# Normalized solutions for a nonlinear Dirac equation

Vittorio Coti Zelati<EMAIL_ADDRESS>Dipartimento di Matematica
Pura e Applicata “R. Caccioppoli”
Università di Napoli “Federico II”
via Cintia, M.S. Angelo
80126 Napoli (NA), Italy  and  Margherita Nolasco<EMAIL_ADDRESS>Dipartimento di Ingegneria e Scienze dell’informazione e Matematica
Università dell’Aquila
via Vetoio, Loc. Coppito
67010 L’Aquila (AQ) Italia

###### Abstract.

We prove the existence of a normalized, stationary solution
$\psi\colon\mathbb{R}^{3}\to\mathbb{C}^{4}$ with frequency $\omega>0$ of the
nonlinear Dirac equation. The result covers the case in which the nonlinearity
is the gradient of a function of the form

$F(\Psi)=a\lvert(\Psi,\gamma^{0}\Psi)\rvert^{\frac{\alpha}{2}}+b\lvert(\Psi,\gamma^{1}\gamma^{2}\gamma^{3}\Psi)\rvert^{\frac{\alpha}{2}}$

with $\alpha\in(2,\frac{8}{3}]$, $b\geq 0$ and $a>0$ sufficiently small. Here
$\gamma^{i}$, $i=0,\ldots,3$ are the $4\times 4$ Dirac’s matrices.

We find the solution as a critical point of a suitable functional restricted
to the unit sphere in $L^{2}$, and $\omega$ turns out to be the corresponding
Lagrange multiplier.

###### Key words and phrases:

Nonlinear Dirac equation, Critical point theory, Min-Max methods, Normalized
solutions

###### 1991 Mathematics Subject Classification:

35Q40 35P30 47J10 49J35

##  1\. Introduction

The nonlinear Dirac equation is a simplified model describing self-interacting
fermions (electrons). The equation we consider is the following:

(1.1) $(-i\gamma^{\mu}\partial_{\mu}+m)\Psi=\gamma\beta\nabla
F(\Psi)\qquad\text{in }\mathbb{R}\times\mathbb{R}^{3}$

where $\Psi\colon\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{C}^{4}$, $m>0$ is
the mass of the electron, $\gamma^{\mu}$ the $4\times 4$ Dirac matrices

$\gamma^{0}=\beta=\begin{pmatrix}I&0\\\
0&-I\end{pmatrix}\qquad\gamma^{k}=\beta\alpha^{k}\begin{pmatrix}0&\sigma^{k}\\\
-\sigma^{k}&0\end{pmatrix},\quad k=1,\ldots,3,$

$\sigma^{k}$ the $2\times 2$ Pauli matrices

$\sigma^{1}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},\qquad\sigma^{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\sigma^{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$

and $\gamma\in\mathbb{R}$ is a constant, $\gamma>0$. The nonlinear self-
interaction of the electron is described by the function $\gamma\nabla
F(\Psi)$. One can find a discussion of the motivation and history of the
models leading to the nonlinear Dirac equation in [21] where in particular the
model nonlinearity

$F(\Psi)=\lvert(\Psi,\beta\Psi)\rvert^{\frac{\alpha}{2}}+b\lvert(\Psi,\gamma^{1}\gamma^{2}\gamma^{3}\Psi)\rvert^{\frac{\alpha}{2}}$

is discussed with $\alpha=4$ (in case $b=0$ this is the so called Soler
model).

We prove existence of localized stationary solutions of fixed $L^{2}$ norm for
a class of nonlinearities which includes those of the form above provided
$\alpha\in(2,\frac{8}{3}]$ (see below the precise assumptions).

A normalized stationary solution is a solution of the form

$\Psi(t,x)=\operatorname{e}^{-i\omega
t}\psi(x),\qquad\int\lvert\Psi\rvert^{2}=1.$

Assuming $F(\operatorname{e}^{i\theta}\phi)=F(\phi)$ for all
$\theta\in\mathbb{R}$, $\phi\in\mathbb{C}^{4}$ we have that $\psi$ solves

(1.2)
$\begin{cases}\left(-i\boldsymbol{\alpha}\cdot\nabla+m\beta-\omega\right)\psi=\gamma\nabla
F(\psi)&\\\ \int\lvert\psi\rvert^{2}=1&\end{cases}.$

We will find solutions for problem (1.2) as critical points of a suitable
functional under the constrain of fixed $L^{2}$ norm. The frequency $\omega$
will arise as the Lagrange multiplier associated to the constrain.

The use of variational methods in the study of the existence of stationary
solutions of nonlinear equations and systems involving the Dirac operator goes
back to the pioneering work of Esteban and Séré in [12], and several later
developments, see for example [1, 8]. Also related problems like the Dirac-
Fock, Maxwell-Dirac and Klein-Gordon-Dirac problems have been tackled using
variational methods: see the review paper [11] where many of these results are
presented. In all these works the frequency $\omega$ is assigned, and the
typical result is that for any $\omega\in(0,m)$ there is a solution of the
equation.

The search of normalized solutions, in particular such that
$\int\lvert\psi\rvert^{2}=1$, is quite natural since
$\lvert\Psi(t,x)\rvert^{2}$ can be interpreted as the probability density of
the position of the electron at point $x$ and time $t$. The problem is also an
interesting mathematical one, and a lot of work has been devoted to finding
normalized solutions for nonlinear Schrödinger equations, see [22, 5, 2, 17,
18] and more recently [15, 16, 14, 19, 10]. The main approach to finding
normalized solutions for this class of problems is by minimization (usually a
nontrivial one) or other min-max procedures of a suitable functional on the
unit sphere of $L^{2}$.

Very little is known about the existence via variational methods of normalized
solutions for equations involving the Dirac operator. In this case the
functional associated is strongly indefinite and a direct minimization on the
sphere in $L^{2}$ is not possible. On this problem we recall here the paper by
Esteban and Séré [13] dealing with the Dirac-Fock equation and more recently
the paper [6], dealing with the problem of an electron interacting with a
nuclei and its own electric field, [20] where the existence of a normalized
solutions for the Dirac-Maxwell system is proved, [7] dealing with a Klein-
Gordon-Dirac system and [9], where Ding, Yu and Zhao consider a nonlinearity
of the form $F(\phi)=K(x)\lvert\phi\rvert^{\alpha}$,
$\alpha\in(2,\frac{8}{3})$ with $K(x)\to 0$ as $\lvert x\rvert\to+\infty$. Let
us also recall the paper [4] where normalized solutions for a strongly
indefinite problem – not involving the Dirac operator – are found.

Let us now state our assumptions: we will assume that $F\in
C^{2}(\mathbb{C}^{4},\mathbb{R})$ is such that the following holds for some
$\alpha\in(2,\frac{8}{3}]$, $R>0$ and $\bar{\gamma}>0$

(H1) $\displaystyle\nabla F(0)=D^{2}F(0)=0,$ (H2) $\displaystyle\lvert
D^{2}F(\phi)\rvert\leq\lvert\phi\rvert^{\alpha-2}\quad\text{ for all
}\lvert\phi\rvert\geq R,$ (H3) $\displaystyle(\nabla F(\phi),\phi)\geq\alpha
F(\phi)\geq 0\quad\text{ for all }\phi\in\mathbb{C}^{4},$ (H4)
$\displaystyle(D^{2}F(\phi)\xi,\xi)\geq 0\quad\text{ for all
}\phi\in\mathbb{C}^{4},\ \xi\in\mathbb{C}^{4},$

and also that, for some $\rho>0$ and $\nu\in(\frac{\alpha}{2},\frac{3}{2})$

(H5) $\displaystyle
F(\phi)\geq\bar{\gamma}\lvert(\phi,\beta\phi)\rvert^{\frac{\alpha}{2}}\quad\quad\text{
for all }\phi\in\mathbb{C}^{4},\ \lvert\phi\rvert<\rho,$ (H6)
$\displaystyle\lvert\nabla F(\phi)\rvert\leq\lvert\phi\rvert^{\nu}\quad\text{
for all }\phi\in\mathbb{C}^{4},\ \lvert\phi\rvert<\rho.$

We also assume that $\xi>3$ is such that for all $\zeta>0$ there is a
$C_{\zeta}$ such that for all $\phi\in\mathbb{C}^{4}$

(H7) $\lvert\nabla
F(\phi)\rvert\leq\left(\zeta+C_{\zeta}F(\phi)^{1/\xi}\right)\lvert\phi\rvert.$

######  Remark 1.3.

Let us remark that the assumptions (H1), (H2), (H3), (H7) are the same as in
the paper [12]. The restriction on the exponent $\alpha$ is quite natural when
looking for normalized solutions. The assumption (H5) is similar to assumption
(H9) of the same paper [12]. The convexity assumption (H4) mighty be relaxed.

Let us also remark that if (H6) holds for some $\bar{\nu}>\frac{\alpha}{2}$
than it holds for all $\nu\in(\frac{\alpha}{2},\bar{\nu})$. also holds for all
$\nu\in(\bar{\nu},\frac{3}{2})$. In particular holds if

$\lvert D^{2}F(\phi)\rvert\leq\lvert\phi\rvert^{\alpha-2}\quad\text{ for all
}\phi\in\mathbb{C}^{4}.$

######  Remark 1.4.

Follows from (H1) and (H2) that for all $\epsilon>0$ there are
$\mu_{\epsilon}$ and $\delta_{\epsilon}$ such that, for all $\phi$,
$\varphi\in\mathbb{C}^{4}$

(1.5) $\displaystyle\lvert\nabla
F(\phi)\rvert\leq\epsilon\lvert\phi\rvert+\mu_{\epsilon}\lvert\phi\rvert^{\alpha-1}$
(1.6) $\displaystyle 0\leq
F(\phi)\leq\frac{\epsilon}{2}\lvert\phi\rvert^{2}+\frac{\mu_{\epsilon}}{\alpha}\lvert\phi\rvert^{\alpha}$
(1.7) $\displaystyle\lvert\nabla F(\phi+\varphi)-\nabla
F(\phi)\rvert\leq\left(\epsilon+\delta_{\epsilon}(\lvert\phi\rvert^{\alpha-2}+\lvert\varphi\rvert^{\alpha-2})\right)\lvert\varphi\rvert$

We will write $\mu=\mu_{1}$, $\delta=\delta_{1}$ and use the fact that, for
all $\phi$, $\varphi\in\mathbb{C}^{4}$

(1.8) $\displaystyle\lvert\nabla
F(\phi)\rvert\leq\lvert\phi\rvert+\mu\lvert\phi\rvert^{\alpha-1}$ (1.9)
$\displaystyle 0\leq
F(\phi)\leq\frac{1}{2}\lvert\phi\rvert^{2}+\frac{\mu}{\alpha}\lvert\phi\rvert^{\alpha}$
(1.10) $\displaystyle\lvert\nabla F(\phi+\varphi)-\nabla
F(\phi)\rvert\leq\left(1+\delta(\lvert\phi\rvert^{\alpha-2}+\lvert\varphi\rvert^{\alpha-2})\right)\lvert\varphi\rvert.$

Our final assumptions give the size of the coefficient $\gamma$ in front of
the nonlinear term $\nabla F$. We let that $\gamma_{0}>0$ be such that

(1.11)
$\displaystyle\gamma_{0}\left(S_{2}^{2}+\mu\lambda^{\frac{\alpha-2}{2}}S_{3}^{3(\alpha-2)}S_{2}^{{3\alpha-8}}\right)<\frac{1}{16},$
(1.12)
$\displaystyle\gamma_{0}\left(S_{2}^{2}+\mu\lambda^{\frac{\alpha-2}{2}}S^{2}_{\frac{4}{4-\alpha}}\right)<\frac{1}{4}$

where $m>0$ is the mass of our electron, $\mu$ is the constant in (1.8),
$S_{q}$ is the constant relative to the Sobolev embedding of $H^{1/2}$ into
$L^{q}$ (with the norm (2.1), equivalent to the usual one) and $\alpha$ is the
exponent in (H2).

