\left(\frac{1}{p}-\frac{1}{6}\right)a_{\infty}\int_{\mathbb{R}^{2}}|u_{n}|^{p}\mathrm{d}x.$

Recalling Lemma 3.6 and $1<p<2$, we see that $\\{\|u_{n}\|\\}$ is uniformly
bounded in $n\in\mathbb{N}$. Then, we are derived from
$\|u_{n}\|\|J_{\infty}^{\prime}(u_{n})\|_{E^{-1}}\to 0$ and (2.2) that

$\displaystyle\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{2}}[|\nabla
u_{n}|^{2}+3(A_{1}^{2}+A_{2}^{2})u_{n}^{2}]\mathrm{d}x+a_{\infty}\int_{\mathbb{R}^{2}}|u_{n}|^{p}\mathrm{d}x+o_{n}(1)$
$\displaystyle\leq\int_{\mathbb{R}^{2}}|\nabla
u_{n}|^{2}\mathrm{d}x+a_{\infty}\int_{\mathbb{R}^{2}}|u_{n}|^{p}\mathrm{d}x+3\bar{C}_{r}^{2}\|u_{n}\|^{6}+o_{n}(1)$

implying the desired result. The proof is completed. ∎

Before establishing the existence of ground state solutions for Eq. (1.15), we
shall introduce a new version type of Vanishing lemma with respect to our
variational setting.

######  Theorem 3.8.

Let $1<p<2$ and $r>0$. If $\\{u_{n}\\}$ is bounded in $E$ and suppose that

$\limsup_{n\to\infty}\sup_{y\in\mathbb{R}^{2}}\int_{B_{r}(y)}|u_{n}|^{p}dx=0,$

then $u_{n}\to 0$ in $L^{s}(\mathbb{R}^{2})$ for all $p<s<+\infty$.

###### Proof.

We follow the idea adopted in [38, Lemma 1.21] to conclude the proof and
exhibit it in detail for the convenience of the reader. First of all, we
recall the Gagliardo-Nirenberg inequality in [3] that

(3.13) $|u|_{s}^{s}\leq\mathcal{C}_{s}|\nabla
u|_{2}^{s-p}|u|_{p}^{p},\leavevmode\nobreak\ \forall u\in
E\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ p<s<+\infty,$

where the constant $\mathcal{C}_{s}>0$ only depends on $s$. So, we are derived
form (3.13) that

$\displaystyle\int_{B_{r}(y)}|u_{n}|^{s}dx$
$\displaystyle\leq\mathcal{C}_{s}\left(\int_{B_{r}(y)}|u_{n}|^{p}dx\right)\left(\int_{B_{r}(y)}|\nabla
u_{n}|^{2}dx\right)^{\frac{s-p}{2}}.$

Covering $\mathbb{R}^{2}$ by balls of radius $r$ in such a way that each point
of $\mathbb{R}^{2}$ is contained in at most $3$ balls, we are able to see that

$\int_{\mathbb{R}^{2}}|u_{n}|^{s}dx\leq
3\mathcal{C}_{s}\sup_{y\in\mathbb{R}^{2}}\left(\int_{B_{r}(y)}|u_{n}|^{p}dx\right)^{\frac{p(1-\varpi)}{q^{*}}}\|u_{n}\|^{s-p}.$

Under the assumption of this lemma, it holds that $u_{n}\to 0$ in
$L^{s}(\mathbb{R}^{2})$ for each $p<s<+\infty$. The proof is completed. ∎

We are now in a position to show the proof of Theorem 4.1.

######  _Proof of Theorem 3.1_.

Due to Lemmas 3.2-3.3 and Proposition 3.4, there is a sequence
$\\{u_{n}\\}\subset E$ satisfying (3.5). From Lemma 3.7, $\\{\|u_{n}\|\\}$ is
uniformly bounded in $n\in\mathbb{N}$. Passing to a subsequence if necessary,
using Lemma 2.1, there exists a $u_{\infty}\in E$ such that
$u_{n}\rightharpoonup u_{\infty}$ in $E$, $u_{n}\to u_{\infty}$ in
$L^{s}_{\text{loc}}(\mathbb{R}^{2})$ with $s\in(p,+\infty)$ and $u_{n}\to
u_{\infty}$ a.e. in $\mathbb{R}^{2}$. We claim that, there are
$y\in\mathbb{R}^{2}$ and $r,\tau>0$ such that

