# Semiclassical states for a magnetic nonlinear Schrödinger equation with
exponential critical growth in $\mathbb{R}^{2}$

Pietro d’Avenia
Dipartimento di Meccanica, Matematica e Management
Politecnico di Bari
Via Orabona 4, 70125 Bari, Italy<EMAIL_ADDRESS>and  Chao Ji
Department of Mathematics
East China University of Science and Technology
Shanghai 200237, PR China<EMAIL_ADDRESS>
###### Abstract.

This paper is devoted to the magnetic nonlinear Schrödinger equation

$\Big{(}\frac{\varepsilon}{i}\nabla-A(x)\Big{)}^{2}u+V(x)u=f(|u|^{2})u\text{
in }\mathbb{R}^{2},$

where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow\mathbb{R}$
and $A:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ are continuous functions and
$f:\mathbb{R}\rightarrow\mathbb{R}$ is a $C^{1}$ function having exponential
critical growth. Under a global assumption on the potential $V$, we use
variational methods and Ljusternick-Schnirelmann theory to prove existence,
multiplicity, concentration, and decay of nontrivial solutions for
$\varepsilon>0$ small.

###### Key words and phrases:

Nonlinear Schrödinger equation, Magnetic field, Exponential critical growth,
Trudinger-Moser inequality, Variational methods.

###### 2010 Mathematics Subject Classification:

35J20, 35J60, 35B33.

###### Contents

  1. 1 Introduction and main results
  2. 2 The variational framework and some preliminaries
  3. 3 The limit problem
  4. 4 Proof of Theorem 1.1
  5. 5 Proof of Theorem 1.2
  6. 6 Proof of Theorem 1.3

##  1\. Introduction and main results

In this paper, we consider the following nonlinear Schrödinger equation

(1.1)
$\Big{(}\frac{\varepsilon}{i}\nabla-A(x)\Big{)}^{2}u+V(x)u=f(|u|^{2})u\quad\hbox{in
}\mathbb{R}^{2},$

where $\varepsilon>0$ is a parameter, $u:\mathbb{R}^{2}\to\mathbb{C}$, $A\in
C(\mathbb{R}^{2},\mathbb{R}^{2})$, $V\in C(\mathbb{R}^{2},\mathbb{R})$
satisfies

(V)
$V_{\infty}=\liminf_{|x|\rightarrow+\infty}V(x)>V_{0}=\inf_{x\in\mathbb{R}^{2}}V(x)>0,$

where $V_{\infty}$ can be also $+\infty$, and $f\in
C^{1}(\mathbb{R},\mathbb{R})$ satisfies the following assumptions:

  1. ($f_{1}$)

$f(t)=0$ if $t\leq 0$;

  2. ($f_{2}$)

there holds

$\lim_{t\rightarrow+\infty}\frac{f(t^{2})t}{e^{\alpha
t^{2}}}=\begin{cases}0,&\hbox{for }\alpha>4\pi,\\\ +\infty,&\hbox{for
}0<\alpha<4\pi,\end{cases}$

(exponential critical growth);

  3. ($f_{3}$)

there is a positive constant $\theta>2$ such that, for every $t>0$,

$0<\frac{\theta}{2}F(t)\leq tf(t),$

where $F(t)=\displaystyle\int_{0}^{t}f(s)ds$;

  4. ($f_{4}$)

there exist two constants $p>2$ and

$C_{p}>\Big{(}\frac{8\theta}{\theta-2}\Big{)}^{\frac{p-2}{2}}\Big{(}\frac{p-2}{p}\Big{)}^{\frac{p-2}{2}}V^{*}S_{p}^{p/2}>0$

such that

$f^{\prime}(t)\geq\frac{p-2}{2}C_{p}t^{(p-4)/2}\quad\text{for all}\,\,t>0,$

where

(1.2) $S_{p}=\inf_{u\in
H^{1}(\mathbb{R}^{2},\mathbb{R})\backslash\\{0\\}}\frac{\|\nabla
u\|_{2}^{2}+\|u\|_{2}^{2}}{\|u\|_{p}^{2}}\text{ and
}V^{*}=\left\\{\begin{array}[]{l}V_{\infty},\quad\quad\text{if}\,\,V_{\infty}<+\infty,\\\
V_{0},\quad\quad\,\,\text{if}\,\,V_{\infty}=+\infty;\end{array}\right.$

  5. ($f_{5}$)

$f^{\prime}(t)\leq(e^{4\pi t}-1)$ for any $t\geq 0$.

Equation (1.1) appears in the study of standing wave solutions
$\psi(x,t):=e^{-i\omega/\hbar}u(x)$, with $\omega\in\mathbb{R}$, of the planar
nonlinear Schrödinger equation

$i\hbar\frac{\partial\psi}{\partial
t}=\Big{(}\frac{\hbar}{i}\nabla-A(x)\Big{)}^{2}\psi+U(x)\psi-f(|\psi|^{2})\psi\quad\hbox{in
}\mathbb{R}^{2}\times\mathbb{R}.$

In particular, the operator

$\psi\mapsto\Big{(}i\hbar\frac{\partial}{\partial
t}-U(x)\Big{)}\psi-\Big{(}\frac{\hbar}{i}\nabla-A(x)\Big{)}^{2}\psi$

appears in the study of the interaction of a matter field $\psi$ with an
external electromagnetic field whose electric potential is $U$ and whose
magnetic potential is $A$ (here we are considering an electromagnetic
potential which does not depend on the time variable) though the minimal
coupling rule. Moreover, from a physical point of view, the study of the
existence of solutions of (1.1), in particular of  its  ground states, and of
their behavior as $\varepsilon\to 0^{+}$ (which is equivalent to send the
Planck constant $\hbar$ to zero and it  is known as semiclassical limit) is of
the greatest importance in the theory of superconductivity (see [23, 26] and
references therein) and since the transition from Quantum Mechanics to
Classical Mechanics can be formally performed by this limit.

If the magnetic potential $A\equiv 0$ and $\mathbb{R}^{2}$ is replaced by
$\mathbb{R}^{N}$, then (1.1) reduces to a following nonlinear Schrödinger
equation

(1.3) $-\varepsilon^{2}\Delta u+V(x)u=f(|u|^{2})u\quad\hbox{in
}\mathbb{R}^{N}.$

A vast literature is dedicated to (1.3), in particular concerning existence
and multiplicity of solutions, by several methods. If the potential $V$
satisfies the global assumption (V), introduced by Rabinowitz in [32],
Cingolani and Lazzo in [17] considered multiplicity and concentration of
positive solutions of (1.3) with $f(s)=s^{(p-2)/2}$ with $p>2$ if $N=1,2$ and
$2<p<2N/(N-2)$ if $N\geq 3$. Then, Alves and Figueiredo in [2] generalized the
results in [17] to the $p$-Laplacian operator for a class of nonlinearities
including power growth and in [3] considered multiplicity and concentration of
positive solutions for $N$-Laplacian equation with exponential critical growth
in $\mathbb{R}^{N}$ for $N\geq 2$. Finally, let us mention [1, 5], that are
devoted to critical cases in $\mathbb{R}^{2}$.

Moreover, after the seminal paper [25], also the magnetic nonlinear
Schrödinger equation (1.1) has been extensively investigated by variational
and topological methods (see [9, 15, 16, 19, 21, 29, 30] and references
therein). In particular, in [4], Alves, Figueiredo, and Furtado, using the
penalization method and Ljusternik-Schnirelmann category theory, obtained a
multiplicity result for (1.1) with a subcritical right hand side; in [10],
Barile and Figueiredo considered the existence of a solution of (1.1) with an
exponentially critically growing nonlinearity in $\mathbb{R}^{2}$; in [20], we
got multiplicity and concentration for (1.1) when the potential $V$ satisfies
a local assumption due to del Pino and Felmer [22]. Let us mention also the
recent contributions [7] concerning a multiplicity result for a nonlinear
fractional magnetic Schrödinger equation with exponential critical growth in
the one-dimensional case and [8], where existence and multiplicity for a
nonlinear fractional magnetic Schrödinger equation in $\mathbb{R}^{N}$, $N\geq
3$, in the presence of a subcritical nonlinearity and of a potential $V$
satisfying a global assumption, has been proved.

The main result of this paper is the following.

######  Theorem 1.1.

Assume that (V) and ($f_{1}$)–($f_{5}$) hold. Then, there exists
$\varepsilon_{0}>0$ such that problem (1.1) admits a ground state solution for
any $\varepsilon\in(0,\varepsilon_{0})$.

Moreover, introducing the sets

(1.4) $M:=\\{x\in\mathbb{R}^{2}:V(x)=V_{0}\\}$

and

$M_{\delta}:=\\{x\in\mathbb{R}^{2}:\text{dist}(x,M)\leq\delta\\},\ \delta>0,$

we are able to show, for small $\varepsilon>0$, a multiplicity result for
problem (1.1) and concentration phenomena of such solutions. More precisely,
we have

######  Theorem 1.2.

Assume that (V) and ($f_{1}$)–($f_{5}$) hold. Then, for any $\delta>0$, there
exists $\varepsilon_{\delta}>0$ such that problem (1.1) has at least
$\text{cat}_{M_{\delta}}(M)$ nontrivial solutions, for any
$0<\varepsilon<\varepsilon_{\delta}$.

Finally, for our solutions we are able to prove their concentration and decay.
Indeed we have

######  Theorem 1.3.

Let $\\{\varepsilon_{n}\\}$ be a sequence in $(0,+\infty)$ such that
$\varepsilon_{n}\rightarrow 0^{+}$, and, for $n$ large enough, let
$\\{u_{\varepsilon_{n}}\\}$ be a sequence of solutions found in Theorem 1.1 or
in Theorem 1.2. If $\eta_{n}\in\mathbb{R}^{2}$ is a global maximum point of
$|u_{\varepsilon_{n}}|$, then

$\underset{n\rightarrow\infty}{\lim}V(\eta_{n})=V_{0}.$

Moreover, if, furtherly, $A\in C^{1}(\mathbb{R}^{2},\mathbb{R}^{2})$, then
there exist $C,c>0$ such that

(1.5) $|u_{\varepsilon_{n}}(x)|\leq
Ce^{-c\frac{|x-\eta_{n}|}{\varepsilon_{n}}}\text{ in }\mathbb{R}^{2}.$

We get our results combining variational methods and the Ljusternik-
Schnirelmann theory.

This paper is a completion of our joint research project started with [20],
where we assumed a local condition on $V$. Here the presence of the global
condition (V) and the weaker assumption $A\in
C(\mathbb{R}^{2},\mathbb{R}^{2})$, require more delicate estimates and a
careful and deeper analysis of some technical aspects. Moreover, comparing our
case with known results obtained for $A\equiv 0$, the presence of the magnetic
field leads to consider a complex valued problem and this implies to get more
delicate estimates. We are also able to weaken the assumptions of some
technical lemmas (see, for instance, Lemma 2.6) and to study the behavior of
the maximum points of the modulus of solutions and their decay both for the
ground states obtained in Theorem 1.1 and for solutions found in Theorem 1.2.

The paper is organized as follows. In Section 2 we introduce the functional
setting and we give some preliminaries. We study the limit problem in Section
3. Then, in Section 4, we prove Theorem 1.1. Section 5 is devoted to prove
Theorem 1.2. Finally, in the last section, we prove Theorem 1.3.

### Notation

  * •

$C,C_{1},C_{2},\ldots$ denote positive constants whose exact values can change
from line to line;

  * •

$B_{R}(y)$ denotes the open disk centered at $y\in\mathbb{R}^{2}$ with radius
$R>0$ and $B^{c}_{R}(y)$ denotes its complement in $\mathbb{R}^{2}$;

  * •

$\|\cdot\|$, $\|\cdot\|_{q}$, and $\|\cdot\|_{\infty}$ denote the usual norms
of the spaces $H^{1}(\mathbb{R}^{2},\mathbb{R})$,
$L^{q}(\mathbb{R}^{2},\mathbb{K})$, and
$L^{\infty}(\mathbb{R}^{2},\mathbb{K})$, respectively, where
$\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$.

##  2\. The variational framework and some preliminaries

In this section we introduce the functional spaces that we use and a classical
equivalent version of (1.1). We also present some results for a limit problem
which will be useful for our arguments.

From now on, in the whole paper, we will assume that (V) holds.

For a function $u:\mathbb{R}^{2}\rightarrow\mathbb{C}$, let us denote by

$\nabla_{A}u:=\Big{(}\frac{\nabla}{i}-A\Big{)}u$

and consider the space

$H_{A}^{1}(\mathbb{R}^{2},\mathbb{C}):=\\{u\in
L^{2}(\mathbb{R}^{2},\mathbb{C}):|\nabla_{A}u|\in
L^{2}(\mathbb{R}^{2},\mathbb{R})\\},$

which is an Hilbert space endowed with the scalar product

$\langle
u,v\rangle:=\operatorname{Re}\int_{\mathbb{R}^{2}}\Big{(}\nabla_{A}u\overline{\nabla_{A}v}+u\overline{v}\Big{)}dx,\quad\text{for
any }u,v\in H_{A}^{1}(\mathbb{R}^{2},\mathbb{C}),$

where $\operatorname{Re}$ and the bar denote the real part of a complex number
and the complex conjugation, respectively. Hence, let $\|\cdot\|_{A}$ be the
norm induced by this inner product.

Observe that a useful tool when we work on
$H_{A}^{1}(\mathbb{R}^{2},\mathbb{C})$ is the diamagnetic inequality

(2.1) $|\nabla_{A}u(x)|\geq|\nabla|u(x)||$

(see e.g. [31, Theorem 7.21]).
Moreover, a simple change of variables allows us to write equation (1.1) as

(2.2) $\Big{(}\frac{1}{i}\nabla-
A_{\varepsilon}(x)\Big{)}^{2}u+V_{\varepsilon}(x)u=f(|u|^{2})u\quad\hbox{in
}\mathbb{R}^{2},$

where $A_{\varepsilon}(x)=A(\varepsilon x)$ and
$V_{\varepsilon}(x)=V(\varepsilon x)$.
Thus, we introduce the Hilbert space

$H_{\varepsilon}:=\Big{\\{}u\in
H_{A_{\varepsilon}}^{1}(\mathbb{R}^{2},\mathbb{C}):\int_{\mathbb{R}^{2}}V_{\varepsilon}(x)|u|^{2}dx<\infty\Big{\\}}$

endowed with the scalar product

$\displaystyle\langle
u,v\rangle_{\varepsilon}:=\operatorname{Re}\int_{\mathbb{R}^{2}}\Big{(}\nabla_{A_{\varepsilon}}u\overline{\nabla_{A_{\varepsilon}}v}+V_{\varepsilon}(x)u\overline{v}\Big{)}dx$

and $\|\cdot\|_{\varepsilon}$ is the norm induced by
$\langle\cdot,\cdot\rangle_{\varepsilon}$.

For every $\varepsilon>0$, the functional space $H_{\varepsilon}$ satisfies
the following embeddings. For the sake of completeness here we give some
details about their proof.

######  Lemma 2.1.

The space $H_{\varepsilon}$ is continuously embedded in
$L^{r}(\mathbb{R}^{2},\mathbb{C})$ for $r\geq 2$ and compact embedded in
$L^{r}_{\rm{loc}}(\mathbb{R}^{2},\mathbb{C})$ for $r\geq 1$. Moreover, if
$V_{\infty}=+\infty$, then $H_{\varepsilon}$ is compactly embedded in
$L^{r}(\mathbb{R}^{2},\mathbb{C})$ for $r\geq 2$.

###### Proof.

According to the diamagnetic inequality (2.1), if $u\in H_{\varepsilon}$, then
$|u|\in H^{1}(\mathbb{R}^{2},\mathbb{R})$ and

(2.3) $\||u|\|\leq C\|u\|_{\varepsilon}.$

Thus, the continuous embedding $H_{\varepsilon}\hookrightarrow
L^{r}(\mathbb{R}^{2},\mathbb{C})$ is a consequence of (2.3) and of the
continuous embedding $H^{1}(\mathbb{R}^{2},\mathbb{R})\hookrightarrow
L^{r}(\mathbb{R}^{2},\mathbb{R})$ for $r\geq 2$.
To get the compact embedding of $H_{\varepsilon}$ in
$L^{r}_{\rm{loc}}(\mathbb{R}^{2},\mathbb{C})$ for $r\geq 1$ we can argue in
the following way.
Let $K$ be an open subset of $\mathbb{R}^{2}$ with compact closure.
First of all, we prove that $H_{\varepsilon}$ is continuously embedded in
$H^{1}(K,\mathbb{C})$.
Indeed we have

$\displaystyle\|u\|_{H^{1}(K,\mathbb{C})}^{2}$ $\displaystyle=\int_{K}|\nabla
u|^{2}dx+\int_{K}|u|^{2}dx=\int_{K}\Big{|}\frac{\nabla}{i}u\Big{|}^{2}dx+\int_{K}|u|^{2}dx$
$\displaystyle\leq
C\left(\int_{K}|\nabla_{A_{\varepsilon}}u|^{2}dx+\int_{K}|A_{\varepsilon}u|^{2}dx+\int_{K}V_{\varepsilon}(x)|u|^{2}dx\right)$
$\displaystyle\leq
C\left(\int_{K}|\nabla_{A_{\varepsilon}}u|^{2}dx+\int_{K}V_{\varepsilon}(x)|u|^{2}dx\right)\leq
C\|u\|_{\varepsilon}^{2}.$

Let now $\mathcal{F}$ be a bounded subset of $H_{\varepsilon}$. We want to
prove that $\mathcal{F}|_{K}$, which is a subset of $H^{1}(K,\mathbb{C})$ due
to the continuous embedding proved before, has compact closure in
$L^{r}(K,\mathbb{C})$, $r\geq 1$.
Let $\tilde{\mathcal{P}}$ be an extension operator found in [12, Theorem 9.7]
for $H^{1}(K,\mathbb{R})$ and $\mathcal{P}:=\tilde{\mathcal{P}}({\rm
Re}(u))+i\tilde{\mathcal{P}}({\rm Im}(u))$.
We know that, for each $v\in H^{1}(K,\mathbb{R})$,

$\tilde{\mathcal{P}}v|_{K}=v,\
\|\tilde{\mathcal{P}}v\|_{L^{2}(\mathbb{R}^{2},\mathbb{R})}\leq
C\|v\|_{L^{2}(K,\mathbb{R})}\text{ and
}\|\tilde{\mathcal{P}}v\|_{H^{1}(\mathbb{R}^{2},\mathbb{R})}\leq
C\|v\|_{H^{1}(K,\mathbb{R})}.$

Thus, we get

$\mathcal{P}u|_{K}=\tilde{\mathcal{P}}({\rm
Re}(u))|_{K}+i\tilde{\mathcal{P}}({\rm Im}(u))|_{K}={\rm Re}(u)+i{\rm
Im}(u)=u,$
$\displaystyle\|\mathcal{P}u\|_{L^{2}(\mathbb{R}^{2},\mathbb{C})}^{2}$
$\displaystyle=\|\tilde{\mathcal{P}}({\rm Re}(u))+i\tilde{\mathcal{P}}({\rm
Im}(u))\|_{L^{2}(\mathbb{R}^{2},\mathbb{C})}^{2}\leq C(\|{\rm
Re}(u)\|_{L^{2}(K,\mathbb{R})}^{2}+\|{\rm Im}(u)\|_{L^{2}(K,\mathbb{R})}^{2})$
$\displaystyle=C\|u\|_{L^{2}(K,\mathbb{C})}^{2}$

and, analogously,

$\displaystyle\|\mathcal{P}u\|_{H^{1}(\mathbb{R}^{2},\mathbb{C})}^{2}$
$\displaystyle=\|\tilde{\mathcal{P}}({\rm Re}(u))+i\tilde{\mathcal{P}}({\rm
Im}(u))\|_{H^{1}(\mathbb{R}^{2},\mathbb{C})}^{2}\leq C(\|{\rm
Re}(u)\|_{H^{1}(K,\mathbb{R})}^{2}+\|{\rm Im}(u)\|_{H^{1}(K,\mathbb{R})}^{2})$
$\displaystyle=C\|u\|_{H^{1}(K,\mathbb{C})}^{2}.$

Let now $\mathcal{H}:=\mathcal{P}(\mathcal{F}|_{K})$.
Since $H_{\varepsilon}$ is continuously embedded in $H^{1}(K,\mathbb{C})$, we
have that $\mathcal{H}$ is bounded in $H^{1}(\mathbb{R}^{2},\mathbb{C})$ and
$\mathcal{H}|_{K}=\mathcal{F}|_{K}$.
Thus we can proceed in a classical way, see e.g. [12, Theorem 9.16], using
[12, Theorem 4.26].
Indeed, first observe that, since $K$ is bounded, we may assume $r\geq 2$.
Then, let $u\in C_{c}^{\infty}(\mathbb{R}^{2},\mathbb{C})$ and
$\tau_{h}u:=u(\cdot+h)$, with $h\in\mathbb{R}^{2}$.
For every $x\in\mathbb{R}^{2}$ we have

$\displaystyle|\tau_{h}u(x)-u(x)|$ $\displaystyle=\sqrt{[{\rm Re}(u)(x+h)-{\rm
Re}(u)(x)]^{2}+[{\rm Im}(u)(x+h)-{\rm Im}(u)(x))]^{2}}$
$\displaystyle\leq|{\rm Re}(u)(x+h)-{\rm Re}(u)(x)|+|{\rm Im}(u)(x+h)-{\rm
Im}(u)(x)|$ $\displaystyle\leq\int_{0}^{1}|v_{\rm
R}^{\prime}(t)|dt+\int_{0}^{1}|v_{\rm I}^{\prime}(t)|dt$
$\displaystyle=\int_{0}^{1}|h\cdot\nabla{\rm
Re}(u)(x+th)|dt+\int_{0}^{1}|h\cdot\nabla{\rm Im}(u)(x+th)|dt$
$\displaystyle\leq|h|\left(\int_{0}^{1}(|\nabla{\rm Re}(u)(x+th)|+|\nabla{\rm
Im}(u)(x+th)|)dt\right)$ $\displaystyle\leq\sqrt{2}|h|\int_{0}^{1}|\nabla
u(x+th)|dt$

where $v_{\rm R}(t):={\rm Re}(u)(x+th)$ and $v_{\rm I}(t):={\rm Im}(u)(x+th)$,
and so, using the Hölder inequality,

$|\tau_{h}u(x)-u(x)|^{2}\leq 2|h|^{2}\int_{0}^{1}|\nabla u(x+th)|^{2}dt.$

Hence, using the density of $C_{c}^{\infty}(\mathbb{R}^{2},\mathbb{C})$ in
$H^{1}(\mathbb{R}^{2},\mathbb{C})$, see [25, Proposition 2.1],

$\|\tau_{h}u-u\|_{2}\leq C|h|\|\nabla u\|_{2}\leq C|h|$

uniformly with respect to $u\in\mathcal{H}$.
For $r>2$, if we take $q>r$, there exists $\alpha\in(0,1)$ such that

$\frac{1}{r}=\frac{\alpha}{2}+\frac{1-\alpha}{q},$

so that, for every $u\in\mathcal{H}$,

$\|\tau_{h}u-u\|_{r}\leq\|\tau_{h}u-u\|_{2}^{\alpha}\|\tau_{h}u-u\|_{q}^{1-\alpha}\leq
C|h|^{\alpha}$

with $C$ independent of the elements of $\mathcal{H}$.
Thus, [12, Theorem 4.26] implies that the closure of
$\mathcal{H}|_{K}=\mathcal{F}|_{K}$ is compact in $L^{r}(K,\mathbb{C})$.
Finally, if $V_{\infty}=+\infty$, let $\\{u_{n}\\}\subset H_{\varepsilon}$ be
a sequence weakly convergent to $u$ in $H_{\varepsilon}$ and $v_{n}:=u_{n}-u$.
We have that $v_{n}\to 0$ in $L^{2}(\mathbb{R}^{2},\mathbb{C})$.
Indeed, for every $\eta>0$, there exists $\sigma>0$ large enough such that

$\frac{1}{V(x)}<\eta\text{ in }B_{\sigma}^{c}(0).$

Moreover, since $v_{n}\rightharpoonup 0$ in $H_{\varepsilon}$, then $v_{n}\to
0$ in $L_{\rm loc}^{2}(\mathbb{R}^{2},\mathbb{C})$. Thus, for $n$ large
enough,

$\|v_{n}\|_{2}^{2}<\eta+\frac{\eta}{V_{0}}\int_{B_{\sigma/\varepsilon}^{c}(0)}V_{\varepsilon}(x)|v_{n}|^{2}dx<C\eta.$

Hence, using the Gagliardo-Nirenberg inequality applied to $|v_{n}|$, for
every $r\geq 2$ we have

$\|v_{n}\|_{r}\leq C\|\nabla|v_{n}|\|_{2}^{1-2/r}\|v_{n}\|_{2}^{2/r},$

and, by the diamagnetic inequality (2.1) and the boundedness of $\\{v_{n}\\}$
in $H_{\varepsilon}$, we conclude. ∎

For compact supported functions in $H^{1}(\mathbb{R}^{2},\mathbb{R})$, we have
the following result which will be very important for some estimates below.