Our result is

######  Theorem 1.13.

Let $m>0$, assume $F$ satisfies (H1)–(H6), and let $\gamma\in(0,\gamma_{0}]$.

Then there is $\omega\in(0,m)$ and $\psi\in
H^{1/2}(\mathbb{R}^{3},\mathbb{C}^{4})$ solutions of problem (1.2).

We will find such a solution as a critical point of the functional

$I(\psi)=\frac{1}{2}\int_{\mathbb{R}^{3}}(H\psi,\psi)-\gamma\int_{\mathbb{R}^{3}}F(\psi)$

restricted on the manifold $\left|\psi\right|_{2}^{2}=1$. Here

$H=-i\boldsymbol{\alpha}\cdot\nabla+m\beta.$

The functional $I$ is strongly indefinite, and, following the method
introduced in [6, 20, 7], the solution will be found via a $\min$-$\max$
procedure consisting in minimizing the supremum of $I$ over subspaces of
dimension $1$ in the positive energy subspace of the linear operator $H$, see
proposition 3.15.

##  2\. Notation and background results

We denote with $\lvert u\rvert^{p}_{p}=\int_{\mathbb{R}^{3}}\lvert
u(x)\rvert^{p}$ the norm in $L^{p}(\mathbb{R}^{3},\mathbb{C}^{4})$ and with
$(u\,|\,v)=\int_{\mathbb{R}^{3}}(u(x),v(x))$ the scalar product in
$L^{2}(\mathbb{R}^{3},\mathbb{C}^{4})$, where
$(\xi,\eta)=\sum_{i=1}^{4}\bar{\xi}_{i}\eta_{i}$ and
$\lvert\xi\rvert^{2}=(\xi,\xi)$ are the scalar product and the norm in
$\mathbb{C}^{4}$. With $\Re z$ we denote the real part of $z\in\mathbb{C}$.

We will work in the Hilbert space $X=H^{1/2}(\mathbb{R}^{3},\mathbb{C}^{4})$
with scalar product

$\langle\psi_{1}\,|\,\psi_{2}\rangle=\int_{\mathbb{R}^{3}}\sqrt{\lvert\xi\rvert^{2}+m^{2}}\,(\hat{\psi}_{1}(\xi),\hat{\psi}_{2}(\xi))\,d\xi$

and corresponding norm

(2.1)
$\lVert\psi\rVert^{2}=\int_{\mathbb{R}^{3}}\sqrt{\lvert\xi\rvert^{2}+m^{2}}\,\lvert\hat{\psi}(\xi)\rvert^{2}\,d\xi.$

where $\hat{\psi}(\xi)=\mathcal{F}\psi(\xi)$ is the Fourier transform of
$\psi\in H^{1/2}(\mathbb{R}^{3},\mathbb{C}^{4})$. The norm in (2.1) is
equivalent to the usual one (given by (2.1) with $m=1$).

Let us also recall some properties (see [23]) of the Dirac operator

$H=-i\boldsymbol{\alpha}\cdot\nabla+m\beta.$

$H$ is a first order, self-adjoint operator on
$H^{1}(\mathbb{R}^{3},\mathbb{C}^{4})$ with purely absolutely continuous
spectrum given by

$\sigma(H)=(-\infty,-m]\cup[m,+\infty).$

One can define orthogonal projectors $\Lambda_{\pm}$ on the positive and
negative part of the spectrum of $H$. This projections are such that

$H\Lambda_{\pm}=\Lambda_{\pm}H=\pm\sqrt{-\Delta+m^{2}}\Lambda_{\pm}=\pm\Lambda_{\pm}\sqrt{-\Delta+m^{2}}$

and

$\langle\Lambda_{+}\psi\,|\,\Lambda_{-}\eta\rangle=(\Lambda_{+}\psi\,|\,\Lambda_{-}\eta)=0.$

so that

$\displaystyle\int(\psi(x),H\psi(x))\,dx$
$\displaystyle=\int(\Lambda_{+}\psi(x),\Lambda_{+}H\psi(x))\,dx+\int(\Lambda_{-}\psi(x),\Lambda_{-}H\psi(x))\,dx$
$\displaystyle=\left|(-\Delta+m^{2})^{1/4}\Lambda_{+}\psi\right|_{2}^{2}-\left|(-\Delta+m^{2})^{1/4}\Lambda_{-}\psi\right|_{2}^{2}$
$\displaystyle=\lVert\Lambda_{+}\psi\rVert^{2}-\lVert\Lambda_{-}\psi\rVert^{2}.$

We also let $X_{\pm}=\Lambda_{\pm}X$ and, for $\lambda\in(0,1]$,
$\Sigma^{\lambda}=\bigl{\\{}\,\psi\in
X\bigm{|}\left|\psi\right|_{2}^{2}=\lambda\,\bigr{\\}}$ and
$\Sigma_{\pm}^{\lambda}=\bigl{\\{}\,\psi\in
X_{\pm}\bigm{|}\left|\psi\right|_{2}^{2}=\lambda\,\bigr{\\}}$,
$\Sigma_{\pm}=\Sigma_{\pm}^{1}$.

With $S_{p}$ we will denote the best constant for the Sobolev embedding of
$H^{1/2}(\mathbb{R}^{3},\mathbb{C}^{4})$ (with norm given by (2.1)) in
$L^{p}(\mathbb{R}^{3};\mathbb{C}^{4})$ for $2\leq p\leq 3$:

$\left|\psi\right|_{p}\leq S_{p}\lVert\psi\rVert.$

Let us remark that, with our choice of the norm in $X$, we have that
$m\left|\psi\right|_{2}^{2}\leq\lVert\pi\rVert$ and $s_{2}\leq\frac{1}{m}$.

##  3\. Maximization

Let $I\colon H^{1/2}(\mathbb{R}^{3},\mathbb{C}^{4})\to\mathbb{R}$

$I(\psi)=\frac{1}{2}\lVert\Lambda_{+}\psi\rVert^{2}-\frac{1}{2}\lVert\Lambda_{-}\psi\rVert^{2}-\gamma\int_{\mathbb{R}^{3}}F(\psi).$

We denote, for $\lambda\in(0,1]$

$B_{\lambda}=\bigl{\\{}\,\eta\in
X_{-}\bigm{|}\left|\eta\right|_{2}^{2}<\lambda\,\bigr{\\}}.$

For $\eta\in B_{\lambda}$ and $w\in\Sigma_{+}$ we let

$a(\eta)=\sqrt{\lambda-\left|\eta\right|_{2}^{2}}\qquad\text{ and
}\qquad\psi=a(\eta)w+\eta\in\Sigma^{\lambda}.$

We will look, given $w\in\Sigma_{+}$, for a maximizer of the functional
$J_{w}^{\lambda}$ defined on $B_{\lambda}$

$J_{w}^{\lambda}(\eta)=I(a(\eta)w+\eta)=\frac{1}{2}\lVert
a(\eta)w\rVert^{2}-\frac{1}{2}\lVert\eta\rVert^{2}-\gamma\int_{\mathbb{R}^{3}}F(\psi).$

Since $da(\eta)[\xi]=-a(\eta)^{-1}\Re(\eta\,|\,\xi)$, the derivative of
$J_{w}^{\lambda}$ is given, for all $\xi\in X_{-}$, by

(3.1)
$dJ_{w}^{\lambda}(\eta)[\xi]=dI(a(\eta)w+\eta)[da(\eta)[\xi]w+\xi]=dI(\psi)[h_{\xi}]\\\
=\Re\langle
a(\eta)w\,|\,da(\eta)[\xi]w\rangle-\Re\langle\eta\,|\,\xi\rangle-\gamma\int_{\mathbb{R}^{3}}(\nabla
F(\psi),h_{\xi})\\\ =-\Re(\eta\,|\,\xi)\lVert
w\rVert^{2}-\Re\langle\eta\,|\,\xi\rangle-\gamma\int_{\mathbb{R}^{3}}(\nabla
F(\psi),h_{\xi})$

(here $h_{\xi}=da(\eta)[\xi]w+\xi$) and, in particular

$dJ_{w}^{\lambda}(\eta)[\eta]=-\left|\eta\right|_{2}^{2}\lVert
w\rVert^{2}-\lVert\eta\rVert^{2}-\gamma\int_{\mathbb{R}^{3}}(\nabla
F(\psi),h_{\eta}).$

######  Lemma 3.2.

For all $w\in\Sigma_{+}$ and $\eta\in B_{\lambda}$ we have

(3.3) $\lVert\eta\rVert^{2}\leq a(\eta)^{2}\lVert
w\rVert^{2}-2J_{w}^{\lambda}(\eta),$

and for all $\eta\in B_{\lambda}$ such that $J_{w}^{\lambda}(\eta)\geq 0$ we
have that

(3.4) $\frac{1}{a(\eta)}\left|\int(\nabla F(\psi),w)\right|\leq
C_{\alpha,\lambda}\lVert w\rVert^{2},$

where

$C_{\lambda,\alpha}=4\left(S_{2}^{2}+\mu\lambda^{\frac{\alpha-2}{2}}S_{3}^{3(\alpha-2)}S_{2}^{{3\alpha-8}}\right).$

Moreover, if $J_{w}^{\lambda}(\eta)\geq 0$,
$\left|\eta\right|_{2}^{2}\geq\frac{\lambda}{2}$ we have that

(3.5) $dJ_{w}^{\lambda}(\eta)[\eta]<-\lVert\eta\rVert^{2}<0.$

###### Proof.

We have that for $\eta\in B_{\lambda}$ and $\psi=a(\eta)w+\eta$

$\frac{1}{2}\lVert\eta\rVert^{2}\leq\frac{1}{2}\lVert\eta\rVert^{2}+\gamma\int
F(\psi)=\frac{1}{2}\lVert a(\eta)w\rVert^{2}-J_{w}^{\lambda}(\eta)$

and (3.3) follows.

Let us now assume that $J_{w}^{\lambda}(\eta)\geq 0$, so that
$\lVert\eta\rVert^{2}\leq a(\eta)^{2}\lVert w\rVert^{2}$.