(3.14) $\int_{B_{r}(y)}|u_{n}|^{p}\mathrm{d}x\geq\tau.$

Otherwise, thanks to Theorem 3.8, we obtain that $u_{n}\to 0$ in
$L^{s}(\mathbb{R}^{2})$ for every $s\in(p,+\infty)$. According to Lemma 3.7,
we now take a similar calculations in (2.11) to deduce that

(3.15) $\lim_{n\to\infty}\int_{\mathbb{R}^{2}}F(u_{n})dx=0.$

Our next goal is to show that

(3.16) $\lim_{n\to\infty}\int_{\mathbb{R}^{2}}f(u_{n})u_{n}dx=0.$

Indeed, on the one hand, taking (3.15), $J_{\infty}(u_{n})\to c_{\infty}$ and
Lemma 3.6 into account that $\limsup\limits_{n\to\infty}|\nabla
u_{n}|_{2}^{2}<\frac{4\pi}{\alpha_{0}}$. Thereby, we shall choose
$\alpha>\alpha_{0}$ sufficiently close to $\alpha_{0}$ and $\nu>1$
sufficiently close to 1 in such a way that
$\frac{1}{\nu}+\frac{1}{\nu^{\prime}}=1$ with $\nu^{\prime}>1$ and

$\alpha\nu|\nabla u_{n}|_{2}^{2}<4\pi(1-\epsilon)\leavevmode\nobreak\
\text{for some suitable}\leavevmode\nobreak\ \epsilon\in(0,1).$

We define

$\bar{u}_{n}=\sqrt{\frac{\alpha\nu}{4\pi(1-\epsilon)}}u_{n},\leavevmode\nobreak\
\forall n\in\mathbb{N}.$

Obviously, one sees that $|\nabla\bar{u}_{n}|_{2}^{2}\leq 1$ for all
sufficiently $n\in\mathbb{N}$ and $|\bar{u}_{n}|_{p}^{p}$ is uniformly bounded
in $n\in\mathbb{N}$. On the other hand, we apply (1.14) in (2.8) to get

$\displaystyle\int_{\mathbb{R}^{2}}f(u_{n})u_{n}dx$
$\displaystyle\leq\varepsilon\int_{\mathbb{R}^{2}}|u_{n}|^{2}dx+C_{\varepsilon}\int_{\mathbb{R}^{2}}|u_{n}|^{q}\Phi_{\alpha,j_{0}}(u_{n})dx$
$\displaystyle\leq\varepsilon\int_{\mathbb{R}^{2}}|u_{n}|^{2}dx+C_{\varepsilon}\left(\int_{\mathbb{R}^{2}}|u_{n}|^{q\nu^{\prime}}dx\right)^{\frac{1}{\nu^{\prime}}}\left(\int_{\mathbb{R}^{2}}\Phi_{4\pi(1-\epsilon),j_{0}}(\bar{u}_{n})dx\right)^{\frac{1}{\nu}}$
$\displaystyle\leq\varepsilon\int_{\mathbb{R}^{2}}|u_{n}|^{2}dx+C_{\varepsilon}\mathbb{S}(4\pi)\left(\int_{\mathbb{R}^{2}}|u_{n}|^{q\nu^{\prime}}dx\right)^{\frac{1}{\nu^{\prime}}}.$

Letting $n\to\infty$ and then tending $\varepsilon\to 0^{+}$, we reach the
desired result (3.16). With (3.16) in hands, as a direct consequence of
$J_{\infty}^{\prime}(u_{n})[u_{n}]\to 0$, we derive that $|\nabla
u_{n}|_{2}^{2}\to 0$, $|u_{n}|_{p}^{p}\to 0$ and
$\int_{\mathbb{R}^{2}}(A_{1}^{2}+A_{2}^{2})u_{n}^{2}dx\to 0$. Exploiting
(3.15) and $J_{\infty}(u_{n})\to c_{\infty}$ again, it immediately concludes
that $c_{\infty}\equiv 0$ which contradicts with $c_{\infty}>0$ in Lemma 3.6.
So, we see that (3.14) must hold true.

