A Pohožaev minimization for normalized solutions: fractional sublinear equations of logarithmic type

Abstract

In this paper, we search for normalized solutions to a fractional, nonlinear, and possibly strongly sublinear Schrödinger equation
(-\Delta)^s u + \mu u = g(u) \quad \hbox{in RN\mathbb{R}^N},
under the mass constraint \int_{\mathbb{R}^N} u^2 \, \dx = m>0; here, N≥2N\geq 2, s∈(0,1)s \in (0,1), and μ\mu is a Lagrange multiplier.
We study the case of L2L^2-subcritical nonlinearities g of Berestycki--Lions type, without assuming that g is superlinear at the origin, which allows us to include examples like a logarithmic term g(u)=ulog⁡(u2)g(u)= u\log(u^2) or sublinear powers g(u)=uq−urg(u)=u^q-u^r, 0<r<1<q0<r<1<q.
Due to the generality of g and the fact that the energy functional might be not well-defined, we implement an approximation process in combination with a Lagrangian approach and a new Pohožaev minimization in the product space, finding a solution for large values of m.
In the sublinear case, we are able to find a solution for each m.
Several insights on the concepts of minimality are studied as well. We highlight that some of the results are new even in the local setting s=1 or for g superlinear.

Full text

A Pohoºaev minimization for normalized solutions:
fractional sublinear equations of logarithmic type
Marco Gallo
Università Cattolica del Sacro Cuore
Dipartimento di Matematica e Fisica
Via della Garzetta 48, 25133 Brescia, Italy
marco.gallo1@unicatt.it
Jacopo Schino
University of Warsaw
Faculty of Mathematics, Informatics and Mechanics
ul. Banacha 2, 02-097 Warsaw, Poland
j.schino2@uw.edu.pl
Abstract
In this paper, we search for normalized solutions to a fractional, nonlinear, and possibly
strongly sublinear Schrödinger equation
(−∆)su+µu =g(u)
in
RN,
under the mass constraint
RRNu2dx=m > 0
; here,
N≥2
,
s∈(0,1)
, and
µ
is a Lagrange
multiplier. We study the case of
L2
-subcritical nonlinearities
g
of BerestyckiLions type,
without assuming that
g
is superlinear at the origin, which allows us to include examples
like a logarithmic term
g(u) = ulog(u2)
or sublinear powers
g(u) = uq−ur
,
0< r < 1< q
.
Due to the generality of
g
and the fact that the energy functional might be not well-dened,
we implement an approximation process in combination with a Lagrangian approach and a
new Pohoºaev minimization in the product space, nding a solution for large values of
m
. In
the sublinear case, we are able to nd a solution for each
m
. Several insights on the concepts
of minimality are studied as well. We highlight that some of the results are new even in the
local setting
s= 1
or for
g
superlinear.
Keywords:
NLS equations, Fractional Laplacian, Sublinear nonlinearity, Nonsmooth analysis,
L2
-norm
constraint, Prescribed-mass solutions, Least-energy solutions, Ground states.
AMS Subject Classication:
35B06, 35B09, 35B38, 35D30, 35J20, 35Q40, 35Q55, 35R09, 35R11.
Contents
1 Introduction 2
1.1 Someliterature..................................... 3
1.2 Existenceresults .................................... 5
1.3 Qualitative results and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Preliminaries 13
1

3 The mass-free problem 15
4 Some general properties 20
4.1 Connectionamonginma ............................... 20
4.2
L2
-minima, Pohoºaev identity, and critical points . . . . . . . . . . . . . . . . . . 21
4.3 Nehariidentity ..................................... 23
5 The superlinear (perturbed) problem 23
5.1
ε
xed: some renement for the superlinear problem . . . . . . . . . . . . . . . . 24
5.2 Symmetryresults.................................... 30
6 The sublinear problem 31
6.1
ε
-independentestimates ................................ 31
6.2 Passagetothelimit .................................. 32
7 Further existence results 35
7.1 Existenceforallmasses ................................ 35
7.2 Existence under relaxed assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 37
References 38
1 Introduction
In this paper, we study the fractional
L2
-subcritical equation
(−∆)su+µu =g(u)
in
RN
(1.1)
under the mass constraint
RRNu2dx=m > 0
; here,
s∈(0,1)
,
N > 2s
,
µ∈R
is a suitable
Lagrange multiplier (part of the unknowns), and
g
is a generic BerestyckiLions [8] nonlinearity
which is allowed to be (negatively)
sublinear
at the origin (see (g0)(g4) below). In particular,
G(t) := Rt
0g(τ) dτ
is allowed to have a growth of the type
−|t|1+θ− |t|2∗
s≲G(t)≲|t|2+|t|¯p+1
for
t∈R,
where
θ∈[0,1)
,
2∗
s:= 2N
N−2s
is the fractional Sobolev critical exponent, and
¯p:= 1 + 4s
N;
(1.2)
the number
¯p+ 1
is often referred to as the
L2
-critical exponent
. This case includes, for example,
the well-known
logarithmic
nonlinearity
g(t) = tlog(t2)
(here
θ∈[0,1)
can be arbitrarily chosen),
or
sublinear-power
-type nonlinearities
g(t) = |t|q−1t− |t|r−1t, 0< r < q < ¯p
(here
θ=r
), but
also strongly sublinear nonlinearities like
g(t)∼1
log(|t|)
as
t→0
(here
θ= 0
).
Exploiting variational methods, a key fact of these nonlinearities is that the related energy
functional
K:Hs(RN)→R∪ {+∞}
,
K(u) := 1
2ZRN
|(−∆)s/2u|2dx−ZRN
G(u) dx,
(1.3)
is only lower semicontinuous, while it is well-posed in
Hs(RN)∩L1(RN)
, which we highlight is
not a reexive space. Here
Hs(RN)
denotes the usual fractional Sobolev space.
2