######  Lemma 2.2.

If $u\in H^{1}(\mathbb{R}^{2},\mathbb{R})$ and $u$ has a compact support, then
$\omega:=e^{iA(0)\cdot x}u\in H_{\varepsilon}$.

###### Proof.

Assume that $\text{supp}(u)\subset B_{R}(0)$. Since $V$ is continuous, it is
clear that

$\displaystyle\int_{\mathbb{R}^{2}}V_{\varepsilon}(x)|\omega|^{2}dx=\int_{B_{R}(0)}V_{\varepsilon}(x)|\omega|^{2}dx\leq
C\|u\|_{2}^{2}<+\infty.$

Moreover, since $V$ and $A$ are continuous, we have

$\displaystyle\int_{\mathbb{R}^{2}}|\nabla_{A_{\varepsilon}}\omega|^{2}dx$
$\displaystyle=\int_{\mathbb{R}^{2}}|\nabla\omega|^{2}dx+\int_{\mathbb{R}^{2}}|A_{\varepsilon}(x)|^{2}|\omega|^{2}dx+2\text{Re}\int_{\mathbb{R}^{2}}iA_{\varepsilon}(x)\overline{\omega}\nabla\omega
dx$ $\displaystyle\leq
2\int_{\mathbb{R}^{2}}|\nabla\omega|^{2}dx+2\int_{\mathbb{R}^{2}}|A_{\varepsilon}(x)|^{2}|\omega|^{2}dx$
$\displaystyle\leq C\left[\int_{\mathbb{R}^{2}}|\nabla
u|^{2}dx+\int_{\mathbb{R}^{2}}|u|^{2}dx\right]<+\infty$

and we conclude. ∎

A useful tool when we work with nonlinearities with exponential critical
growth is the following version of Trudinger-Moser inequality as stated e.g.
in [1, Lemma 1.2] (see also [13, Lemma 2.1]).

######  Lemma 2.3.

If $\alpha>0$ and $u\in H^{1}(\mathbb{R}^{2},\mathbb{R})$, then

$\int_{\mathbb{R}^{2}}(e^{\alpha|u|^{2}}-1)dx<+\infty.$

Moreover, if $\|\nabla u\|_{2}^{2}\leq 1$, $\|u\|_{2}\leq M<+\infty$, and
$0<\alpha<4\pi$, then there exists a positive constant $C(M,\alpha)$, which
depends only on $M$ and $\alpha$, such that

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

Observe also that that ($f_{1}$) and ($f_{2}$) imply that, fixed $q>2$, for
any $\zeta>0$ and $\alpha>4\pi$, there exists a constant $C>0$, which depends
on $q$, $\alpha$, $\zeta$, such that

(2.4) $f(t)\leq\zeta+Ct^{(q-2)/2}(e^{\alpha t}-1),\,\text{ for all }t\geq 0$

and, using ($f_{3}$),

(2.5) $F(t)\leq\zeta t+Ct^{q/2}(e^{\alpha t}-1),\,\text{ for all }t\geq 0.$

Then, by (2.4) and (2.5), we have

(2.6) $f(t^{2})t^{2}\leq\zeta t^{2}+C|t|^{q}(e^{\alpha t^{2}}-1),\,\text{ for
all }t\in\mathbb{R}$

and

(2.7) $F(t^{2})\leq\zeta t^{2}+C|t|^{q}(e^{\alpha t^{2}}-1),\,\text{ for all
}t\in\mathbb{R}.$

Moreover ($f_{4}$) and ($f_{5}$) imply that there exist two positive constants
$C_{1}$ and $C_{2}$ such that

$C_{1}t^{(p-4)/2}\leq f^{\prime}(t)\leq C_{2}t,\quad\text{as }t\rightarrow
0^{+}$

and so $p>6$.

For $u\in H_{\varepsilon}$, let

(2.8) $\hat{u}_{j}:=\varphi_{j}u,$

where $j\in\mathbb{N}^{*}$ and $\varphi_{j}(x)=\varphi(2x/j)$ with $\varphi\in
C_{c}^{\infty}(\mathbb{R}^{2},\mathbb{R})$, $0\leq\varphi\leq 1$,
$\varphi(x)=1$ if $|x|\leq 1$, and $\varphi(x)=0$ if $|x|\geq 2$.
Note that $\hat{u}_{j}$ belongs to $H_{\varepsilon}$ and has compact support.
We have that any $u\in H_{\varepsilon}$ can be approximated by $\hat{u}_{j}$
in the following sense. This property is classical, but, for completeness, we
give here some details.

######  Lemma 2.4.

Let $\varepsilon>0$. For any $u\in H_{\varepsilon}$,
$\|\hat{u}_{j}-u\|_{\varepsilon}\rightarrow 0$ as $j\rightarrow+\infty$.

###### Proof.

Since $u\in H_{\varepsilon}$, by the Lebesgue Dominated Convergence Theorem,
it is easy to see that

$\int_{\mathbb{R}^{2}}V_{\varepsilon}(x)|\hat{u}_{j}-u|^{2}dx=o_{j}(1).$

Moreover, since

$\begin{split}\int_{\mathbb{R}^{2}}|\nabla_{A_{\varepsilon}}(\hat{u}_{j}-u)|^{2}dx&=\int_{\mathbb{R}^{2}}|\nabla(\hat{u}_{j}-u)|^{2}dx+\int_{\mathbb{R}^{2}}|A_{\varepsilon}(x)|^{2}|\hat{u}_{j}-u|^{2}dx\\\
&\qquad+2\text{Re}\int_{\mathbb{R}^{2}}i\nabla(\hat{u}_{j}-u)A_{\varepsilon}(x)\overline{(\hat{u}_{j}-u)}dx\\\
&\leq
2\int_{\mathbb{R}^{2}}|\nabla(\hat{u}_{j}-u)|^{2}dx+2\int_{\mathbb{R}^{2}}|A_{\varepsilon}(x)|^{2}|\hat{u}_{j}-u|^{2}dx,\end{split}$

applying again the Lebesgue Dominated Convergence Theorem, we conclude. ∎

In the spirit of [24, Lemma 3.2], we have the following result for bounded
sequences in $H_{\varepsilon}$.

######  Lemma 2.5.

Let $s\geq 2$ and $\\{u_{n}\\}\subset H_{\varepsilon}$ be a bounded sequence.
Then there exists a subsequence $\\{u_{n_{j}}\\}$ such that for any $\sigma>0$
there is $r_{\sigma,s}>0$ such that, for every $r\geq r_{\sigma,s}$,

$\limsup_{j}\int_{B_{j}(0)\backslash B_{r}(0)}|u_{n_{j}}|^{s}dx\leq\sigma.$

###### Proof.

By Lemma 2.1, for each $j\in\mathbb{N}$,

$\int_{B_{j}(0)}|u_{n}|^{s}dx\rightarrow\int_{B_{j}(0)}|u|^{s}dx\text{ as
}n\rightarrow+\infty.$

Thus there exists $\nu_{j}\in\mathbb{N}$ such that

$\Big{|}\int_{B_{j}(0)}(|u_{n}|^{s}-|u|^{s})dx\Big{|}<\frac{1}{j}\quad\text{
for all }n>\nu_{j}$

and, without loss of generality, we can assume that $\nu_{j+1}\geq\nu_{j}$.
In particular, for $n_{j}:=\nu_{j}+j$ we have

$\int_{B_{j}(0)}(|u_{n_{j}}|^{s}-|u|^{s})dx<\frac{1}{j}.$

On the other hand, there exists $r_{\sigma,s}$ such that, for every $r\geq
r_{\sigma,s}$,

$\int_{\mathbb{R}^{2}\backslash B_{r}(0)}|u|^{s}\leq\frac{\sigma}{3}.$

Hence, for $j$ large enough,

$\displaystyle\int_{B_{j}(0)\backslash B_{r}(0)}|u_{n_{j}}|^{s}dx$
$\displaystyle=\int_{B_{j}(0)}(|u_{n_{j}}|^{s}-|u|^{s})dx+\int_{B_{j}(0)\backslash
B_{r}(0)}|u|^{s}dx+\int_{B_{r}(0)}(|u|^{s}-|u_{n_{j}}|^{s})dx$
$\displaystyle\leq\frac{1}{j}+\frac{\sigma}{3}+\frac{1}{j}\leq\sigma.$

∎

In the proofs of our results, the following properties will be useful.

######  Lemma 2.6.

Let $\\{u_{n}\\}\subset H_{\varepsilon}$ be a bounded sequence and
$\\{u_{n_{j}}\\}$ be a subsequence of $\\{u_{n}\\}$ such that
$u_{n_{j}}\rightharpoonup u$ in $H_{\varepsilon}$ and
$\nabla|u_{n_{j}}|\rightarrow\nabla|u|$ a.e. in $\mathbb{R}^{2}$, as
$j\to+\infty$. Moreover let
$\ell^{2}:=\limsup_{j}\|\nabla|u_{n_{j}}|\|_{2}^{2}$ and $\hat{u}_{j}$ be
defined as in (2.8).
Then, up to a subsequence still denoted by $\\{u_{n_{j}}\\}$, we have that, if
$\ell^{2}<1/4$,

(2.9)
$\int_{\mathbb{R}^{2}}\Big{(}F(|u_{n_{j}}|^{2})-F(|u_{n_{j}}-\hat{u}_{j}|^{2})-F(|\hat{u}_{j}|^{2})\Big{)}dx=o_{j}(1)$

and, for every $\phi\in H_{\varepsilon}$, if $\ell^{2}<1/8$,

(2.10)
$\int_{\mathbb{R}^{2}}\Big{|}f(|u_{n_{j}}|^{2})u_{n_{j}}-f(|u_{n_{j}}-\hat{u}_{j}|^{2})(u_{n_{j}}-\hat{u}_{j})-f(|\hat{u}_{j}|^{2})\hat{u}_{j}\Big{|}^{2}=o_{j}(1).$

###### Proof.

Let us prove (2.9).
Using (2.4) and the Young inequality, we have that, fixed $q>2$, for any
$\zeta>0$ and $\alpha>4\pi$, there exists a constant $C>0$, which depends on
$q$, $\alpha$, $\zeta$, such that

$\displaystyle|F(|u_{n_{j}}|^{2})-F(|u_{n_{j}}-\hat{u}_{j}|^{2})|$
$\displaystyle\leq
2\int_{0}^{1}|f(|u_{n_{j}}-t\hat{u}_{j}|^{2})||u_{n_{j}}-t\hat{u}_{j}||\hat{u}_{j}|dt$
$\displaystyle\leq\zeta\int_{0}^{1}|u_{n_{j}}-t\hat{u}_{j}||\hat{u}_{j}|dt+C\int_{0}^{1}|u_{n_{j}}-t\hat{u}_{j}|^{q-1}(e^{\alpha|u_{n_{j}}-t\hat{u}_{j}|^{2}}-1)|\hat{u}_{j}|dt$
$\displaystyle\leq\zeta(|u_{n_{j}}|+|u|)|u|+C(|u_{n_{j}}|+|u|)^{q-1}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)|u|$
$\displaystyle\leq\zeta\big{(}|u_{n_{j}}|^{2}+|u_{n_{j}}|^{q}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)\big{)}+C\big{(}|u|^{2}+|u|^{q}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)\big{)}.$

Then, by (2.7) we have

(2.11)
$\begin{split}|F(|u_{n_{j}}|^{2})-F(|u_{n_{j}}-\hat{u}_{j}|^{2})-F(|\hat{u}_{j}|^{2})|&\leq\zeta\big{(}|u_{n_{j}}|^{2}+|u_{n_{j}}|^{q}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)\big{)}\\\
&\quad+C\big{(}|u|^{2}+|u|^{q}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)\big{)}.\end{split}$

Now, let

$\displaystyle
G_{j}^{\zeta}:=\max\Big{\\{}|F(|u_{n_{j}}|^{2})-F(|u_{n_{j}}-\hat{u}_{j}|^{2})-F(|\hat{u}_{j}|^{2})|-\zeta(|u_{n_{j}}|^{2}+|u_{n_{j}}|^{q}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)),0\Big{\\}}.$

Note that, by (2.11), we have

(2.12) $G_{j}^{\zeta}\leq
C\big{(}|u|^{2}+|u|^{q}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)\big{)}$

and, by Lemma 2.4 and Lemma 2.1,

(2.13) $G_{j}^{\zeta}\rightarrow 0\text{ a.e. in }\mathbb{R}^{2}$

and

(2.14) $e^{\alpha(|u_{n_{j}}|+|u|)^{2}}\to e^{4\alpha|u|^{2}}\text{ a.e. in
}\mathbb{R}^{2}$

as $j\rightarrow+\infty$.
Let us show that

(2.15)
$|u|^{q}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)\rightarrow|u|^{q}(e^{4\alpha|u|^{2}}-1)\quad\text{in}\,\,L^{1}(\mathbb{R}^{2},\mathbb{R}).$

Since $\nabla|u_{n_{j}}|\rightarrow\nabla|u|$ a.e. in $\mathbb{R}^{2}$ as
$j\rightarrow+\infty$, by the Fatou’s Lemma, we deduce that

$\|\nabla|u|\|_{2}^{2}\leq\liminf_{j}\|\nabla|u_{n_{j}}|\|_{2}^{2}\leq\ell^{2}$

and so, for any fixed $\kappa\in(0,1-4\ell^{2})$, we have that for $j$ large
enough,

(2.16) $\|\nabla(|u_{n_{j}}|+|u|)\|_{2}^{2}\leq
2\|\nabla|u_{n_{j}}|\|_{2}^{2}+2\|\nabla|u|\|_{2}^{2}<4\ell^{2}+\kappa.$

Let

$p^{*}\in\Big{(}1,\frac{1}{4\ell^{2}+\kappa}\Big{)},\qquad\alpha\in\Big{(}4\pi,\frac{4\pi}{p^{*}(4\ell^{2}+\kappa)}\Big{)}.$

By the inequality

(2.17) $(e^{t}-1)^{s}\leq e^{ts}-1,\text{ for }s>1\text{ and }t\geq 0,$

inequality (2.16), and Lemma 2.3, we get

(2.18)
$\int_{\mathbb{R}^{2}}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)^{p^{*}}dx\leq\int_{\mathbb{R}^{2}}(e^{(4\ell^{2}+\kappa)p^{*}\alpha\Big{(}\frac{|u_{n_{j}}|+|u|}{\sqrt{4\ell^{2}+\kappa}}\Big{)}^{2}}-1)dx\leq
C.$

Thus, up to a subsequence, and applying the Severini-Egoroff Theorem [12,
Theorem 4.29],

$(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)\rightharpoonup(e^{4\alpha|u|^{2}}-1)\text{
in }L^{p^{*}}(\mathbb{R}^{2},\mathbb{R})$

and, since $|u|^{q}\in L^{q^{*}}(\mathbb{R}^{2})$, where $q^{*}$ is the
conjugate exponent of $p^{*}$, we get (2.15).
Hence, by (2.12), (2.13), (2.14), and (2.15), applying a variant of the
Lebesgue Dominated Convergence Theorem, we deduce that

$\displaystyle\int_{\mathbb{R}^{2}}G_{j}^{\zeta}dx\rightarrow
0\quad\text{as}\,\,j\rightarrow+\infty.$

On the other hand, by the definition of $G_{j}^{\zeta}$,

$\displaystyle|F(|u_{n_{j}}|^{2})-F(|u_{n_{j}}-\hat{u}_{j}|^{2})-F(|\hat{u}_{j}|^{2})|\leq\zeta(|u_{n_{j}}|^{2}+|u_{n_{j}}|^{q}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1))+G_{j}^{\zeta}.$

Using the Hölder inequality, the boundedness of the sequence $\\{u_{n}\\}$,
Lemma 2.1, and (2.18), we have

$\int_{\mathbb{R}^{2}}\Big{(}|u_{n_{j}}|^{q}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)\Big{)}dx\leq\|u_{n_{j}}\|_{qq^{*}}^{q}\left[\int_{\mathbb{R}^{2}}|(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)^{p^{*}}dx\right]^{\frac{1}{p^{*}}}\leq
C.$

Thus, the arbitrariness of $\zeta$ allows to conclude the proof of (2.9).
Now we prove (2.10).
First observe that, by Lemma 2.1 and Lemma 2.4, we have that

$\Big{|}f(|u_{n_{j}}|^{2})u_{n_{j}}-f(|u_{n_{j}}-\hat{u}_{j}|^{2})(u_{n_{j}}-\hat{u}_{j})-f(|\hat{u}_{j}|^{2})\hat{u}_{j}\Big{|}\to
0\text{ a.e. in }\mathbb{R}^{2}.$

Moreover, using also (2.4), we have that fixed $q>2$, for any $\zeta>0$ and
$\alpha>4\pi$, there exists a constant $C>0$, which depends on $q$, $\alpha$,
$\zeta$, such that

$\displaystyle\Big{|}f(|u_{n_{j}}|^{2})u_{n_{j}}-f(|u_{n_{j}}-\hat{u}_{j}|^{2})(u_{n_{j}}-\hat{u}_{j})-f(|\hat{u}_{j}|^{2})\hat{u}_{j}\Big{|}$
$\displaystyle\qquad\leq\zeta|u_{n_{j}}|+C|u_{n_{j}}|^{q-1}(e^{\alpha|u_{n_{j}}|^{2}}-1)+\zeta|u_{n_{j}}-\hat{u}_{j}|+C|u_{n_{j}}-\hat{u}_{j}|^{q-1}(e^{\alpha|u_{n_{j}}-\hat{u}_{j}|^{2}}-1)$
$\displaystyle\qquad\qquad+\zeta|\hat{u}_{j}|+C|\hat{u}_{j}|^{q-1}(e^{\alpha|\hat{u}_{j}|^{2}}-1)$
$\displaystyle\qquad\leq
C\left[|u_{n_{j}}|+|u|+(|u_{n_{j}}|+|u|)^{q-1}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)\right]$

and so

(2.19)
$\begin{split}&\Big{|}f(|u_{n_{j}}|^{2})u_{n_{j}}-f(|u_{n_{j}}-\hat{u}_{j}|^{2})(u_{n_{j}}-\hat{u}_{j})-f(|\hat{u}_{j}|^{2})\hat{u}_{j}\Big{|}^{2}\\\
&\qquad\leq
C\left[|u_{n_{j}}|^{2}+|u|^{2}+(|u_{n_{j}}|+|u|)^{2(q-1)}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)^{2}\right].\end{split}$

Arguing as before, for any fixed $\kappa\in(0,\frac{1}{2}-4\ell^{2})$, if

$p^{*}\in\Big{(}1,\frac{1}{2(4\ell^{2}+\kappa)}\Big{)}\text{ and
}\alpha\in\Big{(}4\pi,\frac{2\pi}{p^{*}(4\ell^{2}+\kappa)}\Big{)},$

we have that

(2.20)
$\int_{\mathbb{R}^{2}}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)^{2p^{*}}dx\leq\int_{\mathbb{R}^{2}}(e^{2(4\ell^{2}+\kappa)p^{*}\alpha\Big{(}\frac{|u_{n_{j}}|+|u|}{\sqrt{4\ell^{2}+\kappa}}\Big{)}^{2}}-1)dx\leq
C.$

By Lemma 2.5 for $s=2$, from $\\{u_{n_{j}}\\}$ we can extract a first
subsequence, still denoted by $\\{u_{n_{j}}\\}$, such that for any $\sigma>0$
there is $r_{\sigma,2}>0$ such that, for every $r\geq r_{\sigma,2}$,