According to (3.14), we define $v_{n}=u_{n}(\cdot+y_{n})$ for every
$n\in\mathbb{N}$. Since both $J_{\infty}$ and $J^{\prime}_{\infty}$ are
translation invariant in $\mathbb{R}^{2}$, one knows that $\\{v_{n}\\}$ is
still a $(C)$ sequence of $J_{\infty}$ at the level $c_{\infty}$. Arguing as
before, passing to a subsequence if necessary, $v_{n}\rightharpoonup v$ in
$E$, $v_{n}\to v$ in $L^{s}_{\text{loc}}(\mathbb{R}^{2})$ with
$s\in(p,+\infty)$ and $v_{n}\to v$ a.e. in $\mathbb{R}^{2}$. Moreover, we can
see that $v\neq 0$ by (3.14). As a consequence, without loss of generality, we
consider the sequence $\\{u_{n}\\}$ instead of $v_{n}$ to suppose that
$u_{\infty}\neq 0$. In view of (2.3) and (2.12), there holds
$J^{\prime}_{\infty}(u_{\infty})=0$ and so
$u_{\infty}\in\mathcal{N}_{\infty}$. We are then derived from (3.5) and the
Fatou’s lemma that

$\displaystyle c_{\infty}$
$\displaystyle=\liminf_{n\to\infty}J_{\infty}(u_{n})=\liminf_{n\to\infty}\left\\{J_{\infty}(u_{n})-\frac{1}{6}J^{\prime}_{\infty}(u_{n})[u_{n}]\right\\}$
$\displaystyle=\liminf_{n\to\infty}\left\\{\frac{1}{3}\int_{\mathbb{R}^{2}}|\nabla
u_{n}|^{2}\mathrm{d}x+\left(\frac{1}{p}-\frac{1}{6}\right)a_{\infty}\int_{\mathbb{R}^{2}}|u_{n}|^{p}\mathrm{d}x+\frac{1}{6}\int_{\mathbb{R}^{2}}[f(u_{n})u_{n}-6F(u_{n})]\mathrm{d}x\right\\}$
$\displaystyle\geq\frac{1}{3}\int_{\mathbb{R}^{2}}|\nabla
u_{\infty}|^{2}\mathrm{d}x+\left(\frac{1}{p}-\frac{1}{6}\right)a_{\infty}\int_{\mathbb{R}^{2}}|u_{\infty}|^{p}\mathrm{d}x+\frac{1}{6}\int_{\mathbb{R}^{2}}[f(u_{\infty})u_{\infty}-6F(u_{\infty})]\mathrm{d}x$
(3.17)
$\displaystyle=J_{\infty}(u_{\infty})-\frac{1}{6}J^{\prime}_{\infty}(u_{\infty})[u_{\infty}]=J_{\infty}(u_{\infty})\geq
m_{\infty}.$

From which, it follows from Lemma 3.5 that $u_{n}\to u_{\infty}$ in $E$ along
a subsequence. In other words, we deduce that $u_{\infty}$ is a solution of
Eq. (1.15) with $J_{\infty}(u_{\infty})=m_{\infty}$. The positivity of
$u_{\infty}$ is trivial, and so we omit it here. The proof is completed. ∎

######  Remark 3.9.

We invite the reader to see that Theorem 1.7 is a direct corollary of Theorem
3.1.

##  4\. Proof od Theorem 1.5

In this section, we are going to investigate the existence of positive ground
state solutions for Eq. (1.1). From the view point of variational method, we
search for critical points of $J_{a}$ defined in (3.1). Recalling the
discussions in Section 2, $J_{a}$ is well-defined and of class of
$C^{1}(E,\mathbb{R})$.

We shall prove the following result.