1.1 Some literature
When
s= 1
, the
frequency
µ∈R
is xed, and
g(t) = tlog(t2)
, the
logarithmic Schrödinger
equation
[9,10]
−∆u+µu =ulog(u2)
in
RN
(1.4)
has received much attention due to its great relevance in applied sciences, such as atomic physics,
high-energy cosmic rays, Cherenkov-type shock waves, quantum hydrodynamical models, and
many others; we refer to [15,33] and the references therein. A key property of (1.4) is, indeed, the
tensorization property
(or
separability of noninteracting subsystems
): namely, if
ui
are solutions
of (1.4) in
RNi
for
i= 1,2
, then
(x1, x2)7→ u1(x1)u2(x2)
is a solution of (1.4) in
RN1×RN2
.
In order to handle the singularity at the origin, several techniques have been developed: a
rst approach, applied by [16], shows that the functional
K
in (1.3) is of class
C1
on the subspace
X:= u∈H1(RN)ZRN
u2log(u2) dx < +∞;
see also [50,56], and [12] for general doubling nonlinearities. We observe that
X⊊H1(RN)
, as
shown by
u(x) := (|x|N/2log(|x|))−1
for
|x|
large. Finally, in [41], a suitable norm is introduced
in order to force the positive solutions far from the singularity.
A second approach has been developed in [24,53]: here, the authors apply the
nonsmooth
analysis
theory of [26,54] and work with a suitable subdierential of the action functional,
possibly splitting it into a smooth part and a convex lower semicontinuous one.
Dierent approaches rely instead on some
approximation schemes
. The idea that
t2+δ−t2
δ→
log(t)
as
δ→0+
was formalized in the framework of PDEs by [58] (see also [31], [15, Section
7]), and then this idea was applied in [59] to get existence of solutions. See also [33], where the
regularization is of the form
log(t+δ)
, and [1], where the authors approximate the domain of
the problem with expanding balls.
We mention that the problem of the singularity of the logarithm arises also in the study of
planar Choquard equations [22,39].
Let us move, now, to the case of sublinear powers. Although nonlinear (fractional) Schrödinger
equations with combined powers have recently gained the mathematical community's attention
[27,42,52,55], such work usually concerns superlinear exponents. Instead, as regards sublinear-
power nonlinearities (again,
s= 1
and
µ
xed), most of the work focuses on problems of the
type
−∆u+V(x)u=a(x)|u|q−1u+h(x)
in
RN,
q∈(0,1)
, where suitable assumptions on
V, a, h
are considered in order to ensure the existence
of solutions (notice, in fact, that when
V≡a≡1
and
h≡0
, no solution exists, see e.g. [6,
Lemma 2.3]). When
a∈ L2
q−2(RN)
, the analysis of the problem is usually set in the Banach
space
H1(RN)∩Lq(RN)
; see [5,6,57].
The only paper we know that deals with
general nonlinearities
in the spirit of [8] is [44]: here,
the author studies
−∆u=f(u)
in
RN,
where the assumptions about
f
include the case
limt→0f(t)
|t|2∗−2t=−∞
. The author employs a
perturbation argument at the origin and obtains a ground state for the equation.
When
s= 1
and
µ
is free, very few results appear in the literature about the existence of
normalized solutions
RRNu2dx=m > 0
. When
g(t) = tlog(t2)
, we see that, considered a
nontrivial solution
u1
of (1.4) for
µ= 1
, then dening
u:= αmu1
with
αm, := m(RRNu2
1dx)−1
,
we have that
u
satises
RRNu2dx=m
and
−∆u+1 + log(α2
m)u=ulog(u2)
in
RN
, (1.5)
3

thus the existence of a normalized solution is always ensured; this was already noticed in [10].
In particular, the frequency
1 + log(αm)
is negative when
m
is suciently small and positive
when
m
is suciently large. Additionally, an explicit
L2
-minimum can be found. Indeed, by the
logarithmic SobolevGrossNelson inequality [32] there exists
C(N)∈R
such that
K(u)≥ − (C(N) + log(m)) m
(1.6)
for every
u∈H1(RN)
with
RRNu2dx=m
; moreover, we know that the equality is attained by
(and only by, up to a translation) the
Gausson
u(x) = γme−1
2|x|2
with
γm:= mRRNe−|x|2dx−1
.
This procedure can be found, for example, in [9,16]. We see that the sign of the minimal energy
over the
L2
-sphere depends on the size of
m
(and this is connected with the sign of the Lagrange
multiplier, see Proposition 1.8).
When
g
does not enjoy scaling properties of this type, we mention the very recent papers
[2,45,51,60]. In [51] and [2], the authors study, respectively, logarithm-plus-power and nonau-
tonomous problems by restricting to the subspace
X
. In [60], via a direct minimization, the
authors study a similar equation, but with a more general
g
satisfying a monotonicity assump-
tion: they nd an
L2
-minimum for every mass
m
, which, in particular, has positive energy for
m
small and negative energy for
m
large. Finally, by exploiting the perturbation techniques
developed in [44], the authors in [45] succeed in removing the monotonicity condition on
g
, and
nd an
L2
-minimum for
m
large, with a positive multiplier
µ > 0
.
When
s∈(0,1)
, the fractional logarithmic equation
(−∆)su+µu =ulog(u2)
in
RN
(1.7)
appears in the study of nonlinear quantized boson elds, [48]. Mathematically, several papers
succeeded in generalizing the results found in the local case for sublinear nonlinearities, exploiting
and adapting to the nonlocal case the aforementioned techniques, see e.g. [4,25,37] for the
logarithmic equation, and [29] for logarithm-plus-power nonlinearities. In particular, we mention
[35]: here, the author, with a perturbation argument similar to the one in [44], nds innitely
many solutions with a general nonlinear term.
Anyway, all the previous results deal with the unconstrained problem with xed frequency.
When
g(t) = tlog(t2)
, the search for a normalized solution can be pursued as above by scaling
(see (1.5)); on the other hand, the search for an
L2
-minimum through an optimal logarithmic
inequality (1.6) has issues. Indeed, although such an inequality still holds in the fractional
framework [17,23], i.e., there exists
C(N, s)∈R
such that
K(u)≥ − (C(N, s) + log(m)) m
for every
u∈Hs(RN)
with
RRNu2dx=m
, it is not known if the equality is attained or not.
To the authors' knowledge, the only articles dealing with normalized solutions to logarithmic
fractional equations are the recent works [38,43], where semiclassical problems are studied.
The goal of this paper is to nd normalized solutions with minimality properties to fractional
sublinear equations under general assumptions about the nonlinear term. When treating such
general terms
g
, as previously commented, the splitting of the functional does not seem an option,
as well as the denition of a space similar to
X
. To obtain the existence of an
L2
-minimum we
rely, thus, on the perturbation technique employed by [44,45]: this perturbation takes the form
gε(t) := g+(t)−ϕε(t)g−(t)
, where
ϕε(t) = |t|/ε
for
|t| ≤ ε
(see Section 5for details). Additionally,
we obtain several properties that are new even for
s= 1
and
g
superlinear.
4