(2.21) $\limsup_{j}\int_{B_{j}(0)\backslash
B_{r}(0)}|u_{n_{j}}|^{2}dx\leq\sigma.$

Applying again Lemma 2.5 for $s=2(q-1)q^{*}$, where $q^{*}$ is the conjugate
exponent of $p^{*}$, from $\\{u_{n_{j}}\\}$ we can extract a further
subsequence, still denoted by $\\{u_{n_{j}}\\}$, such that for any $\sigma>0$
there is $r_{\sigma,2(q-1)q^{*}}>0$ such that, for every $r\geq
r_{\sigma,2(q-1)q^{*}}$,

(2.22) $\limsup_{j}\int_{B_{j}(0)\backslash
B_{r}(0)}|u_{n_{j}}|^{2(q-1)q^{*}}dx\leq\sigma.$

Let now $\sigma>0$ be arbitrary and
$r\geq\max\\{r_{\sigma,2},r_{\sigma,2(q-1)q^{*}}\\}$ large enough such that

(2.23) $\int_{B^{c}_{r}(0)}|u|^{2}dx\leq\sigma\text{ and
}\int_{B^{c}_{r}(0)}|u|^{2(q-1)q^{*}}dx\leq\sigma.$

Since, $2(q-1)q^{*}>2$ and by Lemma 2.1, there exists $w\in
L^{2(q-1)q^{*}}(B_{r}(0))$ such that $|u_{n_{j}}|\leq w$ a.e. in $B_{r}(0)$.
Thus, by (2.19), for every $s\geq 1$ and a.e. $x\in B_{r}(0)$,

$\displaystyle\Big{|}f(|u_{n_{j}}|^{2})u_{n_{j}}-f(|u_{n_{j}}-\hat{u}_{j}|^{2})(u_{n_{j}}-\hat{u}_{j})-f(|\hat{u}_{j}|^{2})\hat{u}_{j}\Big{|}^{2}$
$\displaystyle\qquad\leq
C\left[w^{2}+|u|^{2}+(w+|u|)^{2(q-1)}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)^{2}\right].$

Then, by (2.20), applying the Hölder inequality,

$\int_{B_{r}(0)}(w+|u|)^{2(q-1)}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)^{2}dx\leq
C\left(\int_{B_{r}(0)}(w+|u|)^{2(q-1)q^{*}}dx\right)^{1/q^{*}},$

and so

$w^{2}+|u|^{2}+(w+|u|)^{2(q-1)}(e^{\alpha(|u_{n_{j}}|+|u|)^{2}}-1)^{2}\in
L^{1}(B_{r}(0)).$

Hence, applying the Lebesgue Dominated Convergence Theorem, we deduce that

$\int_{B_{r}(0)}\Big{|}f(|u_{n_{j}}|^{2})u_{n_{j}}-f(|u_{n_{j}}-\hat{u}_{j}|^{2})(u_{n_{j}}-\hat{u}_{j})-f(|\hat{u}_{j}|^{2})\hat{u}_{j}\Big{|}^{2}dx\to
0\text{ as }j\to+\infty.$

On the other hand, by (2.19), (2.20), (2.21), (2.22), and (2.23), for $j$
large enough,

$\displaystyle\int_{B_{j}(0)\backslash
B_{r}(0)}\Big{|}f(|u_{n_{j}}|^{2})u_{n_{j}}-f(|u_{n_{j}}-\hat{u}_{j}|^{2})(u_{n_{j}}-\hat{u}_{j})-f(|\hat{u}_{j}|^{2})\hat{u}_{j}\Big{|}^{2}dx$
$\displaystyle\qquad\leq C\int_{B_{j}(0)\backslash
B_{r}(0)}\Big{(}|u_{n_{j}}|^{2}+|u|^{2}+(|u_{n_{j}}|+|u|)^{2(q-1)}(e^{2\alpha(|u_{n_{j}}|+|u|)^{2}}-1)\Big{)}dx$
$\displaystyle\qquad\leq C\left[\sigma+\Big{(}\int_{B_{j}(0)\backslash
B_{r}(0)}|u_{n_{j}}|^{2(q-1)q^{*}}dx\Big{)}^{1/q^{*}}+\Big{(}\int_{B_{j}(0)\backslash
B_{r}(0)}|u|^{2(q-1)q^{*}}dx\Big{)}^{1/q^{*}}\right]$ $\displaystyle\qquad\leq
C(\sigma+\sigma^{1/q^{*}}).$

Hence, since $\hat{u}_{j}=0$ in $B^{c}_{j}(0)$ for any $j\geq 1$, the
arbitrariness of $\sigma>0$ allows to conclude. ∎

##  3\. The limit problem

As it is usual for this type of questions, we consider the limit problem

(3.1) $\displaystyle-\Delta u+\mu u=f(u^{2})u,\quad
u:\mathbb{R}^{2}\to\mathbb{R},\quad\mu>0.$

We have already considered this problem in [20, Section 2] for a particular
value of $\mu$, namely for $\mu=V_{0}$. Here, due to the assumptions on $V$,
we need some generalizations.

First of all we observe that solutions of (3.1) can be found as critical
points of the $C^{1}$ functional

$I_{\mu}(u):=\frac{1}{2}\int_{\mathbb{R}^{2}}(|\nabla u|^{2}+\mu
u^{2})dx-\frac{1}{2}\int_{\mathbb{R}^{2}}F(u^{2})dx$

defined in $H^{1}(\mathbb{R}^{2},\mathbb{R})$. Moreover, if

$\mathcal{N}_{\mu}:=\\{u\in
H^{1}(\mathbb{R}^{2},\mathbb{R})\setminus\\{0\\}:I^{\prime}_{\mu}(u)[u]=0\\}\quad\text{
and }\quad c_{\mu}:=\inf_{u\in\mathcal{N}_{\mu}}I_{\mu}(u),$

using ($f_{1}$) and ($f_{4}$), we have that for each fixed $u\in
H^{1}(\mathbb{R}^{2},\mathbb{R})\backslash\\{0\\}$, there is a unique $t(u)>0$
such that

(3.2) $I_{\mu}(t(u)u)=\max_{t\geq 0}I_{\mu}(tu)\quad\text{and}\quad
t(u)u\in\mathcal{N}_{\mu}.$

Then, arguing as in [33, Lemma 4.1 and Theorem 4.2], we also have that

(3.3) $0<c_{\mu}=\inf_{u\in
H^{1}(\mathbb{R}^{2},\mathbb{R})\backslash\\{0\\}}\max_{t\geq 0}I_{\mu}(tu).$

Observe finally that, arguing as in [20, Lemma 2.3] where we consider only the
case $\mu=V_{0}$, we can get that there exists a constant $K>0$ such that, for
all $u\in\mathcal{N}_{\mu}$, $\|u\|_{\mu}\geq K$.

In particular, if $\mu\in(0,V^{*}]$, the assumptions on $f$ allow to get the
existence of a positive ground state solution $\omega_{\mu}$ of (3.1), namely
such that $I_{\mu}(\omega_{\mu})\leq I_{\mu}(v)$ for all positive solutions
$v\in H^{1}(\mathbb{R}^{2},\mathbb{R})$ of (3.1).

######  Lemma 3.1.

For every $\mu\in(0,V^{*}]$, there exists a positive and radially symmetric
ground state solution $\omega_{\mu}\in H^{1}(\mathbb{R}^{2},\mathbb{R})$ for
problem (3.1).

###### Proof.

We can argue exactly as in [20, Lemma 2.2], applying [6, Corollary 1.5] to the
function $h(t):=f(t^{2})t/\mu$. Here we observe only that the condition

$\text{there exist }\lambda>0\text{ and }p>2\text{ such that }h(t)\geq\lambda
t^{p-1}\text{ for }t\geq 0\text{ and
}\lambda>\left(\frac{p-2}{p}\right)^{\frac{p-2}{2}}S_{p}^{p/2}$

follows by ($f_{4}$), since

$h(t)\geq\frac{C_{p}}{\mu}t^{p-1}\quad\text{ for all }t\geq 0$

and

$\frac{C_{p}}{\mu}>\Big{(}\frac{8\theta}{\theta-2}\Big{)}^{\frac{p-2}{2}}\Big{(}\frac{p-2}{p}\Big{)}^{\frac{p-2}{2}}S_{p}^{p/2}>8^{\frac{p-2}{2}}\Big{(}\frac{p-2}{p}\Big{)}^{\frac{p-2}{2}}S_{p}^{p/2}>\Big{(}\frac{p-2}{p}\Big{)}^{\frac{p-2}{2}}S_{p}^{p/2}.$

By standard arguments we can assume $\omega_{\mu}$ positive. We refer to [20,
Lemma 2.2] for the remaining details. ∎

Moreover the ground state $\omega_{\mu}$ decays exponentially at infinity with
its gradient and belongs to $C^{2}(\mathbb{R}^{2},\mathbb{R})\cap
L^{\infty}(\mathbb{R}^{2},\mathbb{R})$ (see also [34, Proposition 2.1]) and,
for any $\mu_{1},\mu_{2}\in(0,V^{*}]$ with $0<\mu_{1}<\mu_{2}$, we have

(3.4) $c_{\mu_{1}}=I_{\mu_{1}}(\omega_{\mu_{1}})\leq\max_{t\geq
0}I_{\mu_{1}}(t\omega_{\mu_{2}})<\max_{t\geq
0}I_{\mu_{2}}(t\omega_{\mu_{2}})=I_{\mu_{2}}(\omega_{\mu_{2}})=c_{\mu_{2}}.$

Let now $S_{p}>0$ be defined as in ($f_{4}$) and, for every $\mu>0$,

$S_{p}^{\mu}:=\inf_{u\in
H^{1}(\mathbb{R}^{2},\mathbb{R})\backslash\\{0\\}}\frac{\|\nabla
u\|_{2}^{2}+\mu\|u\|_{2}^{2}}{\|u\|_{p}^{2}}.$

We have

######  Lemma 3.2.

$S_{p}^{\mu}=\mu^{2/p}S_{p}$.

###### Proof.

Let $u\in H^{1}(\mathbb{R}^{2},\mathbb{R})\backslash\\{0\\}$.
If $\tilde{u}:=u(\mu^{-1/2}\cdot)$, we have

$\frac{\|\nabla
u\|_{2}^{2}+\mu\|u\|_{2}^{2}}{\|u\|_{p}^{2}}=\mu^{2/p}\frac{\|\nabla\tilde{u}\|_{2}^{2}+\|\tilde{u}\|_{2}^{2}}{\|\tilde{u}\|_{p}^{2}}\geq\mu^{2/p}S_{p}$

and, passing to the infimum, we get $S_{p}^{\mu}\geq\mu^{2/p}S_{p}$.
On the other hand, if $\tilde{u}:=u(\mu^{1/2}\cdot)$, we have

$\frac{\|\nabla
u\|_{2}^{2}+\|u\|_{2}^{2}}{\|u\|_{p}^{2}}=\mu^{-2/p}\frac{\|\nabla\tilde{u}\|_{2}^{2}+\mu\|\tilde{u}\|_{2}^{2}}{\|\tilde{u}\|_{p}^{2}}\geq\mu^{-2/p}S_{p}^{\mu}$

and, passing to the infimum, we get $S_{p}^{\mu}\leq\mu^{2/p}S_{p}$,
concluding the proof. ∎

######  Remark 3.3.

Observe that, by [33, Theorem 1.35], $S_{p}$ is achieved by a $C^{2}$ positive
and radially symmetric function, and, arguing as in the proof of Lemma 3.2,
this holds for $S_{p}^{\mu}$ too.

Now we give an upper estimate of the ground state energy for (3.1) with
$\mu=V^{*}$, with $V^{*}$ defined in (1.2).

######  Lemma 3.4.

The minimax level $c_{V^{*}}$ belongs to the interval
$\displaystyle\Big{(}0,\frac{\theta-2}{16\theta}\Big{)}$.

###### Proof.

As observed in Remark 3.3, there exists $\omega^{*}\in
H^{1}(\mathbb{R}^{2},\mathbb{R})\setminus\\{0\\}$ such that

$S_{p}^{V^{*}}=\frac{\|\nabla\omega^{*}\|_{2}^{2}+V^{*}\|\omega^{*}\|_{2}^{2}}{\|\omega^{*}\|_{p}^{2}}.$

Thus, using (3.3), ($f_{4}$), and Lemma 3.2, we have

$\displaystyle 0<c_{V^{*}}$ $\displaystyle\leq\max_{t\geq
0}I_{V^{*}}(t\omega^{*})\leq\max_{t\geq
0}\Big{\\{}\frac{t^{2}}{2}(\|\nabla\omega^{*}\|_{2}^{2}+V^{*}\|\omega^{*}\|_{2}^{2})-\frac{C_{p}t^{p}}{p}\|\omega^{*}\|_{p}^{p}\Big{\\}}$
$\displaystyle\qquad=\frac{p-2}{2p}\left[\frac{(S_{p}^{V^{*}})^{p}}{C_{p}^{2}}\right]^{\frac{1}{p-2}}=\frac{p-2}{2p}\left[\frac{({V^{*}})^{2}S_{p}^{p}}{C_{p}^{2}}\right]^{\frac{1}{p-2}}<\frac{\theta-2}{16\theta}.$

∎

Finally we give the following result.

######  Lemma 3.5.

Let $\mu\in(0,V^{*}]$, and $\\{\omega_{n}\\}\subset\mathcal{N}_{\mu}$ be a
sequence of positive functions satisfying $I_{\mu}(\omega_{n})\rightarrow
c_{\mu}$. Then $\\{\omega_{n}\\}$ is bounded in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$ and, up to a subsequence,
$\omega_{n}\rightharpoonup\omega$ in $H^{1}(\mathbb{R}^{2},\mathbb{R})$.
Moreover we have that:

  1. (1)

if $\omega\neq 0$, then $\omega_{n}\rightarrow\omega\in\mathcal{N}_{\mu}$ in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$ and $\omega$ is a ground state for (3.1);

  2. (2)

if $\omega=0$, then there exists $\\{\tilde{y}_{n}\\}\subset\mathbb{R}^{2}$
with $|\tilde{y}_{n}|\rightarrow+\infty$ and
$\tilde{\omega}\in\mathcal{N}_{\mu}$ such that, up to a subsequence,
$\omega_{n}(\cdot+\tilde{y}_{n})\rightarrow\tilde{\omega}$ in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$ and $\tilde{\omega}$ is a ground state for
(3.1).

###### Proof.

By ($f_{3}$) and Lemma 3.4, it follows that

$\displaystyle\frac{\theta-2}{2\theta}\limsup_{n}(\|\nabla\omega_{n}\|_{2}^{2}+\mu\|\omega_{n}\|_{2}^{2})$
$\displaystyle\leq\limsup_{n}\Big{\\{}\Big{(}\frac{1}{2}-\frac{1}{\theta}\Big{)}(\|\nabla\omega_{n}\|_{2}^{2}+\mu\|\omega_{n}\|_{2}^{2})$
$\displaystyle\qquad\qquad\qquad+\int_{\mathbb{R}^{2}}\Big{(}\frac{1}{\theta}f(\omega_{n}^{2})\omega_{n}^{2}-\frac{1}{2}F(\omega_{n}^{2})\Big{)}dx\Big{\\}}$
$\displaystyle=\limsup_{n}\Big{\\{}I_{\mu}(\omega_{n})-\frac{1}{\theta}I^{\prime}_{\mu}(\omega_{n})[\omega_{n}]\Big{\\}}=c_{\mu}\leq
c_{V^{*}}<\frac{\theta-2}{16\theta}.$

Thus,

(3.5)
$\ell^{2}:=\limsup_{n}(\|\nabla\omega_{n}\|_{2}^{2}+\mu\|\omega_{n}\|_{2}^{2})<\frac{1}{8}$

and so there exists $\omega\in H^{1}(\mathbb{R}^{2},\mathbb{R})$ such that, up
to a subsequence, $\omega_{n}\rightharpoonup\omega$ in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$. Moreover $\omega_{n}\rightarrow\omega$ in
$L^{r}_{\rm loc}(\mathbb{R}^{2},\mathbb{R})$, for any $r\geq 1$ and
$\omega_{n}\rightarrow\omega$ a.e. in $x\in\mathbb{R}^{2}$.
Now let us consider two different cases.
Case 1: $\omega\neq 0$.
Observe that, for every $\phi\in C_{c}^{\infty}(\mathbb{R}^{2},\mathbb{R})$,

$f(\omega_{n}^{2})\omega_{n}\phi\to f(\omega^{2})\omega\phi\text{ a.e. in
}\mathbb{R}^{2}\text{ as }n\to+\infty$

and that, by (2.4), we have that for any $\zeta>0$, $q>2$, and $\alpha>4\pi$,
there exists $C>0$ such that

(3.6) $\displaystyle|f(\omega_{n}^{2})\omega_{n}\phi|$
$\displaystyle\leq\zeta\omega_{n}|\phi|+C\omega_{n}^{q-1}(e^{\alpha\omega_{n}^{2}}-1)|\phi|\leq\zeta\omega_{n}|\phi|+C(\omega_{n}^{q}+|\phi|^{q})(e^{\alpha\omega_{n}^{2}}-1),$

with

$\zeta\omega_{n}|\phi|+C(\omega_{n}^{q}+|\phi|^{q})(e^{\alpha\omega_{n}^{2}}-1)\to\zeta\omega|\phi|+C(\omega^{q}+|\phi|^{q})(e^{\alpha\omega^{2}}-1)\text{
a.e. in }\mathbb{R}^{2}\text{ as }n\to+\infty.$

Now, taking into account (3.5), let us fix $\kappa\in(0,1-8\ell^{2})$, and let

$p^{*}\in\Big{(}1,\frac{1}{8\ell^{2}+\kappa}\Big{)},\quad\alpha\in\Big{(}4\pi,\frac{4\pi}{p^{*}(8\ell^{2}+\kappa)}\Big{)}.$

If $\Omega:=\text{supp}(\phi)$, by [33, Lemma A.1], there exists $\upsilon\in
L^{qq^{*}}(\Omega,\mathbb{R})$ such that $\omega_{n}\leq\upsilon$ a.e. in
$\Omega$, where $qq^{*}>2$ and $q^{*}$ is the conjugate exponent of $p^{*}$
and, by (3.6), we have

$\displaystyle|f(\omega_{n}^{2})\omega_{n}\phi|\leq\zeta\omega_{n}|\phi|+C(\upsilon^{q}+|\phi|^{q})(e^{\alpha\omega_{n}^{2}}-1)\to\zeta\omega|\phi|+C(\upsilon^{q}+|\phi|^{q})(e^{\alpha\omega^{2}}-1)\text{
a.e. in }\Omega.$

Arguing as in the proof of Lemma 2.6, we have

$(e^{\alpha\omega_{n}^{2}}-1)\rightharpoonup(e^{\alpha\omega^{2}}-1)\text{ in
}L^{p^{*}}(\mathbb{R}^{2},\mathbb{R}).$

Moreover, since $\omega_{n}\rightharpoonup\omega$ in
$L^{2}(\mathbb{R}^{2},\mathbb{R})$ we have that

$\zeta\int_{\mathbb{R}^{2}}\omega_{n}|\phi|dx+C\int_{\mathbb{R}^{2}}(\upsilon^{q}+|\phi|^{q})(e^{\alpha\omega_{n}^{2}}-1)dx\to\zeta\int_{\mathbb{R}^{2}}\omega|\phi|dx+C\int_{\mathbb{R}^{2}}(\upsilon^{q}+|\phi|^{q})(e^{\alpha\omega^{2}}-1)dx$

as $n\to+\infty$.
Hence, a variant of the Lebesgue Dominated Convergence Theorem implies that

$\int_{\mathbb{R}^{2}}f(\omega_{n}^{2})\omega_{n}\phi
dx\rightarrow\int_{\mathbb{R}^{2}}f(\omega^{2})\omega\phi dx,$

and so $\omega$ is a nontrivial critical point for $I_{\mu}$.
Since, by ($f_{3}$) and the Fatou’s Lemma,

$\int_{\mathbb{R}^{2}}\Big{(}\frac{1}{\theta}f(\omega^{2})\omega^{2}-\frac{1}{2}F(\omega^{2})\Big{)}dx\leq\liminf_{n}\int_{\mathbb{R}^{2}}\Big{(}\frac{1}{\theta}f(\omega_{n}^{2})\omega_{n}^{2}-\frac{1}{2}F(\omega_{n}^{2})\Big{)}dx,$

we have

$\displaystyle c_{\mu}$ $\displaystyle\leq
I_{\mu}(\omega)=I_{\mu}(\omega)-\frac{1}{\theta}I^{\prime}_{\mu}(\omega)[\omega]$
$\displaystyle=\Big{(}\frac{1}{2}-\frac{1}{\theta}\Big{)}(\|\nabla\omega\|_{2}^{2}+\mu\|\omega\|_{2}^{2})+\int_{\mathbb{R}^{2}}\Big{(}\frac{1}{\theta}f(\omega^{2})\omega^{2}-\frac{1}{2}F(\omega^{2})\Big{)}dx$
$\displaystyle\leq\liminf_{n}\Big{\\{}\Big{(}\frac{1}{2}-\frac{1}{\theta}\Big{)}(\|\nabla\omega_{n}\|_{2}^{2}+\mu\|\omega_{n}\|_{2}^{2})+\int_{\mathbb{R}^{2}}\Big{(}\frac{1}{\theta}f(\omega_{n}^{2})\omega_{n}^{2}-\frac{1}{2}F(\omega_{n}^{2})\Big{)}dx\Big{\\}}$
$\displaystyle=\liminf_{n}\Big{\\{}I_{\mu}(\omega_{n})-\frac{1}{\theta}I^{\prime}_{\mu}(\omega_{n})[\omega_{n}]\Big{\\}}=c_{\mu}$

and, by the assumptions,

$\displaystyle 0$
$\displaystyle\leq\liminf_{n}\left[\Big{(}\frac{1}{2}-\frac{1}{\theta}\Big{)}(\|\nabla\omega_{n}\|_{2}^{2}+\mu\|\omega_{n}\|_{2}^{2}-\|\nabla\omega\|_{2}^{2}-\mu\|\omega\|_{2}^{2})\right]$
$\displaystyle\leq\limsup_{n}\left[\Big{(}\frac{1}{2}-\frac{1}{\theta}\Big{)}(\|\nabla\omega_{n}\|_{2}^{2}+\mu\|\omega_{n}\|_{2}^{2}-\|\nabla\omega\|_{2}^{2}-\mu\|\omega\|_{2}^{2})\right]$
$\displaystyle=\limsup_{n}\Big{[}I_{\mu}(\omega_{n})-c_{\mu}+\int_{\mathbb{R}^{2}}\Big{(}\frac{1}{\theta}f(\omega^{2})\omega^{2}-\frac{1}{2}F(\omega^{2})\Big{)}dx-\int_{\mathbb{R}^{2}}\Big{(}\frac{1}{\theta}f(\omega_{n}^{2})\omega_{n}^{2}-\frac{1}{2}F(\omega_{n}^{2})\Big{)}dx\Big{]}$
$\displaystyle=\int_{\mathbb{R}^{2}}\Big{(}\frac{1}{\theta}f(\omega^{2})\omega^{2}-\frac{1}{2}F(\omega^{2})\Big{)}dx-\liminf_{n}\Big{[}\int_{\mathbb{R}^{2}}\Big{(}\frac{1}{\theta}f(\omega_{n}^{2})\omega_{n}^{2}-\frac{1}{2}F(\omega_{n}^{2})\Big{)}dx\Big{]}\leq
0.$