######  Theorem 4.1.

Let $1<p<2$ and suppose $(A_{1})-(A_{2})$. If $f$ satisfies (1.10) and
$(f_{1})-(f_{4})$, then Eq. (1.1) admits at least a positive ground state
solution $u_{a}\in E$ such that

$J_{a}(u_{a})=m_{a}=\inf_{u\in E\backslash\\{0\\}}\max_{t\geq 0}J_{a}(tu).$

Associated with $J_{a}$, we have the Nehari manifold given by

$\mathcal{N}_{a}=\\{u\in E\setminus\\{0\\}\,:\,J^{\prime}_{a}(u)[u]=0\\},$

and define the minimization problem

(4.1) $m_{a}=\min_{u\in\mathcal{N}_{a}}J_{a}(u).$

By definitions of $m_{\infty}$ and $m_{a}$ in (3.2) and (4.1), adopting
$({A}_{2})$, it is easy to check that

(4.2) $m_{a}<m_{\infty}.$

Moreover, because $a$ is a positive and bounded function, it permits us to
repeat the arguments in Section 3 to find a $(C)$ sequence $\\{u_{n}\\}\subset
E$ of $J_{a}$ at the level $c_{a}$, where

(4.3)
$c_{a}\triangleq\inf_{\gamma\in\Gamma_{a}}\max_{t\in[0,1]}J_{a}(\gamma(t))>0$

with
$\Gamma_{a}=\\{\gamma\in\mathcal{C}([0,1],E):\gamma(0)=0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ J_{a}(\gamma(1))<0\\}$. We can also deduce that

(4.4) $m_{a}=c_{a}=d_{a}\triangleq\inf_{u\in E\backslash\\{0\\}}\max_{t\geq
0}J_{a}(tu)$

and

(4.5) $0<c_{a}<\frac{2\pi}{\alpha_{0}}.$

Actually, the essential, or unique, difference between the proof of Theorem
3.1 and that of Theorem 4.1 is that whether the variational functional is
translation invariant in $\mathbb{R}^{2}$. Clearly, we realize that $J_{a}$
does not have such a good property because of the appearance of the
nonconstant potential $a$. Thereby, to reach the proof, it is enough to verify
that the weak limit of $(C)_{c_{a}}$ sequence is nontrivial.

######  Lemma 4.2.

Under the assumptions of Theorem 4.1, if $\\{u_{n}\\}\subset\mathcal{N}_{a}$
denotes a $(C)_{c_{a}}$ sequence of $J_{a}$ and $u_{n}\rightharpoonup u_{a}$
in $E$ along a subsequence, then $u_{a}\neq 0$.

###### Proof.

Assume by contradiction that $u_{a}=0$ and let $t_{n}>0$ such that
$t_{n}u_{n}\in\mathcal{N}_{\infty}$ by Lemma 3.5. The standard calculations
show that $\\{t_{n}\\}$ is bounded, and so,

$c_{a}+o_{n}(1)=J_{a}(u_{n})=\max_{t\geq 0}J_{a}(tu_{n})\geq
J_{a}(t_{n}u_{n})=J_{\infty}(t_{n}u_{n})+\frac{t_{n}^{p}}{p}\int_{\mathbb{R}^{2}}(a(x)-a_{\infty})|u_{n}|^{p}dx,$

where we have used Lemma 3.5 again in the second equality. From this,

$c_{a}+o_{n}(1)\geq
m_{\infty}+\frac{t_{n}^{p}}{p}\int_{\mathbb{R}^{2}}(a(x)-a_{\infty})|u_{n}|^{p}dx.$

In light of the fact that $\\{t_{n}\\}$ is bounded, we can make use of
$(A_{2})$ to have that

$\int_{\mathbb{R}^{2}}(a(x)-a_{\infty})|u_{n}|^{p}dx\to 0$

loading to

$c_{a}\geq m_{\infty},$

which contradicts with (4.2) and (4.4). The proof is completed. ∎

######  _Proof of Theorem 4.1_.

Owing to (4.4), we just need to find a sequence
$\\{u_{n}\\}\subset\mathcal{N}_{a}$ and it is a $(C)_{m_{a}}$ sequence of
$J_{a}$. It is standard, we refer the reader to [30, Theorem 1.1] and so the
proof is done. ∎

##  5\. Proof od Theorem 1.9

In this section, we aim to derive that Eq. (1.15) admits a mountain-pass type
solution whose energy is equal to the mountain-pass level.

The main result in this direction can be stated as follows.