1.2 Existence results
In the present paper, we study the following problem







(−∆)su+µu =g(u)
in
RN,
ZRN
u2dx=m,
(µ, u)∈R×Hs(RN),
(1.8)
with
s∈(0,1)
,
N≥2
,
m > 0
, and
g:R→R
satisfying (set
G(t) := Rt
0g(τ) dτ
)
(g0)
g
is continuous and
g(0) = 0
,
(g1)
lim sup
t→0
g(t)
t≤0
,
(g2)
lim sup
|t|→+∞
|g(t)|
|t|2∗
s−1<∞
,
(g3)
lim sup
|t|→+∞
g(t)
|t|¯p−1t≤0
,
(g4) there exists
t0= 0
such that
G(t0)>0
.
Here and in what follows, we denote
g±:= max{±g, 0}
the standard positive and negative parts
of
g
, and
g+(t)t:= (g(t)t)+, G+(t) := Zt
0
g+(τ) dτ,
(1.9)
g−(t)t:= (g(t)t)−, G−(t) := Zt
0
g−(τ) dτ;
(1.10)
see Section 2. With these notations, we may rewrite (g1) and (g3) as
(g1)
lim
t→0
g+(t)
t= 0
,
(g3)
lim
|t|→+∞
g+(t)
|t|¯p−1t= 0
.
We further notice that, when (g3) holds, then (g2) actually means
lim sup|t|→+∞
g−(t)
|t|2∗
s−2t<∞
.
Before presenting our results, let us recall that in [18, Proposition 4.1] and [14, Proposition
1.1] (see also [21, Corollary 6.4]), when
g
satises BerestyckiLions assumptions, the following
Pohoºaev identity
holds
1
2∗
sZRN
|(−∆)s/2u|2dx+µ
2ZRN
u2dx−ZRN
G(u) dx= 0
(1.11)
whenever
u
is a solution of (1.8) and either
s∈(1
2,1)
or
g∈ C0,α
loc (RN)
with
α∈(1 −2s, 1)
:
this result can be generalized to our assumptions whenever
RRNG(u) dx
is assumed a priori
nite; however, we see that the condition on the Hölder regularity of
g
forces a restriction on
the singularity of
g−
at the origin. It is not known if this identity holds for general continuous
nonlinearities
g
and general values of
s∈(0,1)
. Additionally, we observe that, due to the possible
5

sublinearity of
g
, for a general solution
u
of equation (1.8) we cannot state a priori that it satises
the
Nehari identity
ZRN
|(−∆)s/2u|2dx+µZRN
u2dx−ZRN
g(u)udx= 0.
(1.12)
To state the main theorem, we make use of a
Lagrangian formulation
of the problem (in the
spirit of [20,34]) and introduce, in addition to (1.3), a Lagrangian functional over the product
space:
Im:R×Hs(RN)→R
,
Im(µ, u) := 1
2ZRN
|(−∆)s/2u|2dx+µ
2ZRN
u2dx−m−ZRN
G(u) dx.
(1.13)
Notice that critical points of
Im
are (formally) solutions to (1.8) and
Im(µ, u) = K(u)
if
RRNu2dx=m
.
From now on, to state our results we will use the following convention.
Convention:
whenever we deal with a pair
(µ, u)
(or
(λ, u)
with
eλ=µ
, see (5.5)) and
we say that such a pair is radially symmetric, nonnegative, does not change sign, etc.,
we are
referring to
u
.
We introduce the following notations, standing for the
L2
-sphere in
Hs(RN)
and the Pohoºaev
set in the product space
R×Hs(RN)
:
Sm:= u∈Hs(RN)| |u|2
2=m,
(1.14)
where
|·|q
is the usual Lebesgue norm,
1≤q≤ ∞
, and
P:= (µ, u)∈R×Hs(RN)|u= 0
and
P(µ, u) = 0,
(1.15)
where
P:R×Hs(RN)→R
is the Pohoºaev functional
1
P(µ, u) := ZRN
|(−∆)s/2u|2dx+ 2∗
sZRN
µ
2u2−G(u) dx.
(1.16)
It is clear that
(µ, u)∈ P
if and only if it satises the Pohoºaev identity (1.11). Moreover,
P =∅
because, xed
u∈Hs
rad
(RN)\ {0}
,
P(·, u)
is a continuous function such that
limµ→±∞ P(µ, u) =
±∞
. Next, we set
S
rad
m:= Sm∩Hs
rad
(RN),P
rad
:= P ∩ R×Hs
rad
(RN),
where
Hs(RN)
is the classical fractional Sobolev space, cf. (2.1), and
Hs
rad
(RN)
is its subspace
of radially symmetric functions, and
κm:= inf
S
rad
m
K, dm:= inf
P
rad
Im;
(1.17)
here
K
and
Im
are dened in (1.3) and (1.13). We further dene
µ0:= sup
t=0
G(t)
t2/2;
(1.18)
1
We highlight that the terminology Pohoºaev functional, in the framework of normalized solutions, is often
referred to a suitable combination of the Pohoºaev and Nehari functionals, which allows getting rid of the Lagrange
multiplier
µ
; such a functional appears, in particular, in the framework of
L2
-supercritical problems. Here, instead,
we focus on the proper Pohoºaev identity, considering a two-variable functional.
6