Since

$a(\eta)-da(\eta)[\eta]=a(\eta)+a(\eta)^{-1}\left|\eta\right|_{2}^{2}=\lambda
a(\eta)^{-1}$

we have that

$dJ_{w}^{\lambda}(\eta)[\eta]=-\left|\eta\right|_{2}^{2}\lVert
w\rVert^{2}-\lVert\eta\rVert^{2}-\gamma\int(\nabla
F(\psi),\psi)+\gamma\frac{\lambda}{a(\eta)^{2}}\int(\nabla F(\psi),a(\eta)w).$

Let’s estimate the last term. Using (1.8) we have

$\left|\int(\nabla
F(\psi),a(\eta)w)\right|\leq\int\left(\lvert\psi\rvert+\mu\lvert\psi\rvert^{\alpha-1}\right)\lvert
a(\eta)w\rvert\\\
\leq\left|\psi\right|_{2}\left|a(\eta)w\right|_{2}+\mu\left|\psi\right|_{\alpha}^{\alpha-1}\left|a(\eta)w\right|_{\alpha}.$

Since $\left|\psi\right|_{2}^{2}=\lambda$ and
$\left|a(\eta)w\right|_{2}^{2}\leq\lambda$ we deduce that

$\left|\psi\right|_{\alpha}^{\alpha}=\int\lvert\psi\rvert^{\alpha}\leq\int\lvert\psi\rvert^{3\alpha-6}\lvert\psi\rvert^{6-2\alpha}\leq\left|\psi\right|_{3}^{3\alpha-6}\left|\psi\right|_{2}^{6-2\alpha}=\lambda^{3-\alpha}\left|\psi\right|_{3}^{3\alpha-6}$

and

$\left|a(\eta)w\right|_{\alpha}^{\alpha}\leq\lambda^{3-\alpha}\left|a(\eta)w\right|_{3}^{3\alpha-6}$

so that

$\displaystyle\left|\int(\nabla
F(\psi),a(\eta)w)\right|\leq\left|\psi\right|_{2}\left|a(\eta)w\right|_{2}+\mu\lambda^{(3-\alpha)}\left|\psi\right|_{3}^{\frac{3(\alpha-2)(\alpha-1)}{\alpha}}\left|a(\eta)w\right|_{3}^{\frac{3(\alpha-2)}{\alpha}}$
$\displaystyle\qquad\leq S_{2}^{2}\lVert\psi\rVert\lVert a(\eta)w\rVert+\mu
S_{3}^{3(\alpha-2)}\lambda^{(3-\alpha)}\lVert\psi\rVert^{\frac{3(\alpha-2)(\alpha-1)}{\alpha}}\lVert
a(\eta)w\rVert^{\frac{3(\alpha-2)}{\alpha}}$ $\displaystyle\qquad\leq
S_{2}^{2}\lVert\psi\rVert^{2}+\mu
S_{3}^{3(\alpha-2)}\lambda^{(3-\alpha)}\lVert\psi\rVert^{3(\alpha-2)}.$

Since $\lVert\psi\rVert^{2}\geq
S_{2}^{2}\left|\psi\right|_{2}^{2}=S_{2}^{2}\lambda$ and $3(\alpha-2)\in(0,2]$
for all $\alpha\in(2,\frac{8}{3}]$ we have that

$\lVert\psi\rVert^{3(\alpha-2)}\leq(S_{2}\sqrt{\lambda})^{{3\alpha-8}}\lVert\psi\rVert^{2}\leq
4(S_{2}\sqrt{\lambda})^{{3\alpha-8}}a(\eta)^{2}\lVert w\rVert^{2}$

and

$\frac{\lambda}{a(\eta)^{2}}\left|\int(\nabla F(\psi),a(\eta)w)\right|\leq
4\lambda\left(S_{2}^{2}+\mu\lambda^{\frac{\alpha-2}{2}}S_{3}^{3(\alpha-2)}S_{2}^{{3\alpha-8}}\right)\lVert
w\rVert^{2}=\lambda C_{\alpha,\lambda}\lVert w\rVert^{2}.$

We finally deduce that

$dJ_{w}^{\lambda}(\eta)[\eta]\leq\left(-\left|\eta\right|_{2}^{2}+\gamma\lambda
C_{\alpha,\lambda}\right)\lVert
w\rVert^{2}-\lVert\eta\rVert^{2}-\gamma\int(\nabla
F(\psi),\psi)<-\lVert\eta\rVert^{2}$

if $\left|\eta\right|_{2}^{2}>\frac{\lambda}{2}$ and since by (1.11) $\gamma$
is such that $\gamma C_{\alpha,\lambda}<\frac{1}{4}$ for all
$\lambda\in(0,1]$. ∎

######  Remark 3.6.

It follows from the Lemma 3.2 that if $\eta_{n}$ is a Palais-Smale sequence
for $J_{w}^{\lambda}$ such that $J_{w}^{\lambda}(\eta_{n})\geq 0$, then
$\left|\eta_{n}\right|_{2}^{2}<\frac{\lambda}{2}$ for all $n\in\mathbb{N}$
large enough.

######  Lemma 3.7.

Let $\eta_{n}\in B_{\lambda}$ be a Palais-Smale sequence for
$J_{w}^{\lambda}$, that is

$J_{w}^{\lambda}(\eta_{n})\to c\geq 0,\qquad dJ_{w}^{\lambda}(\eta_{n})\to 0.$

Then $\eta_{n}$ converges, up to a subsequence, to a critical point $\eta$ of
$J_{w}^{\lambda}$.

###### Proof.

Follows form the Lemma 3.2 and Remark 3.6 that
$\left|\eta_{n}\right|_{2}^{2}<\frac{\lambda}{2}$ and that
$\lVert\eta_{n}\rVert\leq a(\eta_{n})\lVert w\rVert\leq\sqrt{\lambda}\lVert
w\rVert$ is bounded, hence $\eta_{n}\rightharpoonup\eta$ (up to a
subsequence).

From

$\displaystyle o(1)$
$\displaystyle=dJ_{w}^{\lambda}(\eta_{n})[\eta_{n}-\eta]=-\Re(\eta_{n}\,|\,\eta_{n}-\eta)\lVert
w\rVert^{2}-\Re\langle\eta_{n}\,|\,\eta_{n}-\eta\rangle$
$\displaystyle\qquad-\gamma\int(\nabla
F(\psi_{n}),da(\eta_{n})[\eta_{n}-\eta]w+\eta_{n}-\eta)$
$\displaystyle=-\left|\eta_{n}-\eta\right|_{2}^{2}\lVert
w\rVert^{2}-\Re(\eta\,|\,\eta_{n}-\eta)\lVert
w\rVert^{2}-\lVert\eta_{n}-\eta\rVert^{2}-\Re\langle\eta\,|\,\eta_{n}-\eta\rangle$
$\displaystyle\qquad-\gamma\int(\nabla F(\psi_{n}),\eta_{n}-\eta)-\gamma
da(\eta_{n})[\eta_{n}-\eta]\int(\nabla F(\psi_{n}),w)$

we deduce, using also the fact that
$\left|\eta_{n}\right|_{2}^{2}<\frac{\lambda}{2}$, that

$\left|\eta_{n}-\eta\right|_{2}^{2}\lVert
w\rVert^{2}+\lVert\eta_{n}-\eta\rVert^{2}\\\ =-\gamma\int(\nabla
F(\psi_{n}),\eta_{n}-\eta)+\gamma
a(\eta_{n})^{-1}\Re(\eta_{n}\,|\,\eta_{n}-\eta)\int(\nabla
F(\psi_{n}),w)+o(1)\\\ =-\gamma\int(\nabla F(\psi_{n}),\eta_{n}-\eta)+\gamma
a(\eta_{n})^{-1}\left|\eta_{n}-\eta\right|_{2}^{2}\int(\nabla
F(\psi_{n}),w)+o(1).$

From the estimate (3.4) follows

$(1-\gamma C_{\alpha,\lambda})\left|\eta_{n}-\eta\right|_{2}^{2}\lVert
w\rVert^{2}+\lVert\eta_{n}-\eta\rVert^{2}\leq-\gamma\Re\int(\nabla
F(\psi_{n}),\eta_{n}-\eta)+o(1).$

We will prove that $\limsup_{n\to+\infty}\Re\int(\nabla
F(\psi_{n}),\eta_{n}-\eta)\geq 0$ applying concentration compactness to the
sequence of functions $\eta_{n}-\eta$.

In case we have vanishing, that is

$\forall
R>0\qquad\lim_{n\to+\infty}\sup_{y\in\mathbb{R}^{3}}\int_{B_{R}(y)}\lvert\eta_{n}-\eta\rvert^{2}=0,$

we have that $\left|\eta_{n}-\eta\right|_{q}\to 0$ as $n\to+\infty$ for all
$q\in(2,3)$ (see [17] or [24, Lemma 1.21]).

Take $\epsilon>0$ and take
$q\in(\max\\{2,\frac{3}{4-\alpha}\\},\min\\{3,\frac{2}{3-\alpha}\\})$ we have
that

$\displaystyle\left|\int(\nabla F(\psi_{n}),\eta_{n}-\eta)\right|$
$\displaystyle\leq\int\left(\epsilon\lvert\psi_{n}\rvert+\mu_{\epsilon}\lvert\psi_{n}\rvert^{\alpha-1}\right)\lvert\eta_{n}-\eta\rvert$
$\displaystyle\leq\epsilon\left|\eta_{n}-\eta\right|_{2}+\mu_{\epsilon}\left|\psi_{n}\right|_{\frac{q(\alpha-1)}{q-1}}^{\alpha-1}\left|\eta_{n}-\eta\right|_{q}\leq\sqrt{2\lambda}\epsilon+o(1)$

so that $\lim_{n\to+\infty}\Re\int(\nabla F(\psi_{n}),\eta_{n}-\eta)=0$ if
vanishing occurs.

If we have dichotomy then for all $\epsilon>0$ there is a sequences
$R_{n}\to+\infty$ such that

$\int_{\mathbb{R}^{3}\setminus
B_{R_{n}}(0)}\left|\eta_{n}-\eta\right|_{2}^{2}\geq\sigma-\epsilon,$

where (up to a subsequence)

$\sigma=\lim_{n\to+\infty}\int_{\mathbb{R}^{3}}\lvert\eta_{n}-\eta\rvert^{2}.$

We have that

$\displaystyle\Re\int$ $\displaystyle(\nabla F(\psi_{n}),\eta_{n}-\eta)$
$\displaystyle=\Re\int_{B_{R_{n}}(0)}(\nabla
F(\psi_{n}),\eta_{n}-\eta)+\Re\int_{\mathbb{R}^{3}\setminus
B_{R_{n}}(0)}(\nabla F(\eta_{n}),\eta_{n})$
$\displaystyle\qquad+\Re\int_{\mathbb{R}^{3}\setminus B_{R_{n}}(0)}(\nabla
F(\psi_{n})-\nabla F(\eta_{n}),\eta_{n})-\Re\int_{\mathbb{R}^{3}\setminus
B_{R_{n}}(0)}(\nabla F(\psi_{n}),\eta)$
$\displaystyle\geq-c(\sqrt{\epsilon}+\epsilon^{\frac{2(3-\alpha)}{\alpha}})$
$\displaystyle\qquad+\Re\int_{\mathbb{R}^{3}\setminus B_{R_{n}}(0)}(\nabla
F(\psi_{n})-\nabla F(\eta_{n}),\eta_{n})-\Re\int_{\mathbb{R}^{3}\setminus
B_{R_{n}}(0)}(\nabla F(\psi_{n}),\eta)$

where we have used the fact that $(\nabla F(\eta_{n}),\eta_{n})\geq 0$ and

$\displaystyle\left|\int_{B_{R_{n}}(0)}(\nabla
F(\psi_{n}),\eta_{n}-\eta)\right|$
$\displaystyle\quad\leq\left|\psi_{n}\right|_{2}\left(\int_{B_{R_{n}}(0)}\lvert\eta_{n}-\eta\rvert^{2}\right)^{\frac{1}{2}}+\mu\left|\psi_{n}\right|_{\alpha}^{\alpha-1}\left(\int_{B_{R_{n}}(0)}\lvert\eta_{n}-\eta\rvert^{\alpha}\right)^{\frac{1}{\alpha}}$
$\displaystyle\quad\leq\left(\int_{B_{R_{n}}(0)}\lvert\eta_{n}-\eta\rvert^{2}\right)^{\frac{1}{2}}+\mu\left|\psi_{n}\right|_{\alpha}^{\alpha-1}\left|\eta_{n}-\eta\right|_{3}^{\frac{3(\alpha-2)}{\alpha}}\left(\int_{B_{R_{n}}(0)}\lvert\eta_{n}-\eta\rvert^{2}\right)^{\frac{2(3-\alpha)}{\alpha}}$
$\displaystyle\quad\leq
c(\sqrt{\epsilon}+\epsilon^{\frac{2(3-\alpha)}{\alpha}})$

for some constant $c>0$.