Case 2: $\omega=0$.
In this case, there exist $R,\eta>0$, and
$(\tilde{y}_{n})\subset\mathbb{R}^{2}$ such that

(3.7) $\lim_{n}\int_{B_{R}(\tilde{y}_{n})}\omega_{n}^{2}dx\geq\eta.$

Indeed, if this does not hold, for any $R>0$, one has

$\lim_{n}\sup_{y\in\mathbb{R}^{2}}\int_{B_{R}(y)}\omega_{n}^{2}dx=0.$

Thus, by [28, Chapter 6, Lemma 8.4], for every $\tau>2$,

$\lim_{n}\|\omega_{n}\|_{\tau}=0.$

Moreover, by (2.6), (2.17), Lemma 2.3, and (3.5), and arguing again as in
Lemma 2.6, for $0<\zeta<\mu/2$, $q>2$ and $\alpha>4\pi$, there exists $C>0$
such that

$\displaystyle\|\nabla\omega_{n}\|_{2}^{2}+\mu\|\omega_{n}\|_{2}^{2}$
$\displaystyle\leq\zeta\|\omega_{n}\|_{2}^{2}+C\int_{\mathbb{R}^{2}}\omega_{n}^{q}(e^{\alpha\omega_{n}^{2}}-1)dx$
$\displaystyle\leq\zeta\|\omega_{n}\|_{2}^{2}+C\|\omega_{n}\|_{qq^{*}}^{q}\Big{(}\int_{\mathbb{R}^{2}}(e^{p^{*}\alpha\omega_{n}^{2}}-1)dx\Big{)}^{1/p^{*}}$
$\displaystyle\leq\zeta\|\omega_{n}\|_{2}^{2}+C\|\omega_{n}\|_{qq^{*}}^{q}=\zeta\|\omega_{n}\|_{2}^{2}+o_{n}(1).$

Hence $\omega_{n}\rightarrow 0$ in $H^{1}(\mathbb{R}^{2},\mathbb{R})$ as
$n\rightarrow+\infty$, and so $I_{\mu}(\omega_{n})\rightarrow 0$ as
$n\rightarrow+\infty$. But this is in contradiction with
$I_{\mu}(\omega_{n})\rightarrow c_{\mu}>0$ as $n\rightarrow+\infty$.
Now we claim that $|\tilde{y}_{n}|\rightarrow+\infty$.
Otherwise, there exists $\bar{R}>0$ such that

$\lim_{n}\int_{B_{\bar{R}}(0)}\omega_{n}^{2}dx\geq\eta$

and so $\omega\neq 0$.
Moreover $\omega_{n}(\cdot+\tilde{y}_{n})\in\mathcal{N}_{\mu}$ and, the
invariance of $I_{\mu}$ and $\|\nabla\cdot\|_{2}^{2}+\mu\|\cdot\|_{2}^{2}$ by
translations implies

$I_{\mu}(\omega_{n}(\cdot+\tilde{y}_{n}))\rightarrow c_{\mu}\text{ and
}\limsup_{n}(\|\nabla\omega_{n}(\cdot+\tilde{y}_{n})\|_{2}^{2}+\mu\|\omega_{n}(\cdot+\tilde{y}_{n})\|_{2}^{2})<\frac{1}{8}.$

Thus, arguing as before, there exists $\tilde{\omega}\in
H^{1}(\mathbb{R}^{2},\mathbb{R})$ which, by (3.7), is nontrivial, such that

$\displaystyle\omega_{n}(\cdot+\tilde{y}_{n})\rightharpoonup\tilde{\omega}\quad\text{in}\quad
H^{1}(\mathbb{R}^{2},\mathbb{R}).$

Hence, repeating the same arguments of Case 1, we conclude. ∎

##  4\. Proof of Theorem 1.1

The goal of this section is to prove Theorem 1.1.

We say that $u\in H_{\varepsilon}$ is a weak solution of (2.2), if for any
$\phi\in H_{\varepsilon}$

$\text{Re}\int_{\mathbb{R}^{2}}(\nabla_{A_{\varepsilon}}u\overline{\nabla_{A_{\varepsilon}}\phi}+V_{\varepsilon}(x)u\overline{\phi})dx=\text{Re}\int_{\mathbb{R}^{2}}f(|u|^{2})u\overline{\phi}dx.$

Such solutions can be found as the critical points of the $C^{1}$ functional
$J_{\varepsilon}:H_{\varepsilon}\rightarrow\mathbb{R}$ given by

$J_{\varepsilon}(u)=\frac{1}{2}\int_{\mathbb{R}^{2}}(|\nabla_{A_{\varepsilon}}u|^{2}+V_{\varepsilon}(x)|u|^{2})dx-\frac{1}{2}\int_{\mathbb{R}^{2}}F(|u|^{2})dx.$

The functional $J_{\varepsilon}$ satisfies the Mountain Pass geometry, namely
we have

######  Lemma 4.1.

For any $\varepsilon>0$,

  1. (i)

there exist $\beta,r>0$ such that $J_{\varepsilon}(u)\geq\beta$ if
$\|u\|_{\varepsilon}=r$;

  2. (ii)

there exists $e\in H_{\varepsilon}$ with $\|e\|_{\varepsilon}>r$ such that
$J_{\varepsilon}(e)<0$.

###### Proof.

By (2.7), the Hölder and Sobolev inequalities, and (2.17), fixed $q>2$, for
any $\zeta>0$ and $\alpha>4\pi$, there exists a constant $C>0$, which depends
on $q$, $\alpha$, $\zeta$, such that

(4.1)
$\begin{split}\int_{\mathbb{R}^{2}}F(|u|^{2})dx&\leq\zeta\|u\|_{2}^{2}+C\int_{\mathbb{R}^{2}}|u|^{q}(e^{\alpha|u|^{2}}-1)dx\\\
&\leq\zeta\|u\|_{2}^{2}+C\|u\|_{2q}^{q}\Big{(}\int_{\mathbb{R}^{2}}(e^{\alpha|u|^{2}}-1)^{2}dx\Big{)}^{1/2}\\\
&\leq\zeta\|u\|_{2}^{2}+C\|u\|_{\varepsilon}^{q}\Big{(}\int_{\mathbb{R}^{2}}(e^{2\alpha|u|^{2}}-1)dx\Big{)}^{1/2}.\end{split}$

Moreover, by the diamagnetic inequality (2.1) and (V), for any $u\in
H_{\varepsilon}\setminus\\{0\\}$,

$\frac{|u|}{\|u\|_{\varepsilon}}\in
H^{1}(\mathbb{R}^{2},\mathbb{R}),\quad\Big{\|}\frac{|u|}{\|u\|_{\varepsilon}}\Big{\|}_{2}^{2}\leq\frac{1}{V_{0}},\quad\Big{\|}\nabla\frac{|u|}{\|u\|_{\varepsilon}}\Big{\|}_{2}^{2}\leq
1.$

Thus, for every $u\in H_{\varepsilon}$ with $\|u\|_{\varepsilon}=r>0$ and
$r^{2}<\pi/\alpha$, using Lemma 2.3, we get that there exists $C>0$
independent of $r$ such that

(4.2)
$\int_{\mathbb{R}^{2}}(e^{2\alpha|u|^{2}}-1)dx=\int_{\mathbb{R}^{2}}(e^{2\alpha
r^{2}\big{(}\frac{|u|}{\|u\|_{\varepsilon}}\big{)}^{2}}-1)dx<\int_{\mathbb{R}^{2}}(e^{2\pi\big{(}\frac{|u|}{\|u\|_{\varepsilon}}\big{)}^{2}}-1)dx\leq
C.$

Hence, by (4.1) and (4.2), we get that for any $\zeta>0$, there exits $C>0$
independent of $r$ such that

$J_{\varepsilon}(u)\geq\frac{1}{2}r^{2}-\frac{\zeta}{V_{0}}\int_{\mathbb{R}^{2}}V_{\varepsilon}(x)|u|^{2}dx-
Cr^{q}\geq\frac{1}{2}\Big{(}1-\frac{\zeta}{V_{0}}\Big{)}r^{2}-Cr^{q}$

and so, since $q>2$, for $r$ small enough we obtain (i).
Finally, property (ii) is an easy consequence of ($f_{4}$), since, fixed
$\varphi\in C_{c}^{\infty}(\mathbb{R}^{2},\mathbb{C})\setminus\\{0\\}$, we
have that

$J_{\varepsilon}(t\varphi)\leq\frac{t^{2}}{2}\|\varphi\|^{2}_{\varepsilon}-\frac{C_{p}}{p}t^{p}\|\varphi\|_{p}^{p}.$

∎

Let now

(4.3)
$c_{\varepsilon}:=\inf_{\gamma\in\Gamma_{\varepsilon}}\max_{t\in[0,1]}J_{\varepsilon}(\gamma(t)),$

where

$\Gamma_{\varepsilon}=\\{\gamma\in C([0,1],H_{\varepsilon}):\gamma(0)=0\text{
and }J_{\varepsilon}(\gamma(1))<0\\}.$

To find ground states and multiple solutions for (1.1), as usual, we consider
the Nehari manifold

$\mathcal{N}_{\varepsilon}:=\\{u\in
H_{\varepsilon}\backslash\\{0\\}:J^{\prime}_{\varepsilon}(u)[u]=0\\}.$

In virtue of ($f_{4}$), we can show that for any $u\in
H_{\varepsilon}\setminus\\{0\\}$ there exists a unique $t_{\varepsilon}>0$
such that $t_{\varepsilon}u\in\mathcal{N}_{\varepsilon}$ and

(4.4) $J_{\varepsilon}(t_{\varepsilon}u)=\max_{t\geq 0}J_{\varepsilon}(tu)$

and, using classical arguments (see e.g. [33, Lemma 4.1 and Theorem 4.2]),

(4.5) $c_{\varepsilon}=\inf_{u\in H_{\varepsilon}\backslash\\{0\\}}\sup_{t\geq
0}J_{\varepsilon}(tu)=\inf_{u\in\mathcal{N}_{\varepsilon}}J_{\varepsilon}(u).$

Moreover we have the following useful preliminary property.

######  Lemma 4.2.

There exist $K_{1},K_{2}>0$ such that for every $\varepsilon>0$ and for every
$u\in\mathcal{N}_{\varepsilon}$

$\|u\|_{\varepsilon}\geq K_{1}\text{ and }J_{\varepsilon}(u)\geq K_{2}.$

###### Proof.

Observe that, for every $\varepsilon>0$, if $u\in\mathcal{N}_{\varepsilon}$,
by (2.6), fixed $q>2$, for any $\zeta>0$ and $\alpha>4\pi$, there exists a
constant $C>0$, which depends on $q$, $\alpha$, $\zeta$, such that

(4.6)
$\|u\|_{\varepsilon}^{2}=\int_{\mathbb{R}^{2}}f(|u|^{2})|u|^{2}\leq\zeta\|u\|_{2}^{2}+C\int_{\mathbb{R}^{2}}|u|^{q}(e^{\alpha|u|^{2}}-1)dx.$

In particular, for $\zeta<V_{0}/2$, by (V), we have that

(4.7)
$\|u\|_{\varepsilon}^{2}-\zeta\|u\|_{2}^{2}\geq\frac{1}{2}\|u\|_{\varepsilon}^{2}+\frac{1}{2}\int_{\mathbb{R}^{2}}[V_{\varepsilon}(x)-V_{0}]|u|^{2}dx\\\
\geq\frac{1}{2}\|u\|_{\varepsilon}^{2}.$

By the Hölder inequality and (2.17), we infer that

(4.8)
$\int_{\mathbb{R}^{2}}|u|^{q}(e^{\alpha|u|^{2}}-1)dx\leq\|u\|_{2q}^{q}\Big{(}\int_{\mathbb{R}^{2}}(e^{\alpha|u|^{2}}-1)^{2}dx\Big{)}^{1/2}\leq
C\|u\|_{\varepsilon}^{q}\Big{(}\int_{\mathbb{R}^{2}}(e^{2\alpha|u|^{2}}-1)dx\Big{)}^{1/2}.$

Now, assume by contradiction that for every $n\in\mathbb{N}$ there exist
$\varepsilon_{n}>0$ and $u_{n}\in\mathcal{N}_{\varepsilon_{n}}$ such that
$\|u_{n}\|_{\varepsilon_{n}}\to 0$ as $n\to+\infty$. Then, using the
diamagnetic inequality (2.1) and (V), also
$\|\nabla|u_{n}|\|_{2}^{2}+V_{0}\|u_{n}\|_{2}^{2}\to 0$ as $n\to+\infty$ and
so, for $n$ large enough, using Lemma 2.3, for some $\bar{\alpha}\in(0,4\pi)$,

(4.9)
$\begin{split}\int_{\mathbb{R}^{2}}(e^{2\alpha|u_{n}|^{2}}-1)dx&=\int_{\mathbb{R}^{2}}(e^{2\alpha(\|\nabla|u_{n}|\|_{2}^{2}+V_{0}\|u_{n}\|_{2}^{2})\left(\frac{|u_{n}|}{\sqrt{\|\nabla|u_{n}|\|_{2}^{2}+V_{0}\|u_{n}\|_{2}^{2}}}\right)^{2}}-1)dx\\\
&\leq\int_{\mathbb{R}^{2}}(e^{\bar{\alpha}\left(\frac{|u_{n}|}{\sqrt{\|\nabla|u_{n}|\|_{2}^{2}+V_{0}\|u_{n}\|_{2}^{2}}}\right)^{2}}-1)dx\leq
C.\end{split}$

Thus, by (4.6), (4.7), (4.8), and (4.9), we get

$\frac{1}{2}\|u_{n}\|_{\varepsilon_{n}}^{2}\leq
C\|u_{n}\|_{\varepsilon_{n}}^{q}$

reach a contradiction.
The second part is an immediate consequence of the first one observing that,
if $u\in\mathcal{N}_{\varepsilon}$, by ($f_{3}$),

$J_{\varepsilon}(u)=J_{\varepsilon}(u)-\frac{1}{\theta}J_{\varepsilon}^{\prime}(u)[u]=\frac{\theta-2}{2\theta}\|u\|_{\varepsilon}^{2}+\frac{1}{\theta}\int_{\mathbb{R}^{2}}\Big{[}f(|u|^{2})|u|^{2}-\frac{\theta}{2}F(|u|^{2})\Big{]}dx\geq\frac{\theta-2}{2\theta}\|u\|_{\varepsilon}^{2}.$

∎

The functional $J_{\varepsilon}$ satisfies the following fundamental property.

######  Lemma 4.3.

If $\\{u_{n}\\}\subset H_{\varepsilon}$ is a $(PS)$ sequence for
$J_{\varepsilon}$, then it is bounded in $H_{\varepsilon}$.

###### Proof.

Let $d:=\lim_{n}J_{\varepsilon}(u_{n})\in\mathbb{R}$. Using ($f_{3}$) we have
that

$\displaystyle\frac{\theta-2}{2\theta}\|u_{n}\|_{\varepsilon}^{2}$
$\displaystyle\leq(\frac{1}{2}-\frac{1}{\theta})\|u_{n}\|_{\varepsilon}^{2}+\int_{\mathbb{R}^{2}}\Big{[}\frac{1}{\theta}f(|u_{n}|^{2})|u_{n}|^{2}-\frac{1}{2}F(|u_{n}|^{2})\Big{]}dx$
$\displaystyle=J_{\varepsilon}(u_{n})-\frac{1}{\theta}J_{\varepsilon}^{\prime}(u_{n})[u_{n}]$
$\displaystyle\leq d+o_{n}(1)+o_{n}(1)\|u_{n}\|_{\varepsilon}$

and we can conclude. ∎

The following lemma is very important for the proof of Theorem 1.1.

######  Lemma 4.4.

There exists $\bar{\varepsilon}>0$ such that, for all
$\varepsilon\in(0,\bar{\varepsilon})$, $c_{\varepsilon}\leq c_{V_{0}}$.

###### Proof.

Assume for simplicity that $V(0)=V_{0}$ and let $\omega\in
H^{1}(\mathbb{R}^{2},\mathbb{R})$ be a positive ground state solution of the
limit problem (3.1) with $\mu=V_{0}$ (see Lemma 3.1) and $\eta\in
C_{c}^{\infty}(\mathbb{R}^{2},[0,1])$ be a cut-off function such that $\eta=1$
in $B_{1}(0)$ and $\eta=0$ in $B^{c}_{2}(0)$. Moreover, let

$\omega_{r}(x):=\eta_{r}(x)\omega(x)e^{iA(0)\cdot
x},\qquad\eta_{r}(x)=\eta(x/r)\text{ for }r>0.$

Observe that $|\omega_{r}|=\eta_{r}\omega$ and $\omega_{r}\in H_{\varepsilon}$
by Lemma 2.2.
Let $\varepsilon_{n}\rightarrow 0^{+}$ as $n\rightarrow+\infty$. Taking
$t_{\varepsilon_{n},r}>0$ such that
$t_{\varepsilon_{n},r}\omega_{r}\in\mathcal{N}_{\varepsilon_{n}}$, by (4.5) we
have

(4.10) $c_{\varepsilon_{n}}\leq
J_{\varepsilon_{n}}(t_{\varepsilon_{n},r}\omega_{r})=\frac{t^{2}_{\varepsilon_{n},r}}{2}\int_{\mathbb{R}^{2}}|\nabla_{A_{\varepsilon_{n}}}\omega_{r}|^{2}dx+\frac{t^{2}_{\varepsilon_{n},r}}{2}\int_{\mathbb{R}^{2}}V_{\varepsilon_{n}}(x)|\omega_{r}|^{2}dx-\frac{1}{2}\int_{\mathbb{R}^{2}}F(t^{2}_{\varepsilon_{n},r}|\omega_{r}|^{2})dx.$

Observe that, for any fixed $r>0$,

(4.11)
$\displaystyle\lim_{n}\int_{\mathbb{R}^{2}}|\nabla_{A_{\varepsilon_{n}}}\omega_{r}|^{2}dx=\int_{\mathbb{R}^{2}}|\nabla(\eta_{r}\omega)|^{2}dx.$

Indeed,

$\displaystyle\int_{\mathbb{R}^{2}}|\nabla_{A_{\varepsilon_{n}}}\omega_{r}|^{2}dx$
$\displaystyle=\int_{\mathbb{R}^{2}}\Big{|}\frac{\nabla}{i}(\eta_{r}\omega)-[A_{\varepsilon_{n}}(x)-A(0)]\eta_{r}\omega\Big{|}^{2}dx$
$\displaystyle=\int_{\mathbb{R}^{2}}|\nabla(\eta_{r}\omega)|^{2}dx+\int_{\mathbb{R}^{2}}|A_{\varepsilon_{n}}(x)-A(0)|^{2}\eta_{r}^{2}\omega^{2}dx$

and

$\int_{\mathbb{R}^{2}}|A_{\varepsilon_{n}}(x)-A(0)|^{2}\eta_{r}^{2}\omega^{2}dx\leq\int_{B_{2r}(0)}|A_{\varepsilon_{n}}(x)-A(0)|^{2}\omega^{2}dx=o_{n}(1).$

Moreover

(4.12)
$\lim_{n}\int_{\mathbb{R}^{2}}V_{\varepsilon_{n}}(x)|\omega_{r}|^{2}dx=V_{0}\int_{\mathbb{R}^{2}}\eta_{r}^{2}\omega^{2}dx$

since

$\Big{|}\int_{\mathbb{R}^{2}}V_{\varepsilon_{n}}(x)|\omega_{r}|^{2}dx-
V_{0}\int_{\mathbb{R}^{2}}\eta_{r}^{2}\omega^{2}dx\Big{|}\leq\int_{\mathbb{R}^{2}}|V_{\varepsilon_{n}}(x)-V_{0}|\eta_{r}^{2}\omega^{2}dx\leq\int_{B_{2r}(0)}|V_{\varepsilon_{n}}(x)-V_{0}|\omega^{2}dx=o_{n}(1).$

We claim that, if $r>0$ is fixed, $\\{t_{\varepsilon_{n},r}\\}$ is bounded.
Indeed, let us assume by contradiction that there exists a subsequence, still
denoted by $t_{\varepsilon_{n},r}$, such that
$t_{\varepsilon_{n},r}\to+\infty$ as $n\rightarrow+\infty$.
Since
$J^{\prime}_{\varepsilon_{n}}(t_{\varepsilon_{n},r}\omega_{r})[t_{\varepsilon_{n},r}\omega_{r}]=0$,
by ($f_{1}$) and ($f_{4}$) we have

$\int_{\mathbb{R}^{2}}|\nabla_{A_{\varepsilon_{n}}}\omega_{r}|^{2}dx+\int_{\mathbb{R}^{2}}V_{\varepsilon_{n}}(x)|\omega_{r}|^{2}dx=\int_{\mathbb{R}^{2}}f(t^{2}_{\varepsilon_{n},r}|\omega_{r}|^{2})|\omega_{r}|^{2}dx\geq\int_{B_{r}(0)}f(t^{2}_{\varepsilon_{n},r}|\omega|^{2})|\omega|^{2}dx\geq
Ct^{p-2}_{\varepsilon_{n},r}\to+\infty$

as $n\rightarrow+\infty$, which is a contradiction with (4.11) and (4.12).
Thus, up to a subsequence, there exists $t_{r}\geq 0$ such that
$t_{\varepsilon_{n},r}\rightarrow t_{r}$ as $n\rightarrow+\infty$ for $r$
fixed.
Therefore, by (4.10),