notice that if (g4) holds, then
µ0∈(0,+∞]
. Moreover, we introduce the
L2
-ball and the
corresponding inmum:
Dm:= u∈Hs(RN)| |u|2
2≤m,D
rad
m:= Dm∩Hs
rad
(RN), m:= inf
D
rad
m
K;
(1.19)
we mention that the use of the
L2
-ball in normalized problems goes back to [11], while it was
utilized for the rst time in the
L2
-subcritical regime in [49]. We can thus state our rst theorem,
see Denition 2.4 for the notion of solution.
Theorem 1.1 (Existence for large masses)
If
(g0)

(g4)
hold, then there exists
m0≥0
such
that for every
m>m0
there exists
(µ0, u0)∈(0,∞)× S
rad
m
with the following properties:
(i)
(µ0, u0)
is a solution to
(1.8)
, and it satises the Nehari identity
(1.12)
;
(ii)
u0
is a minimum (with Lagrange multiplier
µ0
) on the
L2
-sphere and the
L2
-ball, i.e.,
K(u0) = κm=m<0
; moreover, we have the formula
κm= inf
−∞<µ<µ0a(µ)−µm
2=a(µ0)−µ0
m
2,
(1.20)
where
a(µ)
is the least action related to the problem with xed frequency
µ
, see
(3.3)
.
(iii)
(µ0, u0)
is a Pohoºaev minimum on the product space, i.e.,
(µ0, u0)∈ P
rad
and
Im(µ0, u0) =
dm
; in particular,
(µ0, u0)
satises the Pohoºaev identity
(1.11)
.
As a consequence of the above relations, we have
κm=dm
.
Under suitable additional assumptions about
g
, we highlight that the found solution can be
chosen nonnegative and radially nonincreasing, see Proposition 1.12.
Remark 1.2
We observe the following facts.

The mass threshold
m0
is given by formula
(5.12)
.

Notice that, thanks to our method, we are able to state that the found Lagrange multiplier
µ0
is positive and the energy
κm
is negative. At the same time, the found solution
(µ0, u0)
minimizes
Im
in the whole Pohoºaev space over
R×Hs
rad
(RN)
, not only over
(0,+∞)×
Hs
rad
(RN)
.

In Theorem 1.1, the solution is found for
m
suciently large. In Theorem 1.3 below, we
nd a solution for each
m > 0
; on the other hand, in that case, we have no information
on the sign of the Lagrange multiplier or that of the energy, nor do we know whether the
solution minimizes the energy over
D
rad
m
(which is coherent with Proposition 1.5 below).

We highlight that we cannot state that
u0
is a critical point of
Im(µ0,·)
due to the lack of
regularity of the functional, but only that
Im(µ0,·)
is dierentiable at
u0
along test functions
in
Hs(RN)∩L1(RN)
, cf. Lemma 2.3. In particular, the validity of the Nehari identity in
Theorem 1.1 is not straightforward; in this case, thanks to our method, we are able to obtain
such an identity.

We observe that
(1.20)
is a Legendre transform-type formula which relates the
L2
-minimum
value to the Pohoºaev minimum value (with xed frequency); similar formulas were already
obtained in [28,30,34].
7


We notice that, xed
ρ > 0
, we can nd a threshold
mρ≥m0
such that, for each
m > mρ
there exists
(µρ, uρ)∈(ρ, +∞)× S
rad
m
solution of
(1.8)
. Indeed, it is sucient to apply
Theorem 1.1 to the nonlinearity
gρ(t) := g(t)−ρt
, which satises
(g0)

(g4)
.
Unlike [45], in our fractional setting, two main diculties come into play. The rst is due to
the nonlocality of the problem: the fact that the operator
(−∆)s
does not preserve the supports of
functions creates indeed problems in a direct minimization as in [45]. To deal with this issue, we
handle the perturbed problem through a Lagrangian formulation (1.13) as done in [20,34]: this
approach has additional advantages, like showing some geometrical properties of the solutions,
such as the minimality of
P
rad
, formula (1.20), and the form of the threshold
m0
; this last, in
turn, allows us to show the existence of a solution for every mass
m
(see Theorem 1.3 below).
The second diculty is due to the absence of a Pohoºaev identity: as we already pointed
out, it is an open problem to determine if the Pohoºaev identity holds for general continuous
nonlinearities
g
and general values of
s∈(0,1)
. Under our assumptions (g0)(g4), thus, we are
not able to use this identity. As already mentioned, we rst study the superlinear problem (with
an
ε
-perturbation) and pass to the limit: by construction, the solutions
uε
that we build satisfy
the Pohoºaev identity, which gives some information to start with. On the other hand, in the
limiting process, we get
uε u0
, where
u0
can be proved to be a solution, but with no a priori
minimality properties: in [45], it is thus crucially used, in order to pass to the strong limit and
get the existence of an
L2
-minimum, that the Pohoºaev identity holds for arbitrary solutions.
This obstruction is here overcome by a ner analysis on the information of
uε
, in particular,
regarding the interplay between the minimization over the
L2
-sphere and the
L2
-ball.
To deal with the superlinear perturbed prescribed-mass problem, in the spirit of [20,34], we
rst study the free problem (i.e., when
µ
is xed), following essentially [14,18]: here we revise
and rene both the existence results contained in [14,20]. Additionally, since we are interested
in the existence of a single solution, we simplify the argument of [20] by nding the solution
through a
direct minimization on the Pohoºaev set
(1.15): as a matter of fact, the minimality of
the solution in [20] was obtained as a by-product of a minimax approach (see Remark 5.4), which
requires a nontrivial deformation theory. Here we directly study the convergence of a (precisely
chosen) minimizing sequence on
P
rad
: this procedure is totally unrelated with the perturbation
framework (see Remark 5.3), and it seems a new approach in literature in regards of product
spaces, thus it has an interest of its own.
In Theorem 1.1, we show the existence of a minimizer for
m > 0
suciently large; when the
mass is small, by looking at the pure logarithmic case we expect an
L2
-minimum, possibly with
a Lagrange multiplier and an energy with signs dierent from the ones in Theorem 1.1. This is
the content of the next theorem (see also the more general Theorem 7.2). This result gives a
fractional counterpart of [60], as well as a generalization for
s= 1
by dropping the monotonicity
therein.
Theorem 1.3 (Existence for all masses)
If
(g0)