We now observe that

$\displaystyle\Re\int_{\mathbb{R}^{3}\setminus B_{R_{n}}(0)}$
$\displaystyle(\nabla F(\psi_{n})-\nabla
F(\eta_{n}),\eta_{n})\leq\int_{\mathbb{R}^{3}\setminus
B_{R_{n}}(0)}\lvert\nabla F(\psi_{n})-\nabla
F(\eta_{n})\rvert\lvert\eta_{n}\rvert$
$\displaystyle\leq\int_{\mathbb{R}^{3}\setminus B_{R_{n}}(0)}a(\eta_{n})\lvert
w\rvert\lvert\eta_{n}\rvert$
$\displaystyle\qquad+\delta\int_{\mathbb{R}^{3}\setminus
B_{R_{n}}(0)}\left(\lvert\eta_{n}\rvert^{\alpha-2}+a(\eta_{n})^{\alpha-2}\lvert
w\rvert^{\alpha-2}\right)a(\eta_{n})\lvert w\rvert\lvert\eta_{n}\rvert$
$\displaystyle\leq\sqrt{\lambda}\left|\eta_{n}\right|_{2}^{2}\left(\int_{\mathbb{R}^{3}\setminus
B_{R_{n}(0)}}\lvert
w\rvert^{2}\right)^{\frac{1}{2}}+\delta\sqrt{\lambda}\left|\eta_{n}\right|_{\alpha}^{\alpha-1}\left(\int_{\mathbb{R}^{3}\setminus
B_{R_{n}}(0)}\lvert w\rvert^{\alpha}\right)^{1/\alpha}$
$\displaystyle\qquad+\delta\lambda^{\frac{\alpha-1}{2}}\left|\eta_{n}\right|_{\alpha}\left(\int_{\mathbb{R}^{3}\setminus
B_{R_{n}}(0)}\lvert w\rvert^{\alpha}\right)^{(\alpha-1)/\alpha}=o(1)$

where we have used (1.10).

A similar argument shows that

$\int_{\mathbb{R}^{3}\setminus B_{R_{n}}(0)}(\nabla F(\psi_{n}),\eta)=o(1),$

and also when dichotomy occurs we find that

$(1-\gamma C_{\alpha,\lambda})\left|\eta_{n}-\eta\right|_{2}^{2}\lVert
w\rVert^{2}+\lVert\eta_{n}-\eta\rVert^{2}\leq c\epsilon+o(1).$

So in every case we have that and $\eta_{n}\to\eta$, with $\eta$ critical
point of $J_{w}^{\lambda}$. ∎

We now show that the functional $J_{w}^{\lambda}$ is convex in the set where
it is positive in the $L^{2}$-ball of radius $\sqrt{\frac{\lambda}{2}}$.

######  Lemma 3.8.

Let $\eta\in B_{\lambda}$ be such that
$\left|\eta\right|_{2}^{2}<\frac{\lambda}{2}$ and $J_{w}^{\lambda}(\eta)\geq
0$. Then

$d^{2}J_{w}^{\lambda}(\eta)[\xi,\xi]\leq-\lVert\xi\rVert^{2}\qquad\text{ for
all }\xi\in X_{-}.$

###### Proof.

In order to compute the second derivative we denote $\psi=a(\eta)w+\eta$ and
$h_{\xi}=da(\eta)[\xi]w+\xi$ and observe that

$d^{2}a(\eta)[\xi,\zeta]=-a(\eta)^{-1}\left(\Re(\xi\,|\,\zeta)+\frac{\Re(\eta\,|\,\xi)\Re(\eta,\zeta)}{\lambda-\left|\eta\right|_{2}^{2}}\right).$

Then we deduce, using (3.4) and (H4):

$\displaystyle d^{2}J_{w}^{\lambda}(\eta)[\xi,\xi]$
$\displaystyle=-\left|\xi\right|_{2}^{2}\lVert
w\rVert^{2}-\lVert\xi\rVert^{2}-\gamma\int(F^{\prime\prime}(\psi)h_{\xi},h_{\xi})$
$\displaystyle\qquad-\gamma\int(\nabla F(\psi),d^{2}a(\eta)[\xi,\xi]w)$
$\displaystyle=-\left|\xi\right|_{2}^{2}\lVert
w\rVert^{2}-\lVert\xi\rVert^{2}-\gamma\int(F^{\prime\prime}(\psi)h_{\xi},h_{\xi})$
$\displaystyle\qquad+\frac{\gamma}{a(\eta)}\left(\left|\xi\right|_{2}^{2}+\frac{(\eta\,|\,\xi)^{2}}{\lambda-\left|\eta\right|_{2}^{2}}\right)\int(\nabla
F(\psi),w)$ $\displaystyle\leq-\left|\xi\right|_{2}^{2}\lVert
w\rVert^{2}-\lVert\xi\rVert^{2}-\gamma\int(F^{\prime\prime}(\psi)h_{\xi},h_{\xi})$
$\displaystyle\qquad+\frac{\lambda\left|\xi\right|_{2}^{2}-\left|\xi\right|_{2}^{2}\left|\eta\right|_{2}^{2}+(\eta\,|\,\xi)^{2}}{\lambda-\left|\eta\right|_{2}^{2}}\gamma
C_{\alpha,\lambda}\lVert w\rVert^{2}$
$\displaystyle\leq-\left|\xi\right|_{2}^{2}\lVert
w\rVert^{2}-\lVert\xi\rVert^{2}-\gamma\int(F^{\prime\prime}(\psi)h_{\xi},h_{\xi})+2\left|\xi\right|_{2}^{2}\gamma
C_{\alpha,\lambda}\lVert w\rVert^{2}$ $\displaystyle\leq-\left(1-2\gamma
C_{\alpha,\lambda}\right)\left|\xi\right|_{2}^{2}\lVert
w\rVert^{2}-\lVert\xi\rVert^{2}-\gamma\int(F^{\prime\prime}(\psi)h_{\xi},h_{\xi})$
$\displaystyle\leq-\frac{1}{2}\left|\xi\right|_{2}^{2}\lVert
w\rVert^{2}-\lVert\xi\rVert^{2}-\gamma\int(F^{\prime\prime}(\psi)h_{\xi},h_{\xi})\leq-\lVert\xi\rVert^{2}.$

since $2\gamma C_{\alpha,\lambda}<\frac{1}{2}$ by (1.11). ∎

######  Lemma 3.9.

Let $w\in\Sigma_{+}$, $\eta\in B_{\lambda}$ and
$\psi=a(\eta)w+\eta\in\Sigma^{\lambda}$. Then we have that

(3.10) $\int F(\psi)\geq\int F(\sqrt{\lambda}w)+\int(\nabla
F(aw),\eta)-\left(S_{2}^{2}+\mu\lambda^{\frac{\alpha-2}{2}}S^{2}_{\frac{4}{4-\alpha}}\right)\left|\eta\right|_{2}^{2}\lVert
w\rVert^{2}$

###### Proof.

The result follows, using the convexity of $F$, from

$\int(F(\psi)-F(a(\eta)w))\geq\int(\nabla F(a(\eta)w),\eta)$

and

$\displaystyle\int$
$\displaystyle(F(a(\eta)w)-F(\sqrt{\lambda}w))\geq\int(\nabla
F(\sqrt{\lambda}w),(a(\eta)-\sqrt{\lambda})w)$
$\displaystyle\geq-(\sqrt{\lambda}-a(\eta))\left(\sqrt{\lambda}\left|w\right|_{2}^{2}+\mu(\sqrt{\lambda})^{\alpha-1}\int\lvert
w\rvert^{\alpha}\right)$
$\displaystyle\geq-\frac{\lambda-a(\eta)^{2}}{\sqrt{\lambda}+a(\eta)}\left(\sqrt{\lambda}\left|w\right|_{2}^{2}+\mu(\sqrt{\lambda})^{\alpha-1}\left|w\right|_{2}^{(\alpha-2)}\left|w\right|_{4/(4-\alpha)}^{2}\right)$
$\displaystyle\geq-\left(S_{2}^{2}+\mu\lambda^{\frac{\alpha-2}{2}}S^{2}_{\frac{4}{4-\alpha}}\right)\left|\eta\right|_{2}^{2}\lVert
w\rVert^{2}$

∎

We let, for all $w\in\Sigma_{+}$

$\mathcal{E}_{\lambda}(w)=\sup_{\eta\in B_{\lambda}}J_{w}^{\lambda}(\eta).$

######  Lemma 3.11.

For all $w\in\Sigma_{+}$ we have

$0<\frac{\lambda}{2S_{2}^{2}}\left(1-\gamma\left(S_{2}^{2}+2\frac{\mu}{\alpha}\lambda^{\frac{\alpha-2}{2}}S^{2}_{\frac{4}{4-\alpha}}\right)\right)\\\
\leq\frac{\lambda}{2}\left(1-\gamma\left(S_{2}^{2}+2\frac{\mu}{\alpha}\lambda^{\frac{\alpha-2}{2}}S^{2}_{\frac{4}{4-\alpha}}\right)\right)\lVert
w\rVert^{2}\leq\mathcal{E}_{\lambda}(w)\leq\frac{\lambda}{2}\lVert
w\rVert^{2}.$

###### Proof.

We have, using (1.9), that

$\displaystyle\mathcal{E}_{\lambda}(w)$ $\displaystyle\geq
J_{w}^{\lambda}(0)=\frac{1}{2}\lVert\sqrt{\lambda}w\rVert^{2}-\gamma\int
F(\sqrt{\lambda}w)$ $\displaystyle\geq\frac{\lambda}{2}\lVert
w\rVert^{2}-\gamma\frac{\lambda}{2}\left|w\right|_{2}^{2}-\gamma\frac{\mu}{\alpha}\lambda^{\frac{\alpha}{2}}\left|w\right|_{2}^{\alpha-2}\left|w\right|_{\frac{4}{4-\alpha}}^{2}$
$\displaystyle\geq\frac{\lambda}{2}\lVert
w\rVert^{2}-\gamma\lambda\left(\frac{S_{2}^{2}}{2}+\frac{\mu}{\alpha}\lambda^{\frac{\alpha-2}{2}}S^{2}_{\frac{4}{4-\alpha}}\right)\lVert
w\rVert^{2}$
$\displaystyle\geq\frac{\lambda}{2S_{2}^{2}}\left(1-\gamma\left(S_{2}^{2}+2\frac{\mu}{\alpha}\lambda^{\frac{\alpha-2}{2}}S^{2}_{\frac{4}{4-\alpha}}\right)\right)$

which is positive by (1.12) and

$J_{w}^{\lambda}(\eta)=\frac{1}{2}\lVert
a(\eta)w\rVert^{2}-\frac{1}{2}\lVert\eta\rVert^{2}-\gamma\int
F(\psi)\leq\frac{\lambda}{2}\lVert w\rVert^{2}.$

∎

Tho following proposition is analogous to [6, Proposition 4.5], [20,
Proposition 3.6] and [7, Proposition 2.9]. Since in the present situation the
functional $J_{w}^{\lambda}$ is, by lemma 3.8, convex in the region where
$J_{w}^{\lambda}(\eta)>0$ and $\left|\eta\right|_{2}^{2}<\frac{\lambda}{2}$
the proof is actually simpler.