(4.13)
$\limsup_{n}c_{\varepsilon_{n}}\leq\frac{t^{2}_{r}}{2}\int_{\mathbb{R}^{2}}|\nabla(\eta_{r}\omega)|^{2}dx+\frac{t^{2}_{r}}{2}V_{0}\int_{\mathbb{R}^{2}}|\eta_{r}\omega|^{2}dx-\frac{1}{2}\int_{\mathbb{R}^{2}}F(t^{2}_{r}|\eta_{r}\omega|^{2})dx=I_{V_{0}}(t_{r}\eta_{r}\omega).$

Let us consider now $r_{m}\rightarrow+\infty$ as $m\rightarrow+\infty$.
Since

$\int_{\mathbb{R}^{2}}|\nabla(\eta_{r_{m}}\omega-\omega)|^{2}dx=\int_{\mathbb{R}^{2}}|\nabla\eta_{r_{m}}|^{2}\omega^{2}dx+2\int_{\mathbb{R}^{2}}(\eta_{r_{m}}-1)\omega\nabla\eta_{r_{m}}\nabla\omega
dx+\int_{\mathbb{R}^{2}}(\eta_{r_{m}}-1)^{2}|\nabla\omega|^{2}dx,$
$\int_{\mathbb{R}^{2}}|\nabla\eta_{r_{m}}|^{2}\omega^{2}dx\leq\frac{C}{r_{m}^{2}}\int_{B_{r_{m}}^{c}}\omega^{2}dx=o_{m}(1),$
$\Big{|}\int_{\mathbb{R}^{2}}(\eta_{r_{m}}-1)\omega\nabla\eta_{r_{m}}\nabla\omega
dx\Big{|}\leq\frac{C}{r_{m}}\Big{(}\int_{B_{r_{m}}^{c}}\omega^{2}dx\Big{)}^{1/2}\Big{(}\int_{B_{r_{m}}^{c}}|\nabla\omega|^{2}dx\Big{)}^{1/2}=o_{m}(1),$
$\int_{\mathbb{R}^{2}}(\eta_{r_{m}}-1)^{2}|\nabla\omega|^{2}dx\leq\int_{B_{r_{m}}^{c}}|\nabla\omega|^{2}dx=o_{m}(1),$
$\int_{\mathbb{R}^{2}}(\eta_{r_{m}}-1)^{2}\omega^{2}dx\leq\int_{B_{r_{m}}^{c}}\omega^{2}dx=o_{m}(1),$

then $\eta_{r_{m}}\omega\rightarrow\omega$ in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$ as $m\rightarrow+\infty$.
Now, let $\tau_{r_{m}}>0$ such that
$\tau_{r_{m}}\eta_{r_{m}}\omega\in\mathcal{N}_{V_{0}}$.
Arguing as before, since

(4.14)
$\int_{\mathbb{R}^{2}}|\nabla(\eta_{r_{m}}\omega)|^{2}dx+V_{0}\int_{\mathbb{R}^{2}}\eta_{r_{m}}^{2}\omega^{2}dx=\int_{\mathbb{R}^{2}}f(\tau^{2}_{r_{m}}|\eta_{r_{m}}\omega|^{2})\eta_{r_{m}}^{2}\omega^{2}dx,$

we have that $\\{\tau_{r_{m}}\\}$ is bounded.
Thus, up to a subsequence, $\tau_{r_{m}}\rightarrow\bar{\tau}\geq 0$ as
$m\rightarrow+\infty$ and, by (4.14), the uniqueness of the projection in
$\mathcal{N}_{V_{0}}$, and since $\omega\in\mathcal{N}_{V_{0}}$ we get that
$\bar{\tau}=1$.
Hence, by (4.13) and (3.2),

$\limsup_{n}c_{\varepsilon_{n}}\leq I_{V_{0}}(t_{r_{m}}\eta_{r_{m}}\omega)\leq
I_{V_{0}}(\tau_{r_{m}}\eta_{r_{m}}\omega)$

and so, passing to the limit as $m\to+\infty$, we get

$\limsup_{n}c_{\varepsilon_{n}}\leq I_{V_{0}}(\omega)=c_{V_{0}}.$

∎

In our arguments we will use the following result that is a version of the
celebrated Lions Lemma (see e.g. [33] and [20, Lemma 3.3]).

######  Lemma 4.5.

Let $\\{u_{n}\\}\subset H_{\varepsilon}$ satisfy
$J_{\varepsilon}^{\prime}(u_{n})[u_{n}]\to 0$, $u_{n}\rightharpoonup 0$ in
$H_{\varepsilon}$, as $n\to+\infty$, and
$\ell^{2}:=\limsup_{n}\|\nabla|u_{n}|\|_{2}^{2}<1$. Then, one of the following
alternatives occurs:

  1. (i)

$u_{n}\rightarrow 0$ in $H_{\varepsilon}$ as $n\to+\infty$;

  2. (ii)

there are a sequence $\\{y_{n}\\}\subset\mathbb{R}^{2}$ and constants $R$,
$\beta>0$ such that

$\liminf_{n}\int_{B_{R}(y_{n})}|u_{n}|^{2}dx\geq\beta.$

###### Proof.

If (ii) does not hold, then, for every $R>0$,

$\lim_{n}\sup_{y\in\mathbb{R}^{2}}\int_{B_{R}(y)}|u_{n}|^{2}dx=0.$

Since $\\{|u_{n}|\\}$ is bounded in $H^{1}(\mathbb{R}^{2})$, [28, Chapter 6,
Lemma 8.4] implies that

(4.15) $\|u_{n}\|_{\tau}\rightarrow 0\text{ as }n\rightarrow+\infty\text{ for
all }\tau>2.$

Moreover, using (2.6), we have that, fixed $q>2$, for any $\zeta>0$ and
$\alpha>4\pi$, there exists $C>0$ such that

(4.16)
$\begin{split}0\leq\|u_{n}\|_{\varepsilon}^{2}&=\int_{\mathbb{R}^{2}}f(|u_{n}|^{2})|u_{n}|^{2}dx+o_{n}(1)\\\
&\leq\zeta\|u_{n}\|_{2}^{2}+C\int_{\mathbb{R}^{2}}|u_{n}|^{q}(e^{\alpha|u_{n}|^{2}}-1)dx+o_{n}(1)\\\
&\leq\frac{\zeta}{V_{0}}\|u_{n}\|_{\varepsilon}^{2}+C\int_{\mathbb{R}^{2}}|u_{n}|^{q}(e^{\alpha|u_{n}|^{2}}-1)dx+o_{n}(1).\end{split}$

Let us fix $\kappa\in(0,1-\ell^{2})$ and take

$r\in\Big{(}1,\frac{1}{\ell^{2}+\kappa}\Big{)},\qquad\alpha\in\Big{(}4\pi,\frac{4\pi}{r(\ell^{2}+\kappa)}\Big{)}.$

Thus, the Hölder inequality, (2.17), and Lemma 2.3, imply that, for $n$ large
enough (such that $\|\nabla|u_{n}|\|_{2}^{2}<\ell^{2}+\kappa$),

$\int_{\mathbb{R}^{2}}|u_{n}|^{q}(e^{\alpha|u_{n}|^{2}}-1)dx\leq\|u_{n}\|_{qr^{\prime}}^{q}\Big{(}\int_{\mathbb{R}^{2}}(e^{r\alpha(\ell^{2}+\kappa)\Big{(}\frac{|u_{n}|}{\sqrt{\ell^{2}+\kappa}}\Big{)}^{2}}-1)dx\Big{)}^{1/r}dx\leq
C\|u_{n}\|_{qr^{\prime}}^{q}$

and so, using (4.15) and (4.16) we can conclude. ∎

Now we prove a fundamental result on the $(PS)_{d}$ sequences for
$J_{\varepsilon}$ in the case $V_{\infty}<\infty$.

######  Lemma 4.6.

Let $d\in\mathbb{R}$, $V_{\infty}<+\infty$ and $\\{u_{n}\\}\subset
H_{\varepsilon}$ be a $(PS)_{d}$ sequence for $J_{\varepsilon}$ such that
$\ell^{2}:=\limsup_{n}\|\nabla|u_{n}|\|_{2}^{2}<1$ and $u_{n}\rightharpoonup
0$ in $H_{\varepsilon}$. If $u_{n}\not\rightarrow 0$ in $H_{\varepsilon}$,
then $d\geq c_{V_{\infty}}$.

###### Proof.

For every $n\in\mathbb{N}$, let $t_{n}:=t(|u_{n}|)\in(0,\infty)$ be such that
$t_{n}|u_{n}|\in\mathcal{N}_{V_{\infty}}$ (see the first part of Section 3).
Firstly we show that $\lim\sup_{n}t_{n}\leq 1$.
Assume by contradiction that there exist $\delta>0$ and a subsequence, still
denoted by $\\{t_{n}\\}$, such that, for every $n\in\mathbb{N}$,

(4.17) $t_{n}\geq 1+\delta.$

Since $\\{u_{n}\\}$ is a $(PS)_{d}$ sequence for $J_{\varepsilon}$, we have

(4.18)
$\int_{\mathbb{R}^{2}}(|\nabla_{A_{\varepsilon}}u_{n}|^{2}+V_{\varepsilon}(x)|u_{n}|^{2})dx=\int_{\mathbb{R}^{2}}f(|u_{n}|^{2})|u_{n}|^{2}dx+o_{n}(1).$

On the other hand, $t_{n}|u_{n}|\in\mathcal{N}_{V_{\infty}}$, then one has

(4.19)
$\int_{\mathbb{R}^{2}}\Big{(}|\nabla|u_{n}||^{2}+V_{\infty}|u_{n}|^{2}\Big{)}dx=\int_{\mathbb{R}^{2}}f(t_{n}^{2}|u_{n}|^{2})|u_{n}|^{2}dx.$

Putting together (4.18), (4.19) and using the diamagnetic inequality (2.1) we
obtain

$\int_{\mathbb{R}^{2}}\Big{(}f(t_{n}^{2}|u_{n}|^{2})-f(|u_{n}|^{2})\Big{)}|u_{n}|^{2}dx\leq\int_{\mathbb{R}^{2}}(V_{\infty}-V_{\varepsilon}(x))|u_{n}|^{2}dx+o_{n}(1).$

Now, by the assumption (V), we can see that for every $\eta>0$ there exists
$R=R(\eta)>0$ such that, for every $|x|\geq R/\varepsilon$,

$V_{\infty}-V_{\varepsilon}(x)\leq\eta.$

Then, using the boundedness of $\\{u_{n}\\}$ in $H_{\varepsilon}$ and that
$u_{n}\rightarrow 0$ in $L^{2}(B_{R/\varepsilon}(0),\mathbb{C})$ by Lemma 2.1,

$\int_{\mathbb{R}^{2}}(V_{\infty}-V(\varepsilon x))|u_{n}|^{2}dx\leq
V_{\infty}\int_{B_{R/\varepsilon}(0)}|u_{n}|^{2}dx+\eta\int_{B_{R}^{c}(0)}|u_{n}|^{2}dx\leq
o_{n}(1)+C\eta.$

Thus,

(4.20)
$\int_{\mathbb{R}^{2}}\Big{(}f(t_{n}^{2}|u_{n}|^{2})-f(|u_{n}|^{2})\Big{)}|u_{n}|^{2}dx\leq
o_{n}(1)+C\eta.$

Moreover, since $u_{n}\not\rightarrow 0$ in $H_{\varepsilon}$, by Lemma 4.5,
there exist a sequence $\\{y_{n}\\}\subset\mathbb{R}^{2}$, and two positive
numbers $R$, $\beta$ such that

(4.21)
$\liminf_{n\rightarrow\infty}\int_{B_{R}(y_{n})}|u_{n}|^{2}dx\geq\beta.$

Now, let us consider $v_{n}:=|u_{n}|(\cdot+y_{n})$. Taking into account (V),
the diamagnetic inequality (2.1), and the boundedness of $\\{u_{n}\\}$ in
$H_{\varepsilon}$, we can see that

$\|\nabla
v_{n}\|_{2}^{2}+V_{0}\|v_{n}\|_{2}^{2}=\|\nabla|u_{n}|\|_{2}^{2}+V_{0}\|u_{n}\|_{2}^{2}\leq\|u_{n}\|_{\varepsilon}^{2}\leq
C.$

Therefore $v_{n}\rightharpoonup v$ in $H^{1}(\mathbb{R}^{2},\mathbb{R})$,
$v_{n}\rightarrow v$ in $L^{r}_{\text{loc}}(\mathbb{R}^{2},\mathbb{R})$ for
all $r\geq 1$, and, by (4.21),

$\int_{B_{R}(0)}|v|^{2}dx=\lim_{n\rightarrow\infty}\int_{B_{R}(0)}|v_{n}|^{2}dx\geq\beta.$

Thus there exists $\Omega\subset B_{R}(0)$ with positive measure such that
$v\not\equiv 0$ in $\Omega$.
Moreover, by (4.17), ($f_{4}$), and (4.20), we obtain

$\int_{\mathbb{R}^{2}}\Big{(}f((1+\delta)^{2}v_{n}^{2})-f(v_{n}^{2})\Big{)}v_{n}^{2}dx\leq
o_{n}(1)+C\eta,$

and so, ($f_{4}$) and the Fatou’s Lemma imply

$0<\int_{\Omega}\Big{(}f((1+\delta)^{2}v^{2})-f(v^{2})\Big{)}v^{2}dx\leq
C\eta.$

Thus, by the arbitrariness of $\eta>0$, we get a contradiction.
Now, two cases can occur.
Case 1: $\limsup_{n}t_{n}=1$.
In this case there exists a subsequence still denoted by $\\{t_{n}\\}$ such
that $t_{n}\rightarrow 1$.
Taking into account that $\\{u_{n}\\}$ is a $(PS)_{d}$ sequence for
$J_{\varepsilon}$, (3.2), (3.3), and the diamagnetic inequality (2.1), we have

(4.22) $\begin{split}d+o_{n}(1)&=J_{\varepsilon}(u_{n})\geq
J_{\varepsilon}(u_{n})-I_{V_{\infty}}(t_{n}|u_{n}|)+c_{V_{\infty}}\\\
&\geq\frac{1-t_{n}^{2}}{2}\int_{\mathbb{R}^{2}}|\nabla|u_{n}||^{2}dx+\frac{1}{2}\int_{\mathbb{R}^{2}}\Big{(}V_{\varepsilon}(x)-t_{n}^{2}V_{\infty}\Big{)}|u_{n}|^{2}dx\\\
&\qquad+\frac{1}{2}\int_{\mathbb{R}^{2}}\Big{(}F(t_{n}^{2}|u_{n}|^{2})-F(|u_{n}|^{2})\Big{)}dx+c_{V_{\infty}}.\end{split}$

Since $\\{|u_{n}|\\}$ is bounded in $H^{1}(\mathbb{R}^{2},\mathbb{R})$ and
$t_{n}\rightarrow 1$, it is easy to see that

(4.23)
$\frac{1-t_{n}^{2}}{2}\int_{\mathbb{R}^{2}}|\nabla|u_{n}||^{2}dx=o_{n}(1).$

Moreover, by (V), we have that for every $\eta>0$ there exists $R=R(\zeta)>0$
such that for any $x\in\mathbb{R}^{2}$ with $|x|>R/\varepsilon$, it holds

$V_{\varepsilon}(x)-t_{n}^{2}V_{\infty}=(V_{\varepsilon}(x)-V_{\infty})+(1-t_{n}^{2})V_{\infty}\geq-\eta+(1-t_{n}^{2})V_{\infty}.$

Thus, using as before the boundedness of $\\{u_{n}\\}$ in $H_{\varepsilon}$,
that $|u_{n}|\rightarrow 0$ in $L^{r}(B_{R/\varepsilon})$ for $r\geq 1$, and
$t_{n}\rightarrow 1$, for $n$ large enough we get

(4.24)
$\begin{split}\int_{\mathbb{R}^{2}}\Big{(}V_{\varepsilon}(x)-t_{n}^{2}V_{\infty}\Big{)}|u_{n}|^{2}dx&\geq(V_{0}-t_{n}^{2}V_{\infty})\int_{B_{R/\varepsilon}(0)}|u_{n}|^{2}dx\\\
&\qquad+((1-t_{n}^{2})V_{\infty}-\eta)\int_{B_{R/\varepsilon}^{c}(0)}|u_{n}|^{2}dx\\\
&\geq o_{n}(1)-C\eta.\end{split}$

Finally let $\vartheta,l>1$ such that $l\vartheta\ell^{2}<1$. Since $t_{n}\to
1$, for $n$ large enough, applying the Mean Value Theorem and using also
($f_{4}$), we get

$|F(t_{n}^{2}|u_{n}|^{2})-F(|u_{n}|^{2})|\leq|t_{n}^{2}-1|f(\theta_{n}(x)|u_{n}|^{2})|u_{n}|^{2}\leq|t_{n}^{2}-1|f(\vartheta|u_{n}|^{2})|u_{n}|^{2}.$

Then

(4.25)
$\Big{|}\int_{\mathbb{R}^{2}}\Big{(}F(t_{n}^{2}|u_{n}|^{2})-F(|u_{n}|^{2})\Big{)}dx\Big{|}\leq|t_{n}^{2}-1|\int_{\mathbb{R}^{2}}f(\vartheta|u_{n}|^{2})|u_{n}|^{2}dx=o_{n}(1).$

Indeed, $t_{n}\rightarrow 1$. Moreover, if we take

$\kappa\in\Big{(}0,\frac{1}{l\vartheta}-\ell^{2}\Big{)}\text{ and
}\alpha\in\Big{(}4\pi,\frac{4\pi}{l\vartheta(\ell^{2}+\kappa)}\Big{)},$

by (2.4), fixed $q>2$, for any $\zeta>0$, there exists a constant $C>0$ such
that

$\int_{\mathbb{R}^{2}}f(\vartheta|u_{n}|^{2})|u_{n}|^{2}dx\leq\zeta\|u_{n}\|_{2}^{2}+C\int_{\mathbb{R}^{2}}|u_{n}|^{q}(e^{\vartheta\alpha|u_{n}|^{2}}-1)dx$

and the right hand side is bounded due to the boundedness of $\\{u_{n}\\}$ and
since the Hölder inequality, (2.17), Lemma 2.3, and

$\|\nabla|u_{n}|\|_{2}^{2}<\ell^{2}+\kappa<\frac{1}{l\vartheta}<1\text{ for
}n\text{ large enough},$

imply that

$\displaystyle\int_{\mathbb{R}^{2}}|u_{n}|^{q}(e^{\vartheta\alpha|u_{n}|^{2}}-1)dx$
$\displaystyle\leq\|u_{n}\|_{ql^{\prime}}^{q}\big{(}\int_{\mathbb{R}^{2}}(e^{l\vartheta\alpha|u_{n}|^{2}}-1)dx\big{)}^{1/l}$
$\displaystyle\leq\|u_{n}\|_{ql^{\prime}}^{q}\big{(}\int_{\mathbb{R}^{2}}(e^{l\vartheta\alpha(\ell^{2}+\kappa)\big{(}\frac{|u_{n}|}{\sqrt{\ell^{2}+\kappa}}\big{)}^{2}}-1)dx\big{)}^{1/l}\leq
C,$

where $l^{\prime}>1$ is the conjugate exponent of $l$.
Hence, putting together (4.22), (4.23), (4.24), and (4.25), we can obtain

$\displaystyle d+C\eta+o_{n}(1)\geq c_{V_{\infty}},$

and, due to the arbitrariness of $\eta$, taking the limit as
$n\rightarrow\infty$, we conclude.
Case 2: $\lim\sup_{n}t_{n}=\bar{t}<1$.
In this case there exists a subsequence still denoted by $\\{t_{n}\\}$, such
that $t_{n}\rightarrow\bar{t}$ and $t_{n}<1$ for $n$ large enough.
Since $\\{u_{n}\\}$ is a bounded $(PS)_{d}$ sequence for $J_{\varepsilon}$ in
$H_{\varepsilon}$, we have

(4.26)
$d+o_{n}(1)=J_{\varepsilon}(u_{n})-\frac{1}{2}J^{\prime}_{\varepsilon}(u_{n})[u_{n}]=\frac{1}{2}\int_{\mathbb{R}^{2}}\Big{(}f(|u_{n}|^{2})|u_{n}|^{2}-F(|u_{n}|^{2})\Big{)}dx.$

Note that, by ($f_{3}$), the map $t\mapsto f(t)t-F(t)$ is increasing for
$t>0$. Hence, this fact combined with
$t_{n}|u_{n}|\in\mathcal{N}_{V_{\infty}}$, $t_{n}<1$, and (4.26), allows us to
get

$\displaystyle c_{V_{\infty}}$ $\displaystyle\leq
I_{V_{\infty}}(t_{n}|u_{n}|)=I_{V_{\infty}}(t_{n}|u_{n}|)-\frac{t_{n}}{2}I^{\prime}_{V_{\infty}}(t_{n}|u_{n}|)[|u_{n}|]$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{2}}\Big{(}f(t_{n}^{2}|u_{n}|^{2})t_{n}^{2}|u_{n}|^{2}-F(t_{n}^{2}|u_{n}|^{2})\Big{)}dx$
$\displaystyle\leq\frac{1}{2}\int_{\mathbb{R}^{2}}\Big{(}f(|u_{n}|^{2})|u_{n}|^{2}-F(|u_{n}|^{2})\Big{)}dx$
$\displaystyle=d+o_{n}(1).$

Passing to the limit as $n\rightarrow+\infty$, we complete the proof. ∎

We shall also use the next result about the $(PS)$ sequences.

######  Lemma 4.7.

Let $\\{u_{n}\\}\subset H_{\varepsilon}$ be a (PS) sequence for
$J_{\varepsilon}$ such that
$\ell^{2}:=\limsup_{n}\|u_{n}\|_{\varepsilon}^{2}<1$. Then, up to a
subsequence, $\nabla|u_{n}|\rightarrow\nabla|u|$ a.e. in $\mathbb{R}^{2}$.