(g4)
hold and, moreover,
lim
t→0
g(t)
t=−∞,
(1.21)
then for every
m > 0
there exists a solution
(µ0, u0)∈R× S
rad
m
of
(1.8)
satisfying the Nehari
and Pohoºaev identities and such that
Im(µ0, u0) = K(u0) = a(µ0)−µ0
2m=κm=dm= inf
−∞<µ<µ0a(µ)−µ
2m.
Finally, if
κm≤0
, then
µ0>0
.
8

We remark that in Theorem 1.3, we are not able to determine the sign of the Lagrange multiplier
µ0
and of the energy
κm
. To obtain Theorem 1.3 we make use of a shifted problem, which still
falls into the assumptions of Theorem 1.1 (here the generality of (g1) is essential), and rely on
the representation formula (5.12) for the mass threshold
m0
.
Remark 1.4 (On the multiplicity)
We highlight that the present approach seems suitable for
generalizations in the
L2
-critical (see [19]) and supercritical cases. Moreover, by following the
minimax ideas in [20], one may have the additional advantage to possibly obtain a multiplicity
result: indeed, for each
ε > 0
, with the same techniques (and genus arguments, see Remark 5.4)
we can get the existence of multiple solutions
(uε
k)k∈N
; on the other hand, at the moment, we are
not able to exclude the possibility that these solutions collapse to the same solution of the original
problem when
ε→0+
; we expect that some ner topological information on the genus of these
solutions is needed.
1.3 Qualitative results and examples
We present now some outcomes about the interplay among the various concepts of minimality
we introduced, as well as some symmetry results. Most of them will not require (g3).
As highlighted in Subsection 1.1, we recall that, in some cases, the Lagrange multiplier of
the equation might be negative. In the following, thus, for
ρ∈[−∞,+∞)
we dene
P(ρ):= (µ, u)∈(ρ, +∞)×Hs(RN)|u= 0
and
P(µ, u)=0
(1.22)
and
P
rad
(ρ):= P(ρ)∩Hs
rad
(RN)
. Observe that
P(−∞)=P
.
In the following propositions, we are not claiming that the mentioned inma are attained or
nite merely under (g0)(g2). On the other hand, minimizers can exist under dierent assump-
tions than those in Theorems 1.1 or 1.3: for instance, with proper hypotheses on
m
and
g
, they
exist when
lim sup|t|→+∞g(t)/(|t|¯p−1t)<+∞
(which is weaker than (g3)); see, e.g., [42].
Under suitable assumptions, the several notions of minimality we introduced coincide; notice
that when
m>m0
, we have
κm<0
, which is a particular case of such assumptions.
Proposition 1.5 (Connections among inma)
Assume
(g0)

(g2)
. We have
inf
Sm
K= inf
PIm.
(1.23)
If
ρ∈R
and
infSmK < −ρ
2m
, then
inf
Sm
K= inf
PIm= inf
P(ρ)
Im.
If
ρ≥0
and
infDmK < −ρ
2m
, then
inf
Dm
K= inf
Sm
K= inf
PIm= inf
P(ρ)
Im.
In particular, if
infDmK < 0
we have
infDmK= infSmK.
Finally, for every
ρ∈[−∞,+∞)
, we have
inf
P(ρ)
Im= inf Im(µ, u)|(µ, u)∈(ρ, +∞)×Hs(RN), u = 0, P (µ, u)≤0.
We show now a generalization of (1.20).
9

Proposition 1.6 (Legendre transform formula)
Assume
(g0)

(g2)
and let
ρ∈[−∞, µ0)
.
Then,
inf
P(ρ)
Im= inf
ρ<µ<µ0inf
Pµ
Jµ−µ
2m.
We move on to show that Pohoºaev minima are indeed always weak solutions.
Proposition 1.7 (Pohoºaev minima are solutions)
Assume
(g0)

(g2)
and let
ρ∈[−∞,+∞)
.
If
(µ, u)∈ P(ρ)
satises
Im(µ, u) = minP(ρ)Im
, then it is a solution to
(1.8)
. Moreover, if
minP(ρ)Im<−ρ
2m
, then
Im(µ, u) = K(u) = infSmK
.
We show now that if
u
is an
L2
-minimum, then
u
satises the Pohoºaev identity; this is
obtained without assuming extra regularity. See also Proposition 4.3 for some results on the
Nehari identity.
Proposition 1.8 (Pohoºaev identity for
L2
-minima)
Assume
(g0)

(g2)
. Let
u∈ Sm
be
such that
K(u) = infSmK
. Then,
u
is a solution of
(1.1)
for some Lagrange multiplier
µ∈R
,
and
(µ, u)
satises the Pohoºaev identity. Moreover, for every
ρ∈[−∞,−2
minfSmK)
, we have
K(u) = Im(µ, u) = infP(ρ)Im
.
As a consequence of Propositions 1.7 and 1.8 we have the following statement.
Corollary 1.9
Under
(g0)