######  Theorem 5.1.

Let $1<p<2$ and suppose $(A_{1})-(A_{2})$. If $f$ satisfies (1.10) and
$(f_{1})-(f^{\prime}_{2})$ as well as $(f_{3})-(f_{4})$, then Eq. (1.15) has a
positive mountain-pass type solution in $u\in E$ with
$J_{\infty}(u)=c_{\infty}$, where $J_{\infty}$ and $c_{\infty}$ are defined by
(3.1) and (3.6), respectively.

As we have pointed out in the Introduction, when $(f_{2})$ is absence, we
cannot restore the compactness as what we have done in the Sections 3 and 4.
Speaking it clearly, let $\\{u_{n}\\}\subset E$ be a $(C)_{c_{\infty}}$
sequence of $J_{\infty}$, it is impossible to conclude that $\\{u_{n}\\}$
admits a strongly convergent subsequence in $E$ by (3). The existence of such
a sequence is guaranteed by adopting some very similar calculations in Section
3.

Whereas, since the whole space $\mathbb{R}^{2}$ itself also results in the
lack of compactness, we shall always restrict ourselves in the radially
symmetric subspace of $E$. In other words, in this section, we prefer to take
advantage of $E_{r}$ to be the work space, instead of $E$.

Now, we are able to verify that the variational functional $J_{\infty}$
satisfies the $(C)$ condition at the level $c_{\infty}$.

######  Lemma 5.2.

Under the assumptions of Theorem 5.1, if $\\{u_{n}\\}\subset E_{r}$ is a
$(C)_{c_{\infty}}$ sequence of $J_{\infty}$, then there is a $u_{0}\in E$ such
that $u_{n}\to u_{0}$ in $E_{r}$ along a subsequence.

###### Proof.

Proceeding as the proof of Lemma 3.7, $\\{\|u_{n}\|\\}$ is uniformly bounded
in $n\in\mathbb{N}$. Passing to a subsequence if necessary, there is a
$u_{0}\in E_{r}$ such that $u_{n}\rightharpoonup u_{0}$ in $E_{r}$, $u_{n}\to
u_{0}$ in $L^{s}(\mathbb{R}^{N})$ with $s>p$ and $u_{n}\to u_{0}$ a.e. in
$\mathbb{R}^{N}$. Combining (2.3) and (2.12), one has
$J^{\prime}_{\infty}(u_{0})=0$ which implies that

(5.1)
$J_{\infty}(u_{0})=J_{\infty}(u_{0})-\frac{1}{6}J^{\prime}_{\infty}(u_{0})[u_{0}]\geq
0.$

Moreover, it follows from the Brézis-Lieb lemma, (2.4), (2.11) and (5.1) that

$\displaystyle c_{\infty}$ $\displaystyle=\frac{1}{2}|\nabla
u_{n}|_{2}^{2}+\frac{a_{\infty}}{p}|u_{n}|_{p}^{p}+\frac{1}{2}\int_{\mathbb{R}^{2}}(A_{1}^{2}[u_{n}]+A_{2}^{2}[u_{n}])u_{n}^{2}dx-\int_{\mathbb{R}^{2}}F(u_{n})dx+o_{n}(1)$
$\displaystyle=\frac{1}{2}|\nabla u_{n}-\nabla
u_{0}|_{2}^{2}+\frac{a_{\infty}}{p}|u_{n}-u_{0}|_{p}^{p}+\frac{1}{2}\int_{\mathbb{R}^{2}}(A_{1}^{2}[u_{n}-u_{0}]+A_{2}^{2}[u_{n}-u_{0}])(u_{n}-u_{0})^{2}dx$
$\displaystyle\ \ \ \ +\frac{1}{2}|\nabla
u_{0}|_{2}^{2}+\frac{a_{\infty}}{p}|u_{0}|_{p}^{p}+\frac{1}{2}\int_{\mathbb{R}^{2}}(A_{1}^{2}[u_{0}]+A_{2}^{2}[u_{0}])u_{0}^{2}dx-\int_{\mathbb{R}^{2}}F(u_{0})dx+o_{n}(1)$
$\displaystyle\geq\frac{1}{2}|\nabla u_{n}-\nabla
u_{0}|_{2}^{2}+J_{\infty}(u_{0})+o_{n}(1)\geq\frac{1}{2}|\nabla u_{n}-\nabla
u_{0}|_{2}^{2}+o_{n}(1).$