###### Proof.

Let $\\{u_{n}\\}\subset H_{\varepsilon}$ be a (PS) sequence for
$J_{\varepsilon}$ with $\limsup_{n}\|u_{n}\|_{\varepsilon}^{2}<1$. By Lemma
4.3 we have that $\\{u_{n}\\}$ is bounded in $H_{\varepsilon}$ and so, up to a
subsequence, $u_{n}\rightharpoonup u$ in $H_{\varepsilon}$, $u_{n}\rightarrow
u$ in $L^{r}_{\rm loc}(\mathbb{R}^{2},\mathbb{C})$ for any $r\geq 1$ and
$u_{n}\rightarrow u$ a.e. in $\mathbb{R}^{2}$. Since the norm is weakly lower
semicontinuous, it follows that $\|u\|_{\varepsilon}^{2}<1$.
Now observe that, for every $\phi\in
C_{c}^{\infty}(\mathbb{R}^{2},\mathbb{C})$,

$f(|u_{n}|^{2})u_{n}\overline{\phi}\rightarrow
f(|u|^{2})u\overline{\phi}\,\text{ a.e. in }\,\mathbb{R}^{2}\,\text{ as
}\,n\to+\infty$

and, by (2.4), fixed $q>2$, for any $\zeta>0$ and $\alpha>4\pi$, there exists
$C>0$ such that, for every $\phi\in
C_{c}^{\infty}(\mathbb{R}^{2},\mathbb{R})$,

$|{\rm
Re}[f(|u_{n}^{2}|)u_{n}\overline{\phi}]|\leq|f(|u_{n}^{2}|)u_{n}\overline{\phi}|\leq\zeta|u_{n}||\phi|+C|u_{n}|^{q-1}(e^{\alpha|u_{n}|^{2}}-1)|\phi|$

with

$\zeta|u_{n}||\phi|+C|u_{n}|^{q-1}(e^{\alpha|u_{n}|^{2}}-1)|\phi|\to\zeta|u||\phi|+C|u|^{q-1}(e^{\alpha|u|^{2}}-1)|\phi|\text{
a.e. in }\mathbb{R}^{2}\text{ as }n\to+\infty.$

Moreover, let $r^{*},\tau>1$ such that $r^{*}\tau\ell^{2}<1$. Using the Hölder
inequality, (2.17), Sobolev inequality, Lemma 2.3, and taking

$\kappa\in\Big{(}0,\frac{1}{r^{*}\tau}-\ell^{2}\Big{)}\text{ and
}\alpha\in\Big{(}4\pi,\frac{4\pi}{r^{*}\tau(\ell^{2}+\kappa)}\Big{)},$

we have that, for $n$ large enough, such that
$\|\nabla|u_{n}|\|_{2}^{2}\leq\ell^{2}+\kappa$,

(4.27)
$\begin{split}\int_{\mathbb{R}^{2}}\Big{[}|u_{n}|^{q-1}(e^{\alpha|u_{n}|^{2}}-1)\Big{]}^{r^{*}}dx&\leq\|u_{n}\|_{\tau^{\prime}r^{*}(q-1)}^{r^{*}(q-1)}\Big{(}\int_{\mathbb{R}^{2}}(e^{\alpha
r^{*}\tau|u_{n}|^{2}}-1)dx\Big{)}^{1/\tau}\\\ &\leq
C\|u_{n}\|_{\varepsilon}^{r^{*}(q-1)}\Big{(}\int_{\mathbb{R}^{2}}(e^{\alpha
r^{*}\tau(\ell^{2}+\kappa)\Big{(}\frac{|u_{n}|}{\sqrt{\ell^{2}+\kappa}}\Big{)}^{2}}-1)dx\Big{)}^{1/\tau}\leq
C,\end{split}$

where $\tau^{\prime}$ is the conjugate exponent of $\tau$.
Then, up to a subsequence and applying the Severini-Egoroff Theorem, we get

$|u_{n}|^{q-1}(e^{\alpha|u_{n}|^{2}}-1)\rightharpoonup|u|^{q-1}(e^{\alpha|u|^{2}}-1)\text{
in }L^{r}(\mathbb{R}^{2},\mathbb{R}).$

Thus, since, up to a subsequence, also $|u_{n}|\rightharpoonup|u|$ in
$L^{2}(\mathbb{R}^{2},\mathbb{R})$, we have, as $n\to+\infty$,

$\zeta\int_{\mathbb{R}^{2}}|u_{n}||\phi|dx+C\int_{\mathbb{R}^{2}}|u_{n}|^{q-1}(e^{\alpha|u_{n}|^{2}}-1)|\phi|dx\to\zeta\int_{\mathbb{R}^{2}}|u||\phi|dx+C\int_{\mathbb{R}^{2}}|u|^{q-1}(e^{\alpha|u|^{2}}-1)|\phi|dx.$

Hence, by a variant of the Lebesgue Dominated Convergence Theorem, we obtain
that, for every $\phi\in C_{c}^{\infty}(\mathbb{R}^{2},\mathbb{C})$

$\text{Re}\int_{\mathbb{R}^{2}}f(|u_{n}|^{2})u_{n}\overline{\phi}dx\rightarrow\text{Re}\int_{\mathbb{R}^{2}}f(|u|^{2})u\overline{\phi}dx,$

from which we get $J^{\prime}_{\varepsilon}(u)=0$.
Let now $R>0$ be arbitrary and $\psi$ be a
$C_{c}^{\infty}(\mathbb{R}^{2},[0,1])$ function such that $\psi\equiv 1$ in
$B_{R}(0)$ and $\psi\equiv 0$ in $B_{2R}^{c}(0)$.
First we observe that, since for every $v\in H_{\varepsilon}$,

$\nabla_{A_{\varepsilon}}(v\psi)=\psi\nabla_{A_{\varepsilon}}v-iv\nabla\psi,$

and then, by the Hölder inequality,

$\|\nabla_{A_{\varepsilon}}(v\psi)\|_{2}^{2}=\|\psi\nabla_{A_{\varepsilon}}v\|_{2}^{2}+2\text{Re}\int_{\mathbb{R}^{2}}i\overline{v}\psi\nabla_{A_{\varepsilon}}v\nabla\psi
dx+\||v||\nabla\psi|\|_{2}^{2}\leq
C(\|\nabla_{A_{\varepsilon}}v\|_{2}^{2}+\|v\|_{2}^{2}),$

we have that $v\psi\in H_{\varepsilon}$ and that the boundedness of
$\\{u_{n}\\}$ in $H_{\varepsilon}$ implies the boundedness of
$\\{u_{n}\psi\\}$ in $H_{\varepsilon}$.
By (2.4), the Hölder and diamagnetic inequalities, (2.17), Lemma 2.1, and
Lemma 2.3, for $q>2$ and $\alpha>4\pi$,

$\begin{split}\left|\text{Re}\int_{\mathbb{R}^{2}}f(|u|^{2})u\overline{(u-u_{n})}\psi
dx\right|&\leq\int_{\mathbb{R}^{2}}f(|u|^{2})|u||u-u_{n}|\psi dx\\\
&\leq\zeta\int_{\mathbb{R}^{2}}|u||u-u_{n}|\psi dx\\\
&\qquad+C\int_{\mathbb{R}^{2}}|u|^{q-1}(e^{\alpha|u|^{2}}-1)|u-u_{n}|\psi
dx\\\
&\leq\zeta\|u\|_{2}\left(\int_{B_{2R}(0)}|u-u_{n}|^{2}dx\right)^{\frac{1}{2}}\\\
&\qquad+C\left(\int_{\mathbb{R}^{2}}|u|^{2(q-1)}(e^{2\alpha|u|^{2}}-1)dx\right)^{\frac{1}{2}}\left(\int_{B_{2R}(0)}|u-u_{n}|^{2}dx\right)^{\frac{1}{2}}\\\
&\leq
o_{n}(1)+C\|u\|_{4(q-1)}^{q-1}\left(\int_{\mathbb{R}^{2}}(e^{4\alpha|u|^{2}}-1)dx\right)^{\frac{1}{4}}o_{n}(1)\\\
&=o_{n}(1).\end{split}$

Moreover, by Lemma 2.1 and the Hölder inequality,

$\left|\text{Re}\int_{\mathbb{R}^{2}}V_{\varepsilon}(x)u\overline{(u-u_{n})}\psi
dx\right|\leq
C\|u\|_{2}\left(\int_{B_{2R}(0)}|u-u_{n}|^{2}dx\right)^{\frac{1}{2}}=o_{n}(1)$

and

$\left|\text{Re}\int_{\mathbb{R}^{2}}i\overline{(u-u_{n})}\nabla_{A_{\varepsilon}}u\nabla\psi
dx\right|\leq C\|\nabla_{A_{\varepsilon}}u\|_{2}\left(\int_{B_{2R}(0)\setminus
B_{R}(0)}|u-u_{n}|^{2}dx\right)^{\frac{1}{2}}=o_{n}(1).$

Thus

(4.28)
$\begin{split}\text{Re}\int_{\mathbb{R}^{2}}\nabla_{A_{\varepsilon}}u\overline{\nabla_{A_{\varepsilon}}(u-u_{n})}\psi
dx&=J^{\prime}_{\varepsilon}(u)[(u-u_{n})\psi]-\text{Re}\int_{\mathbb{R}^{2}}V_{\varepsilon}(x)u\overline{(u-u_{n})}\psi
dx\\\
&\quad-\text{Re}\int_{\mathbb{R}^{2}}i\overline{(u-u_{n})}\nabla_{A_{\varepsilon}}u\nabla\psi
dx+\text{Re}\int_{\mathbb{R}^{2}}f(|u|^{2})u\overline{(u-u_{n})}\psi dx\\\
&=o_{n}(1).\end{split}$

Now, using (4.28), $J^{\prime}_{\varepsilon}(u_{n})\rightarrow 0$ as
$n\rightarrow+\infty$, and the boundedness of the sequence $\\{u_{n}\psi\\}$
in $H_{\varepsilon}$, we obtain

(4.29)
$\begin{split}0\leq\int_{B_{R}(0)}|\nabla_{A_{\varepsilon}}u_{n}-\nabla_{A_{\varepsilon}}u|^{2}dx&\leq\int_{\mathbb{R}^{2}}|\nabla_{A_{\varepsilon}}u_{n}-\nabla_{A_{\varepsilon}}u|^{2}\psi
dx\\\
&=\text{Re}\int_{\mathbb{R}^{2}}\nabla_{A_{\varepsilon}}u_{n}\overline{\nabla_{A_{\varepsilon}}(u_{n}-u)}\psi
dx+o_{n}(1)\\\
&=J^{\prime}_{\varepsilon}(u_{n})[(u_{n}-u)\psi]+K_{1,n}+K_{2,n}+K_{3,n}+o_{n}(1)\\\
&=K_{1,n}+K_{2,n}+K_{3,n}+o_{n}(1),\end{split}$

where

$\displaystyle K_{1,n}$
$\displaystyle=\text{Re}\int_{\mathbb{R}^{2}}i\overline{(u-u_{n})}\nabla_{A_{\varepsilon}}u_{n}\nabla\psi
dx,$ $\displaystyle K_{2,n}$
$\displaystyle=\text{Re}\int_{\mathbb{R}^{2}}V_{\varepsilon}(x)u_{n}\overline{(u-u_{n})}\psi
dx,$ $\displaystyle K_{3,n}$
$\displaystyle=\text{Re}\int_{\mathbb{R}^{2}}f(|u_{n}|^{2})u_{n}\overline{(u_{n}-u)}\psi
dx.$

By Lemma 2.1, the Hölder inequality, and the boundedness of $\\{u_{n}\\}$ in
$H_{\varepsilon}$, we get

$|K_{1,n}|\leq\int_{B_{2R}(0)\backslash
B_{R}(0)}|\nabla_{A_{\varepsilon}}u_{n}||u-u_{n}||\nabla\psi|dx\leq
C\|u_{n}\|_{\varepsilon}\Big{(}\int_{B_{2R}(0)\backslash
B_{R}(0)}|u-u_{n}|^{2}dx\Big{)}^{1/2}=o_{n}(1)$

and

$|K_{2,n}|\leq\int_{B_{2R}(0)}V_{\varepsilon}(x)|u_{n}||u-u_{n}|dx\leq
C\|u_{n}\|_{2}\left(\int_{B_{2R}(0)}|u-u_{n}|^{2}dx\right)^{1/2}=o_{n}(1).$

Moreover, by (2.4), the Hölder inequality and the boundedness of $\\{u_{n}\\}$
in $H_{\varepsilon}$, for $q>2$ and suitable $\tau>1$ and $\alpha>4\pi$, if
$\tau^{\prime}$ is the conjugate exponent of $\tau$,

$\begin{split}|K_{3,n}|&\leq\int_{B_{2R}(0)}|f(|u_{n}|^{2})||u_{n}||u_{n}-u|dx\\\
&\leq\zeta\|u_{n}\|_{2}\left(\int_{B_{2R}(0)}|u_{n}-u|^{2}dx\right)^{1/2}+C\int_{B_{2R}(0)}|u_{n}-u||u_{n}|^{q-1}(e^{\alpha|u_{n}|^{2}}-1)dx\\\
&\leq
o_{n}(1)+C\left(\int_{B_{2R}(0)}|u_{n}-u|^{\tau^{\prime}}dx\right)^{1/\tau^{\prime}}\left(\int_{\mathbb{R}^{2}}\Big{[}|u_{n}|^{q-1}(e^{\alpha|u_{n}|^{2}}-1)\Big{]}^{\tau}dx\right)^{1/\tau}\\\
&=o_{n}(1),\end{split}$

since, arguing as in (4.27), for $n$ large enough,

$\int_{\mathbb{R}^{2}}\Big{[}|u_{n}|^{q-1}(e^{\alpha|u_{n}|^{2}}-1)\Big{]}^{\tau}dx\leq
C.$

Hence, by (4.29), we obtain that, for any $R>0$,

$\int_{B_{R}(0)}|\nabla_{A_{\varepsilon}}u_{n}-\nabla_{A_{\varepsilon}}u|^{2}dx\rightarrow
0,\,\,\text{as}\,\,n\rightarrow+\infty,$

which implies that $\nabla_{A_{\varepsilon}}u_{n}\to\nabla_{A_{\varepsilon}}u$
a.e. in $\mathbb{R}^{2}$. Thus, using also that $u_{n}\to u$ a.e. in
$\mathbb{R}^{2}$, we have

$\nabla u_{n}\to\nabla u\text{ a.e. in }\mathbb{R}^{2}$

and we can conclude. ∎

Now we provide a range of levels in which the functional $J_{\varepsilon}$
satisfies the $(PS)$ condition.

######  Proposition 4.8.

Let $c\in\mathbb{R}$ and $\\{u_{n}\\}$ be a $(PS)_{c}$ sequence for
$J_{\varepsilon}$ such that
$\lim\sup_{n\rightarrow\infty}\|u_{n}\|_{\varepsilon}^{2}<1/8$. Assume that
$c<c_{V_{\infty}}$ if $V_{\infty}<+\infty$ or $c$ is arbitrary if
$V_{\infty}=+\infty$. Then, $\\{u_{n}\\}$ admits a convergent subsequence in
$H_{\varepsilon}$.

###### Proof.

Since the sequence $\\{u_{n}\\}$ is bounded in $H_{\varepsilon}$, up to a
subsequence, $u_{n}\rightharpoonup u$ in $H_{\varepsilon}$ and
$u_{n}\rightarrow u$ in $L^{r}_{\rm loc}(\mathbb{R}^{2},\mathbb{C})$ for any
$r\geq 1$. Moreover, arguing as in Lemma 4.7, we also have that, for any
$\phi\in H_{\varepsilon}$,

$\text{Re}\int_{\mathbb{R}^{2}}f(|u_{n}|^{2})u_{n}\overline{\phi}dx\rightarrow\text{Re}\int_{\mathbb{R}^{2}}f(|u|^{2})u\overline{\phi}dx\text{
as }n\rightarrow+\infty.$

Thus $u$ is a critical point of $J_{\varepsilon}$ and, using ($f_{3}$),

(4.30)
$J_{\varepsilon}(u)=J_{\varepsilon}(u)-\frac{1}{\theta}J^{\prime}_{\varepsilon}(u)[u]=\frac{\theta-2}{2\theta}\|u\|_{\varepsilon}^{2}+\frac{1}{\theta}\int_{\mathbb{R}^{2}}\Big{(}f(|u|^{2})|u|^{2}-\frac{\theta}{2}F(|u|^{2})\Big{)}dx\geq
0.$

Now, let $\\{u_{n_{j}}\\}$ be a subsequence of $\\{u_{n}\\}$ satisfying the
assumptions of Lemma 4.7 and Lemma 2.6 and let $v_{j}:=u_{n_{j}}-\hat{u}_{j}$
where $\hat{u}_{j}=\varphi_{j}u$, with $\varphi_{j}(x)=\varphi(2x/j)$,
$\varphi\in C_{c}^{\infty}(\mathbb{R}^{2},\mathbb{R})$, $0\leq\varphi\leq 1$,
$\varphi(x)=1$ if $|x|\leq 1$, and $\varphi(x)=0$ if $|x|\geq 2$. We claim
that

(4.31) $J_{\varepsilon}(v_{j})=c-J_{\varepsilon}(u)+o_{j}(1),$

and

(4.32) $J^{\prime}_{\varepsilon}(v_{j})=o_{j}(1).$

In order to prove (4.31), let us observe that

$J_{\varepsilon}(v_{j})-J_{\varepsilon}(u_{n_{j}})+J_{\varepsilon}(\hat{u}_{j})=\|\hat{u}_{j}\|_{\varepsilon}^{2}-\langle
u_{n_{j}},\hat{u}_{j}\rangle_{\varepsilon}+\frac{1}{2}\int_{\mathbb{R}^{2}}(F(|u_{n_{j}}|^{2})-F(|v_{j}|^{2})-F(|\hat{u}_{j}|^{2}))dx.$

In view of the weak convergence of $\\{u_{n_{j}}\\}$ to $u$ in
$H_{\varepsilon}$ and using Lemma 2.4, we can see that

$\|\hat{u}_{j}\|_{\varepsilon}^{2}-\langle
u_{n_{j}},\hat{u}_{j}\rangle_{\varepsilon}=o_{j}(1).$

Moreover, by Lemma 4.7 and (2.9) in Lemma 2.6, we have that

$\int_{\mathbb{R}^{2}}(F(|u_{n_{j}}|^{2})-F(|v_{j}|^{2})-F(|\hat{u}_{j}|^{2}))dx=o_{j}(1),$

getting (4.31).
To prove (4.32), we observe that, for any $\phi\in H_{\varepsilon}$,

$\displaystyle|(J^{\prime}_{\varepsilon}(v_{j})-J^{\prime}_{\varepsilon}(u_{n_{j}})+J^{\prime}_{\varepsilon}(\hat{u}_{j}))[\phi]|$
$\displaystyle=\Big{|}\text{Re}\int_{\mathbb{R}^{2}}(f(|u_{n_{j}}|^{2})u_{n_{j}}-f(|v_{j}|^{2})v_{j}-f(|\hat{u}_{j}|^{2})\hat{u}_{j})\overline{\phi}dx\Big{|}$
$\displaystyle\leq\int_{\mathbb{R}^{2}}|f(|u_{n_{j}}|^{2})u_{n_{j}}-f(|v_{j}|^{2})v_{j}-f(|\hat{u}_{j}|^{2})\hat{u}_{j}||\phi|dx$

and so, by Lemma 4.7 and (2.10) in Lemma 2.6, applying the diamagnetic
inequality (2.1), we have that

$\sup_{\|\phi\|_{\varepsilon}\leq
1}|(J^{\prime}_{\varepsilon}(v_{j})-J^{\prime}_{\varepsilon}(u_{n_{j}})+J^{\prime}_{\varepsilon}(\hat{u}_{j}))[\phi]|\rightarrow
0.$

Thus, since $J^{\prime}_{\varepsilon}(u_{n_{j}})\rightarrow 0$ and
$J^{\prime}_{\varepsilon}(\hat{u}_{j})\rightarrow
J^{\prime}_{\varepsilon}(u)=0$, (4.32) holds.
Now, if $V_{\infty}<+\infty$ and $c<c_{V_{\infty}}$, by (4.30), we have that

(4.33) $c-J_{\varepsilon}(u)\leq c<c_{V_{\infty}}.$

Moreover, by (4.31) and (4.32), $\\{v_{j}\\}$ is a
$(PS)_{c-J_{\varepsilon}(u)}$ sequence for $J_{\varepsilon}$ with
$v_{j}\rightharpoonup 0$ in $H_{\varepsilon}$.
Note also that, due to the weakly lower semicontinuity of the norm,

$\|u\|_{\varepsilon}^{2}\leq\liminf_{j}\|u_{n_{j}}\|_{\varepsilon}^{2}.$

Then, using also the diamagnetic inequality and Lemma 2.4,

(4.34)
$\limsup_{j}(\|\nabla|v_{j}|\|_{2}^{2}+V_{0}\|v_{j}\|_{2}^{2})\leq\limsup_{j}\|u_{n_{j}}-\hat{u}_{j}\|_{\varepsilon}^{2}\\\
\leq
2(\limsup_{j}\|u_{n_{j}}\|_{\varepsilon}^{2}+\limsup_{j}\|\hat{u}_{j}\|_{\varepsilon}^{2})<\frac{1}{2}.$

Hence, (4.33) and Lemma 4.6 imply that $v_{j}\rightarrow 0$ in
$H_{\varepsilon}$ and so, by Lemma 2.4, we get that $u_{n_{j}}\rightarrow u$
in $H_{\varepsilon}$ as $j\rightarrow+\infty$.
If $V_{\infty}=+\infty$, then, by Lemma 2.4 and Lemma 2.1, $v_{j}\rightarrow
0$ in $L^{r}(\mathbb{R}^{2},\mathbb{C})$ for any $r\geq 2$. Arguing as in
(4.34), we also have

$\limsup_{j}(\|\nabla|v_{j}|\|_{2}^{2}+V_{0}\|v_{j}\|_{2}^{2})<1/2<1$

and so, by (2.6), the Hölder inequality, (2.17), and Lemma 2.3, we deduce
that, fixed $q>2$, for suitable $\tau>1$ and $\alpha>4\pi$ such that
$\alpha\tau<8\pi$, for $j$ large enough,

(4.35)
$\begin{split}\|v_{j}\|_{\varepsilon}^{2}&=\int_{\mathbb{R}^{2}}f(|v_{j}|^{2})|v_{j}|^{2}dx+o_{j}(1)\leq
C\int_{\mathbb{R}^{2}}|v_{j}|^{q}(e^{\alpha|v_{j}|^{2}}-1)dx+o_{j}(1)\\\ &\leq
C\|v_{j}\|_{q\tau^{\prime}}^{q}\Big{(}\int_{\mathbb{R}^{2}}(e^{\alpha|v_{j}|^{2}}-1)^{\tau}dx\Big{)}^{1/\tau}+o_{j}(1)\\\
&\leq
C\|v_{j}\|_{q\tau^{\prime}}^{q}\Big{(}\int_{\mathbb{R}^{2}}\Big{(}e^{\frac{\alpha\tau}{2}(\sqrt{2}|v_{j}|)^{2}}-1\Big{)}dx\Big{)}^{1/\tau}+o_{j}(1)=o_{j}(1),\end{split}$

where $\tau^{\prime}$ is the conjugate exponent of $\tau$.
Hence $u_{n_{j}}\rightarrow u$ in $H_{\varepsilon}$ as $j\rightarrow+\infty$
and the proof is completed. ∎

To show that $\mathcal{N}_{\varepsilon}$ is a natural constraint, namely that
the constrained critical points of the functional $J_{\varepsilon}$ on
$\mathcal{N}_{\varepsilon}$ are critical points of $J_{\varepsilon}$ in
$H_{\varepsilon}$, we prove the following preliminary result.