######  Proposition 3.12.

For every $w\in\Sigma_{+}$ and $\lambda>0$ there is a unique $\eta(w)\in
B_{\lambda}$ such that

$J_{w}^{\lambda}(\eta(w))=\max_{\eta\in
B_{\lambda}}J_{w}^{\lambda}(\eta)=\mathcal{E}_{\lambda}(w).$

$\eta(w)$ is a critical point of $J_{w}^{\lambda}$ on $B_{\lambda}$ such that
$\left|\eta(w)\right|_{2}^{2}<\frac{\lambda}{2}$.

Moreover the map

$w\in X_{+}\setminus\\{0\\}\mapsto\gamma(w)=\eta(\left|w\right|_{2}^{-1}w)\in
B_{\lambda}$

is smooth.

###### Proof.

We can find, by lemma 3.11 and using Ekeland’s variational principle a
maximizing Palais-Smale sequence $\eta_{n}$ at positive level.

Then, by lemma 3.7, $\eta_{n}\to\eta$ (up to a subsequence), with

$dJ_{w}^{\lambda}(\eta)=0,\qquad
J_{w}^{\lambda}(\eta)=\mathcal{E}_{\lambda}(w)>0.$

The strict convexity of $J_{w}^{\lambda}$ in the region
$J_{w}^{\lambda}(\eta)>0$ and $\left|\eta\right|_{2}^{2}<\frac{\lambda}{2}$
imply that such a maximum is unique.

The properties of the map $w\mapsto\gamma(w)=\eta(\left|w\right|_{2}^{-1}w)$
follows exactly as in [7, Proposition 2.9]. ∎

Exactly as in [7] we can consider the smooth functional
$\mathcal{E}_{\lambda}\colon X_{+}\setminus\\{0\\}\to\mathbb{R}$ defined as

$\mathcal{E}_{\lambda}(w)=J_{P(w)}^{\lambda}(\gamma(w))=\sup_{\eta\in
B_{\lambda}}J_{P(w)}^{\lambda}(\eta),$

where, as in the proof of 3.12, $P(w)=\frac{w}{\left|w\right|_{2}}$ and
$\gamma(w)=\eta(Pw)$ and deduce that for all $w\in\Sigma_{+}$ and $v\in
X_{+}$.

(3.13)
$d\mathcal{E}_{\lambda}(w)[v]=a(\gamma(w))dI(\psi_{w})[v]-a(\gamma(w))^{2}\omega(\psi_{w})(w\,|\,v)$

where

(3.14) $\omega(\psi_{w})=a(\gamma(w))^{-1}dI(\psi_{w})[w].$

We also deduce that, for all $h=v+\xi\in X$, $v\in X_{+}$, $\xi\in X_{-}$ and
with $\psi$, $a$ and $\omega$ as above we have

$\displaystyle dI(\psi)[h]-\omega(\psi\,|\,h)$
$\displaystyle=dI(\psi)[v]+dI(\psi)[\xi]-\omega(a(\eta)w+\eta\,|\,v+\xi)$
$\displaystyle=dI(\psi)[v]+\left(-da(\eta)[\xi]-(w\,|\,v)-\frac{(\eta\,|\,\xi)}{a(\eta)}\right)dI(\psi)[w]$
$\displaystyle=dI(\psi)[v-(w\,|\,v)w]=\frac{1}{a(\eta)}d\mathcal{E}_{\lambda}(w)[v]$

which shows that $w$ is a critical point of $\mathcal{E}_{\lambda}$ if and
only if $\psi=a(\eta(w))w+\eta(w)$ is a critical point for $I$ under the
constrain $\left|\psi\right|_{2}^{2}=\lambda$.

The following is essentially Proposition 2.13 of [7]:

######  Proposition 3.15.

Let $w_{0}\in\Sigma_{+}$ be a critical points of $\mathcal{E}_{\lambda}$
restricted on the manifold $\Sigma_{+}$. Then $w_{0}$ is a critical points for
$\mathcal{E}_{\lambda}$ on $X_{+}$ and the function

$\psi_{0}=a(\eta(w_{0}))w_{0}+\eta(w_{0})\in\Sigma^{\lambda}$

is a critical point for $I$ on the manifold $\Sigma$ and satisfies

(3.16) $dI(\psi_{0})[h]=\omega(\psi_{0}\,|\,h)\qquad\text{for all }h\in X$

where $\omega=\omega(\psi_{0})\in\mathbb{R}$,

$(1-\gamma C_{\alpha,\lambda})\lVert w_{0}\rVert^{2}\leq\omega(\psi_{0})\leq
2I(\psi_{w_{0}})=2\mathcal{E}_{\lambda}(w_{0}).$

Moreover, if $\psi_{0}\in\Sigma^{\lambda}$ satisfies (3.16) for some
$\omega>0$, then
$w=\left|\Lambda_{+}\psi_{0}\right|_{2}^{-1}\Lambda_{+}\psi_{0}$ is a critical
point for $\mathcal{E}_{\lambda}(w)$ .

###### Proof.

We only have to prove the estimate on the Lagrange multiplier. The other
points follow as in proof of Proposition 2.13 of [7]. Using (3.4) we
immediately deduce

$\displaystyle\omega(\psi_{0})$
$\displaystyle=a(\eta(w_{0}))^{-1}dI(\psi_{0})[w_{0}]\geq(1-\gamma
C_{\alpha,\lambda})\lVert w_{0}\rVert^{2}.$

We also have, using (H3), that

$\omega(\psi_{0})=dI(\psi_{0})[\psi_{0}]\leq 2I(\psi_{0}).$

∎

##  4\. The constrained minimization

We introduce now the following minimization problem:

$e(\lambda)=\inf_{w\in\Sigma_{+}}\mathcal{E}_{\lambda}(w)=\inf_{w\in\Sigma_{+}}\left\\{\frac{1}{2}a(\eta(w))^{2}\lVert
w\rVert^{2}-\frac{1}{2}\lVert\eta(w)\rVert^{2}-\gamma\int
F(\psi_{w})\right\\}$

where $\eta(w)$ is given in proposition 3.12,
$a(\eta(w))=\sqrt{\lambda-\left|\eta(w)\right|_{2}^{2}}$ and
$\psi_{w}=a(\eta(w))w+\eta(w)$.

The next lemma contains an estimate wich will be essential in proving (via the
concentration compactness lemma [17, 18]) convergence of minimizing sequences
for $\mathcal{E}_{\lambda}$.

######  Lemma 4.1.

For all $\lambda\in(0,1]$ we have that $0<e(\lambda)<\frac{\lambda m}{2}$.

###### Proof.

From lemma 3.11 we have that

$e(\lambda)\geq\frac{\lambda}{2S_{2}^{2}}\left(1-\gamma\left(S_{2}^{2}+2\frac{\mu}{\alpha}\lambda^{\frac{\alpha-2}{2}}S^{2}_{\frac{4}{4-\alpha}}\right)\right)>0.$

Using lemma 3.9 we deduce that

$\displaystyle\mathcal{E}_{\lambda}(w)$
$\displaystyle=I(\psi_{w})=\frac{1}{2}a(\eta(w))^{2}\lVert
w\rVert^{2}-\frac{1}{2}\lVert\eta(w)\rVert^{2}-\gamma\int F(\psi_{w})$
$\displaystyle\leq\frac{\lambda}{2}\lVert
w\rVert^{2}-\frac{1}{2}\left|\eta(w)\right|_{2}^{2}\lVert
w\rVert^{2}-\frac{1}{2}\lVert\eta(w)\rVert^{2}-\gamma\int F(\sqrt{\lambda}w)$
$\displaystyle\qquad-\gamma\int(\nabla
F(aw),\eta)+\gamma\left(S_{2}^{2}+\mu\lambda^{\frac{\alpha-2}{2}}S^{2}_{\frac{4}{4-\alpha}}\right)\left|\eta\right|_{2}^{2}\lVert
w\rVert^{2}$

Since
$\gamma\left(S_{2}^{2}+\mu\lambda^{\frac{\alpha-2}{2}}S^{2}_{\frac{4}{4-\alpha}}\right)<\frac{1}{4}$
by (1.12) we have, for all $w\in\Sigma_{+}$:

$\mathcal{E}_{\lambda}(w)\leq\frac{\lambda}{2}\lVert
w\rVert^{2}-\frac{1}{4}\left|\eta(w)\right|_{2}^{2}\lVert
w\rVert^{2}-\frac{1}{2}\lVert\eta(w)\rVert^{2}-\gamma\int
F(\sqrt{\lambda}w)-\gamma\int(\nabla F(aw),\eta).$

Fix $w_{1}\in H^{1}(\mathbb{R}^{3},\mathbb{C})\cap
C^{1}(\mathbb{R}^{3},\mathbb{C})$ such that $\left|w_{1}\right|_{2}=1$ and let
$w=\left(\begin{smallmatrix}w_{1}\\\
0\end{smallmatrix}\right)\in\mathbb{C}^{4}$ and
$w_{\epsilon}(x)=\epsilon^{3/2}w(\epsilon x)$. Then
$\left|w_{\epsilon}\right|_{2}=1$ for all $\epsilon>0$ and
$\left|w_{\epsilon}(x)\right|_{\infty}\to 0$ as $\epsilon\to 0$. We also have
that $\lvert(w(x),\beta w(x))\rvert=\lvert w_{1}(x)\rvert^{2}$.

The same computation of Lemma 2.16 in [7] show that

$\displaystyle\lVert
w_{\epsilon}\rVert^{2}-m\left|w_{\epsilon}\right|_{2}^{2}\leq\frac{\epsilon^{2}}{2m}\int\lvert
q\rvert^{2}\lvert\hat{w}_{1}(q)\rvert^{2}$ $\displaystyle\lVert
w_{\epsilon}\rVert^{2}\leq m+C\epsilon^{2}$ $\displaystyle\lVert
w_{\epsilon}-\Lambda_{+}w_{\epsilon}\rVert^{2}\leq\frac{\epsilon^{2}}{4m}\int\lvert
p\rvert^{2}\lvert\hat{w}_{1}(p)\rvert^{2}$
$\displaystyle\left|1-\left|\Lambda_{+}w_{\epsilon}\right|_{2}\right|^{2}\leq\frac{\epsilon^{2}}{4m}\int\lvert
p\rvert^{2}\lvert\hat{w}_{1}(p)\rvert^{2}.$

We deduce from this that for $\epsilon>0$ small enough
$\left|\Lambda_{+}w_{\epsilon}\right|_{2}>\frac{1}{2}$.