From which and Lemma 3.6, then $\limsup\limits_{m\to\infty}|\nabla
u_{n}-\nabla u_{0}|_{2}^{2}<\frac{4\pi}{\alpha_{0}}$. Consequently, we shall
choose $\alpha>\alpha_{0}$ sufficiently close to $\alpha_{0}$ and $\nu>1$
sufficiently close to 1 in such a way that
$\frac{1}{\nu}+\frac{1}{\nu^{\prime}}=1$ with $\nu^{\prime}>1$ and

$\alpha\nu|\nabla u_{n}-\nabla
u_{0}|_{2}^{2}<4\pi(1-\hat{\epsilon})\leavevmode\nobreak\ \text{for some
suitable}\leavevmode\nobreak\ \epsilon\in(0,1).$

We define

$\hat{u}_{n}=\sqrt{\frac{\alpha\nu}{4\pi(1-\hat{\epsilon})}}(u_{n}-u_{0}),\leavevmode\nobreak\
\forall n\in\mathbb{N}.$

Obviously, one sees that $|\nabla\hat{u}_{n}|_{2}^{2}\leq 1$ for all
sufficiently $n\in\mathbb{N}$ and $|\hat{u}_{n}|_{p}^{p}$ is uniformly bounded
in $n\in\mathbb{N}$. Besides, for the above fixed $\hat{\epsilon}\in(0,1)$, we
need the following two types of Young’s inequality

$|a+b|^{2}\leq(1+\hat{\epsilon})|a|^{2}+(1+\hat{\epsilon}^{-1})|b|^{2},\leavevmode\nobreak\
\forall a,b\in\mathbb{R}$

and

$e^{a+b}-d\leq\frac{1}{1+\hat{\epsilon}}\big{[}e^{(1+\hat{\epsilon})a}-d\big{]}+\frac{\hat{\epsilon}}{1+\hat{\epsilon}}\big{[}e^{(1+\hat{\epsilon}^{-1})b}-d\big{]},\leavevmode\nobreak\
\forall a,b,d\in\mathbb{R}.$

By means of the above facts together with (1.14), we derive

$\displaystyle\int_{\mathbb{R}^{2}}\Phi_{\alpha\nu,j_{0}}(u_{n})dx$
$\displaystyle\leq\frac{1}{1+\bar{\epsilon}}\int_{\mathbb{R}^{2}}\Phi_{4\pi(1+\bar{\epsilon})^{-2},j_{0}}(\hat{u}_{n})dx+\frac{\bar{\epsilon}}{1+\bar{\epsilon}}\int_{\mathbb{R}^{2}}\Phi_{\nu\alpha(1+\bar{\epsilon}^{-1})^{2},j_{0}}(u_{0})dx$
$\displaystyle\leq\frac{\mathbb{S}(4\pi)}{1+\bar{\epsilon}}+\frac{C_{2}\bar{\epsilon}}{1+\bar{\epsilon}}\leq
C_{3}<+\infty,\leavevmode\nobreak\ \forall n\in\mathbb{N}^{+}.$

As a consequence, by (2.8), we obtain

$\displaystyle\bigg{|}\int_{\mathbb{R}^{2}}f(u_{n})(u_{n}-u_{0})dx\bigg{|}$
$\displaystyle\leq\bigg{(}\int_{\mathbb{R}^{2}}|u_{n}|^{2}dx\bigg{)}^{\frac{1}{2}}\bigg{(}\int_{\mathbb{R}^{2}}|u_{n}-u_{0}|^{2}dx\bigg{)}^{\frac{1}{2}}$
$\displaystyle\ \ \ \
+C|u_{n}|_{2(q-1)\nu^{\prime}}^{(q-1)}|u_{n}-u_{0}|_{2\nu^{\prime}}\bigg{(}\int_{\mathbb{R}^{2}}\Phi_{\alpha\nu,j_{0}}(u_{n})dx\bigg{)}^{\frac{1}{\nu}}$
(5.2) $\displaystyle=o_{n}(1).$

It is simple to see that

(5.3) $\int_{\mathbb{R}^{2}}f(u_{0})(u_{n}-u_{0})dx=o_{n}(1).$

For all $1<r<2$, thanks to the significant inequality [36, (2.2)] which can be
stated as follows