(g2)
, a function
u∈Hs(RN)
is an
L2
-minimum with Lagrange
multiplier
µ∈R
if and only if
(µ, u)
is a Pohoºaev minimum over
P
.
Finally, we present a fractional and possibly strongly sublinear counterpart of [36]: notice
that here we deal with Pohoºaev minima instead of classical ground state solutions, which are
indeed equivalent notions in the case treated in [36].
Proposition 1.10 (Least energy versus least action)
Assume
(g0)

(g2)
.
(i) Let
u∈ Sm
be such that
K(u) = infSmK
with Lagrange multiplier
µ∈R
. Then
u∈ Pµ
is
such that
Jµ(u) = infPµJµ
and
inf
Sm
K= inf
Pµ
Jµ−µ
2m.
(1.24)
(ii) Let
µ∈nν∈R∃v∈ Sm
s.t.
K(v) = infSmK
with Lagrange multiplier
νo.
Let now
u∈ Pµ
be such that
Jµ(u) = infPµJµ
. Then
u∈ Sm
and
K(u) = infSmK
.
Remark 1.11
We highlight that with similar proofs we may show the same results of Propositions
1.51.8,1.10, and Corollary 1.9 in the radial setting, extending what we found in Theorem 1.1
for
m>m0
.
We move now to some qualitative properties of the solutions. We highlight that, dealing
with general solutions of a nonregular problem, their proofs are not standard and will require
some additional arguments. Since, in Theorem 1.1, we restricted to the radial framework, we
investigate when this property is naturally achieved, together with positivity.
Proposition 1.12 (Symmetry and sign)
Assume
(g0)

(g2)
and let
ρ∈[−∞,+∞)
.
10

(i) All minimizers of
Im
over
P(ρ)
and all minimizers of
K
over
Sm
are radial about a point.
(ii) All nonnegative minimizers of
Im
over
P(ρ)
or
P
rad
(ρ)
and all nonnegative minimizers of
K
over
Sm
or
S
rad
m
are Schwarz-symmetric up to a translation (see Denition 2.5).
(iii) Let
(µ, u)
be a solution to
(1.8)
such that
RRNg−(u)udx < +∞
, and assume
2
g(t)−µt ≥0
for each
t≤0
. Then
u≥0
.
(iv) Assume, in addition,
(g3)
and
(g4)
. If
g|(−∞,0) = 0
or
g
is odd, then for any
m > m0
there exists a Schwarz-symmetric minimizer
(µ0, u0)
of
Im
over
P
, where
µ0>0
is also a
minimizer of
µ∈(−∞, µ0)7→ infPµJµ−µ
2m∈R
and
u0
is also a minimizer of
K
over
Sm
and
Dm
with Lagrange multiplier
µ0
. In particular,
inf
PIm=dm,inf
Sm
K=κm,inf
Dm
K=m,
inf
−∞<µ<µ0inf
Pµ
Jµ−µm
2= inf
−∞<µ<µ0a(µ)−µm
2,
and all the above quantities are indeed equal and negative. This minimizer is a solution to
(1.8)
and satises the Nehari identity.
Remark 1.13
In a way analogous to the relation between Theorems 1.1 and 1.3, if
(g0)

(g4)
and
(1.21)
hold, then the outcome of Proposition 1.12 (iv), except for the signs of
µ0
and
K(u0)
and the statement on
m
, holds for every
m > 0
.
When searching for a normalized solution without necessarily minimality properties, we can
drop (g2); moreover, we can require the solution to be Schwarz symmetric (up to the sign).
Proposition 1.14 (Existence and symmetry under relaxed assumptions)
Assume
(g0)
,
(g1)
,
(g3)
, and
(g4)
. Then, for every
m
suciently large, there exists a solution
(µ0, u0)∈
(0,+∞)× S
rad
m
of
(1.8)
such that
u0
or
−u0
is Schwarz symmetric
3
. This solution satises the
Nehari identity
(1.12)
and the Pohoºaev identity
(1.11)
. Finally, if
(1.21)
holds as well, then
such a solution exists with no restriction on
m
.
We end this section by giving some examples of nonlinearities to which the previous results
apply. In these cases, we also show nonexistence of solutions.
Corollary 1.15 (Logarithmic case)
Let
m > 0
and
g(t) = αt log(t2) + β|t|q−1t
with
α > 0
,
β∈R
, and
0< q ≤2∗
s−1
. Then the following facts hold.
(i) If
q > 1
and
β≤ −α(q+1)
q−1e−(q+1)/2
, then there exist no solutions
(µ, u)∈[0,+∞)× Sm
to
(1.8)
that satisfy the Pohoºaev identity; in particular, no
L2
-minima with nonnegative
Lagrange multipliers
µ
exist.
(ii) If
q∈(1,¯p)
and
β > 0
2
Notice that if
µ≥0
, this is implied by
g|(−∞,0) ≥0
.
3
u0
if
t0>0
,
−u0
if
t0<0
, where
t0
is introduced in (g4).
11