Let

$\varphi_{\epsilon}(x)=\left|\Lambda_{+}w_{\epsilon}\right|_{2}^{-1}\Lambda_{+}w_{\epsilon}(x)\in\Sigma_{+}.$

We have that

$\displaystyle\lVert\varphi_{\epsilon}\rVert\leq\left|\Lambda_{+}w_{\epsilon}\right|_{2}^{-1}\lVert
w_{\epsilon}\rVert\leq\sqrt{m}+C\epsilon$ $\displaystyle\lVert
w_{\epsilon}-\varphi_{\epsilon}\rVert\leq\frac{\epsilon}{2\sqrt{m}}(2+\lVert\varphi_{\epsilon}\rVert)\left(\int\lvert
p\rvert^{2}\lvert\hat{w}_{1}(p)\rvert^{2}\right)^{1/2}=\frac{\epsilon}{\sqrt{m}}(2+\lVert\varphi_{\epsilon}\rVert)\left|\nabla
w_{1}\right|_{2}$
$\displaystyle\left|\varphi_{\epsilon}-w_{\epsilon}\right|_{2}\leq\frac{\epsilon}{\sqrt{m}}\left(\int\lvert
p\rvert^{2}\lvert\hat{w}_{1}(p)\rvert^{2}\right)^{1/2}=\frac{\epsilon}{\sqrt{m}}\left|\nabla
w_{1}\right|_{2}$

and hence, for $\epsilon$ small enough

$\lVert\varphi_{\epsilon}\rVert\leq\sqrt{m}+1,\qquad\lVert\eta(\varphi_{\epsilon})\rVert\leq
a(\eta(\varphi_{\epsilon}))\lVert\varphi_{\epsilon}\rVert\leq\sqrt{\lambda}(\sqrt{m}+1).$

We now estimate the functional for small $\epsilon>0$:

$\mathcal{E}_{\lambda}(\varphi_{\epsilon})\leq\frac{\lambda
m}{2}+\frac{\lambda}{2}(\lVert\varphi_{\epsilon}\rVert^{2}-m\left|\varphi_{\epsilon}\right|_{2}^{2})-\frac{1}{4}\left|\eta(\varphi_{\epsilon})\right|_{2}^{2}\lVert\varphi_{\epsilon}\rVert^{2}\\\
-\frac{1}{2}\lVert\eta(\varphi_{\epsilon})\rVert^{2}-\gamma\int
F(\sqrt{\lambda}\varphi_{\epsilon})-\gamma\int(\nabla
F(a\varphi_{\epsilon}),\eta(\varphi_{\epsilon}))$

The first term can be estimated as follows:

$\lVert\varphi_{\epsilon}\rVert^{2}-m\left|\varphi_{\epsilon}\right|_{2}^{2}\leq\frac{\epsilon^{2}}{2m}(3+\lVert\varphi_{\epsilon}\rVert)^{2}\left|\nabla
w_{1}\right|_{2}^{2}\leq\frac{\epsilon^{2}}{2m}(4+\sqrt{m})^{2}\left|\nabla
w_{1}\right|_{2}^{2}.$

For the next term we have, using (H5) and (H6) since
$\left|w_{\epsilon}\right|_{\infty}\to 0$, that

$\displaystyle\int F(\sqrt{\lambda}\varphi_{\epsilon})$ $\displaystyle\geq\int
F(\sqrt{\lambda}w_{\epsilon})+\int(\nabla
F(\sqrt{\lambda}w_{\epsilon}),\varphi_{\epsilon}-w_{\epsilon})$
$\displaystyle\geq\bar{\gamma}\lambda^{\frac{\alpha}{2}}\int\lvert(w_{\epsilon},\beta
w_{\epsilon})\rvert^{\frac{\alpha}{2}}-\lambda^{\frac{\nu}{2}}\left|w_{\epsilon}\right|_{2\nu}^{\nu}\left|\varphi_{\epsilon}-w_{\epsilon}\right|_{2}$
$\displaystyle\geq\epsilon^{\frac{3(\alpha-2)}{2}}\bar{\gamma}\lambda^{\frac{\alpha}{2}}\int\lvert
w_{1}\rvert^{\alpha}-\lambda^{\frac{\nu}{2}}\epsilon^{\frac{3(\nu-1)}{2}}\left|w_{1}\right|_{2\nu}^{\nu}\left|\varphi_{\epsilon}-w_{\epsilon}\right|_{2}$
$\displaystyle\geq\epsilon^{\frac{3(\alpha-2)}{2}}\bar{\gamma}\lambda^{\frac{\alpha}{2}}\int\lvert
w_{1}\rvert^{\alpha}-\epsilon^{\frac{3\nu-1}{2}}\frac{\lambda^{\frac{\nu}{2}}}{\sqrt{m}}\left|w_{1}\right|_{2\nu}^{\nu}\left|\nabla
w_{1}\right|_{2}$

(we can always assume $\nu<\frac{3}{2}$, see remark 1.3).

For the last term we observe that

$\int(\nabla
F(a\varphi_{\epsilon}),\eta(\varphi_{\epsilon}))\leq\int\lvert\nabla
F(a\varphi_{\epsilon})-\nabla
F(aw_{\epsilon})\rvert\lvert\eta(\varphi_{\epsilon})\rvert+\int\lvert\nabla
F(aw_{\epsilon})\rvert\lvert\eta(\varphi_{\epsilon})\rvert.$

Let us analyze the first term in this expression: we have

$\displaystyle\int\lvert\nabla F(a\varphi_{\epsilon})-\nabla
F(aw_{\epsilon})\rvert\lvert\eta(\varphi_{\epsilon})\rvert$
$\displaystyle\quad\leq\int\left(\lvert
a(\varphi_{\epsilon}-w_{\epsilon})\rvert+\lvert
aw_{\epsilon}\rvert+\delta\lvert
a(\varphi_{\epsilon}-w_{\epsilon})\rvert^{\alpha-2}+\delta\lvert
aw_{\epsilon}\rvert^{\alpha-2}\right)a\lvert\varphi_{\epsilon}-w_{\epsilon}\rvert\lvert\eta(\varphi_{\epsilon})\rvert$
$\displaystyle\quad\leq
a^{2}\int\left(\lvert(\varphi_{\epsilon}-w_{\epsilon})\rvert^{2}+\lvert
w_{\epsilon}\rvert\lvert\varphi_{\epsilon}-w_{\epsilon}\rvert\right)\lvert\eta(\varphi_{\epsilon})\rvert$
$\displaystyle\qquad+\delta
a^{\alpha-1}\int\left(\lvert\varphi_{\epsilon}-w_{\epsilon}\rvert^{\alpha-1}+\lvert
w_{\epsilon}\rvert^{\alpha-2}\lvert\varphi_{\epsilon}-w_{\epsilon}\rvert\right)\lvert\eta(\varphi_{\epsilon})\rvert$
$\displaystyle\quad\leq
a^{2}\left|\varphi_{\epsilon}-w_{\epsilon}\right|_{3}^{2}\left|\eta(\varphi_{\epsilon})\right|_{3}+a^{2}\left|w_{\epsilon}\right|_{3}\left|\varphi_{\epsilon}-w_{\epsilon}\right|_{3}\left|\eta(\varphi_{\epsilon})\right|_{3}$
$\displaystyle\qquad+\delta
a^{\alpha-1}\left|\varphi_{\epsilon}-w_{\epsilon}\right|_{\alpha}^{\alpha-1}\left|\eta(\varphi_{\epsilon})\right|_{\alpha}+\delta
a^{\alpha-1}\epsilon^{\frac{3(\alpha-2)}{2}}\left|w_{1}\right|_{\infty}^{\alpha-2}\left|\varphi_{\epsilon}-w_{\epsilon}\right|_{2}\left|\eta(\varphi_{\epsilon})\right|_{2}$
$\displaystyle\quad\leq
a^{2}\left|\eta(\varphi_{\epsilon})\right|_{3}\left(S_{3}^{2}\lVert\varphi_{\epsilon}-w_{\epsilon}\rVert^{2}+\left|w_{\epsilon}\right|_{3}S_{3}\lVert\varphi_{\epsilon}-w_{\epsilon}\rVert\right)$
$\displaystyle\qquad+\delta
a^{\alpha-1}S_{\alpha}^{\alpha}\lVert\eta(\varphi_{\epsilon})\rVert\lVert\varphi_{\epsilon}-w_{\epsilon}\rVert^{\alpha-1}$
$\displaystyle\qquad+\delta
a^{\alpha-1}\epsilon^{\frac{3(\alpha-2)}{2}}\left|w_{1}\right|_{\infty}^{\alpha-2}\left|\varphi_{\epsilon}-w_{\epsilon}\right|_{2}\left|\eta(\varphi_{\epsilon})\right|_{2}$
$\displaystyle\quad\leq\lambda^{\frac{3}{2}}S_{3}^{2}(\sqrt{m}+1)\left(\epsilon^{2}S_{3}\frac{(3+\sqrt{m})^{2}}{m}\left|\nabla
w_{1}\right|_{2}^{2}+\epsilon^{\frac{3}{2}}\left|w_{1}\right|_{3}\frac{3+\sqrt{m}}{\sqrt{m}}\left|\nabla
w_{1}\right|_{2}\right)$
$\displaystyle\qquad+\epsilon^{\alpha-1}\delta\lambda^{\frac{2\alpha-1}{2}}S_{\alpha}^{\alpha}(\sqrt{m}+1)\left(\frac{3+\sqrt{m}}{\sqrt{m}}\right)^{\alpha-1}\left|\nabla
w\right|_{2}^{\alpha-1}$
$\displaystyle\qquad+\epsilon^{\frac{3\alpha-4}{2}}\delta\lambda^{\frac{2\alpha-1}{2}}S_{2}^{2}\left|w_{1}\right|_{\infty}^{\alpha-2}\frac{(\sqrt{m}+1)}{\sqrt{m}}\left|\nabla
w_{1}\right|_{2}$

while for the second term we have

$\displaystyle\int\lvert\nabla
F(aw_{\epsilon})\rvert\lvert\eta(\varphi_{\epsilon})\rvert$ $\displaystyle\leq
a^{\nu}\int\lvert
w_{\epsilon}\rvert^{\nu}\lvert\eta(\varphi_{\epsilon})\rvert\leq\frac{1}{2}a^{2\nu}\left|w_{\epsilon}\right|_{2\nu}^{2\nu}+\frac{1}{2}\left|\eta(\varphi_{\epsilon})\right|_{2}^{2}$
$\displaystyle\leq\epsilon^{3(\nu-1)}\frac{\lambda^{\nu}}{2}\left|w_{1}\right|_{2\nu}^{2\nu}+\frac{S_{2}^{2}}{2}\lVert\eta(\varphi_{\epsilon})\rVert^{2}$

We therefore deduce that

$\displaystyle\mathcal{E}_{\lambda}(w_{\epsilon})\leq\frac{\lambda
m}{2}+\epsilon^{2}\frac{\lambda}{4m}(4+\sqrt{m})^{2}\left|\nabla
w_{1}\right|_{2}^{2}-\frac{1}{4}\left|\eta(\varphi_{\epsilon})\right|_{2}^{2}\lVert
w\rVert^{2}-\frac{1}{2}\lVert\eta(\varphi_{\epsilon})\rVert^{2}$
$\displaystyle\quad-\epsilon^{\frac{3(\alpha-2)}{2}}\gamma\bar{\gamma}\lambda^{\frac{\alpha}{2}}\int\lvert
w_{1}\rvert^{\alpha}+\epsilon^{\frac{3\nu-1}{2}}\gamma\frac{\lambda^{\frac{\nu}{2}}}{\sqrt{m}}\left|w_{1}\right|_{2\nu}^{\nu}\left|\nabla
w_{1}\right|_{2}$
$\displaystyle\quad+\epsilon^{\frac{3}{2}}\gamma\lambda^{\frac{3}{2}}S_{3}(\sqrt{m}+1)\left(\sqrt{\epsilon}S_{3}^{2}\frac{(3+\sqrt{m})^{2}}{m}\left|\nabla
w_{1}\right|_{2}^{2}+\left|w_{1}\right|_{3}S_{3}\frac{3+\sqrt{m}}{\sqrt{m}}\left|\nabla
w_{1}\right|_{2}\right)$
$\displaystyle\quad+\epsilon^{\alpha-1}\delta\lambda^{\frac{2\alpha-1}{2}}S_{\alpha}^{\alpha}(\sqrt{m}+1)\left(\frac{3+\sqrt{m}}{\sqrt{m}}\right)^{\alpha-1}\left|\nabla
w\right|_{2}^{\alpha-1}$
$\displaystyle\quad+\epsilon^{\frac{3\alpha-4}{2}}\delta\lambda^{\frac{2\alpha-1}{2}}S_{2}^{2}\left|w_{1}\right|_{\infty}^{\alpha-2}\frac{(\sqrt{m}+1)}{\sqrt{m}}\left|\nabla
w_{1}\right|_{2}$
$\displaystyle\quad+\epsilon^{3(\nu-1)}\gamma\frac{\lambda^{\nu}}{2}\left|w_{1}\right|_{2\nu}^{2\nu}+\gamma\frac{S_{2}^{2}}{2}\lambda(\sqrt{m}+1)^{2}$

