# Normalized solutions for a Schrödinger equation with critical growth in
$\mathbb{R}^{N}$

Claudianor O. Alves,111C.O. Alves was partially supported by CNPq/Brazil grant
304804/2017-7. Chao Ji222Corresponding author 333C. Ji was partially supported
by Natural Science Foundation of Shanghai(20ZR1413900,18ZR1409100). and
Olimpio H. Miyagaki444O. H. Miyagaki was supported by FAPESP/Brazil grant
2019/24901-3 and CNPq/Brazil grant 307061/2018-3.

###### Abstract

In this paper we study the existence of normalized solutions to the following
nonlinear Schrödinger equation with critical growth

$\displaystyle\left\\{\begin{aligned} &-\Delta u=\lambda
u+f(u),\quad\quad\hbox{in }\mathbb{R}^{N},\\\
&\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2},\end{aligned}\right.$

where $a>0$, $\lambda\in\mathbb{R}$ and $f$ has an exponential critical growth
when $N=2$, and $f(t)=\mu|u|^{q-2}u+|u|^{2^{*}-2}u$ with
$q\in(2+\frac{4}{N},2^{*})$, $\mu>0$ and $2^{*}=\frac{2N}{N-2}$ when $N\geq
3$. Our main results complement some recent results for $N\geq 3$ and it is
totally new for $N=2$.

2010 Mathematics Subject Classification: 35A15, 35J10, 35B09, 35B33.

Keywords: Normalized solutions, Nonlinear Schrödinger equation, Variational
methods, Critical exponents.

##  1 Introduction

This paper concerns the existence of normalized solutions to the following
nonlinear Schrödinger equation with critical growth

$\displaystyle\left\\{\begin{aligned} &-\Delta u=\lambda
u+f(u),\quad\quad\hbox{in }\mathbb{R}^{N},\\\
&\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2},\end{aligned}\right.$ (1.1)

where $a>0$, $\lambda\in\mathbb{R}$ and $f$ has an exponential critical growth
when $N=2$, and $f(t)=\mu|u|^{q-2}u+|u|^{2^{*}-2}u$ with
$q\in(2+\frac{4}{N},2^{*})$, $\mu>0$ and $2^{*}=\frac{2N}{N-2}$ when $N\geq
3$.

The equation (1.1) arises when ones look for the solutions with prescribed
mass for the nonlinear Schrödinger equation

$i\frac{\partial\psi}{\partial
t}+\triangle\psi+h(|\psi|^{2})\psi=0\quad\hbox{in }\mathbb{R}^{N}.$

A stationary wave solution is a solution of the form $\psi(t,x)=e^{-i\lambda
t}u(x)$, where $\lambda\in\mathbb{R}$ and
$u:\mathbb{R}^{N}\rightarrow\mathbb{R}$ is a time-independent that must solve
the elliptic problem

$-\Delta u=\lambda u+g(u),\quad\mbox{in}\quad\mathbb{R}^{N},$ (1.2)

where $g(u)=h(|u|^{2})u$. For some values of $\lambda$ the existence of
nontrivial solutions for (1.2) are obtained as the critical points of the
action functional $J_{\lambda}:H^{1}(\mathbb{R}^{N})\to\mathbb{R}$ given by

$J_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}-\lambda|u|^{2})\,dx-\int_{\mathbb{R}^{N}}G(u)\,dx,$

where $G(t)=\int_{0}^{t}g(s)\,ds$. In this case the particular attention is
devoted to the least action solutions, namely, the minimizing solutions of
$J_{\lambda}$ among all non-trivial solutions.

Another important way to find the nontrivial solutions for (1.2) is to search
for solutions with prescribed mass, and in this case $\lambda\in\mathbb{R}$ is
part of the unknown. This approach seems to be particularly meaningful from
the physical point of view, because there is a conservation of mass.

The present paper has been motivated by a seminal paper due to Jeanjean [26]
that studied the existence of normalized solutions for a large class of
Schrödinger equations of the type

$\displaystyle\left\\{\begin{aligned} &-\Delta u=\lambda
u+g(u),\quad\quad\hbox{in }\mathbb{R}^{N},\\\
&\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2},\end{aligned}\right.$ (1.3)

with $N\geq 2$, where function $g:\mathbb{R}\to\mathbb{R}$ is an odd
continuous function with subcritical growth that satisfies some technical
conditions. One of these conditions is the following:
$\exists(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}$ satisfying

$\left\\{\begin{array}[]{l}\frac{2N+4}{N}<\alpha\leq\beta<\frac{2N}{N-2},\quad\mbox{for}\,\,N\geq
3,\\\ \mbox{}\\\
\frac{2N+4}{N}<\alpha\leq\beta,\quad\mbox{for}\,\,N=1,2,\end{array}\right.$

such that

$\alpha G(s)\leq g(s)s\leq\beta G(s)\quad\mbox{with}\quad
G(s)=\int_{0}^{s}g(t)\,dt.$

As an example of a function $g$ that satisfies the above condition is
$g(s)=|s|^{q-2}s$ with $q\in(2+\frac{4}{N},2^{*})$ when $N\geq 3$ and $q>4$ if
$N=2$. In order to overcome the loss of compactness of the Sobolev embedding
in whole $\mathbb{R}^{N}$, the author worked on the space
$H_{rad}^{1}(\mathbb{R}^{N})$ to get some compactness results. However the
most important and interesting point, in our opinion, is the fact that
Jeanjean did not work directly with the energy functional
$I:H^{1}(\mathbb{R}^{N})\to\mathbb{R}$ associated with the problem (1.3) given
by

$I(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\int_{\mathbb{R}^{N}}G(u)dx.$

In his approach, he considered the functional
$\widetilde{I}:H^{1}(\mathbb{R}^{N})\times\mathbb{R}\to\mathbb{R}$ given by

$\widetilde{I}(u,s)=\frac{e^{2s}}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{1}{e^{Ns}}\int_{\mathbb{R}^{N}}G(e^{\frac{Ns}{2}}u(x))\,dx.$

After a careful analysis, it was proved that $I$ and $\widetilde{I}$ satisfy
the mountain pass geometry on the manifold

$S(a)=\\{u\in H^{1,2}(\mathbb{R}^{N})\,:\,|u|_{2}=a\,\\},$

and their mountain pass levels are equal, which we denote by $\gamma(a)$.
Moreover, using the properties of $\widetilde{I}(u,s)$, it was obtained a
$(PS)$ sequence $(u_{n})$ to $I$ associated with the mountain pass level
$\gamma(a)$ which is bounded in $H_{rad}^{1}(\mathbb{R}^{N})$. Finally, after
some estimates, the author was able to prove that the weak limit of $(u_{n})$,
denoted by $u$, is nontrivial, $u\in S(a)$ and it verifies

$-\Delta u-g(u)=\lambda_{a}u\quad\mbox{in}\quad\mathbb{R}^{N},$

for some $\lambda_{a}<0$. An example of a nonlinearity explored in [26] we
cite

$g(t)=\mu|t|^{q-2}t\quad t\in\mathbb{R},$ $None$

where $\mu>0$ and $q\in(2+\frac{4}{N},2^{*})$. We recall that the study of the
normalized problem despite being more convenient in the application, this
brings some difficulties such as Nehari manifold method can not be applied
because the constant $\lambda_{a}$ is unknown in the problem; it is necessary
to prove that the weak limit belongs to the constrained manifold; and also it
brings some difficult to apply some usual approach for obtaining the
boundedness of the Palais Smale sequence.

We recall that the number $\bar{q}:=2+\frac{4}{N}$ is called in the literature
as $L^{2}$-critical exponent, which come from Gagliardo-Nirenberg inequality
(see [15, Theorem 1.3.7, page 9]. If $g$ is of the form $(g_{0})$ with
$q\in(2,\bar{q})$, we say that the problem is $L^{2}$-subcritical, while in
the case $q\in(\bar{q},2^{*})$ the problem is $L^{2}$-supercritical.
Associated with the $L^{2}$-supercritical problem, we would like to cite [10],
where the authors studied a problem involving vanishing potential. In the
purely $L^{2}$-critical case, that is, $q=2+\frac{4}{N}$, related problems
were studied in [16, 32].

In [36], Soave studied the normalized solutions for the nonlinear Schrödinger
equation (1.1) with combined power nonlinearities of the type

$f(t)=\mu|t|^{q-2}t+|t|^{p-2}t,\quad t\in\mathbb{R},$ $None$

where

$2<q\leq 2+\frac{4}{N}\leq p<2^{*},\,\,p\neq
q\quad\text{and}\,\,\mu\in\mathbb{R}.$

He showed that interplay between subcritical, critical and supercritical
nonlinearities strongly affects the geometry of the functional as well as the
existence and properties of ground states.

Recently some authors have considered the problem (1.1) with $f$ of the form
$(f_{0})$ but with $p=2^{*}$, which implies that $f$ has a critical growth in
the Sobolev sense. In [27], the existence of a ground state normalized
solution is obtained as minimizer of the constrained functional assuming that
$q\in(2,2+\frac{4}{N})$. While in [29] a multiplicity result was established,
where the second solution is not a ground state. For the general case
$q\in(2,2^{*})$, we would like to mention Soave [37], where the existence
result was obtained by imposing that $\mu a^{(1-\gamma_{p})q}<\alpha$, where
$\alpha$ is a specific constant that depends on $N$ and $q$ and
$\gamma_{p}=\frac{N(p-2)}{2p}$. We have seen that the results in the paper are
deeply dependent on the assumptions about $a$ and $\mu$, because by the
Pohozaev identidy the problem (1.1) does not have any solution if
$\lambda=-a\geq 0$ and $\mu>0.$ Still related to the case
$q\in(2+\frac{4}{N},2^{*})$, we would like to refer [2, 3] where the existence
of least action solutions was proved with $\mu>0$ and $N\geq 4,$ and for $N=3$
by supposing a technical on the constants $\lambda$ and $\mu$.

We recall that elliptic problems involving critical Sobolev exponent were
studied by many researchers after appeared the pioneering paper by Brezis and
Nirenberg [13], which have had many progresses in several directions. We would
like to mention the excellent book [42], for a review on this subject. In our
setting, since if $\lambda=0$, the problem (1.1) does not have any solution
for any $\mu$, then taking $\lambda$ as a Lagrange multiplier, it is able to
get solution, by combining the arguments made in [13] with the concentration
compactness principle. For the reader interested in normalized solutions for
the Schrödinger equations, we would also like to refer [9], [11], [12], [17],
[25], [28], [31], [34], [38], [39], [41] and references therein.

Our main result for the Sobolev critical case is the following:

######  Theorem 1.1.

Assume that $f$ is of the form $(f_{0})$ with $p=2^{*}$ and
$q\in(2+\frac{4}{N},2^{*})$. Then, there exists $\mu^{*}=\mu^{*}(a)>0$ such
that the problem (1.1) admits a couple $(u_{a},\lambda_{a})\in
H^{1}(\mathbb{R}^{N})\times\mathbb{R}$ of weak solutions such that
$\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}$ and $\lambda_{a}<0$ for all
$\mu\geq\mu^{*}$.