######  Proposition 4.9.

Let $c\in\mathbb{R}$ and $\\{u_{n}\\}$ be a $(PS)_{c}$ sequence for
$J_{\varepsilon}$ restricted to $\mathcal{N}_{\varepsilon}$ with
$\limsup_{n\rightarrow\infty}\|u_{n}\|_{\varepsilon}^{2}<1/8$. Assume that
$c<c_{V_{\infty}}$ if $V_{\infty}<+\infty$ or $c$ is arbitrary if
$V_{\infty}=+\infty$. Then, $\\{u_{n}\\}$ has a convergent subsequence in
$\mathcal{N}_{\varepsilon}$.

###### Proof.

Let $\\{u_{n}\\}\subset\mathcal{N}_{\varepsilon}$ be a $(PS)_{c}$ sequence of
$J_{\varepsilon}$ restricted to $\mathcal{N}_{\varepsilon}$. Then,
$J_{\varepsilon}(u_{n})\rightarrow c$ as $n\rightarrow+\infty$ and there
exists $\\{\lambda_{n}\\}\subset\mathbb{R}$ such that

(4.36) $\displaystyle
J^{\prime}_{\varepsilon}(u_{n})=\lambda_{n}T^{\prime}_{\varepsilon}(u_{n})+o_{n}(1),$

with

$T_{\varepsilon}:H_{\varepsilon}\rightarrow\mathbb{R},\qquad
T_{\varepsilon}(u):=\|u\|_{\varepsilon}^{2}-\int_{\mathbb{R}^{2}}f(|u|^{2})|u|^{2}dx.$

Using ($f_{4}$), we get

$\displaystyle T^{\prime}_{\varepsilon}(u_{n})[u_{n}]$
$\displaystyle=2\|u_{n}\|_{\varepsilon}^{2}-2\int_{\mathbb{R}^{2}}f(|u_{n}|^{2})|u_{n}|^{2}dx-2\int_{\mathbb{R}^{2}}f^{\prime}(|u_{n}|^{2})|u_{n}|^{4}dx$
$\displaystyle=-2\int_{\mathbb{R}^{2}}f^{\prime}(|u_{n}|^{2})|u_{n}|^{4}dx\leq-(p-2)C_{p}\int_{\mathbb{R}^{2}}|u_{n}|^{p}dx\leq
0.$

Moreover $\\{T^{\prime}_{\varepsilon}(u_{n})\\}$ is bounded. Indeed, using the
Hölder inequality, (2.4), and ($f_{5}$),

$\displaystyle|T^{\prime}_{\varepsilon}(u_{n})[\phi]|$ $\displaystyle\leq
2\|u_{n}\|_{\varepsilon}\|\phi\|_{\varepsilon}+2\int_{\mathbb{R}^{2}}f(|u_{n}|^{2})|u_{n}||\phi|dx+2\int_{\mathbb{R}^{2}}f^{\prime}(|u_{n}|^{2})|u_{n}|^{3}|\phi|dx$
$\displaystyle\leq
C\left[\|u_{n}\|_{\varepsilon}\|\phi\|_{\varepsilon}+\int_{\mathbb{R}^{2}}|u_{n}|^{q-1}|\phi|(e^{\alpha|u_{n}|^{2}}-1)dx+\int_{\mathbb{R}^{2}}|u_{n}|^{3}|\phi|(e^{4\pi|u_{n}|^{2}}-1)dx\right].$

Then, using similar arguments as in (4.35) and taking $\alpha>4\pi$ and $r>1$
with $\alpha r<32\pi$, for $n$ large enough,

$\displaystyle\int_{\mathbb{R}^{2}}|u_{n}|^{q-1}|\phi|(e^{\alpha|u_{n}|^{2}}-1)dx$
$\displaystyle\leq\Big{(}\int_{\mathbb{R}^{2}}|u_{n}|^{r^{\prime}(q-1)}|\phi|^{r^{\prime}}dx\Big{)}^{1/r^{\prime}}\Big{(}\int_{\mathbb{R}^{2}}(e^{\frac{\alpha
r}{8}(2\sqrt{2}|u_{n}|)^{2}}-1)dx\Big{)}^{1/r}$ $\displaystyle\leq
C\|u_{n}\|_{\varepsilon}^{q-1}\|\phi\|_{\varepsilon},$

where $r^{\prime}$ is the conjugate exponent of $r$.
Analogously

$\int_{\mathbb{R}^{2}}|u_{n}|^{3}|\phi|(e^{4\pi|u_{n}|^{2}}-1)dx\leq
C\|u_{n}\|_{\varepsilon}^{3}\|\phi\|_{\varepsilon}.$

Thus, using the boundedness of $\\{u_{n}\\}$, for every $\phi\in
H_{\varepsilon}$ we have that

$|T^{\prime}_{\varepsilon}(u_{n})[\phi]|\leq
C(\|u_{n}\|_{\varepsilon}+\|u_{n}\|_{\varepsilon}^{q-1}+\|u_{n}\|_{\varepsilon}^{3})\|\phi\|_{\varepsilon}\leq
C\|\phi\|_{\varepsilon}.$

Thus, up to a subsequence,
$T^{\prime}_{\varepsilon}(u_{n})[u_{n}]\rightarrow\varsigma\leq 0$.
We claim that $T^{\prime}_{\varepsilon}(u_{n})[u_{n}]\not\rightarrow 0$.
Indeed, if $T^{\prime}_{\varepsilon}(u_{n})[u_{n}]\rightarrow 0$, then also
$u_{n}\rightarrow 0$ in $L^{p}(\mathbb{R}^{2},\mathbb{C})$, since

$o_{n}(1)=|T^{\prime}_{\varepsilon}(u_{n})[u_{n}]|\geq
C\int_{\mathbb{R}^{2}}|u_{n}|^{p}dx.$

Thus, by interpolation, $u_{n}\rightarrow 0$ in
$L^{\tau}(\mathbb{R}^{2},\mathbb{C})$, for all $\tau>2$.
Moreover, using similar arguments as before and taking $\alpha>4\pi$ and $r>1$
with $\alpha r<32\pi$, we get that, for every $\zeta>0$ and for $n$ large
enough,

$\|u_{n}\|_{\varepsilon}^{2}=\int_{\mathbb{R}^{2}}f(|u_{n}|^{2})|u_{n}|^{2}dx\leq\frac{\zeta}{V_{0}}\|u_{n}\|_{\varepsilon}^{2}+C\|u_{n}\|_{qr^{\prime}}^{q}\Big{(}\int_{\mathbb{R}^{2}}(e^{\frac{\alpha
r}{8}(2\sqrt{2}|u_{n}|)^{2}}-1)dx\Big{)}^{1/r^{\prime}}=\frac{\zeta}{V_{0}}\|u_{n}\|_{\varepsilon}^{2}+o_{n}(1)$

and so $u_{n}\rightarrow 0$ in $H_{\varepsilon}$, which is in contradiction
with Lemma 4.2.
Hence, since $\\{u_{n}\\}\subset\mathcal{N}_{\varepsilon}$, by (4.36) we get
$\lambda_{n}=o_{n}(1)$ and so $J^{\prime}_{\varepsilon}(u_{n})=o_{n}(1)$.
Then, applying Proposition 4.8, we conclude. ∎

Arguing as in the first part of the proof of Proposition 4.9, we have

######  Corollary 4.10.

The constrained critical points of the functional $J_{\varepsilon}$ on
$\mathcal{N}_{\varepsilon}$ are critical points of $J_{\varepsilon}$ in
$H_{\varepsilon}$.

Now we can complete the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

By Lemma 4.1, we know that the functional $J_{\varepsilon}$ has the Mountain
Pass geometry. Thus, there exists a $(PS)_{c_{\varepsilon}}$ sequence
$\\{u_{n}\\}\subset H_{\varepsilon}$ for $J_{\varepsilon}$, with
$c_{\varepsilon}$ defined in (4.3). By Lemma 4.3 we have that $\\{u_{n}\\}$ is
bounded in $H_{\varepsilon}$ and, by ($f_{3}$), Lemma 4.4, and (3.4), for
$\varepsilon$ small enough,

$\displaystyle\frac{\theta-2}{2\theta}\|u_{n}\|_{\varepsilon}^{2}$
$\displaystyle\leq\Big{(}\frac{1}{2}-\frac{1}{\theta}\Big{)}\|u_{n}\|_{\varepsilon}^{2}+\frac{1}{\theta}\int_{\mathbb{R}^{2}}\Big{(}f(|u_{n}|^{2})|u_{n}|^{2}-\frac{\theta}{2}F(|u_{n}|^{2})\Big{)}dx=J_{\varepsilon}(u_{n})-\frac{1}{\theta}J_{\varepsilon}^{\prime}(u_{n})[u_{n}]$
$\displaystyle\leq
c_{\varepsilon}+o_{n}(1)+o_{n}(1)\|u_{n}\|_{\varepsilon}\leq
c_{V_{0}}+o_{n}(1)+o_{n}(1)\|u_{n}\|_{\varepsilon}\leq
c_{V^{*}}+o_{n}(1)+o_{n}(1)\|u_{n}\|_{\varepsilon},$

Thus, using Lemma 3.4, for $n$ large enough we have

$\frac{\theta-2}{2\theta}\|u_{n}\|_{\varepsilon}^{2}<\frac{\theta-2}{16\theta}$

and so $\limsup_{n}\|u_{n}\|_{\varepsilon}^{2}<1/8$. Hence, (3.4) and
Proposition 4.8 allow us to conclude. ∎

We conclude this section showing the behavior of the energy levels
$c_{\varepsilon}$ of the solutions found in Theorem 1.1.

######  Proposition 4.11.

Let $\\{\varepsilon_{n}\\}\subset(0,+\infty)$ be such that $\varepsilon_{n}\to
0^{+}$ as $n\to+\infty$ and, for $n$ large enough, $u_{n}$ be a respective
ground state found in Theorem 1.1 for $\varepsilon=\varepsilon_{n}$ with
energy $J_{\varepsilon_{n}}(u_{n})=c_{\varepsilon_{n}}$. Up to a subsequence,
we have that

$c_{\varepsilon_{n}}\rightarrow c_{V_{0}}.$

###### Proof.

By Lemma 4.2 and Lemma 4.4, $0<K_{2}\leq c_{\varepsilon_{n}}\leq c_{V_{0}}$.
Thus, up to a subsequence, $c_{\varepsilon_{n}}\rightarrow\bar{c}>0$.
Since, using ($f_{3}$), Lemma 4.4, the monotonicity of $c_{\mu}$’s, and Lemma
3.4,

$\displaystyle\frac{\theta-2}{2\theta}\limsup_{n}\|u_{n}\|_{\varepsilon_{n}}^{2}$
$\displaystyle\leq\limsup_{n}\Big{[}\Big{(}\frac{1}{2}-\frac{1}{\theta}\Big{)}\|u_{n}\|_{\varepsilon_{n}}^{2}+\frac{1}{\theta}\int_{\mathbb{R}^{2}}\Big{(}f(|u_{n}|^{2})|u_{n}|^{2}-\frac{\theta}{2}F(|u_{n}|^{2})\Big{)}dx\Big{]}$
$\displaystyle=\lim_{n}\Big{[}J_{\varepsilon_{n}}(u_{n})-\frac{1}{\theta}J_{\varepsilon_{n}}^{\prime}(u_{n})[u_{n}]\Big{]}=\bar{c}\leq
c_{V_{0}}<\frac{\theta-2}{16\theta}$

and so, by the diamagnetic inequality (2.1),

(4.37)
$\limsup_{n}\|\nabla|u_{n}|\|_{2}^{2}\leq\limsup_{n}(\|\nabla|u_{n}|\|_{2}^{2}+V_{0}\|u_{n}\|_{2}^{2})\leq\limsup_{n}\|u_{n}\|_{\varepsilon_{n}}^{2}<\frac{1}{8}.$

Moreover, since $\bar{c}>0$, we have that
$\|u_{n}\|_{\varepsilon_{n}}\not\rightarrow 0$. Thus, arguing as in Lemma 4.5,
we get that there exist a sequence $(\tilde{y}_{n})\subset\mathbb{R}^{2}$ and
constants $R$, $\gamma>0$ such that

(4.38) $\liminf_{n}\int_{B_{R}(\tilde{y}_{n})}|u_{n}|^{2}dx\geq\gamma.$

Let $v_{n}:=u_{n}(\cdot+\tilde{y}_{n})$.
By (4.37) we have that $\\{|v_{n}|\\}$ is bounded in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$ and so, up to a subsequence,
$|v_{n}|\rightharpoonup v$ in $H^{1}(\mathbb{R}^{2},\mathbb{R})$, with $v\neq
0$ by (4.38).
Let now $t_{n}>0$ be such that $t_{n}|v_{n}|\in\mathcal{N}_{V_{0}}$ and
$y_{n}:=\varepsilon_{n}\tilde{y}_{n}$.
Using the diamagnetic inequality (2.1)

$c_{V_{0}}\leq I_{V_{0}}(t_{n}|v_{n}|)=I_{V_{0}}(t_{n}|u_{n}|)\leq
J_{\varepsilon_{n}}(t_{n}u_{n})\leq\max_{t\geq
0}J_{\varepsilon_{n}}(tu_{n})=J_{\varepsilon_{n}}(u_{n})=c_{\varepsilon_{n}}\leq
c_{V_{0}},$

and we conclude. ∎

##  5\. Proof of Theorem 1.2

In this section, by Ljusternik-Schnirelmann category theory, we prove a
multiplicity result for problem (2.2).

First, let us provide some useful preliminaries.

Let us start with the following useful compactness result.

######  Proposition 5.1.

Let $\varepsilon_{n}\rightarrow 0^{+}$ and
$\\{u_{n}\\}\subset\mathcal{N}_{\varepsilon_{n}}$ be such that
$J_{\varepsilon_{n}}(u_{n})\rightarrow c_{V_{0}}$. Then there exists
$\\{\tilde{y}_{n}\\}\subset\mathbb{R}^{2}$ such that the sequence
$\\{|v_{n}|\\}\subset H^{1}(\mathbb{R}^{2},\mathbb{R})$, where
$v_{n}(x):=u_{n}(x+\tilde{y}_{n})$, has a convergent subsequence in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$. Moreover, up to a subsequence,
$y_{n}:=\varepsilon_{n}\tilde{y}_{n}\rightarrow y\in M$ as $n\to+\infty$.

###### Proof.

Since $\\{u_{n}\\}\subset\mathcal{N}_{\varepsilon_{n}}$ and
$J_{\varepsilon_{n}}(u_{n})\rightarrow c_{V_{0}}$, arguing as in the first
part of the proof of Lemma 3.5 and using the monotonicity of $c_{\mu}$’s, we
have ${\limsup_{n}}\|u_{n}\|_{\varepsilon_{n}}^{2}<1/8$ and so, by the
diamagnetic inequality (2.1),

(5.1)
$\limsup_{n}\|\nabla|u_{n}|\|_{2}^{2}\leq\limsup_{n}(\|\nabla|u_{n}|\|_{2}^{2}+V_{0}\|u_{n}\|_{2}^{2})<\frac{1}{8}.$

Moreover, since $c_{V_{0}}>0$, we have that
$\|u_{n}\|_{\varepsilon_{n}}\not\rightarrow 0$. Thus, arguing as in Lemma 4.5,
we get that there exist a sequence $(\tilde{y}_{n})\subset\mathbb{R}^{2}$ and
constants $R$, $\beta>0$ such that

(5.2) $\liminf_{n}\int_{B_{R}(\tilde{y}_{n})}|u_{n}|^{2}dx\geq\beta.$

Let $v_{n}(x)=u_{n}(x+\tilde{y}_{n})$. By (5.1) we have that $\\{|v_{n}|\\}$
is bounded in $H^{1}(\mathbb{R}^{2},\mathbb{R})$ and so, up to a subsequence,
$|v_{n}|\rightharpoonup v$ in $H^{1}(\mathbb{R}^{2},\mathbb{R})$, with $v\neq
0$ by (5.2).
Let now $t_{n}>0$ be such that
$\tilde{v}_{n}:=t_{n}|v_{n}|\in\mathcal{N}_{V_{0}}$ and
$y_{n}:=\varepsilon_{n}\tilde{y}_{n}$.
By the diamagnetic inequality (2.1)

$c_{V_{0}}\leq I_{V_{0}}(\tilde{v}_{n})=I_{V_{0}}(t_{n}|u_{n}|)\leq
J_{\varepsilon_{n}}(t_{n}u_{n})\leq\max_{t\geq
0}J_{\varepsilon_{n}}(tu_{n})=J_{\varepsilon_{n}}(u_{n})=c_{V_{0}}+o_{n}(1),$

and so $I_{V_{0}}(\tilde{v}_{n})\rightarrow c_{V_{0}}$ as $n\to+\infty$.
Thus, by Lemma 3.5 for $\mu=V_{0}$, $\\{\tilde{v}_{n}\\}$ is bounded in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$ and, since $|v_{n}|\not\rightarrow 0$ in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$, then $\\{t_{n}\\}$ is bounded too and so,
up to a subsequence, $t_{n}\rightarrow t_{0}\geq 0$.
Actually we have that $t_{0}>0$.
Indeed, if $t_{0}=0$, then, due to the boundedness of $\\{|v_{n}|\\}$ in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$, $\tilde{v}_{n}\rightarrow 0$ in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$, so that
$I_{V_{0}}(\tilde{v}_{n})\rightarrow 0$, in contradiction with $c_{V_{0}}>0$.
Hence, up to a subsequence, $\tilde{v}_{n}\rightharpoonup\tilde{v}:=t_{0}v\neq
0$ in $H^{1}(\mathbb{R}^{2},\mathbb{R})$ and so, by Lemma 3.5,
$\tilde{v}_{n}\rightarrow\tilde{v}$ in $H^{1}(\mathbb{R}^{2},\mathbb{R})$,
which gives $|v_{n}|\rightarrow v$ in $H^{1}(\mathbb{R}^{2},\mathbb{R})$.
Let us show now that $\\{y_{n}\\}$ is bounded.
Assume by contradiction that $\\{y_{n}\\}$ is not bounded. Then there exists a
subsequence such that $|y_{n}|\rightarrow+\infty$.
If $V_{\infty}=+\infty$, since, by (V),

$\inf_{B_{R}(\tilde{y}_{n})}V_{\varepsilon_{n}}=\inf_{B_{R\varepsilon_{n}}(y_{n})}V\to+\infty,$

then

$\displaystyle\int_{\mathbb{R}^{2}}V(\varepsilon_{n}x+y_{n})|v_{n}|^{2}dx$
$\displaystyle\geq\int_{B_{R}(0)}V(\varepsilon_{n}x+y_{n})|v_{n}|^{2}dx=\int_{B_{R}(0)}V(\varepsilon_{n}x+y_{n})|u_{n}(x+\tilde{y}_{n})|^{2}dx$
$\displaystyle=\int_{B_{R}(\tilde{y}_{n})}V(\varepsilon_{n}z)|u_{n}(z)|^{2}dz\geq\inf_{B_{R}(\tilde{y}_{n})}V_{\varepsilon_{n}}\int_{B_{R}(\tilde{y}_{n})}|u_{n}(z)|^{2}dz\to+\infty,$

in contradiction with

$\int_{\mathbb{R}^{2}}V(\varepsilon_{n}x+y_{n})|v_{n}|^{2}dx\leq\|u_{n}\|_{\varepsilon_{n}}^{2}\leq
C.$

If $V_{\infty}\in\mathbb{R}$, since $\tilde{v}_{n}\rightarrow\tilde{v}$ in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$, by the Fatou’s Lemma and since
$H^{1}(\mathbb{R}^{2},\mathbb{R})$ is continuously embedded in
$L^{2}(\mathbb{R}^{2},\mathbb{R})$,

$V_{\infty}\lim_{n}\int_{\mathbb{R}^{2}}|\tilde{v}_{n}|^{2}dx=V_{\infty}\int_{\mathbb{R}^{2}}|\tilde{v}|^{2}dx\leq\liminf_{n}\int_{\mathbb{R}^{2}}V(\varepsilon_{n}x+y_{n})|\tilde{v}_{n}|^{2}dx,$

and so, by the diamagnetic inequality (2.1) and (4.4),

(5.3)
$\begin{split}c_{V_{0}}&=I_{V_{0}}(\tilde{v})<I_{V_{\infty}}(\tilde{v})=\lim_{n}I_{V_{\infty}}(\tilde{v}_{n})\\\
&\leq\liminf_{n}\frac{1}{2}\int_{\mathbb{R}^{2}}V(\varepsilon_{n}x+y_{n})|\tilde{v}_{n}|^{2}dx-\frac{V_{\infty}}{2}\int_{\mathbb{R}^{2}}|\tilde{v}|^{2}dx+\lim_{n}I_{V_{\infty}}(\tilde{v}_{n})\\\
&\leq{\liminf_{n}}\Big{(}\frac{1}{2}\int_{\mathbb{R}^{2}}(|\nabla\tilde{v}_{n}|^{2}+V(\varepsilon_{n}x+y_{n})|\tilde{v}_{n}|^{2})dx-\frac{1}{2}\int_{\mathbb{R}^{2}}F(|\tilde{v}_{n}|^{2})dx\Big{)}\\\
&={\liminf_{n}}\Big{(}\frac{t^{2}_{n}}{2}\int_{\mathbb{R}^{2}}(|\nabla|u_{n}||^{2}+V_{\varepsilon_{n}}(z)|u_{n}|^{2})dz-\frac{1}{2}\int_{\mathbb{R}^{2}}F(t^{2}_{n}|u_{n}|^{2})dz\Big{)}\\\
&\leq{\liminf_{n}}\,J_{\varepsilon_{n}}(t_{n}u_{n})\leq{\liminf_{n}}\,J_{\varepsilon_{n}}(u_{n})=c_{V_{0}},\end{split}$

which gives a contradiction.
Hence, up to a subsequence, $y_{n}\rightarrow y\in\mathbb{R}^{2}$.
If $y\not\in M$, then $V(y)>V_{0}$ and we can argue as in (5.3) to get a
contradiction and so the proof is complete. ∎

Let now $\delta>0$ be fixed, $\omega\in H^{1}(\mathbb{R}^{2},\mathbb{R})$ be a
positive ground state solution of problem (3.1) for $\mu=V_{0}$, and $\eta\in
C^{\infty}([0,+\infty),[0,1])$ be a non-increasing cut-off function such that
$\eta(t)=1$ if $0\leq t\leq\delta/{2}$ and $\eta(t)=0$ if $t\geq\delta$.