Since $\gamma S_{2}^{2}<1$ by (1.12) and

$\frac{3(\alpha-2)}{2}<\min\left\\{2,\frac{3\nu-1}{2},\frac{3}{2},\alpha-1,\frac{3\alpha-4}{2},3(\nu-1)\right\\}\qquad\text{for
all }\alpha\in(2,\frac{8}{3}]$

we deduce that

$\mathcal{E}_{\lambda}(w_{\epsilon})<\frac{\lambda m}{2}$

for $\epsilon$ small enough and for all $\lambda\in(0,1]$ and hence
$e(\lambda)<\frac{\lambda m}{2}$. ∎

######  Proposition 4.2.

For all $\lambda\in(0,1)$ and $\theta>1$ such that $\theta\lambda<1$ we have
that

$e(\theta\lambda)<\theta e(\lambda).$

###### Proof.

Let $\theta>1$ be such that $\lambda\theta\in(0,1)$. Take $w\in\Sigma_{+}$ and
let $\eta_{\theta\lambda}(w)\in B_{\theta\lambda}$ be the function whose
existence follows from proposition 3.12. Let
$\psi_{\theta\lambda}(w)=a_{\theta\lambda}(\eta_{\theta\lambda}(w))w+\eta_{\theta\lambda}(w)\in\Sigma_{\theta\lambda}$.
We have that

$\mathcal{E}_{\theta\lambda}(w)=\frac{1}{2}\lVert
a_{\theta\lambda}(\eta_{\theta\lambda}(w))w\rVert^{2}-\frac{1}{2}\lVert\eta_{\theta\lambda}(w)\rVert^{2}-\gamma\int
F(\psi_{\theta\lambda}(w)).$

We recall that from assumption (H3) follows that

$F(\sigma
x)\geq\sigma^{\alpha}F(x)\geq\sigma^{2}F(x)+\left(\sigma^{\alpha}-\sigma^{2}\right)F(x)\qquad\text{for
all }x\in\mathbb{C}^{4},\sigma\geq 1.$

We have that for all $\eta\in B_{\theta\lambda}$

$a_{\lambda\theta}(\eta)=\sqrt{\lambda\theta-\left|\eta\right|_{2}^{2}}=\sqrt{\theta}\sqrt{\lambda-\left|\frac{\eta}{\sqrt{\theta}}\right|_{2}^{2}}=\sqrt{\theta}a_{\lambda}(\frac{\eta}{\sqrt{\theta}})$

and hence we have that

$\displaystyle e(\theta\lambda)$
$\displaystyle\leq\mathcal{E}_{\theta\lambda}(w)=\theta\left[\frac{1}{2}\lVert
a_{\lambda}(\frac{\eta_{\theta\lambda}(w)}{\sqrt{\theta}})w\rVert^{2}-\frac{1}{2}\lVert\frac{\eta_{\theta\lambda}(w)}{\sqrt{\theta}}\rVert^{2}-\frac{\gamma}{\theta}\int
F\left(\sqrt{\theta}\frac{\psi_{\theta\lambda}(w)}{\sqrt{\theta}}\right)\right]$
$\displaystyle\leq\theta\left[\frac{1}{2}\lVert
a_{\lambda}(\frac{\eta_{\theta\lambda}(w)}{\sqrt{\theta}})w\rVert^{2}-\frac{1}{2}\lVert\frac{\eta_{\theta\lambda}(w)}{\sqrt{\theta}}\rVert^{2}-\gamma\int
F\left(\frac{\psi_{\theta\lambda}(w)}{\sqrt{\theta}}\right)\right]$
$\displaystyle\qquad-\gamma\theta\left(\theta^{(\alpha-2)/2}-1\right)\int
F\left(\frac{\psi_{\theta\lambda}(w)}{\sqrt{\theta}}\right)$
$\displaystyle\leq\theta\mathcal{E}_{\lambda}(w)-\gamma\theta\left(\theta^{(\alpha-2)/2}-1\right)\int
F\left(\frac{\psi_{\theta\lambda}(w)}{\sqrt{\theta}}\right).$

We claim that for all $w\in\Sigma_{+}$ such that
$\mathcal{E}_{\lambda}(w)<\frac{1}{2}(\frac{\lambda m}{2}+e(\lambda))$

$\int
F\left(\frac{\psi_{\theta\lambda}(w)}{\sqrt{\theta}}\right)\geq\delta>0.$

Assuming the claim, let $w_{n}\in\Sigma_{+}$ be a sequence which minimize
$\mathcal{E}_{\lambda}(w)$, $\mathcal{E}_{\lambda}(w_{n})\to
e(\lambda)<\frac{1}{2}(\frac{\lambda m}{2}+e(\lambda))$. We have that for all
$\theta>1$

$e(\theta\lambda)\leq\mathcal{E}_{\theta\lambda}(w_{n})\leq\theta\mathcal{E}_{\lambda}(w_{n})-\theta\left(\theta^{(\alpha-2)/2}-1\right)\delta\\\
=\theta
e(\lambda)-\theta\left(\theta^{(\alpha-2)/2}-1\right)\delta+o(1)<\theta
e(\lambda)$

and the proposition follows.

To prove the claim, we assume it does not hold and that there is a sequence
$w_{n}\in\Sigma_{+}$ such that
$\mathcal{E}_{\lambda}(w_{n})<\frac{1}{2}(\frac{\lambda m}{2}+e(\lambda))$ and

$\int F\left(\frac{\psi_{\theta\lambda}(w_{n})}{\sqrt{\theta}}\right)\to 0.$

Then also $\int F(\psi_{\theta\lambda}(w_{n}))\to 0$ and we deduce from (H7)
that for all $\zeta>0$

$\left|\int(\nabla
F(\psi_{\theta\lambda}(w_{n})),w_{n})\right|\leq\int(\zeta+C_{\zeta}F(\psi_{\theta\lambda}(w_{n}))^{\frac{1}{\xi}})\lvert\psi_{\theta\lambda}(w_{n})\rvert\lvert
w_{n}\rvert\\\
\leq\zeta\left|\psi_{\theta\lambda}(w_{n})\right|_{2}\left|w_{n}\right|_{2}+C_{\zeta}\left|\psi_{\theta\lambda}(w_{n})\right|_{\frac{2\xi}{\xi-1}}\left|w_{n}\right|_{\frac{2\xi}{\xi-1}}\int
F(\psi_{\theta\lambda}(w_{n}))\leq C\zeta+o(1)$

which implies that

$\lim_{n\to+\infty}\left|\int(\nabla
F(\psi_{\theta\lambda}(w_{n})),w_{n})\right|=0$

and similarly

$\lim_{n\to+\infty}\left|\int(\nabla
F(\psi_{\theta\lambda}(w_{n})),\eta_{\theta\lambda}(w_{n}))\right|=0$

Since

$0=-\left|\eta_{\theta\lambda}(w_{n})\right|_{2}^{2}\lVert
w_{n}\rVert^{2}-\lVert\eta_{\theta\lambda}(w_{n})\rVert^{2}\\\
-\gamma\int(\nabla
F(\psi_{\theta\lambda}(w_{n})),-\frac{\left|\eta_{\theta\lambda}(w_{n})\right|_{2}^{2}}{a_{\theta\lambda}(\eta_{\theta\lambda}(w_{n}))}w_{n}+\eta_{\theta\lambda}(w_{n}))$

we deduce from the boundedness of $\lVert w_{n}\rVert$ and
$\lVert\eta_{\theta\lambda}(w_{n})\rVert$ that
$\lVert\eta_{\theta\lambda}(w_{n})\rVert\to 0$ and hence

$\displaystyle\frac{\theta}{2}(\frac{\lambda m}{2}$
$\displaystyle+e(\lambda))>\theta\mathcal{E}_{\lambda}(w_{n})\geq\mathcal{E}_{\theta\lambda}(w_{n})$
$\displaystyle=\frac{\theta\lambda}{2}\lVert
w_{n}\rVert^{2}-\frac{1}{2}\left|\eta_{\theta\lambda}(w_{n})\right|_{2}^{2}\lVert
w_{n}\rVert^{2}-\frac{1}{2}\lVert\eta_{\theta\lambda}(w_{n})\rVert^{2}-\gamma\int
F(\psi_{\theta\lambda}(w_{n}))$ $\displaystyle\geq\frac{\theta\lambda
m}{2}+o(1),$

a contradiction which proves the claim. ∎

######  Lemma 4.3.

Let $\omega\in(0,m)$, $C>0$ and $\psi\in X$ be such that

(4.4) $\begin{cases}dI(\psi)[h]=\omega(\psi\,|\,h)\qquad\text{for all }h\in
X\\\ \lVert\psi\rVert\leq C\end{cases}$

Then there exists two positive constants, $c_{1}$ and $c_{2}$, which depend
only on $\omega$ and $C$, such that

$\lVert\psi\rVert\geq c_{1}>0,\qquad I(\psi)\geq c_{2}>0.$

###### Proof.

Taking $h=\Lambda_{+}\psi-\Lambda_{-}\psi$ in (4.4) and $\epsilon>0$ we have

$\displaystyle\lVert\Lambda_{+}\psi\rVert^{2}$
$\displaystyle+\lVert\Lambda_{-}\psi\rVert^{2}+\omega\left|\Lambda_{-}\psi\right|_{2}^{2}=\omega\left|\Lambda_{+}\psi\right|_{2}^{2}+\int(\nabla
F(\psi),\Lambda_{+}\psi-\Lambda_{-}\psi)$
$\displaystyle\leq\frac{\omega}{m}\lVert\Lambda_{+}\psi\rVert^{2}+\int(\epsilon\lvert\psi\rvert+\mu_{\epsilon}\lvert\psi\rvert^{\alpha-1})(\lvert\Lambda_{+}\psi\rvert+\lvert\Lambda_{-}\psi\rvert)$
$\displaystyle\leq\frac{\omega}{m}\lVert\Lambda_{+}\psi\rVert^{2}+2\epsilon\left|\psi\right|_{2}^{2}+\mu_{\epsilon}\left|\psi\right|_{\alpha}^{\alpha-1}(\left|\Lambda_{+}\psi\right|_{\alpha}+\left|\Lambda_{-}\psi\right|_{\alpha})$
$\displaystyle\leq\frac{\omega}{m}\lVert\Lambda_{+}\psi\rVert^{2}+2\epsilon
C_{2}^{2}\lVert\psi\rVert^{2}+\mu_{\epsilon}C_{\alpha}^{\alpha}\lVert\psi\rVert^{\alpha}.$