The above theorem complements the results found in [37] for the
$L^{2}$-supercritical case, because in that paper
$\mu\in(0,a^{-(1-\gamma_{p})q}\alpha)$ for some $\alpha>0$, then $\mu$ cannot
be large enough, while in our paper $\mu$ can be large enough, because
$\mu\in[\mu^{*}(a),+\infty)$. Here, we used a different approach from that
explored in [37], because we work directly with the mountain pass geometry and
concentration-compactness principle due to Lions [30], while in [37], Soave
employed minimization technique and used the properties of the Pohozaev
manifold.

Motivated by the research made in the critical Sobolev case, in this paper we
also study the exponential critical growth for $N=2$, which is a novelty for
this type of problem. To the best our knowledge we have not found any
reference involving normalizing problem with the exponential critical growth.
We recall that in $\mathbb{R}^{2}$, the natural growth restriction on the
function $f$ is given by the inequality of Trudinger and Moser [33, 40]. More
precisely, we say that a function $f$ has an exponential critical growth if
there is $\alpha_{0}>0$ such that

$\lim_{|s|\to\infty}\frac{|f(s)|}{e^{\alpha
s^{2}}}=0\,\,\,\forall\,\alpha>\alpha_{0}\quad\mbox{and}\quad\lim_{|s|\to\infty}\frac{|f(s)|}{e^{\alpha
s^{2}}}=+\infty\,\,\,\forall\,\alpha<\alpha_{0}.$

We would like to mention that problems involving exponential critical growth
have received a special attention at last years, see for example, [4, 5, 7, 8,
14, 18, 19, 20, 21] for semilinear elliptic equations, and [1, 6, 22, 23] for
quasilinear equations.

In this case, we assume that $f$ is a continuous function that satisfies the
following conditions:

  * ($f_{1}$)

$\displaystyle\lim_{t\to 0}\frac{|f(t)|}{|t|^{\tau}}=0$ as $t\rightarrow 0$,
for some $\tau>3$;

  * ($f_{2}$)

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

  * ($f_{3}$)

there exists a positive constant $\theta>4$ such that

$0<\theta F(t)\leq tf(t),\,\,\forall\,t\not=0,\,\hbox{where
}F(t)=\int_{0}^{t}f(s)ds;$

  * ($f_{4}$)

there exist constants $p>4$ and $\mu>0$ such that

$sgn(t)f(t)\geq\mu\,|t|^{p-1}\quad\text{for all}\,\,t\not=0,$

where $sgn:\mathbb{R}\setminus\\{0\\}\to\mathbb{R}$ is given by

$sgn(t)=\left\\{\begin{array}[]{l}1,\quad\mbox{if}\quad t>0\mbox{}\\\
-1,\quad\mbox{if}\quad t<0.\end{array}\right.$

Our main result is as follows:

######  Theorem 1.2.

Assume that $f$ satisfies $(f_{1})-(f_{4})$. If $a\in(0,1)$, then there exists
$\mu^{*}=\mu^{*}(a)>0$ such that the problem (1.1) admits a couple
$(u_{a},\lambda_{a})\in H^{1}(\mathbb{R}^{2})\times\mathbb{R}$ of weak
solutions with $\int_{\mathbb{R}^{2}}|u|^{2}dx=a^{2}$ and $\lambda_{a}<0$ for
all $\mu\geq\mu^{*}$.

In the proof of Theorem 1.1 and Theorem 1.2 we borrow the ideas developed in
Jeanjean [26]. The main difficulty in the proof of these theorems is
associated with the fact that we are working with critical nonlinearities in
whole $\mathbb{R}^{N}$. As above mentioned, in the proof of Theorem 1.1, the
concentration-compactness principle due to Lions [30] is crucial in our
arguments, while in the proof of Theorem 1.2, the Trundiger-Moser inequality
developed by Cao [14] plays an important role in a lot of estimates. Moreover,
in the proofs of these theorems we shall work on the space
$H^{1}_{rad}(\mathbb{R}^{N})$, because it has very nice compact embeedings.
Moreover, by Palais’ principle of symmetric criticality, see [35], it is well
known that the solutions in $H^{1}_{rad}(\mathbb{R}^{N})$ are in fact
solutions in whole $H^{1}(\mathbb{R}^{N})$.

Notation: From now on in this paper, otherwise mentioned, we use the following
notations:

  * •

$B_{r}(u)$ is an open ball centered at $u$ with radius $r>0$,
$B_{r}=B_{r}(0)$.

  * •

$C,C_{1},C_{2},...$ denote any positive constant, whose value is not relevant.

  * •

$|\,\,\,|_{p}$ denotes the usual norm of the Lebesgue space
$L^{p}(\mathbb{R}^{N})$, for $p\in[1,+\infty]$, $\|\,\,\,\|$ denotes the usual
norm of the Sobolev space $H^{1}(\mathbb{R}^{N})$.

  * •

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

##  2 Normalized solutions: The Sobolev critical case for $N\geq 3$

In order to follow the same strategy in [26], we need the following
definitions to introduce our variational procedure.

  * (1)

$S(a)=\\{u\in H^{1}(\mathbb{R}^{N})\,:\,|u|_{2}=a\,\\}$ is the sphere of
radius $a>0$ defined with the norm $|\,\,\,\,|_{2}$.

  * (2)

$J:H^{1}(\mathbb{R}^{N})\rightarrow\mathbb{R}$ with

$J(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\int_{\mathbb{R}^{N}}F(u)dx,$

where

$F(t)=\frac{\mu}{q}|t|^{q}+\frac{1}{2^{*}}|t|^{2^{*}},\quad t\in\mathbb{R}.$

Hereafter, $H=H^{1}(\mathbb{R}^{N})\times\mathbb{R}$ is equipped with the
scalar product

$\langle\cdot,\cdot\rangle_{H}=\langle\cdot,\cdot\rangle_{H^{1}(\mathbb{R}^{N})}+\langle\cdot,\cdot\rangle_{\mathbb{R}}$

and corresponding norm

$\|\cdot\|_{H}=(\|\cdot\|_{H}^{2}+|\cdot|_{\mathbb{R}}^{2})^{1/2}.$

In this section, $f$ denotes the function $f(t)=\mu|t|^{q-2}t+|t|^{2^{*}-2}t$
with $t\in\mathbb{R}$, and so, $F(t)=\int_{0}^{t}f(s)\,ds$.

  * (3)

$\mathcal{H}:H\rightarrow H^{1}(\mathbb{R}^{N})$ with

$\mathcal{H}(u,s)(x)=e^{\frac{Ns}{2}}u(e^{s}x).$

  * (4)

$\tilde{J}:H\rightarrow\mathbb{R}$ with

$\tilde{J}(u,s)=\frac{e^{2s}}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{1}{e^{Ns}}\int_{\mathbb{R}^{N}}F(e^{\frac{Ns}{2}}u(x))\,dx$

or

$\tilde{J}(u,s)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
v|^{2}dx-\int_{\mathbb{R}^{N}}F(v(x))\,dx=J(v)\quad\mbox{for}\
v=\mathcal{H}(u,s)(x).$

Throughout this section, $S$ denotes the following constant

$S=\inf_{{\footnotesize{\begin{array}[]{l}u\in D^{1,2}(\mathbb{R}^{N})\\\
u\not=0\end{array}}}}\frac{\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,dx}{\left(\int_{\mathbb{R}^{N}}|u|^{2^{*}}\,dx\right)^{\frac{2}{2^{*}}}},$
(2.1)

where $2^{*}=\frac{2N}{N-2}$ for $N\geq 3$, and $D^{1,2}(\mathbb{R}^{N})$ is
the Banach space given by

$D^{1,2}(\mathbb{R}^{N})=\left\\{u\in L^{2^{*}}(\mathbb{R}^{N})\;:\;|\nabla
u|^{2}\in L^{2}(\mathbb{R}^{N})\right\\}$

endowed with the norm

$\|u\|_{D^{1,2}(\mathbb{R}^{N})}=\left(\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,dx\right)^{\frac{1}{2}}.$

It is well known that the embedding $D^{1,2}(\mathbb{R}^{N})\hookrightarrow
L^{2^{*}}(\mathbb{R}^{N})$ is continuous.

###  2.1 The minimax approach

We shall prove that $\tilde{J}$ on $S(a)\times\mathbb{R}$ possesses a kind of
mountain-pass geometrical structure.

######  Lemma 2.1.

Let $u\in S(a)$ be arbitrary but fixed. Then we have:
(i) $|\nabla\mathcal{H}(u,s)|_{2}\rightarrow 0$ and
$J(\mathcal{H}(u,s))\rightarrow 0$ as $s\rightarrow-\infty$;
(ii) $|\nabla\mathcal{H}(u,s)|_{2}\rightarrow+\infty$ and
$J(\mathcal{H}(u,s))\rightarrow-\infty$ as $s\rightarrow+\infty$.

###### Proof.

By a straightforward calculation, it follows that

$\int_{\mathbb{R}^{N}}|\mathcal{H}(u,s)(x)|^{2}\,dx=a^{2},\quad\int_{\mathbb{R}^{N}}|\mathcal{H}(u,s)(x)|^{\xi}\,dx=e^{\frac{(\xi-2)Ns}{2}}\int_{\mathbb{R}^{N}}|u(x)|^{\xi}\,dx,\quad\forall\xi\geq
2,$ (2.2)

and

$\int_{\mathbb{R}^{N}}|\nabla\mathcal{H}(u,s)(x)|^{2}\,dx=e^{2s}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx.$ (2.3)

From the above equalities, fixing $\xi>2$, we have

$|\nabla\mathcal{H}(u,s)|_{2}\rightarrow
0\quad\mbox{and}\quad|\mathcal{H}(u,s)|_{\xi}\to 0\quad\mbox{as}\quad
s\to-\infty.$ (2.4)

Hence,

$\int_{\mathbb{R}^{N}}|F(\mathcal{H}(u,s))|\,dx\leq
C_{1}\int_{\mathbb{R}^{N}}|\mathcal{H}(u,s)|^{q}\,dx+C_{2}\int_{\mathbb{R}^{N}}|\mathcal{H}(u,s)|^{2^{*}}\,dx\to
0\quad\mbox{as}\quad s\to-\infty,$

from where it follows that

$J(\mathcal{H}(u,s))\rightarrow 0\quad\mbox{as}\quad s\rightarrow-\infty,$

showing $(i)$.