or
q∈(1,2∗
s−1]
and
−α(q+ 1)
q−1e−(q+1)/2< β ≤0
or
q∈(0,1]
and
β≤0,
then there exists
(µ0, u0)∈R× S
rad
m
, a Schwarz-symmetric minimizer of
K
over
Sm
, which
is a solution of
(1.8)
, a Pohoºaev minimum, and it satises the Nehari identity. Moreover,
the following Legendre-transform formula holds:
κm= inf
µ∈Ra(µ)−µm
2=a(µ0)−µ0
m
2.
Finally, if
m0
is as in Theorem 1.1 and
m>m0
, then
µ0>0
and
m=κm<0
.
As already mentioned in Subsection 1.1, we highlight that also the pure logarithmic case
α= 1
and
β= 0
, that is, (1.7), has an interest of its own: indeed, even if the existence of a
normalized solution can be achieved by scaling arguments, in the fractional setting the existence
of an
L2
-minimum is not straightforward as in the local case, due to the lack of extremals for
the fractional logarithmic Sobolev inequality.
Corollary 1.16 (Low-power case)
Let
g(t) = −γ|t|r−1t+β|t|q−1t
with
γ, β > 0
,
0< r < 1< q < ¯p
. Then, Corollary 1.15 (ii) holds
4
.
The results apply also to nonlinearities of the type
αt log(t2)−γ|t|r−1t+β|t|q−1t
, but in this
case the exact range of coecients for existence is not explicitly available.
Remark 1.17
We notice that all the results we obtain are valid in the local case
s= 1
,
N≥3
,
and this leads to some improvements of [45]: in particular, in Theorem 1.1, the minimality on
P
rad
of the solution, the Nehari identity, and formula
(1.20)
are new; moreover, Theorem 1.3
and all the results of Subsection 1.3 are new as well. With some slight adaptations, Theorems
1.1 and 1.3 and most of the results of Subsection 1.3 can be generalized also to
s= 1
and
N= 2
(notice that the existence of an
L2
-minimum follows from [45, Theorem 1.1]). The only result that
cannot be proved via a mere adaptation of the arguments herein is Proposition 1.12 (i) because
Lemma 3.3 is false, being
RG(u)−µ
2u2dx= 0
from the Pohoºaev identity, cf. [8, Remark 3.1]
and [7]. We leave the details to the interested reader.
Remark 1.18
Some of the results contained in this paper are new even in the superlinear setting,
extending or further developing [20]. First, the growth of
g
at innity is slightly generalized (see
(g2)
-
(g3)
). Then, the techniques involved for the existence  that is, the direct minimization
over the Pohoºaev set  are new as well. Furthermore, new results appear, such as the fact that
every Pohoºaev minimum is a critical point (see Proposition 1.7), or the precise relation between
L2
-minima and unconstrained ground states (see Proposition 1.10), together with other results in
Subsection 1.3.
4
The results in Corollary 1.16 hold also for
0< r < q ≤1
. In this case, in the Legendre transform, we have
0< µ < µ0
, with
µ0= 2(q−r)(1+r)(1−q)
γ
1−q
q−rβ
(1−r)(1+q)
1−r
q−r
if
q < 1
and
µ0=β
if
q= 1
.
12

Outline of the paper.
In Section 2, we prove some preliminary properties that will be used
throughout the paper. In Section 3, we recall and rene some results about the free-mass problem
(that is, with xed
µ
). In Section 4, together with Subsection 5.2, we prove most of the results
from Subsection 1.3. In Section 5, we nd a solution to the perturbed problem; then, in Section
6, passing to the limit, we obtain a solution to (1.8) and prove Theorem 1.1. Finally, in Section
7, we prove Theorem 1.3, Proposition 1.14, and Corollaries 1.15 and 1.16.
Notations.
The denition of (weak) solution is given in Denition 2.4, while the denition of
Schwarz-symmetric function is given in Denition 2.5. The fractional Sobolev space
Hs(RN)
is
dened in (2.1), while the restrictions over the radial subspace will be denoted with rad. The
L2
-sphere
Sm
and the
L2
-ball
Dm
are given in (1.14) and (1.19), while the
L2
-critical exponent
¯p+1
is dened in (1.2). The perturbed functions
gε
(and
ϕε
,
Gε
−
,
gε
−
,
Gε
) are given in (5.2)(5.3);
the quantities related to the perturbed problem will appear with an 
ε
. The energy functional
K(u)
and the
L2
-minimal value (over the sphere or the ball)
κm
,
m
are given in (1.3), (1.17),
(1.19). The (action, Pohoºaev) functionals
Jµ(u)
,
Pµ(u)
over
Hs(RN)
are dened in (2.3), (3.2),
while the (Lagrangian, Pohoºaev) functionals
Im(µ, u)
,
P(µ, u)
over
R×Hs(RN)
are given in
(1.13), (1.16). For technical reasons, the change of variable
µ=eλ
in the frequency will be
used when
µ > 0
, and sometimes an abuse of notation will be employed. In particular, we will
write (leading to no ambiguity)
Im(λ, u)≡Im(eλ, u)
,
P(λ, u)≡P(eλ, u)
. Thus, the (action,
Lagrangian, Pohoºaev) functionals
J(λ, u)
,
Im(λ, u)
,
P(λ, u)
over
R×Hs(RN)
are given in (5.8)
and at the beginning of Subsection 5.1. The Pohoºaev (set, minimal action)
Pµ
,
a(µ)
are given
in (3.1), (3.3), while
P
,
P(ρ)
,
P+
,
dm
,
dm
+
can be found in (1.15), (1.22), (5.6), (1.17), and (5.7).
The frequency thresholds
µ0
,
λ0
appear in (1.18) and (5.10). The Legendre-type minimum
bm
is given in (5.11), while the mass threshold
m0
appears in (5.12).
2 Preliminaries
We rst re-write here
g±
and
G±
, introduced in (1.9)-(1.10), in dierent terms:
G+(t) := 