If $\epsilon>0$ is such that $2\epsilon
C_{2}^{2}<\frac{1-\frac{\omega}{m}}{2}$ we deduce that

$\frac{1}{2}\left(1-\frac{\omega}{m}\right)\lVert\psi\rVert^{2}\leq\mu_{\epsilon}C_{\alpha}^{\alpha}\lVert\psi\rVert^{\alpha}$

and

$\lVert\psi\rVert^{\alpha-2}\geq\frac{1}{2\mu_{\epsilon}C_{\alpha}^{\alpha}}\left(1-\frac{\omega}{m}\right).$

Let us now estimate the critical level:

$\displaystyle I(\psi)$
$\displaystyle=I(\psi)-\frac{1}{2}dI(\psi)[\psi]+\frac{1}{2}\omega\left|\psi\right|_{2}^{2}$
$\displaystyle=\int\left(\frac{1}{2}(\nabla
F(\psi),\psi)-F(\psi)\right)+\frac{1}{2}\omega\left|\psi\right|_{2}^{2}$
$\displaystyle\geq\left(\frac{\alpha}{2}-1\right)\int
F(\psi)+\frac{1}{2}\omega\left|\psi\right|_{2}^{2}.$

We claim that there is $\delta>0$ scuht that $\int F(\psi)\geq\delta>0$ for
all $\psi$ for which (4.4) holds. If not, we find a sequence $\psi_{n}$ of
solutions such that $\int F(\psi_{n})\to 0$. From

$0=dI(\psi_{n})[\Lambda_{\pm}\psi_{n}]-\omega\left|\Lambda_{\pm}\psi_{n}\right|_{2}^{2}$

we deduce that

$\displaystyle\lVert\Lambda_{+}\psi_{n}\rVert^{2}-\omega\left|\Lambda_{+}\psi_{n}\right|_{2}^{2}=\int(\nabla
F(\psi_{n}),\Lambda_{-}\psi_{n})$
$\displaystyle\lVert\Lambda_{-}\psi_{n}\rVert^{2}+\omega\left|\Lambda_{-}\psi_{n}\right|_{2}^{2}=-\int(\nabla
F(\psi_{n}),\Lambda_{-}\psi_{n}).$

As in the proof of 4.2, using (H7) we can deduce from $\int F(\psi_{n})\to 0$
that

$\int(\nabla F(\psi_{n}),\Lambda_{\pm}\psi_{n})\to 0$

and hence also

$\lVert\psi_{n}\rVert\to 0,$

contradiction which proves the claim. ∎

###### Proof of theorem.

By Ekeland’s variational principle, there exists a sequence
$w_{n}\in\Sigma_{+}$ such that

$\mathcal{E}_{1}(w_{n})\to
e(1),\qquad\sup_{v\in\Sigma_{+}}\left|d\mathcal{E}_{1}(w_{n})[v]\right|\to 0.$

Since $\mathcal{E}_{1}(w_{n})\to e(1)$ we deduce from lemma 3.11 that the
sequence $w_{n}$ is bounded. Follows from proposition 3.2 that also
$\eta_{n}=\eta(w_{n})$ and $\psi_{n}=a(\eta_{n})w_{n}+\eta_{n}$ are bounded in
$X$. Letting
$\omega_{n}=a(\eta_{n})^{-1}dI(\psi_{n})[w_{n}]\geq(1-C_{\alpha,1})$ we have
that

$dI(\psi_{n})[h]-\omega_{n}(\psi_{n}\,|\,h)=\frac{1}{a(\eta_{n})}d\mathcal{E}_{\lambda}(w_{n})[v]\to
0\qquad\text{for all }h=v+\xi\in X.$

We can assume that (up to a subsequence) $\psi_{n}\rightharpoonup\psi$ in $X$
and that $\omega_{n}\to\omega$, with
$\omega\in(1-C_{\alpha,1},2\mathcal{E}_{1}(w_{n}))\subset(1-C_{\alpha,1},m)$
for $n$ sufficiently large. Then we have that for all $h\in X$

$dI(\psi_{n})[h]-\omega_{n}(\psi_{n}\,|\,h)=\langle\psi_{n}\,|\,\Lambda_{+}h\rangle-\langle\psi_{n}\,|\,\Lambda_{-}h\rangle\\\
-\gamma\int(\nabla F(\psi_{n}),h)-\omega_{n}(\psi_{n}\,|\,h)\to 0$

since we have that

$\int(\nabla F(\psi_{n})-\nabla F(\psi),h)\to 0$

we deduce that

$dI(\psi)[h]-\omega(\psi\,|\,h)=0\qquad\text{for all }h\in X.$

The weak convergence does not, however, preserve the $L^{2}$ norm, so we only
know that $\left|\psi\right|_{2}\leq\left|\psi_{n}\right|_{2}=1$ (it could
even be that $\psi=0$).

To conclude we will now apply the concentration-compactness principle, see
[17, 18]. First of all let us show that no vanishing occurs. By contradiction,
assume that

$\limsup_{n\to+\infty}\sup_{y\in\mathbb{R}^{3}}\int_{B(y,1)}\lvert\psi_{n}\rvert^{2}=0.$

Then we know, see [17] or [24, Lemma 1.21], that $\psi_{n}\to 0$ in
$L^{p}(\mathbb{R}^{3})$ for $2<p<3$. Taking
$q\in(\max\\{2,\frac{3}{4-\alpha}\\},\min\\{3,\frac{2}{3-\alpha}\\})$ we have
that for any $\epsilon>0$

$\left|\int(\nabla
F(\psi_{n}),w_{n})\right|\leq\epsilon\int\lvert\psi_{n}\rvert\lvert
w_{n}\rvert+\mu_{\epsilon}\int\lvert\psi_{n}\rvert^{\alpha-1}\lvert
w_{n}\rvert\leq\epsilon+\mu_{\epsilon}\left|\psi_{n}\right|_{\frac{q(\alpha-1)}{q-1}}^{\alpha-1}\left|w_{n}\right|_{q}$

and

$\lim_{n\to+\infty}\left|\int(\nabla F(\psi_{n}),w_{n})\right|=0.$

We deduce from

$\epsilon_{n}=dI(\psi_{n})[w_{n}]-\omega_{n}a(\eta_{n})\left|w_{n}\right|_{2}^{2}=a(\eta_{n})(\lVert
w_{n}\rVert^{2}-\omega_{n}\left|w_{n}\right|_{2}^{2})-\int(\nabla
F(\psi_{n}),w_{n})$

that for all $\lambda\in(0,1]$

$\epsilon_{n}=a(\eta_{n})(\lVert
w_{n}\rVert^{2}-\omega_{n}\left|w_{n}\right|_{2}^{2})\geq
a(\eta_{n})(m-\omega_{n})\left|w_{n}\right|_{2}^{2}\geq\sqrt{\frac{1}{2}}(m-\omega_{n})>0$

a contradiction since, by proposition 3.15 and lemma 4.1 we have that
$\omega_{n}\leq 2\mathcal{E}_{1}(w_{n})$ and $e(1)<\frac{m}{2}$. Hence
vanishing does not occur.

Since $\lVert\psi\rVert\leq\lVert\psi_{n}\rVert$ and $I(\psi)\geq c_{2}$ (by
lemma 4.3) we deduce from the concentration compactness principle that there
exists $p\geq 1$ functions $\phi_{1}$, …, $\phi_{p}\in X$, critical points for
$I$ under the constraint $\left|\psi\right|_{2}^{2}=\mu_{i}\in(0,1]$
(satisfying (3.16) with $\omega=\lim_{n}\omega_{n}>0$) and $p$ sequences of
points $x_{i,n}\in\mathbb{R}^{3}$, $i=1$, …, $p$ such that $\lvert
x_{i,n}-x_{j,n}\rvert\to+\infty$ for all $i\neq j$ as $n\to+\infty$ and

$\lVert\psi_{n}-\sum_{i=i}^{{}^{p}}\phi_{i}(\cdot-x_{i,n})\rVert\to
0\quad\text{as }n\to+\infty.$

From this follows also that
$\left|\psi_{n}\right|_{2}^{2}=1=\sum_{i=1}^{p}\mu_{i}$.

We then observe that

$\displaystyle\lVert\Lambda_{+}\psi_{n}\rVert^{2}-$
$\displaystyle\lVert\Lambda_{-}\psi_{n}\rVert^{2}=\langle\psi_{n}\,|\,\Lambda_{+}\psi_{n}-\Lambda_{-}\psi_{n}\rangle$
$\displaystyle=\langle\psi_{n}-\sum_{i=i}^{{}^{p}}\phi_{i}(\cdot-
x_{i,n})\,|\,\Lambda_{+}\psi_{n}-\Lambda_{-}\psi_{n}\rangle$
$\displaystyle\qquad+\sum_{i=i}^{{}^{p}}\langle\phi_{i}(\cdot-
x_{i,n})\,|\,\Lambda_{+}\psi_{n}-\Lambda_{-}\psi_{n}\rangle$
$\displaystyle=\sum_{i=i}^{{}^{p}}\left(\langle\Lambda_{+}\phi_{i}(\cdot-
x_{i,n})\,|\,\psi_{n}\rangle-\langle\Lambda_{-}\phi_{i}(\cdot-
x_{i,n})\,|\,\psi_{n}\rangle\right)+o(1)$
$\displaystyle=\sum_{i=i}^{{}^{p}}\left(\lVert\Lambda_{+}\phi_{i}\rVert^{2}-\lVert\Lambda_{-}\phi_{i}\rVert^{2}\right)+o(1)$

and also

$\int F(\psi_{n})=\sum_{i=1}^{p}\int F(\phi_{i}(x))+o(1).$

(this can be deduced from the Brezis-Lieb lemma, see [3, Theorem 2]). Finally
we have that

(4.5) $e(1)=I(\psi_{n})+o(1)=\sum_{i=1}^{p}I(\phi_{i})+o(1)$

and

$0=dI(\phi_{i})[h]-\omega(\phi_{i}\,|\,h)\qquad\text{for all }h\in X,$

with $\omega>0$. Follows from proposition 3.15 that
$w_{i}=\left|\Lambda_{+}\phi_{i}\right|_{2}^{-1}\Lambda_{+}\phi_{i}\in\Sigma_{+}$
is a critical point for $\mathcal{E}_{\mu_{i}}$ and
$\mathcal{E}_{\mu_{i}}(w_{i})=I(\phi_{i})$.

Since

$\mathcal{E}_{\mu_{i}}(w_{i})\geq e(\mu_{i})$

we deduce from (4.2) that if $p>1$ (i.e. if dichotomy occurs)

$e(1)=\sum_{i=1}^{p}I(\phi_{i})+o(1)=\sum_{i=1}^{p}\mathcal{E}_{\mu_{i}}(w_{i})+o(1)\geq\sum_{i=1}^{p}e(\mu_{i})+o(1)\\\
>\sum_{i=1}^{p}\mu_{i}e(\frac{1}{\mu_{i}}\mu_{i})+o(1)=e(1)\sum_{i=1}^{p}\mu_{i}+o(1),$

a contradiction.

Since there is no vanishing or dichotomy, our sequence $\psi_{n}$ converges –
up to a translation – strongly in $X$ to a critical point $\psi\in X$ of
(3.16) such that $\left|\psi\right|_{2}=1$ and the theorem follows.

∎

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