In order to show $(ii)$, note that by (2.3),

$|\nabla\mathcal{H}(u,s)|_{2}\rightarrow+\infty\quad\mbox{as}\quad
s\to+\infty.$

On the other hand,

$J(\mathcal{H}(u,s))\leq\frac{1}{2}|\nabla\mathcal{H}(u,s)|_{2}^{2}-\frac{\mu}{q}\int_{\mathbb{R}^{N}}|\mathcal{H}(u,s)|^{q}\,dx=\frac{e^{2s}}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{\mu
e^{\frac{(q-2)Ns}{2}}}{q}\int_{\mathbb{R}^{N}}|u(x)|^{q}\,dx.$

Since $q>2+\frac{4}{N}$, the last inequality yields

$J(\mathcal{H}(u,s))\rightarrow-\infty\quad\mbox{as}\quad
s\rightarrow+\infty.$

∎

######  Lemma 2.2.

There exists $K(a)>0$ small enough such that

$0<\sup_{u\in A}J(u)<\inf_{u\in B}J(u)$

with

$A=\left\\{u\in S(a),\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\leq
K(a)\right\\}\quad\mbox{and}\quad B=\left\\{u\in
S(a),\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx=2K(a)\right\\}.$

###### Proof.

We will need the following Gagliardo-Nirenberg inequality: for any $\xi>2$,

$|u|_{\xi}\leq C(\xi,N)|\nabla u|_{2}^{\gamma}|u|_{2}^{1-\gamma},$

where $\gamma=N(\frac{1}{2}-\frac{1}{\xi})$. If we fix $|\nabla u|^{2}_{2}\leq
K(a)$ and $|\nabla v|^{2}_{2}=2K(a)$, we derive that

$\int_{\mathbb{R}^{N}}F(u)\,dx\leq C_{1}|u|^{q}_{q}+C_{2}|u|^{2^{*}}_{2^{*}}.$

Then, by the Gagliardo-Nirenberg inequality,

$\int_{\mathbb{R}^{N}}F(v)\,dx\leq C_{1}(|\nabla
v|_{2}^{2})^{N(\frac{q-2}{4})}+C_{2}(|\nabla
v|_{2}^{2})^{N(\frac{2^{*}-2}{4})}.$

Since $F(u)\geq 0$ for any $u\in H^{1}(\mathbb{R}^{N})$, we have

$\displaystyle J(v)-J(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
v|^{2}\,dx-\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,dx-\int_{\mathbb{R}^{N}}F(v)\,dx+\int_{\mathbb{R}^{N}}F(u)\,dx$
$\displaystyle\geq$ $\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
v|^{2}\,dx-\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,dx-\int_{\mathbb{R}^{N}}F(v)\,dx,$

and so,

$J(v)-J(u)\geq\frac{1}{2}K(a)-C_{3}K(a)^{N(\frac{q-2}{4})}-C_{4}K(a)^{N(\frac{2^{*}-2}{4})}.$

Thereby, fixing $K(a)$ small enough of such way that,

$\frac{1}{2}K(a)-C_{3}K(a)^{N(\frac{q-2}{4})}-C_{4}K(a)^{N(\frac{2^{*}-2}{4})}>0,$

we get the desired result. ∎

As a byproduct of the last lemma is the following corollary.

######  Corollary 2.1.

There exists $K(a)>0$ such that if $u\in S(a)$ and $|\nabla u|^{2}_{2}\leq
K(a)$, then $J(u)>0$.

###### Proof.

Arguing as in the last lemma,

$J(u)\geq\frac{1}{2}|\nabla u|_{2}^{2}-C_{1}|\nabla
u|_{2}^{N(\frac{q-2}{2})}-C_{2}|\nabla u|_{2}^{N(\frac{2^{*}-2}{2})}>0,$

for $K(a)$ small enough. ∎

In what follows, we fix $u_{0}\in S(a)$ and apply Lemma 2.1 to get two numbers
$s_{1}<0$ and $s_{2}>0$, of such way that the functions
$u_{1}=\mathcal{H}(s_{1},u_{0})$ and $u_{2}=\mathcal{H}(s_{2},u_{0})$ satisfy

$|\nabla u_{1}|^{2}_{2}<\frac{K(a)}{2},\,\,|\nabla
u_{2}|_{2}^{2}>2K(a),\,\,J(u_{1})>0\quad\mbox{and}\quad J(u_{2})<0.$

Now, following the ideas from Jeanjean [26], we fix the following mountain
pass level given by

$\gamma_{\mu}(a)=\inf_{h\in\Gamma}\max_{t\in[0,1]}J(h(t))$

where

$\Gamma=\left\\{h\in
C([0,1],S(a)):h(0)=u_{1}\,\,\mbox{and}\,\,h(1)=u_{2}\right\\}.$

From Lemma 2.2,

$\max_{t\in[0,1]}J(h(t))>\max\left\\{J(u_{1}),J(u_{2})\right\\}.$

######  Lemma 2.3.

There holds $\displaystyle\lim_{\mu\to+\infty}\gamma_{\mu}(a)=0$.

###### Proof.

In what follows we set the path
$h_{0}(t)=\mathcal{H}((1-t)s_{1}+ts_{2},u_{0})\in\Gamma$. Then,

$\gamma_{\mu}(a)\leq\max_{t\in[0,1]}J(h_{0}(t))\leq\max_{r\geq
0}\left\\{\frac{r^{2}}{2}|\nabla
u_{0}|_{2}^{2}-\frac{\mu}{q}r^{\frac{N(q-2)}{2}}|u_{0}|_{q}^{q}\right\\},$

and so, for some positive constant $C_{2}$,

$\gamma_{\mu}(a)\leq C_{2}\left(\frac{1}{\mu}\right)^{\frac{4}{N(q-2)-4}}\to
0\quad\mbox{as}\quad\mu\to+\infty.$

Here, we have used the fact that $q>2+\frac{4}{N}$.

∎

In what follows $(u_{n})$ denotes the $(PS)$ sequence associated with the
level $\gamma_{\mu}(a)$, which is obtained by making
$u_{n}=\mathcal{H}(v_{n},s_{n})$, where $(v_{n},s_{n})$ is the $(PS)$ sequence
for $\tilde{J}$ obtained by [26, Proposition 2.2 ], associated with the level
$\gamma_{\mu}(a).$ More precisely, we have

$J(u_{n})\to\gamma_{\mu}(a)\quad\mbox{as}\quad n\to+\infty,$ (2.5)

and

$\|J|^{\prime}_{S(a)}(u_{n})\|\to 0\quad\mbox{as}\quad n\to+\infty.$

Setting the functional $\Psi:H^{1}(\mathbb{R}^{N})\to\mathbb{R}$ given by

$\Psi(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|u|^{2}\,dx,$

it follows that $S(a)=\Psi^{-1}(\\{a^{2}/2\\})$. Then, by Willem [42,
Proposition 5.12], there exists $(\lambda_{n})\subset\mathbb{R}$ such that

$||J^{\prime}(u_{n})-\lambda_{n}\Psi^{\prime}(u_{n})||_{H^{-1}}\to
0\quad\mbox{as}\quad n\to+\infty.$

Hence,

$-\Delta u_{n}-f(u_{n})=\lambda_{n}u_{n}\
+o_{n}(1)\quad\mbox{in}\quad(H^{1}(\mathbb{R}^{N}))^{*}.$ (2.6)

Moreover, another important limit involving the sequence $(u_{n})$ is

$Q(u_{n})=\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{2}dx+N\int_{\mathbb{R}^{N}}F(u_{n})dx-\frac{N}{2}\int_{\mathbb{R}^{N}}f(u_{n})u_{n}dx\to
0\quad\mbox{as}\quad n\to+\infty,$ (2.7)

which is obtained using the limit below

${\partial_{s}}\tilde{J}(v_{n},s_{n})\to 0\quad\mbox{as}\quad n\to+\infty,$

that was also proved in [26].

Arguing as in [26, Lemmas 2.3 and 2.4], we know that $(u_{n})$ is a bounded
sequence, and so, the number $\lambda_{n}$ must satisfy the equality below

$\lambda_{n}=\frac{1}{|u_{n}|^{2}_{2}}\left\\{|\nabla
u_{n}|^{2}_{2}-\int_{\mathbb{R}^{N}}f(u_{n})u_{n}dx\right\\}+o_{n}(1),$

or equivalently,

$\lambda_{n}=\frac{1}{a^{2}}\left\\{|\nabla
u_{n}|^{2}_{2}-\int_{\mathbb{R}^{N}}f(u_{n})u_{n}dx\right\\}+o_{n}(1).$ (2.8)

######  Lemma 2.4.

There exists $C>0$ such that

$\limsup_{n\to+\infty}\int_{\mathbb{R}^{N}}F(u_{n})\,dx\leq C\gamma_{\mu}(a)$

and

$\limsup_{n\to+\infty}\int_{\mathbb{R}^{N}}f(u_{n})u_{n}\,dx\leq
C\gamma_{\mu}(a).$

###### Proof.

From (2.5) and (2.7)

$N{J}(u_{n})+Q(u_{n})=N\gamma_{\mu}(a)+o_{n}(1),$

then

$\frac{N+2}{2}\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{2}dx-\frac{N}{2}\int_{\mathbb{R}^{N}}f(u_{n})u_{n}dx=N\gamma_{\mu}(a)+o_{n}(1).$

Using again (2.5), we get

$\frac{N+2}{2}\left(2\int_{\mathbb{R}^{N}}F(u_{n})dx+2\gamma_{\mu}(a)+o_{n}(1)\right)-\frac{N}{2}\int_{\mathbb{R}^{N}}f(u_{n})u_{n}dx=N\gamma_{\mu}(a)+o_{n}(1),$

that is,

$-(N+2)\int_{\mathbb{R}^{N}}F(u_{n})dx+\frac{N}{2}\int_{\mathbb{R}^{N}}f(u_{n})u_{n}dx=2\gamma_{\mu}(a)+o_{n}(1).$
(2.9)

Since $q\in(2+\frac{4}{N},2^{*})$ and
$F(t)=\frac{\mu}{q}|t|^{q}+\frac{1}{2^{*}}|t|^{2^{*}},\,\,\forall\,t\in\mathbb{R}^{N},$
we obtain

$qF(t)\leq f(t)t,\,\,t\in\mathbb{R}.$ (2.10)

This together with (2.9) yields

$\left(\frac{qN}{2}-(N+2)\right)\int_{\mathbb{R}^{N}}F(u_{n})dx\leq
2\gamma_{\mu}(a)+o_{n}(1),$

and so,

$\limsup_{n\to+\infty}\int_{\mathbb{R}^{N}}F(u_{n})\,dx\leq C\gamma_{\mu}(a).$

This inequality combined again with (2.9) ensures that

$\limsup_{n\to+\infty}\int_{\mathbb{R}^{N}}f(u_{n})u_{n}\,dx\leq
C\gamma_{\mu}(a).$

∎

######  Lemma 2.5.

$\displaystyle\limsup_{n\to+\infty}|\nabla u_{n}|_{2}^{2}\leq
C\gamma_{\mu}(a)$.

###### Proof.

First of all, let us recall that

$\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{2}\,dx=2\gamma_{\mu}(a)+2\int_{\mathbb{R}^{N}}F(u_{n})\,dx+o_{n}(1).$

Then, from Lemma 2.4,

$\limsup_{n\to+\infty}|\nabla u_{n}|_{2}^{2}\leq(2+C_{1})\gamma_{\mu}(a).$

∎

Now, from (2.9), the sequence $(\int_{\mathbb{R}^{N}}F(u_{n})\,dx)$ is bounded
away from zero, otherwise we would have

$\int_{\mathbb{R}^{N}}F(u_{n})\,dx\to 0\quad\mbox{as}\quad n\to+\infty,$

which leads to

$\int_{\mathbb{R}^{N}}f(u_{n})u_{n}\,dx\to 0\quad\mbox{as}\quad n\to+\infty.$

These limits combined with (2.9) imply that $\gamma_{\mu}(a)=0$, which is
absurd. From this, in what follows we can assume that

$\int_{\mathbb{R}^{N}}F(u_{n})\,dx\rightarrow C_{1}>0,\ \mbox{as}\
n\rightarrow\infty.$ (2.11)

######  Lemma 2.6.

The sequence $(\lambda_{n})$ is bounded with

$\lambda_{n}=-\frac{\mu}{a^{2}}\Big{(}\frac{N}{q}-\frac{N-2}{2}\Big{)}\int_{\mathbb{R}^{N}}|u_{n}|^{q}\,dx+o_{n}(1)$

and

$\limsup_{n\to+\infty}|\lambda_{n}|\leq\frac{C}{a^{2}}\gamma_{\mu}(a),$

for some $C>0$.