$(|y_{2}|^{r-2}y_{2}-|y_{1}|^{r-2}y_{1})\cdot(y_{2}-y_{1})\geq\hat{C}_{r}\cdot\frac{|y_{2}-y_{1}|^{2}}{(|y_{2}|+|y_{1}|)^{2-r}}.$

From which, using $J_{\infty}^{\prime}(u_{n})=o_{n}(1)$ and
$J_{\infty}^{\prime}(u_{0})=0$ as well as (2.3) and (5)-(5.3), it holds that

$\displaystyle o_{n}(1)$
$\displaystyle=J_{\infty}^{\prime}(u_{n})[u_{n}-u_{0}]-J_{\infty}^{\prime}(u_{0})[u_{n}-u_{0}]$
$\displaystyle=\int_{\mathbb{R}^{2}}\big{[}|\nabla u_{n}-\nabla
u_{0}|^{2}+(|u_{n}|^{p-2}u_{n}-|u_{0}|^{p-2}u_{0})(u_{n}-u_{0})\big{]}dx$
$\displaystyle\ \ \ \
+\int_{\mathbb{R}^{2}}(A_{1}^{2}[u_{n}]u_{n}+A_{2}^{2}[u_{n}]u_{n})(u_{n}-u)dx+\int_{\mathbb{R}^{2}}(A_{1}^{2}[u_{0}]u_{0}+A_{2}^{2}[u_{0}]u_{0})(u_{n}-u_{0})dx$
$\displaystyle\ \ \ \
+\int_{\mathbb{R}^{2}}f(u_{n})(u_{n}-u_{0})dx+\int_{\mathbb{R}^{2}}f(u_{0})(u_{n}-u_{0})dx$
$\displaystyle=\int_{\mathbb{R}^{2}}\big{[}|\nabla u_{n}-\nabla
u_{0}|^{2}+(|u_{n}|^{p-2}u_{n}-|u_{0}|^{p-2}u_{0})(u_{n}-u_{0})\big{]}dx+o_{n}(1)$
$\displaystyle\geq o_{n}(1)$

yielding that

$|\nabla u_{n}-\nabla u_{0}|^{2}_{2}=o_{n}(1)\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\
\int_{\mathbb{R}^{2}}(|u_{n}|^{p-2}u_{n}-|u_{0}|^{p-2}u_{0})(u_{n}-u_{0})dx.$

At this stage, we apply the Hölder’s inequality to get

$\displaystyle\int_{\mathbb{R}^{2}}|u_{n}-u_{0}|^{p}dx$
$\displaystyle\leq\hat{C}_{p}^{-\frac{p}{2}}\int_{\mathbb{R}^{2}}|(|u_{n}|^{p-2}u_{n}-|u_{0}|^{p-2}u_{0})(u_{n}-u_{0})|^{\frac{p}{2}}(|u_{n}|+|u_{0}|)^{\frac{p(2-p)}{2}}dx$
$\displaystyle\leq\hat{C}_{p}^{-\frac{p}{2}}\bigg{(}\int_{\mathbb{R}^{2}}|(|u_{n}|^{p-2}u_{n}-|u_{0}|^{p-2}u_{0})(u_{n}-u_{0})|dx\bigg{)}^{{\frac{p}{2}}}\bigg{(}\int_{\mathbb{R}^{2}}(|u_{n}|+|u_{0}|)^{p}dx\bigg{)}^{{\frac{2-p}{2}}}$
$\displaystyle\leq
C\bigg{(}\int_{\mathbb{R}^{2}}|(|u_{n}|^{p-2}u_{n}-|u_{0}|^{p-2}u_{0})(u_{n}-u_{0})|dx\bigg{)}^{{\frac{p}{2}}}=o_{n}(1).$

Thus, we can derive that $u_{n}\to u_{0}$ in $E_{r}$ as $n\to\infty$. The
proof is completed. ∎

######  _Proof of Theorem 5.1_.

In view of Section 3, there is a $(C)_{c_{\infty}}$ sequence
$\\{u_{n}\\}\subset E_{r}$ of $J_{\infty}$. So, we can finish the proof by
Lemma 5.2. This proof also concludes Theorem 1.9. ∎

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