For any $y\in M$, let

$\Psi_{\varepsilon,y}(x):=\eta(|\varepsilon
x-y|)\omega\Big{(}\frac{\varepsilon
x-y}{\varepsilon}\Big{)}\exp\Big{(}i\tau_{y}\Big{(}\frac{\varepsilon
x-y}{\varepsilon}\Big{)}\Big{)},$

where $M$ is defined in (1.4) and $\tau_{y}(x):=A_{1}(y)x_{1}+A_{2}(y)x_{2}$.

Let $t_{\varepsilon,y}>0$ be the unique positive number such that
$t_{\varepsilon,y}\Psi_{\varepsilon,y}\in\mathcal{N}_{\varepsilon}$ (and so
$\max_{t\geq
0}J_{\varepsilon}(t\Psi_{\varepsilon,y})=J_{\varepsilon,y}(t_{\varepsilon,y}\Psi_{\varepsilon,y})$)
and let us consider

$\Phi_{\varepsilon}:=y\in M\mapsto
t_{\varepsilon,y}\Psi_{\varepsilon,y}\in\mathcal{N}_{\varepsilon}.$

Observe that, by the assumptions on $V$, $M$ is compact and, for any $y\in M$,
$\Phi_{\varepsilon}(y)$ has compact support.
Moreover we have

######  Lemma 5.2.

As $\varepsilon\rightarrow 0^{+}$,

$\|\Psi_{\varepsilon,y}\|_{\varepsilon}^{2}\rightarrow\int_{\mathbb{R}^{2}}(|\nabla\omega|^{2}+V_{0}\omega^{2})dx$

uniformly with respect to $y\in M$.

###### Proof.

First we observe that

$\sup_{y\in
M}\Big{|}\int_{\mathbb{R}^{2}}V_{\varepsilon}(x)|\Psi_{\varepsilon,y}|^{2}dx-
V_{0}\int_{\mathbb{R}^{2}}\omega^{2}dx\Big{|}\to 0\text{ as
}\varepsilon\rightarrow 0^{+}.$

Indeed, taking $z=(\varepsilon x-y)/\varepsilon$,

$\sup_{y\in
M}\Big{|}\int_{\mathbb{R}^{2}}|\Psi_{\varepsilon,y}|^{2}dx-\int_{\mathbb{R}^{2}}\omega^{2}dx\Big{|}\leq\int_{\mathbb{R}^{2}}|\eta^{2}(|\varepsilon
z|)-1|\omega^{2}(z)dz\leq\int_{B_{\delta/(2\varepsilon)}^{c}(0)}\omega^{2}(z)dz\to
0.$

Moreover

$\displaystyle\sup_{y\in
M}\Big{|}\int_{\mathbb{R}^{2}}[V_{\varepsilon}(x)-V_{0}]|\Psi_{\varepsilon,y}|^{2}dx\Big{|}$
$\displaystyle\leq\sup_{y\in M}\int_{\mathbb{R}^{2}}|V(\varepsilon
z+y)-V_{0}|\eta^{2}(|\varepsilon z|)\omega^{2}(z)dz$
$\displaystyle\leq\int_{B_{\delta/\varepsilon}(0)}\sup_{y\in M}|V(\varepsilon
z+y)-V_{0}|\omega^{2}(z)dz$ $\displaystyle=\int_{\mathbb{R}^{2}}\sup_{y\in
M}|V(\varepsilon
z+y)-V_{0}|\chi_{B_{\delta/\varepsilon}(0)}(z)\omega^{2}(z)dz\to 0,$

applying the Lebesgue Dominated Convergence Theorem, since, if $|\varepsilon
z|<\delta$, then, due to the compactness of $M$, $|\varepsilon z+y|$ is
bounded when $y\in M$,

$\sup_{y\in M}|V(\varepsilon
z+y)-V_{0}|\chi_{B_{\delta/\varepsilon}(0)}(z)\omega^{2}(z)\leq
C\omega^{2}(z)\in L^{1}(\mathbb{R}^{2},\mathbb{R}),$

and $\sup_{y\in M}|V(\varepsilon
z+y)-V_{0}|\chi_{B_{\delta/\varepsilon}(0)}(z)\omega^{2}(z)$ converges to $0$
a.e. in $\mathbb{R}^{2}$ as $\varepsilon\rightarrow 0^{+}$.
Then, since, taking again $z=(\varepsilon x-y)/\varepsilon$,

$\displaystyle\int_{\mathbb{R}^{2}}|\nabla_{A_{\varepsilon}}\Psi_{\varepsilon,y}|^{2}dx$
$\displaystyle=\varepsilon^{2}\int_{\mathbb{R}^{2}}|\eta^{\prime}(|\varepsilon
z|)|^{2}\omega^{2}(z)dz+\int_{\mathbb{R}^{2}}\eta^{2}(|\varepsilon
z|)|\nabla\omega(z)|^{2}dz$
$\displaystyle\qquad+\int_{\mathbb{R}^{2}}|A(y)-A(\varepsilon
z+y)|^{2}\eta^{2}(|\varepsilon z|)\omega^{2}(z)dz$
$\displaystyle\qquad+2\varepsilon\int_{\mathbb{R}^{2}}\eta(|\varepsilon
z|)\eta^{\prime}(|\varepsilon
z|)\omega(z)\nabla\omega(z)\cdot\frac{z}{|z|}dz,$

and,

$\Big{|}\int_{\mathbb{R}^{2}}\eta^{2}(|\varepsilon
z|)|\nabla\omega(z)|^{2}dz-\int_{\mathbb{R}^{2}}|\nabla\omega(z)|^{2}dz\Big{|}\leq\int_{\mathbb{R}^{2}}|\eta^{2}(|\varepsilon
z|)-1||\nabla\omega(z)|^{2}dz\leq\int_{B_{\delta/(2\varepsilon)}^{c}(0)}|\nabla\omega(z)|^{2}dz,$
$\int_{\mathbb{R}^{2}}|\eta^{\prime}(|\varepsilon z|)|^{2}\omega^{2}(z)dz\leq
C\int_{B_{\delta/(2\varepsilon)}^{c}(0)}\omega^{2}(z)dz,$
$\int_{\mathbb{R}^{2}}|\eta(|\varepsilon z|)\eta^{\prime}(|\varepsilon
z|)\omega(z)\nabla\omega(z)|dz\leq
C\int_{B_{\delta/(2\varepsilon)}^{c}(0)}\omega(z)|\nabla\omega|dz,$

$\sup_{y\in M}\int_{\mathbb{R}^{2}}|A(y)-A(\varepsilon
z+y)|^{2}\eta^{2}(|\varepsilon
z|)\omega^{2}(z)dz\leq\int_{B_{\delta/\varepsilon}(0)}\sup_{y\in
M}|A(y)-A(\varepsilon z+y)|^{2}\omega^{2}(z)dz\to 0,$

due to the continuity of $A$ and a similar argument as before.
Thus we infer

$\sup_{y\in
M}\Big{|}\int_{\mathbb{R}^{2}}|\nabla_{A_{\varepsilon}}\Psi_{\varepsilon,y}|^{2}dx-\int_{\mathbb{R}^{2}}|\nabla\omega|^{2}dx\Big{|}\to
0$

as $\varepsilon\rightarrow 0^{+}$ and we conclude. ∎

Hence we can conclude in the following way.

###### Proof of Theorem 1.2.

Arguing as in [20, Lemma 4.1], and using Lemma 5.2, we can prove that
$J_{\varepsilon}(\Phi_{\varepsilon}(y))\to c_{V_{0}}$ uniformly with respect
to $y\in M$ as $\varepsilon\rightarrow 0^{+}$.
Then, given $\delta>0$, if $\rho>0$ with $M_{\delta}\subset B_{\rho}$, let

$\Upsilon:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2},\qquad\Upsilon(x):=\begin{cases}x,&\text{if
}|x|<\rho,\\\ \rho x/|x|,&\text{if }|x|\geq\rho\end{cases}$

and then let us define the barycenter map as

$\beta_{\varepsilon}:=u\in\mathcal{N}_{\varepsilon}\mapsto\frac{1}{\|u\|_{2}^{2}}\int_{\mathbb{R}^{2}}\Upsilon(\varepsilon
x)|u(x)|^{2}dx\in\mathbb{R}^{2}.$

By [20, Lemma 4.2] we have that $\beta_{\varepsilon}(\Phi_{\varepsilon}(y))\to
y$ as $\varepsilon\rightarrow 0^{+}$ uniformly in $y\in M$.
Let now $h:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be any positive function
such that $h(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0^{+}$ and

$\tilde{\mathcal{N}}_{\varepsilon}:=\\{u\in\mathcal{N}_{\varepsilon}:J_{\varepsilon}(u)\leq
c_{V_{0}}+h(\varepsilon)\\}.$

Since $|J_{\varepsilon}(\Phi_{\varepsilon}(y))-c_{V_{0}}|\rightarrow 0$
uniformly in $y\in M$ as $\varepsilon\rightarrow 0^{+}$, then
$\tilde{\mathcal{N}}_{\varepsilon}\neq\emptyset$ for $\varepsilon>0$ small
enough.
Thus, arguing as in [20, Lemma 4.4],

$\lim_{\varepsilon\rightarrow
0^{+}}\sup_{u\in\tilde{\mathcal{N}}_{\varepsilon}}\operatorname{dist}(\beta_{\varepsilon}(u),M_{\delta})=0.$

Hence, as in [17, Section 6], we can find $\tilde{\varepsilon}_{\delta}>0$
such that for any $\varepsilon\in(0,\tilde{\varepsilon}_{\delta})$, the
following diagram

$M\xrightarrow{\Phi_{\varepsilon}}\tilde{\mathcal{N}}_{\varepsilon}\xrightarrow{\beta_{\varepsilon}}M_{\delta}$

is well defined and $\beta_{\varepsilon}\circ\Phi_{\varepsilon}$ is
homotopically equivalent to the embedding $\iota:M\rightarrow M_{\delta}$.
Thus, [11, Lemma 4.3] (see also [18, Lemma 2.2]) implies that

(5.4)
$\text{cat}_{\tilde{\mathcal{N}}_{\varepsilon}}(\tilde{\mathcal{N}}_{\varepsilon})\geq\text{cat}_{M_{\delta}}(M).$

Now observe that $J_{\varepsilon}$ is bounded from below on
$\mathcal{N}_{\varepsilon}$ (see the preliminary results in Section 4 and in
particular (4.5)). Moreover, using, respectively, Lemma 4.4, Lemma 3.4, and
(3.4), for $\varepsilon$ small enough,

$c_{\varepsilon}\leq c_{V_{0}}<c_{V_{0}}+h(\varepsilon),$
$c_{V^{*}}+h(\varepsilon)<\frac{\theta-2}{16\theta},$

and, if $V_{\infty}<+\infty$,

(5.5) $c_{V_{0}}+h(\varepsilon)<c_{V_{\infty}}.$

Thus, if $\\{u_{n}\\}$ is a $(PS)_{c}$ sequence for
$J_{\varepsilon}|_{\mathcal{N}_{\varepsilon}}$ with
$c\in[c_{\varepsilon},c_{V_{0}}+h(\varepsilon)]$, then, by ($f_{3}$) and
(3.4),

$\displaystyle\frac{\theta-2}{2\theta}\|u_{n}\|_{\varepsilon}^{2}$
$\displaystyle\leq\Big{(}\frac{1}{2}-\frac{1}{\theta}\Big{)}\|u_{n}\|_{\varepsilon}^{2}+\frac{1}{\theta}\int_{\mathbb{R}^{2}}\Big{(}f(|u_{n}|^{2})|u_{n}|^{2}-\frac{\theta}{2}F(|u_{n}|^{2})\Big{)}dx$
$\displaystyle=J_{\varepsilon}(u_{n})-\frac{1}{\theta}J_{\varepsilon}^{\prime}(u_{n})[u_{n}]=c+o_{n}(1)$
$\displaystyle<c_{V_{0}}+h(\varepsilon)+o_{n}(1)\leq
c_{V^{*}}+h(\varepsilon)+o_{n}(1).$

Then, passing to the $\limsup$ on $n$ we have

$\frac{\theta-2}{2\theta}\limsup_{n}\|u_{n}\|_{\varepsilon}^{2}\leq
c_{V^{*}}+h(\varepsilon)<\frac{\theta-2}{16\theta}$

for $\varepsilon$ small enough and so

$\limsup_{n}\|u_{n}\|_{\varepsilon}^{2}<\frac{1}{8}.$

Hence, using (5.5) if $V_{\infty}<+\infty$, we can apply Proposition 4.9,
obtaining that $J_{\varepsilon}|_{\mathcal{N}_{\varepsilon}}$ satisfies the
$(PS)_{c}$ condition for $c\in[c_{\varepsilon},c_{V_{0}}+h(\varepsilon)]$.
Then, by [33, Theorem 5.20], the Ljusternik-Schnirelmann theory for $C^{1}$
functionals implies that $\tilde{\mathcal{N}}_{\varepsilon}$ contains at least
$\text{cat}_{\tilde{\mathcal{N}}_{\varepsilon}}(\tilde{\mathcal{N}}_{\varepsilon})$
critical points of $J_{\varepsilon}|_{\mathcal{N}_{\varepsilon}}$. Observing
that, due to Corollary 4.10, such critical points are critical points for
$J_{\varepsilon}$ and using (5.4), we conclude the proof of the multiplicity
result. ∎

##  6\. Proof of Theorem 1.3

This last section is dedicated to the study of the behavior of the maximum
points of the modulus of the solutions obtained in Theorem 1.1 and in Theorem
1.2 and to their the decay property.

###### Proof of Theorem 1.3.

Let $\varepsilon_{n}\rightarrow 0^{+}$ as $n\rightarrow+\infty$ and let
$u_{\varepsilon_{n}}$ be a nontrivial solution of (2.2) found in Theorem 1.1
or in Theorem 1.2. For ground states found in Theorem 1.1, by Proposition
4.11, up to a subsequence, we have that
$J_{\varepsilon_{n}}(u_{\varepsilon_{n}})=c_{\varepsilon_{n}}\to c_{V_{0}}$
and, for solutions found in Theorem 1.2 we have that
$J_{\varepsilon_{n}}(u_{\varepsilon_{n}})\leq c_{V_{0}}+h(\varepsilon_{n})$.
Thus, Proposition 5.1 applies and so we can consider
$\\{\tilde{y}_{n}\\}\subset\mathbb{R}^{2}$, and
$v_{\varepsilon_{n}}:=u_{\varepsilon_{n}}(\cdot+\tilde{y}_{n})$, which is a
solution of

$\Big{(}\frac{1}{i}\nabla-
A_{\varepsilon_{n}}(x+\tilde{y}_{n})\Big{)}^{2}v_{\varepsilon_{n}}+V_{\varepsilon_{n}}(x+\tilde{y}_{n})v_{\varepsilon_{n}}=f(|v_{\varepsilon_{n}}|^{2})v_{\varepsilon_{n}}\quad\hbox{in
}\mathbb{R}^{2},$

such that, up to a subsequence, $|v_{\varepsilon_{n}}|\rightarrow v$ in
$H^{1}(\mathbb{R}^{2},\mathbb{R})$ and
$y_{n}=\varepsilon_{n}\tilde{y}_{n}\rightarrow y\in M$.
Arguing as in [20, Lemma 5.1], we have that $\\{|v_{\varepsilon_{n}}|\\}$ is
bounded in $L^{\infty}(\mathbb{R}^{2},\mathbb{R})$ and

(6.1)
$\lim_{|x|\rightarrow+\infty}|v_{\varepsilon_{n}}(x)|=0\quad\text{uniformly in
}n\in\mathbb{N}.$

In addition, by the first part of the proof of Proposition 5.1, there exist
$R,\beta>0$ such that

${\liminf_{n}}\int_{B_{R}(\tilde{y}_{n})}|u_{\varepsilon_{n}}|^{2}dx\geq\beta.$

Then, if $\kappa>0$ such that $0<\pi\kappa^{2}R^{2}<\beta$, for $n$ large
enough,

$\pi\kappa^{2}R^{2}\leq\int_{B_{R}(\tilde{y}_{n})}|u_{\varepsilon_{n}}|^{2}dx\leq\pi
R^{2}\|v_{\varepsilon_{n}}\|^{2}_{\infty},$

and so

(6.2) $\|v_{\varepsilon_{n}}\|_{\infty}\geq\kappa.$

Let us show the concentration of global maximum points $p_{n}$ of
$|v_{\varepsilon_{n}}|$.
If $z_{n}:=p_{n}+\tilde{y}_{n}$, by the definition of $v_{\varepsilon_{n}}$,
we have that each $z_{n}$ is a global maximum point of
$|u_{\varepsilon_{n}}|$. Moreover, since by (6.1) and (6.2), $p_{n}\in
B_{R}(0)$ for some $R>0$,
$\eta_{n}:=\varepsilon_{n}z_{n}=\varepsilon_{n}p_{n}+\varepsilon_{n}\tilde{y}_{n}\rightarrow
y\in M$ and so

$\lim_{n}V(\eta_{n})=V_{0}.$

Now let us assume that $A\in C^{1}(\mathbb{R}^{2},\mathbb{R}^{2})$ and let us
show decay of the modulus of our solutions.
Arguing, for instance, as in [16, Remark 1.2], $v_{\varepsilon_{n}}\in
C^{1}(\mathbb{R}^{2},\mathbb{C})$. Moreover, by (6.1) and ($f_{1}$), there
exists a $R_{1}>0$ such that

(6.3) $f(|v_{\varepsilon_{n}}|^{2})\leq\frac{V_{0}}{2}\text{ for all }|x|\geq
R_{1}\text{ and uniformly in }n\in\mathbb{N}.$

Since $v_{\varepsilon_{n}}\in C^{1}(\mathbb{R}^{2},\mathbb{C})$ and (6.1)
holds, there exist $c,C>0$ with $c^{2}<V_{0}/2$ such that

$|v_{\varepsilon_{n}}(x)|\leq Ce^{-cR_{1}}\text{ for }|x|=R_{1}.$

Let $\varphi(x):=Ce^{-c|x|}$ in $\mathbb{R}^{2}$.
Observe that

(6.4) $\Delta\varphi\leq c^{2}\varphi\text{ in
}\mathbb{R}^{2}\setminus\\{0\\}.$

Moreover, by the Kato’s inequality (see [27, Lemma A]) and (V),
$|v_{\varepsilon_{n}}|$ satisfies (in the weak sense)

$-\Delta|v_{\varepsilon_{n}}|+V_{0}|v_{\varepsilon_{n}}|\leq
f(|v_{\varepsilon_{n}}|^{2})|v_{\varepsilon_{n}}|\hbox{ in }\mathbb{R}^{2}.$

Thus, by (6.3), we have that, uniformly in $n\in\mathbb{N}$,

(6.5) $-\Delta|v_{\varepsilon_{n}}|+\frac{V_{0}}{2}|v_{\varepsilon_{n}}|\leq
0\text{ for }|x|\geq R_{1}.$

Let, now, $\varphi_{n}:=\varphi-|v_{\varepsilon_{n}}|$.
By (6.4) and (6.5), we have

$\begin{cases}\displaystyle-\Delta\varphi_{n}+\frac{V_{0}}{2}\varphi_{n}\geq
0&\text{ for }|x|\geq R_{1},\\\ \varphi_{n}\geq 0&\text{ for }|x|=R_{1},\\\
\displaystyle\lim_{|x|\rightarrow+\infty}\varphi_{n}(x)=0.\end{cases}$

The classical maximum principle (see e.g. [14, Theorem 7.4.2]) implies that
$\varphi_{n}\geq 0$ for $|x|\geq R_{1}$ and so we obtain that

$|u_{\varepsilon_{n}}(x)|=|v_{\varepsilon_{n}}(x-\tilde{y}_{n})|\leq
Ce^{-c|x-\tilde{y}_{n}|}\text{ for all }|x|\geq R_{1}\text{ and uniformly in
}n\in\mathbb{N}.$

Observe that, due to the change of variable introduced in Section 2 to write
(1.1) in the equivalent form (2.2), the points $\eta_{n}$ are maximizers of
solutions of (1.1) and (1.5) holds. ∎

## Acknowledgments

P. d’Avenia is supported by PRIN project 2017JPCAPN Qualitative and
quantitative aspects of nonlinear PDEs. C. Ji is partially supported by
Shanghai Natural Science Foundation (20ZR1413900).

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