###### Proof.

The boundedness of $(u_{n})$ yields that $(\lambda_{n})$ is bounded, because

$\lambda_{n}a^{2}=\lambda_{n}|u_{n}|^{2}_{2}=|\nabla
u_{n}|_{2}^{2}-\int_{\mathbb{R}^{N}}f(u_{n})u_{n}\,dx+o_{n}(1),$ (2.12)

and so,

$\displaystyle|\lambda_{n}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{a^{2}}\left(|\nabla
u_{n}|_{2}^{2}+\int_{\mathbb{R}^{N}}f(u_{n})u_{n}\,dx\right)+o_{n}(1)$
$\displaystyle\leq$ $\displaystyle\frac{C}{a^{2}}\gamma_{\mu}(a)+o_{n}(1).$

This guarantees the boundedness of $(\lambda_{n})$ and the second inequality
is proved.

In order to prove the first inequality, we know by (2.7) that

$|\nabla
u_{n}|_{2}^{2}=\frac{N}{2}\int_{\mathbb{R}^{N}}f(u_{n})u_{n}\,dx-N\int_{\mathbb{R}^{N}}F(u_{n})\,dx+o_{n}(1).$

Inserting this equality in (2.12), we obtain

$\lambda_{n}a^{2}=-\mu\Big{(}\frac{N}{q}-\frac{N-2}{2}\Big{)}\int_{\mathbb{R}^{N}}|u_{n}|^{q}\,dx+o_{n}(1),$

showing the first inequality. ∎

In the sequel, we restrict our study to the space
$H_{rad}^{1}(\mathbb{R}^{N})$. Then, it is well known that

$\lim_{n\to+\infty}\int_{\mathbb{R}^{N}}|u_{n}|^{q}\,dx=\int_{\mathbb{R}^{N}}|u|^{q}\,dx,$
(2.13)

where $u_{n}\rightharpoonup u$ in $H_{rad}^{1}(\mathbb{R}^{N})$, because
$q\in(2+\frac{4}{N},2^{*})$.

######  Lemma 2.7.

There exists $\mu^{*}>0$ such that $u\not=0$ for all $\mu\geq\mu^{*}>0$.

###### Proof.

Seeking for a contradiction, let us assume that $u=0$. Then,

$\lim_{n\to+\infty}\int_{\mathbb{R}^{N}}|u_{n}|^{q}\,dx=0,$ (2.14)

and by Lemma 2.6,

$\limsup_{n\to+\infty}\lambda_{n}=0.$ (2.15)

The equality

$a^{2}\lambda_{n}=|\nabla
u_{n}|^{2}_{2}-\int_{\mathbb{R}^{N}}f(u_{n})u_{n}\,dx+o_{n}(1)$

together with (2.14) and (2.15) leads to

$|\nabla u_{n}|_{2}^{2}-|u_{n}|_{2^{*}}^{2^{*}}=o_{n}(1).$ (2.16)

In what follows, going to a subsequence, we assume that

$|\nabla
u_{n}|_{2}^{2}=L+o_{n}(1)\quad\mbox{and}\quad|u_{n}|_{2^{*}}^{2^{*}}=L+o_{n}(1).$

We claim that $L>0$, otherwise if $L=0$, we must have

$|\nabla u_{n}|_{2}^{2}=o_{n}(1),$ (2.17)

then,

$|\nabla u_{n}|^{2}_{2}\to 0,$

which is absurd, because $\gamma_{\mu}(a)>0$.

Since $L>0$, by definition of $S$ in (2.1),

$S\leq\frac{\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{2}\,dx}{\left(\int_{\mathbb{R}^{N}}|u_{n}|^{2^{*}}\,dx\right)^{\frac{2}{2^{*}}}}.$

Taking the $limsup$ as $n\to+\infty$, we obtain

$S\leq\frac{L}{L^{\frac{2}{2^{*}}}},$

that is,

$L\geq S^{\frac{N}{2}}.$

On the other hand

$o_{n}(1)+\gamma_{\mu}(a)-\frac{a^{2}\lambda_{n}}{2}=\frac{1}{2}(|\nabla
u_{n}|_{2}^{2}-a^{2}\lambda_{n})-\frac{\mu}{q}|u_{n}|^{q}_{q}-\frac{1}{2^{*}}|u_{n}|^{2^{*}}_{2^{*}}=\frac{1}{N}L+o_{n}(1).$

Recalling that
$\displaystyle\limsup_{n\to+\infty}|\lambda_{n}|\leq\frac{C}{a^{2}}\gamma_{\mu}(a)$,
it follows that

$\frac{1}{N}S^{\frac{N}{2}}\leq C\gamma_{\mu}(a).$

Now, fixing $\mu^{*}$ large enough of a such way that

$C\gamma_{\mu}(a)<\frac{1}{N}S^{\frac{N}{2}},\quad\forall\mu\geq\mu^{*},$

we get a new contradiction. This proves that $u\not=0$ for $\mu>0$ large
enough.

∎

######  Lemma 2.8.

Increasing if necessary $\mu^{*}$, we have $u_{n}\to u$ in
$L^{2^{*}}(\mathbb{R}^{N})$ for all $\mu\geq\mu^{*}$.

###### Proof.

Using the concentration-compactness principle due to Lions [30], we can find
an at most countable index $\mathcal{J}$, sequences
$(x_{i})\subset\mathbb{R}^{N},(\kappa_{i}),(\nu_{i})\subset(0,\infty)$ such
that
$(i)$ $|\nabla u_{n}|^{2}\to\kappa$ weakly-∗ in the sense of measure
and
$(ii)$ $|u_{n}|^{2^{*}}\to\nu$ weakly-∗ in the sense of measure,
and

$\left\\{\begin{array}[]{l}(a)\quad\nu=|u|^{2^{*}}+\sum_{j\in
J}\nu_{j}\delta_{x_{j}},\\\ (b)\quad\kappa\geq|\nabla u|^{2}+\sum_{j\in
J}\kappa_{j}\delta_{x_{j}},\\\ (c)\quad
S\nu_{j}^{\frac{2}{2^{*}}}\leq\kappa_{j},\,\,\forall j\in\mathcal{J},\\\
\end{array}\right.$

where $\delta_{x_{j}}$ is the Dirac mass at the point $x_{j}$. Since

$-\Delta
u_{n}-f(u_{n})=\lambda_{n}u_{n}+o_{n}(1)\quad\mbox{in}\quad(H^{1}(\mathbb{R}^{N}))^{*},$

we derive that

$\int_{\mathbb{R}^{N}}\nabla
u_{n}\nabla\phi\,dx-\lambda_{n}\int_{\mathbb{R}^{N}}u_{n}\phi\,dx=\mu\int_{\mathbb{R}^{N}}|u_{n}|^{q-2}u_{n}\phi\,dx+\int_{\mathbb{R}^{N}}|u_{n}|^{2^{*}-2}u_{n}\phi\,dx,\quad\forall\phi\in
H^{1}(\mathbb{R}^{N}).$

Now, arguing as in [24, Lemma 2.3], $\mathcal{J}$ is empty or otherwise
$\mathcal{J}$ is nonempty but finite. In the case that $\mathcal{J}$ is
nonempty but finite, we must have

$\kappa_{j}\geq{S^{\frac{N}{2}}},\quad\forall j\in\mathcal{J}.$

However, by Lemma 2.5,

$\limsup_{n\to+\infty}|\nabla u_{n}|_{2}^{2}\leq C\gamma_{\mu}(a).$

Then, if $\mu^{*}>0$ is fixed of such way that

$C\gamma_{\mu}(a)<\frac{1}{2}{S^{\frac{N}{2}}},$

we get a contradiction, and so, $\mathcal{J}=\emptyset$. From this,

$u_{n}\to u\quad\mbox{in}\quad L^{2^{*}}_{loc}(\mathbb{R}^{N}).$ (2.18)

######  Claim 2.1.

For each $R>0$, we have

$u_{n}\to u\quad\mbox{in}\quad L^{2^{*}}(\mathbb{R}^{N}\setminus B_{R}(0)).$

Indeed, as $u_{n}\in H_{rad}^{1}(\mathbb{R}^{N})$, we know that

$|u_{n}(x)|\leq\frac{\|u_{n}\|}{|x|^{\frac{N-1}{2}}},\quad\mbox{a.e.
in}\quad\mathbb{R}^{N}.$

Since $(u_{n})$ is a bounded sequence in $H^{1}(\mathbb{R}^{N})$, we obtain

$|u_{n}(x)|\leq\frac{C}{|x|^{\frac{N-1}{2}}},\quad\mbox{a.e.
in}\quad\mathbb{R}^{N},$

and so,

$|u_{n}(x)|^{2^{*}}\leq\frac{C_{1}}{|x|^{\frac{N(N-1)}{N-2}}},\quad\mbox{a.e.
in}\quad\mathbb{R}^{N}.$

Recalling that $\frac{C_{1}}{|\,\cdot\,|^{\frac{N(N-1)}{N-2}}}\in
L^{1}(\mathbb{R}^{N}\setminus B_{R}(0))$ and $u_{n}(x)\to u(x)$ a.e. in
$\mathbb{R}^{N}\setminus B_{R}(0)$, the Lebesgue’s Theorem gives

$u_{n}\to u\quad\mbox{in}\quad L^{2^{*}}(\mathbb{R}^{N}\setminus B_{R}(0)),$

showing the Claim 2.1. Now, the Claim 2.1 combined with (2.18) ensures that

$u_{n}\to u\quad\mbox{in}\quad L^{2^{*}}(\mathbb{R}^{N}).$

∎

##  3 Proof of Theorem 1.1

From the previous analysis the weak limit $u$ of $(u_{n})$ is nontrivial.
Therefore, by Lemma 2.6

$\lim_{n\to+\infty}\lambda_{n}=-\frac{\mu}{a^{2}}\Big{(}\frac{N}{q}-\frac{N-2}{2}\Big{)}\lim_{n\to+\infty}\int_{\mathbb{R}^{N}}|u_{n}|^{q}\,dx=-\frac{\mu}{a^{2}}\Big{(}\frac{N}{q}-\frac{N-2}{2}\Big{)}\int_{\mathbb{R}^{N}}|u|^{q}\,dx<0.$

So, we can assume without loss of generality that

$\lambda_{n}\to\lambda_{a}<0\quad\mbox{as}\quad n\to+\infty.$ (3.1)

Now, the equality (2.6) and (3.1) imply that

$-\Delta u-f(u)=\lambda_{a}u,\quad\mbox{in}\quad\mathbb{R}^{N}.$ (3.2)

Thus,

$|\nabla u|_{2}^{2}-\lambda_{a}|u|_{2}^{2}=\int_{\mathbb{R}^{N}}f(u)u\,dx.$

On the other hand,

$|\nabla
u_{n}|_{2}^{2}-\lambda_{n}|u_{n}|_{2}^{2}=\int_{\mathbb{R}^{N}}f(u_{n})u_{n}\,dx+o_{n}(1),$

or yet,

$|\nabla
u_{n}|_{2}^{2}-\lambda_{a}|u_{n}|_{2}^{2}=\int_{\mathbb{R}^{N}}f(u_{n})u_{n}\,dx+o_{n}(1).$

Recalling that by Lemma 2.8

$u_{n}\to u\quad\mbox{in}\quad L^{2^{*}}(\mathbb{R}^{N}),$

and using the below limit

$u_{n}\to u\quad\mbox{in}\quad L^{q}(\mathbb{R}^{N}),$

we obtain

$\lim_{n\to+\infty}\int_{\mathbb{R}^{N}}f(u_{n})u_{n}\,dx=\int_{\mathbb{R}^{N}}f(u)u\,dx,$

from where it follows that

$\lim_{n\to+\infty}(|\nabla u_{n}|_{2}^{2}-\lambda_{a}|u_{n}|_{2}^{2})=|\nabla
u|_{2}^{2}-\lambda_{a}|u|_{2}^{2}.$

Since $\lambda_{a}<0$, the last equality implies that

$u_{n}\to u\quad\mbox{in}\quad H^{1}(\mathbb{R}^{N}),$

implying that $|u|_{2}^{2}=a$. This establishes the desired result.

##  4 Normalized solutions: The exponential critical growth case for $N=2$

In this section we shall deal with the case $N=2$, where $f$ has an
exponential critical growth and $a\in(0,1)$. We start our study recalling that
by $(f_{1})$ and $(f_{2})$, we know 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

$|f(t)|\leq\zeta|t|^{\tau}+C|t|^{q-1}(e^{\alpha t^{2}}-1)\text{ for all
}t\in\mathbb{R}$ (4.1)

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

$|F(t)|\leq\zeta|t|^{\tau+1}+C|t|^{q}(e^{\alpha t^{2}}-1)\text{ for all
}t\in\mathbb{R}.$ (4.2)

Moreover, it is easy to see that, by (4.1),

$|f(t)t|\leq\zeta|t|^{\tau+1}+C|t|^{q}(e^{\alpha t^{2}}-1)\text{ for all
}t\in\mathbb{R}.$ (4.3)

Finally, let us recall the following version of Trudinger-Moser inequality as
stated e.g. in [14].

######  Lemma 4.1.

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

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

######  Lemma 4.2.

Let $(u_{n})$ be a sequence in $H^{1}(\mathbb{R}^{2})$ with $u_{n}\in S(a)$
and

$\limsup_{n\to+\infty}|\nabla u_{n}|_{2}^{2}<1-a^{2}.$

Then, there exist $t>1$, $t$ close to 1, and $C>0$ satisfying

$\int_{\mathbb{R}^{2}}\left(e^{4\pi|u_{n}|^{2}}-1\right)^{t}dx\leq
C,\,\,\,\,\forall\,n\in\mathbb{N}.$

###### Proof.

As

$\limsup_{n\rightarrow\infty}|\nabla
u_{n}|_{2}^{2}<1-a^{2}\quad\mbox{and}\quad|u_{n}|_{2}^{2}=a^{2}<1,$

there exist $m>0$ and $n_{0}\in\mathbb{N}$ verifying

$\|u_{n}\|^{2}<m<1,\,\,\,\text{for any}\,\,\,n\geq n_{0}.$

Fix $t>1$, with $t$ close to $1$ such that $tm<1$ and

$\int_{\mathbb{R}^{2}}\left(e^{4\pi|u_{n}|^{2}}-1\right)^{t}dx\leq\int_{\mathbb{R}^{2}}\left(e^{4tm\pi(\frac{|u_{n}|}{||u_{n}||})^{2}}-1\right)dx,\,\,\text{for
any}\,\,n\geq n_{0},$

where we have used the inequality

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

Hence, by Lemma 4.1, there exists $C_{1}=C_{1}(t,m,a)>0$

$\int_{\mathbb{R}^{2}}\left(e^{4\pi|u_{n}|^{2}}-1\right)^{t}dx\leq
C_{1}\,\,\,\,\,\forall\,n\geq n_{0}.$

Now, the lemma follows fixing

$C=\max\left\\{C_{1},\int_{\mathbb{R}^{2}}\left(e^{4\pi|u_{1}|^{2}}-1\right)^{t}dx,....,\int_{\mathbb{R}^{2}}\left(e^{4\pi|u_{n_{0}}|^{2}}-1\right)^{t}dx\right\\}.$

∎

######  Corollary 4.1.

Let $(u_{n})$ be a sequence in $H^{1}(\mathbb{R}^{2})$ with $u_{n}\in S(a)$
and

$\limsup_{n\to+\infty}|\nabla u_{n}|_{2}^{2}<1-a^{2}.$

If $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{2})$ and $u_{n}(x)\to u(x)$
a.e in $\mathbb{R}^{2}$, then

$F(u_{n})\to F(u)\,\,\mbox{in}\,\,L^{1}(B_{R}(0)),\,\,\text{for any}\,\,R>0.$

###### Proof.

By (4.2), 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

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

from where it follows that,

$|F(u_{n})|\leq\zeta|u_{n}|^{\tau+1}+C|u_{n}|^{q}(e^{\alpha|u_{n}|^{2}}-1),\quad\forall
n\in\mathbb{N}.$ (4.4)

with

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

Similar to arguments in Lemma 4.2, there exist $m>0$ and $n_{0}\in\mathbb{N}$
verifying

$\|u_{n}\|^{2}<m<1,\,\,\,\text{for any}\,\,\,n\geq n_{0}.$

Choose $\alpha>4\pi$ close to 4$\pi$, $t>1$ close to $1$ with $\alpha
mt<4\pi$, and by [42, Lemma A.1], there exists $\omega\in
L^{qt^{\prime}}(B_{R}(0))$ such that $|u_{n}|\leq\omega$ a.e. in $B_{R}(0)$,
where $t^{\prime}$ is the conjugate exponent of $t$. Thus,

$|u_{n}|^{q}(e^{\alpha|u_{n}|^{2}}-1)\leq\omega^{q}(e^{\alpha|u_{n}|^{2}}-1),\quad\text{a.e.
in}\,B_{R}(0),$

and

$|F(u_{n})|\leq\zeta|u_{n}|^{\tau+1}+C\omega^{q}(e^{\alpha|u_{n}|^{2}}-1),\quad\text{a.e.
in}\,B_{R}(0).$ (4.5)

Setting

$h_{n}(x)=C(e^{\alpha|u_{n}|^{2}}-1),$

We can argue as in the proof of Lemma 4.2, there exists $C>0$ such that

$h_{n}\in
L^{t}(\mathbb{R}^{2})\quad\mbox{and}\quad\sup_{n\in\mathbb{N}}|h_{n}|_{t}<+\infty,$

Therefore, for some subequence of $(u_{n})$, still denoted by itself, we
derive that

$h_{n}\rightharpoonup
h=C(e^{\alpha|u|^{2}}-1),\,\,\mbox{in}\,\,L^{t}(\mathbb{R}^{2}).$ (4.6)

######  Claim 4.1.

Now we show that

$\omega^{q}h_{n}\to\omega^{q}h\quad\mbox{in}\quad L^{1}(B_{R}(0)),\quad\forall
R>0.$

Indeed, for each $R>0$, we consider the characteristic function $\chi_{R}$
associated with $B_{R}(0)\subset\mathbb{R}^{2}$, that is,

$\chi_{R}(x)=\left\\{\begin{array}[]{l}1,\quad x\in B_{R}(0),\\\ 0,\quad
x\in\mathbb{R}^{N}\setminus B_{R}(0),\end{array}\right.$

which belongs to $L^{qt^{\prime}}(\mathbb{R}^{2})$. Thus, by the weak limit
(4.6),

$\int_{\mathbb{R}^{2}}\omega^{q}\chi_{R}h_{n}\,dx\to\int_{\mathbb{R}^{2}}\omega^{q}\chi_{R}h\,dx,$

or equivalently,

$\int_{B_{R}(0)}\omega^{q}h_{n}\,dx\to\int_{B_{R}(0)}\omega^{q}h\,dx.$ (4.7)

Hence, by $u_{n}\to u$ in $L^{\tau+1}(B_{R})$, (4.5) and (4.7), applying a
variant of the Lebesgue Dominated Convergence Theorem, we deduce that

$F(u_{n})\to F(u)\quad\mbox{in}\quad L^{1}(B_{R}(0)).$

∎

The next lemma is crucial in our argument.

######  Lemma 4.3.

Let $(u_{n})\subset H_{rad}^{1}(\mathbb{R}^{2})$ be a sequence with $u_{n}\in
S(a)$ and

$\limsup_{n\to+\infty}|\nabla u_{n}|_{2}^{2}<1-a^{2}.$

Then, there exists $\alpha$ close to $4\pi$, such that for all $q>2$,

$|u_{n}|^{q}(e^{\alpha|u_{n}(x)|^{2}}-1)\to|u|^{q}(e^{\alpha|u(x)|^{2}}-1)\,\,\mbox{in}\,\,L^{1}(\mathbb{R}^{N}).$

###### Proof.

Arguing as in Corollary 4.1, there are $\alpha>4\pi$ close to $4\pi$ and $t>1$
close to $1$ such that the sequence

$h_{n}(x)=(e^{\alpha|u_{n}(x)|^{2}}-1),$

is a bounded sequence in $L^{t}(\mathbb{R}^{N})$. Therefore, for some
subsequence of $(h_{n})$, still denoted by itself, we derive that

$h_{n}\rightharpoonup h=(e^{\alpha|u|^{2}}-1)\quad\mbox{in}\quad
L^{t}(\mathbb{R}^{2}).$

For $t^{\prime}=\frac{t}{t-1}$, we know that the embedding
$H_{rad}^{1}(\mathbb{R}^{N})\hookrightarrow L^{qt^{\prime}}(\mathbb{R}^{N})$
is compact, then

$u_{n}\to u\quad\mbox{in}\quad L^{qt^{\prime}}(\mathbb{R}^{N}),$

and so,

$|u_{n}|^{q}\to|u|^{q}\quad\mbox{in}\quad L^{t^{\prime}}(\mathbb{R}^{N}).$

Thus,

$\lim_{n\to+\infty}\int_{\mathbb{R}^{2}}|u_{n}|^{q}h_{n}(x)\,dx=\int_{\mathbb{R}^{2}}|u|^{q}h(x)\,dx,$

that is,

$\lim_{n\to+\infty}\int_{\mathbb{R}^{2}}|u_{n}|^{q}(e^{\alpha|u_{n}(x)|^{2}}-1)\,dx=\int_{\mathbb{R}^{2}}|u|^{q}(e^{\alpha|u(x)|^{2}}-1)\,dx.$

Since

$|u_{n}|^{q}(e^{\alpha|u_{n}(x)|^{2}}-1)\geq
0\quad\mbox{and}\quad|u|^{q}(e^{\alpha|u(x)|^{2}}-1)\geq 0,$

the last limit gives

$|u_{n}|^{q}(e^{\alpha|u_{n}(x)|^{2}}-1)\to|u|^{q}(e^{\alpha|u(x)|^{2}}-1)\quad\mbox{in}\quad
L^{1}(\mathbb{R}^{2}).$

∎

######  Corollary 4.2.

Let $(u_{n})$ be a sequence in $H_{rad}^{1}(\mathbb{R}^{2})$ with $u_{n}\in
S(a)$ and

$\limsup_{n\to+\infty}|\nabla u_{n}|_{2}^{2}<1-a^{2}.$

If $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{2})$ and $u_{n}(x)\to u(x)$
a.e in $\mathbb{R}^{2}$, then

$F(u_{n})\to F(u)\quad\mbox{and}\quad f(u_{n})u_{n}\to
f(u)u\,\,\mbox{in}\,\,L^{1}(\mathbb{R}^{2}).$

###### Proof.

By (4.2),

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

where $\alpha>4\pi$ close to $4\pi$ and $q>2$ as in the last Lemma 4.3.
Therefore,

$|F(u_{n})|\leq\zeta|u_{n}|^{\tau+1}+C|u_{n}|^{q}(e^{\alpha|u_{n}|^{2}}-1).$
(4.8)

By Lemma 4.3,

$|u_{n}|^{q}(e^{\alpha|u_{n}(x)|^{2}}-1)\to|u|^{q}(e^{\alpha|u(x)|^{2}}-1)\quad\mbox{in}\quad
L^{1}(\mathbb{R}^{2}),$

and by the compact embedding $H_{rad}^{1}(\mathbb{R}^{N})\hookrightarrow
L^{\tau+1}(\mathbb{R}^{N})$,

$u_{n}\to u\quad\mbox{in}\quad L^{\tau+1}(\mathbb{R}^{2}).$

Now, we can use the Lebesgue’s Theorem to conclude that

$F(u_{n})\to F(u)\quad\mbox{in}\quad L^{1}(\mathbb{R}^{2}).$

A similar argument works to show that

$f(u_{n})u_{n}\to f(u)u\quad\mbox{in}\quad L^{1}(\mathbb{R}^{2}).$

∎

From now on, we will use the same notations introduced in Section 2 to apply
our variational procedure, more precisely

  * (1)

$S(a)=\\{u\in H^{1}(\mathbb{R}^{2})\,:\,|u|_{2}=a\,\\}$ is the sphere of
radius $a>0$ defined with the norm $|\,\,\,\,|_{2}$.

  * (2)

$J:H^{1}(\mathbb{R}^{2})\rightarrow\mathbb{R}$ with

$J(u)=\frac{1}{2}\int_{\mathbb{R}^{2}}|\nabla
u|^{2}dx-\int_{\mathbb{R}^{2}}F(u)dx.$

  * (3)

$\mathcal{H}:H\rightarrow\mathbb{R}$ with

$\mathcal{H}(u,s)(x)=e^{s}u(e^{s}x).$

  * (4)

$\tilde{J}:H\rightarrow\mathbb{R}$ with

$\tilde{J}(u,s)=\frac{e^{2s}}{2}\int_{\mathbb{R}^{2}}|\nabla
u|^{2}dx-\frac{1}{e^{2s}}\int_{\mathbb{R}^{2}}F(e^{s}u(x))dx.$

###  4.1 The minimax approach

We will prove that $\tilde{J}$ on $S(a)\times\mathbb{R}$ possesses a kind of
mountain-pass geometrical structure.

######  Lemma 4.4.

Assume that $(f_{1})-(f_{2})$ and ($f_{3}$) hold and let $u\in S(a)$ be
arbitrary but fixed. Then we have:
(i) $|\nabla\mathcal{H}(u,s)|_{2}\rightarrow 0$ and
$J(\mathcal{H}(u,s))\rightarrow 0$ as $s\rightarrow-\infty$;
(ii) $|\nabla\mathcal{H}(u,s)|_{2}\rightarrow+\infty$ and
$J(\mathcal{H}(u,s))\rightarrow-\infty$ as $s\rightarrow+\infty$.

###### Proof.

By a straightforward calculation, it follows that

$\int_{\mathbb{R}^{2}}|\mathcal{H}(u,s)(x)|^{2}\,dx=a^{2},\quad\int_{\mathbb{R}^{2}}|\mathcal{H}(u,s)(x)|^{\xi}\,dx=e^{(\xi-2)s}\int_{\mathbb{R}^{2}}|u(x)|^{\xi}\,dx,\quad\forall\xi\geq
2,$ (4.9)

and

$\int_{\mathbb{R}^{2}}|\nabla\mathcal{H}(u,s)(x)|^{2}\,dx=e^{2s}\int_{\mathbb{R}^{2}}|\nabla
u|^{2}dx.$ (4.10)

From the above equalities, fixing $\xi>2$, we have

$|\nabla\mathcal{H}(u,s)|_{2}\rightarrow
0\quad\mbox{and}\quad|\mathcal{H}(u,s)|_{\xi}\to 0\quad\mbox{as}\quad
s\to-\infty.$ (4.11)

Thus, there are $s_{1}<0$ and $m\in(0,1)$ such that

$\|\mathcal{H}(u,s)\|^{2}\leq m,\quad\forall s\in(-\infty_{1},s_{1}].$

By (4.2),

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

where $\alpha>4\pi$ close to $4\pi$ and $q>2$ as in the last Lemma 4.3. Hence,

$|F(\mathcal{H}(u,s))|\leq\zeta|\mathcal{H}(u,s)|^{\tau+1}+C|\mathcal{H}(u,s)|^{q}(e^{\alpha|\mathcal{H}(u,s)|^{2}}-1),\quad\forall
s\in(-\infty,s_{1}].$

Using the Hölder’s inequality together with Lemma 4.1, there exists
$C=C(u,m)>0$ such that

$\int_{\mathbb{R}^{2}}(e^{\alpha|\mathcal{H}(u,s)|^{2}}-1)^{t}dx\leq C,$

and so,

$\int_{\mathbb{R}^{2}}|F(\mathcal{H}(u,s))|\,dx\leq\zeta\int_{\mathbb{R}^{2}}|\mathcal{H}(u,s)|^{\tau+1}\,dx+C_{1}\Big{(}\int_{\mathbb{R}^{2}}|\mathcal{H}(u,s)|^{qt^{\prime}}\,dx\Big{)}^{1/t^{\prime}},\quad\forall
s\in(-\infty,s_{1}],$

where $t^{\prime}=\frac{t}{t-1}$, and $t>1$ is close to 1. Now, by using
(4.11),

$\int_{\mathbb{R}^{2}}|F(\mathcal{H}(u,s))|\to 0\quad\mbox{as}\quad
s\to-\infty,$

from where it follows that

$J(\mathcal{H}(u,s))\rightarrow 0\quad\mbox{as}\quad s\rightarrow-\infty,$

showing $(i)$.

In order to show $(ii)$, note that by (4.10),

$|\nabla\mathcal{H}(u,s)|_{2}\rightarrow+\infty\quad\mbox{as}\quad
s\to+\infty.$

On the other hand, by $(f_{4})$,

$J(\mathcal{H}(u,s))\leq\frac{1}{2}|\nabla\mathcal{H}(u,s)|_{2}^{2}-\frac{\mu}{p}\int_{\mathbb{R}^{2}}|\mathcal{H}(u,s)|^{p}\,dx=e^{2s}\int_{\mathbb{R}^{2}}|\nabla
u|^{2}dx-\frac{\mu e^{(p-2)s}}{p}\int_{\mathbb{R}^{2}}|u(x)|^{p}\,dx.$

Since $p>4$, the last inequality yields

$J(\mathcal{H}(u,s))\rightarrow-\infty\,\,\mbox{as}\,\,s\rightarrow+\infty.$

∎

######  Lemma 4.5.

Assume that $(f_{1})-(f_{3})$ hold. Then there exists $K(a)>0$ small enough
such that

$0<\sup_{u\in A}J(u)<\inf_{u\in B}J(u)$

with

$A=\left\\{u\in S(a),\int_{\mathbb{R}^{2}}|\nabla u|^{2}dx\leq
K(a)\right\\}\quad\mbox{and}\quad B=\left\\{u\in
S(a),\int_{\mathbb{R}^{2}}|\nabla u|^{2}dx=2K(a)\right\\}.$

###### Proof.

We will need the following Gagliardo-Sobolev inequality: for any $\xi>2$,

$|u|_{\xi}\leq C(\xi,2)|\nabla u|_{2}^{\gamma}|u|_{2}^{1-\gamma},$

where $\gamma=2(\frac{1}{2}-\frac{1}{\xi})$. If we fix
$K(a)<\frac{1-a^{2}}{2}$, $|\nabla u|^{2}_{2}\leq K(a)$ and $|\nabla
v|^{2}_{2}=2K(a)$, the conditions $(f_{1})-(f_{2})$ combined with Lemma 4.1
ensure that

$\int_{\mathbb{R}^{2}}F(u)\,dx\leq\zeta|u|^{\tau+1}_{\tau+1}+C_{2}|u|^{q}_{qt^{\prime}}$

where $q>2$, $t^{\prime}=\frac{t}{t-1}>1$, $t$ closed to 1. Then, by the
Gagliardo-Sobolev inequality,

$\int_{\mathbb{R}^{2}}F(v)\,dx\leq C_{1}|\nabla v|_{2}^{\tau-1}+C_{2}|\nabla
v|^{(q-2/t^{\prime})}_{2}.$

From $(f_{3})$, $F(u)\geq 0$ for any $u\in H^{1}(\mathbb{R}^{2})$, then

$\displaystyle J(v)-J(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{2}}|\nabla
v|^{2}\,dx-\frac{1}{2}\int_{\mathbb{R}^{2}}|\nabla
u|^{2}\,dx-\int_{\mathbb{R}^{2}}F(v)\,dx+\int_{\mathbb{R}^{2}}F(u)\,dx$
$\displaystyle\geq$ $\displaystyle\frac{1}{2}\int_{\mathbb{R}^{2}}|\nabla
v|^{2}\,dx-\frac{1}{2}\int_{\mathbb{R}^{2}}|\nabla
u|^{2}\,dx-\int_{\mathbb{R}^{2}}F(v)\,dx,$

and so,

$J(v)-J(u)\geq\frac{1}{2}K(a)-C_{3}K(a)^{\frac{\tau-1}{2}}-C_{4}K(a)^{(\frac{q}{2}-\frac{1}{t^{\prime}})}.$

Since $\tau>3$ and $t^{\prime}>0$ with $\frac{q}{2}-\frac{1}{t^{\prime}}>1$,
decreasing $K(a)$ if necessary, it follows that

$\frac{1}{2}K(a)-C_{3}K(a)^{\frac{\tau-1}{2}}-C_{4}K(a)^{(\frac{q}{2}-\frac{1}{t^{\prime}})}>0,$

showing the desired result. ∎

As a byproduct of the last lemma is the following corollary.

######  Corollary 4.3.

There exists $K(a)>0$ small enough such that if $u\in S(a)$ and $|\nabla
u|^{2}_{2}\leq K(a)$, then $J(u)>0$.

###### Proof.

Arguing as in the last lemma,

$J(u)\geq\frac{1}{2}|\nabla u|_{2}^{2}-C_{1}|\nabla
u|_{2}^{\tau-1}-C_{2}|\nabla u|^{(q-\frac{2}{q^{*}})}_{2}>0,$

for $K(a)>0$ small enough. ∎

In what follows, we fix $u_{0}\in S(a)$ and apply Lemma 4.4 and Corollary 4.3
to get two numbers $s_{1}<0$ and $s_{2}>0$, of such way that the functions
$u_{1}=\mathcal{H}(u_{0},s_{1})$ and $u_{2}=\mathcal{H}(u_{0},s_{2})$ satisfy

$|\nabla u_{1}|^{2}_{2}<\frac{K(a)}{2},\,\,|\nabla
u_{2}|_{2}^{2}>2K(a),\,\,J(u_{1})>0\,\,\mbox{and}\,\,J(u_{2})<0.$

Now, following the ideas from Jeanjean [26], we fix the following mountain
pass level given by

$\gamma_{\mu}(a)=\inf_{h\in\Gamma}\max_{t\in[0,1]}J(h(t))$

where

$\Gamma=\left\\{h\in
C([0,1],S(a))\,:\,h(0)=u_{1}\,\,\mbox{and}\,\,h(1)=u_{2}\right\\}.$

From Lemma 4.5,

$\max_{t\in[0,1]}J(h(t))>\max\left\\{J(u_{1}),J(u_{2})\right\\}.$

######  Lemma 4.6.

There holds $\displaystyle\lim_{\mu\to+\infty}\gamma_{\mu}(a)=0$.

###### Proof.

In what follow we set the path
$h_{0}(t)=\mathcal{H}\big{(}u_{0},(1-t)s_{1}+ts_{2}\big{)}\in\Gamma$. Then, by
$(f_{4})$,

$\gamma_{\mu}(a)\leq\max_{t\in[0,1]}J(h_{0}(t))\leq\max_{r\geq
0}\left\\{\frac{r^{2}}{2}|\nabla
u_{0}|_{2}^{2}-\frac{\mu}{p}r^{p-2}|u_{0}|_{p}^{p}\right\\}$

and so,

$\gamma_{\mu}(a)\leq C_{2}\left(\frac{1}{\mu}\right)^{\frac{2}{p-4}}\to
0\quad\mbox{as}\quad\mu\to+\infty,$

for some $C_{2}>0$. Here, we have used the fact that $p>4$.

∎

Arguing as Section 2, in what follows $(u_{n})$ denotes the $(PS)$ sequence
associated with the level $\gamma_{\mu}(a)$, which satisfies:

$J(u_{n})\to\gamma_{\mu}(a),\,\,\mbox{as}\,\,n\to+\infty,$ (4.12) $-\Delta
u_{n}-f(u_{n})=\lambda_{n}u_{n}\
+o_{n}(1),\,\,\mbox{in}\,\,(H^{1}(\mathbb{R}^{2}))^{*},$ (4.13)

for some sequence $(\lambda_{n})\subset\mathbb{R}$, and

$Q(u_{n})=\int_{\mathbb{R}^{2}}|\nabla
u_{n}|^{2}dx+2\int_{\mathbb{R}^{2}}F(u_{n})dx-\int_{\mathbb{R}^{2}}f(u_{n})u_{n}dx\to
0,\,\,\mbox{as}\,\,n\to+\infty.$ (4.14)

Moreover, $(u_{n})$ is a bounded sequence, and so, the number $\lambda_{n}$
must satisfy the equality below

$\lambda_{n}=\frac{1}{a^{2}}\left\\{|\nabla
u_{n}|^{2}_{2}-\int_{\mathbb{R}^{2}}f(u_{n})u_{n}dx\right\\}+o_{n}(1).$ (4.15)

######  Lemma 4.7.

There holds

$\limsup_{n\to+\infty}\int_{\mathbb{R}^{2}}F(u_{n})\,dx\leq\frac{2}{\theta-4}\gamma_{\mu}(a).$

###### Proof.

Using the fact that $J(u_{n})=\gamma_{\mu}(a)+o_{n}(1)$ and
$Q(u_{n})=o_{n}(1)$, it follows that

$2{J}(u_{n})+Q(u_{n})=2\gamma_{\mu}(a)+o_{n}(1),$

and so,

$\displaystyle 2|\nabla
u_{n}|_{2}^{2}-\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx=2\gamma_{\mu}(a)+o_{n}(1).$
(4.16)

Using that $J(u_{n})=\gamma_{\mu}(a)+o_{n}(1)$, we get

$4\int_{\mathbb{R}^{2}}F(u_{n})dx+4\gamma_{\mu}(a)+o_{n}(1)-\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx=2\gamma_{\mu}(a)+o_{n}(1).$

Hence, by $(f_{3})$,

$2\gamma_{\mu}(a)+o_{n}(1)=\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx-4\int_{\mathbb{R}^{2}}F(u_{n})dx\geq(\theta-4)\int_{\mathbb{R}^{2}}F(u_{n})dx.$

Since $\theta>4$, we have

$\limsup_{n\to+\infty}\int_{\mathbb{R}^{2}}F(u_{n})\,dx\leq\frac{2}{\theta-4}\gamma_{\mu}(a).$

∎

######  Lemma 4.8.

The sequence $(u_{n})$ satisfies $\displaystyle\limsup_{n\to+\infty}|\nabla
u_{n}|_{2}^{2}\leq\frac{2(\theta-2)}{\theta-4}\gamma_{\mu}(a)$. Hence, there
exists $\mu^{*}>0$ such that

$\limsup_{n\to+\infty}|\nabla u_{n}|_{2}^{2}<1-a^{2},\,\,\text{for
any}\,\,\,\mu\geq\mu^{*}.$

###### Proof.

Since $J(u_{n})=\gamma_{\mu}(a)+o_{n}(1)$, we have

$\int_{\mathbb{R}^{2}}|\nabla
u_{n}|^{2}\,dx=2\gamma_{\mu}(a)+2\int_{\mathbb{R}^{2}}F(u_{n})\,dx+o_{n}(1).$

Thereby, by Lemma 4.7,

$\limsup_{n\to+\infty}|\nabla
u_{n}|_{2}^{2}\leq\frac{2(\theta-2)}{\theta-4}\gamma_{\mu}(a).$

∎

######  Lemma 4.9.

Fix $\mu\geq\mu^{*}$, where $\mu^{*}$ is given in Lemma 4.8. Then,
$(\lambda_{n})$ is a bounded sequence with

$\limsup_{n\to+\infty}|\lambda_{n}|\leq\frac{2\theta}{a^{2}(\theta-4)}\gamma_{\mu}(a)\quad\mbox{and}\quad\limsup_{n\to+\infty}\lambda_{n}=-\frac{2}{a^{2}}\liminf_{n\to+\infty}\int_{\mathbb{R}^{2}}F(u_{n})\,dx.$

###### Proof.

The boundedness of $(u_{n})$ yields that $(\lambda_{n})$ is bounded, because

$\lambda_{n}|u_{n}|_{2}^{2}=|\nabla
u_{n}|_{2}^{2}-\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx+o_{n}(1),$

and as $|u_{n}|_{2}^{2}=a^{2}$, we have

$\lambda_{n}a^{2}=|\nabla
u_{n}|_{2}^{2}-\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx+o_{n}(1).$

Hence,

$|\lambda_{n}|a^{2}\leq|\nabla
u_{n}|_{2}^{2}+\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx+o_{n}(1).$

The limit (4.14) together with Lemmas 4.7 and 4.8 ensures that
$(\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx)$ is bounded with

$\limsup_{n\to+\infty}\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx\leq\frac{2\theta}{\theta-4}\gamma_{\mu}(a).$

This is enough to conclude that $(\lambda_{n})$ is a bounded sequence with

$\limsup_{n\to+\infty}|\lambda_{n}|\leq\frac{2\theta}{a^{2}(\theta-4)}\gamma_{\mu}(a).$

In order to prove the second inequality, the equality

$\lambda_{n}a^{2}=|\nabla
u_{n}|_{2}^{2}-\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx+o_{n}(1)$

together with the limit (4.14) leads to

$\lambda_{n}a^{2}=-2\int_{\mathbb{R}^{2}}F(u_{n})\,dx+o_{n}(1),$

showing the desired result. ∎

Now, we restrict our study to the space $H_{rad}^{1}(\mathbb{R}^{2})$. For any
$\mu\geq\mu^{*}$, using Lemmas 4.3 and 4.8, Corollary 4.2, it follows that

$\lim_{n\to+\infty}\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx=\int_{\mathbb{R}^{2}}f(u)u\,dx,$

and

$\lim_{n\to+\infty}\int_{\mathbb{R}^{2}}F(u_{n})\,dx=\int_{\mathbb{R}^{2}}F(u)\,dx,$

where $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{2})$. The last limit
implies that $u\not=0$, because otherwise, Corollary 4.2 gives

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

and by Lemma 4.9,

$\limsup_{n\to+\infty}\lambda_{n}\leq 0.$

Since $(u_{n})$ is bounded in $H^{1}(\mathbb{R}^{2})$ and
$\displaystyle\limsup_{n\to+\infty}|\nabla u_{n}|_{2}^{2}<1-a^{2}$ if
$\mu\geq\mu^{*}$, Corollary 4.2 together with $(f_{1})-(f_{2})$ and the
equality below

$\lambda_{n}|u_{n}|_{2}^{2}=|\nabla
u_{n}|^{2}_{2}-\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx+o_{n}(1),$

lead to

$\lambda_{n}a^{2}=|\nabla u_{n}|_{2}^{2}+o_{n}(1).$ (4.17)

From this,

$0\geq\limsup_{n\to+\infty}\lambda_{n}a^{2}=\limsup_{n\to+\infty}|\nabla
u_{n}|^{2}_{2}\geq\liminf_{n\to+\infty}|\nabla u_{n}|^{2}_{2}\geq 0,$

then

$|\nabla u_{n}|^{2}_{2}\to 0,$

which is absurd, because $\gamma_{\mu}(a)>0$.

##  5 Proof of Theorem 1.2

The above analysis ensures that the weak limit $u$ of $(u_{n})$ is nontrivial.
Moreover, the equality

$\limsup_{n\to+\infty}\lambda_{n}=-\frac{2}{a^{2}}\liminf_{n\to+\infty}\int_{\mathbb{R}^{2}}F(u_{n})\,dx$

ensures that

$\limsup_{n\to+\infty}\lambda_{n}=-\frac{2}{a^{2}}\liminf_{n\to+\infty}\int_{\mathbb{R}^{2}}F(u)\,dx<0.$

From this, for some subsequence, still denoted by $(\lambda_{n})$, we can
assume that

$\lambda_{n}\to\lambda_{a}<0\quad\mbox{as}\quad n\to+\infty.$

Now, the equality (4.13) implies that

$-\Delta u-f(u)=\lambda_{a}u\quad\mbox{in}\quad\mathbb{R}^{2}.$ (5.1)

Thus,

$|\nabla u|_{2}^{2}-\lambda_{a}|u|_{2}^{2}=\int_{\mathbb{R}^{2}}f(u)u\,dx.$

On the other hand,

$|\nabla
u_{n}|_{2}^{2}-\lambda_{n}|u_{n}|_{2}^{2}=\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx+o_{n}(1),$

or yet,

$|\nabla
u_{n}|_{2}^{2}-\lambda_{a}|u_{n}|_{2}^{2}=\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx+o_{n}(1).$

Recalling that

$\lim_{n\to+\infty}\int_{\mathbb{R}^{2}}f(u_{n})u_{n}\,dx=\int_{\mathbb{R}^{2}}f(u)u\,dx,$

we derive that

$\lim_{n\to+\infty}(|\nabla u_{n}|_{2}^{2}-\lambda_{a}|u_{n}|_{2}^{2})=|\nabla
u|_{2}^{2}-\lambda_{a}|u|_{2}^{2}.$

Since $\lambda_{a}<0$, the last limit implies that

$u_{n}\to u\quad\mbox{in}\quad H^{1}(\mathbb{R}^{2}),$

implying that $|u|_{2}^{2}=a$. This establishes the desired result.

## References

  * [1] A. Adimurthi. Existence of Positive solutions of the semilinear Dirichlet problem with critical growth for the $N$-Laplacian. Ann. Sc. Norm. Super. Pisa 17 (1990), 393-413.
  * [2] T. Akahori, S. Ibrahim, H. Kikuchi, and H. Nawa. Existence of a ground state and scattering for a nonlinear Schrödinger equation with critical growth. Selecta Math. (N.S.) 19(2)(2013), 545-609.
  * [3] T. Akahori, S. Ibrahim, H. Kikuchi, and H. Nawa. Global dynamics above the ground state energy for the combined power type nonlinear Schrödinger equations with energy critical growth at low frequencies. arXiv.1510.08034, 2019.
  * [4] C. O. Alves. Multiplicity of solutions for a class of elliptic problem in $\mathbb{R}^{2}$ with Neumann conditions. J. Differential Equations 219 (2005), 20–39.
  * [5] C. O. Alves, J. M. B. do Ó and O. H. Miyagaki. On nonlinear perturbations of a periodic elliptic problem in $\mathbb{R}^{2}$ involving critical growth, Nonlinear Anal. 56 (2004), 781–791.
  * [6] C.O. Alves and G.M. Figueiredo.On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in $\mathbb{R}^{N}$. J. Differential Equations 219 (2009), 1288-1311.
  * [7] C.O. Alves and S.H.M. Soares. Nodal solutions for singularly perturbed equations with critical exponential growth. J. Differential Equations 234 (2007), 464-484.
  * [8] C. O. Alves, M. A. S. Souto and M. Montenegro. Existence of a ground state solution for a nonlinear scalar field equation with critical growth.  Calc. Var. Partial Differential Equations 43(3-4)(2012), 537-554.
  * [9] T. Bartsch, L. Jeanjean and N. Soave. Normalized solutions for a system of coupled cubic Schrödinger equations on $\mathbb{R}^{3}$. J. Math. Pures Appl. (9) 106(4)(2016), 583-614.
  * [10] T. Bartsch, R. Molle, M. Rizzi and G.Verzini. Normalized solutions of mass supercritical Schrödinger equations with potential. arXiv:2008.07431V1, 2020.
  * [11] T. Bartsch and N. Soave. Multiple normalized solutions for a competing system of Schrödinger equations. Calc. Var. Partial Differential Equations 58(1) (2019), art 22, pp.24.
  * [12] J. Bellazzini, L. Jeanjean and T. Luo. Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations. Proc. Lond. Math. Soc. (3), 107(2)(2013), 303-339.
  * [13] H. Brezis and L. Nirenberg. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.
  * [14] D. M. Cao. Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb{R}^{2}$. Comm. Partial Differential Equation 17 (1992),407–435.
  * [15] T. Cazenave,_Semilinear Schrödinger equations,_ Courant Lecture Notes in Mathematics 10 (New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003, 323 pp. ISBN: 0-8218-3399-5.
  * [16] X. Cheng, C.X. Miao and L.F. Zhao. Global well-posedness and scattering for nonlinear Schrodinger equations with combined nonlinearities in the radial case.  J. Differential Equations 261(6)(2016), 2881-2934.
  * [17] S. Cingolani and L. Jeanjean. Stationary waves with prescribed $L^{2}$-norm for the planar Schrödinger-Poisson system. SIAM J. Math. Anal. 51(4)(2019), 3533-3568.
  * [18] D.G. de Figueiredo, O.H. Miyagaki and B. Ruf. Elliptic equations in $\mathbb{R}^{2}$ with nonlinearities in the critical growth range. Calc. Var. Partial Differential Equations 3 (1995), 139-153.
  * [19] D.G. de Figueiredo, João Marcos do Ó and B. Ruf. On an inequality by N. Trudinger and J. Moser and related elliptic equations. Comm. Pure Appl. Math. 55 (2002), 1-18.
  * [20] J. M. B. do Ó and M.A.S. Souto. On a class of nonlinear Schrödinger equations in $\mathbb{R}^{2}$ involving critical growth. J. Differential Equations 174 (2001), 289-311.
  * [21] J. M. B. do Ó and B. Ruf. On a Schrödinger equation with periodic potential and critical growth in $\mathbb{R}^{2}$. Nonlinear Differential Equations Appl. 13 (2006), 167-192.
  * [22] J. M. B. do Ó. Quasilinear elliptic equations with exponential nonlinearities. Comm. Appl. Nonlin. Anal. 2 (1995), 63-72.
  * [23] J. M. B. do Ó, M. de Souza, E. de Medeiros and U. Severo. An improvement for the Trudinger-Moser inequality and applications. J. Differential Equations 256 (2014), 1317-1349.
  * [24] J. Garcia Azorero and I. Peral Alonso. Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term. Trans. Amer. Math. Soc. 2 (1991), 877-895.
  * [25] T.X. Gou and L. Jeanjean. Multiple positive normalized solutions for nonlinear Schrödinger systems.  Nonlinearity 31(5)(2018), 2319-2345.
  * [26] L. Jeanjean. Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. 28 (1997), 1633-1659.
  * [27] L. Jeanjean, J. Jendrej, T. T. Le and N. Visciglia. Orbital stability of ground states for a Sobolev critical Schrödinger equation. arXiv.2008.12084, 2020.
  * [28] L. Jeanjean and S.S. Lu. Nonradial normalized solutions for nonlinear scalar field equations. Nonlinearity 32(12)(2019), 4942- 4966\.
  * [29] L. Jeanjean and T. T. Le. Multiple normalized solutions for a Sobolev critical Schrödinger equations. arXiv:2011.02945v1, 2020.
  * [30] P.L. Lions. The concentration-compactness principle in the calculus of variations. The locally compact case, part 2. Anal. Inst. H. Poincaré, Sect. C 1 (1984), 223-283.
  * [31] J. Mederski and J. Schino. Least energy solutions to a cooperative system of Schrödinger equations with prescribed $L^{2}$-bounds: At least $L^{2}$-crtical growth.  arXiv:2101.02611v1, 2021
  * [32] C. X. Miao, G. X. Xu and L. F. Zhao. The dynamics of the 3D radial NLS with the combined terms. Comm. Math. Phys. 318(3)(2013), 767-808.
  * [33] J. Moser. A sharp form of an inequality by N. Trudinger. Ind. Univ. Math. J. (20) (1971), 1077–1092.
  * [34] B. Noris, H. Tavares and G. Verzini. Normalized solutions for nonlinear Schrödinger systems on bounded domains. Nonlinearity 32(3)(2019), 10441072.
  * [35] R.S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), 19-30.
  * [36] N. Soave. Normalized ground states for the NLS equation with combined nonlinearities. J. Differential Equations 269(9)(2020), 6941- 6987\.
  * [37] N. Soave. Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case. J. Funct. Anal. 279(6)(2020), 108610, 43.
  * [38] A. Stefanov. On the normalized ground states of second order PDE’s with mixed power non-linearities. Comm. Math. Phys. 369(3)(2019), 929-971.
  * [39] T. Tao, M. Visan and X. Zhang. The nonlinear Schrödinger equation with combined power-type nonlinearities. Comm. Partial Differential Equations 32(7-9)(2007), 1281-1343.
  * [40] N. S. Trudinger. On imbedding into Orlicz spaces and some application. J. Math Mech. 17 (1967), 473–484.
  * [41] W. Wang, Q. Li, J. Zhou and Y. Li. Normalized solutions for p-Laplacian equations with a $L^{2}$-supercritical growth. Ann. Funct. Anal. 12(1) (2020), doi:10.1007/s43034-020-00101-w.
  * [42] M. Willem. Minimax Theorems, Birkhauser, 1996.

Claudianor O. Alves
Unidade Acadêmica de Matemática
Universidade Federal de Campina Grande
Campina Grande, PB, CEP:58429-900, Brazil
<EMAIL_ADDRESS>
and
Chao Ji
Department of Mathematics
East China University of Science and Technology
Shanghai 200237, PR China
<EMAIL_ADDRESS>
and
Olímpio Hiroshi, Miyagaki
Departamento de Matemática
Universidade Federal de São Carlos
São Carlos, SP, CEP:13565-905, Brazil
<EMAIL_ADDRESS>


* Hillar and Sullivant (2012) Christopher J. Hillar and Seth Sullivant. 2012.  Finite Gröbner bases in infinite dimensional polynomial rings and applications.  _Advances in Mathematics_ 229, 1 (2012), 1–25.
  * Hodges (1997) Wilfrid Hodges. 1997\.  _A shorter model theory_.  Cambridge University Press.
  * Kriz and Sgall (1991) I. Kriz and J. Sgall. 1991\.  Well-quasi-ordering depends on labels.  _Acta Scientarium Mathematicarum_ 55 (1991), 55–69.
  * Lasota (2016) Slawomir Lasota. 2016\.  Decidability Border for Petri Nets with Data: WQO Dichotomy Conjecture. In _Proc. Petri Nets 2016_ _(Lecture Notes in Computer Science, Vol. 9698)_. Springer, 20–36.
  * Lazic et al. (2007) Ranko Lazic, Thomas Christopher Newcomb, Joël Ouaknine, A. W. Roscoe, and James Worrell. 2007.  Nets with Tokens Which Carry Data. In _Proc. ICATPN 2007_ _(Lecture Notes in Computer Science, Vol. 4546)_. Springer, 301–320.
  * Mayr and Meyer (1982) Ernst W. Mayr and Albert R. Meyer. 1982.  The Complexity of the Word Problems for Commutative Semigroups and Polynomial Ideals.  _Advances in Mathematics_ 46 (1982), 305–329.
  * Nash-Williams (1965) C. St. J. A. Nash-Williams. 1965\.  On well-quasi-ordering transfinite sequences.  _Mathematical Proceedings of the Cambridge Philosophical Society_ 61 (1965), 33–39.  Issue 1.
  * Pouzet (2020) Maurice Pouzet. 2020\.  Well quasi ordering and embeddability of relational structures. In _Proc. ALGOS 2020_.  https://hal.science/hal-02918958/document
  * Stríbrná (1997) Jitka Stríbrná. 1997\.  Decidability of strong bisimulation of basic parallel processes using Hilbert’s basis theorem. In _Proc. Infinity 1997_ _(Electronic Notes in Theoretical Computer Science, Vol. 9)_ , Faron Moller (Ed.). Elsevier, 44.