Zt
0
g+(τ) dτ
if
t≥0,
Z0
t
g−(τ) dτ
if
t < 0,
G−(t) := 






Zt
0
g−(τ) dτ
if
t≥0,
Z0
t
g+(τ) dτ
if
t < 0,
g+(t) := G′
+(t) = (g+(t)
if
t≥0,
−g−(t)
if
t≤0,g−(t) := G′
−(t) = (g−(t)
if
t≥0,
−g+(t)
if
t≤0.
Clearly, we have
G±(t)≥0
and
g±(t)t≥0
for each
t∈R
; moreover,
G=G+−G−, g =g+−g−,
and
G+(t)≥G(t)≥ −G−(t), g−(t)t≤g(t)t≤g+(t)t
for
t∈R.
Generally,
G±
do not coincide with
G±
, and unlike
G±
, we have
G±∈ C1(R)
. Finally, we notice
that, if
g
is odd, then
G+, G−
are even.
We briey recall that the fractional Laplacian
(−∆)su:= F−1(|ξ|2sF(u))
,
s∈(0,1)
, when-
ever
u
is regular enough, can be expressed pointwise as
(−∆)su(x) = CN,s ZRN
u(x)−u(y)
|x−y|N+2sdy,
13

for some suitable
CN,s >0
; we refer to [30, Section 1.2]. We recall the fractional Sobolev space
Hs(RN) := u∈L2(RN)| | · |sF(u)∈L2(RN)
(2.1)
and dene
⟨u, v⟩:= ZRN
(−∆)s/2u(−∆)s/2vdx=CN,s ZRNu(x)−u(y)v(x)−v(y)
|x−y|N+2sdxdy.
It is well-known that
Hs(RN)→Lq(RN)
,
q∈[2,2∗
s]
, while
Hs
rad
(RN)→Lq(RN)
is compact for
q∈(2,2∗
s)
; here
Hs
rad
(RN)
stands for the subspace of
Hs(RN)
composed of radially symmetric
functions. We recall, moreover, the Gagliardo-Nirenberg inequality [47]
|u|¯p+1
¯p+1 ≤C
GN
|(−∆)s/2u|2
2|u|¯p−1
2,
(2.2)
where
C
GN
>0
is optimal. We also recall some basic properties (see, e.g., [30, Lemma 1.4.1]).
Lemma 2.1
Let
u∈Hs(RN)
. Then
u±,|u| ∈ Hs(RN)
with
|(−∆)s/2(u±)|2≤ |(−∆)s/2u|2
and
|(−∆)s/2|u||2≤ |(−∆)s/2u|2
. Finally,
⟨u, u+⟩ ≥ ⟨u+, u+⟩
and
⟨u, u−⟩≤⟨u−, u−⟩
.
We show the following elementary convergence result.
Lemma 2.2
Let
v∈Hs(RN)
. Let
vR:= vϕR
, where
ϕR:= ϕ(·/R)
and
ϕ∈ C∞
c(RN)
is chosen
0≤ϕ≤1
,
ϕ= 1
in
B1(0)
,
ϕ= 0
in
RN\B2(0)
. Then,
ZRN
(−∆)s/2u(−∆)s/2vRdx→ZRN
(−∆)s/2u(−∆)s/2vdx
as
R→+∞
for each
u∈Hs(RN)
.
Proof.
We compute
1
CN,s ZRN
(−∆)s/2u(−∆)s/2(vR−v) dx
=ZR2Nu(x)−u(y)(ϕR(x)−1)v(x)−(ϕR(y)−1)v(y)
|x−y|N+2sdxdy
=ZR2N
(ϕR(x)−1)u(x)−u(y)v(x)−v(y)
|x−y|N+2sdxdy
+ZR2Nu(x)−u(y)ϕR(x)−ϕR(y)
|x−y|N+2sv(y) dxdy=: (I1)+(I2).
As regards
(I1)
, we observe that
u(x)−u(y)v(x)−v(y)
|x−y|N+2s= u(x)−u(y)2
|x−y|N+2s!1/2 v(x)−v(y)2
|x−y|N+2s!1/2
∈L1(R2N),
thus we can apply the dominated convergence theorem. Regarding
(I2)
, we have
ZR2Nu(x)−u(y)ϕR(x)−ϕR(y)
|x−y|N+2sv(y) dxdy
≤ ZRNu(x)−u(y)2
|x−y|N+2sdxdy!1
2 ZRNϕR(x)−ϕR(y)2
|x−y|N+2sv2(y) dxdy!1
2
,
and we can apply [3, Lemma 1.4.5].
14

Lemma 2.3
Assume
(g0)

(g2)
. Then the functional
u∈Hs(RN)7→ ZRN
G(u) dx∈R
is dierentiable at every
u∈Hs(RN)
along every direction
ϕ∈Hs(RN)∩L1(RN)
.
Proof.
We study
limt→0RRN
G(u+tφ)−G(u)
tdx
and then apply Vitali's convergence theorem. We
estimate thus

G(u+tϕ)−G(u)
t=
1
tZu+tφ
u
g(τ) dτ≤
1
tZu+tφ
u
1 + |τ|2∗
s−1dτ
≤ |ϕ|+
1
tZu+tφ
u
|τ|2∗
s−1dτ≤ |ϕ|+
1
tZu+tφ
u
(|u|+|ϕ|)2∗
s−1dτ
=|ϕ|+ (|u|+|ϕ|)2∗
s−1|ϕ|≲|ϕ|+|u|2∗
s−1|ϕ|+|ϕ|2∗
s,
where we used the convexity of
t2∗
s−1
. Then, xed
A⊂RN
measurable, we have
ZA
G(u+tϕ)−G(u)
tdx≲ZA
|ϕ|dx+|u|2∗
s−1
2∗
sZA
|ϕ|2∗
sdx1
2∗
s+ZA
|ϕ|2∗
sdx
and thus we get both the uniform integrability and the tightness. This allows us to pass to the
limit and get the claim.
Denition 2.4
By
weak solution
(or simply
solution
) of
(1.1)
we mean
u∈Hs(RN)
such that
ZRN
(−∆)s/2u(−∆)s/2ϕdx+µZRN
uϕ dx=ZRN
g(u)ϕdx
for each
ϕ∈Hs(RN)∩L1(RN)
; in particular, it holds for every
ϕ∈ C∞
c(RN)
.
We highlight that, generally, a solution
u
of (1.1) is not a critical point of the action functional
Jµ:Hs(RN)→R
,
Jµ(u) := 1
2ZRN
|(−∆)s/2u|2dx+µ
2ZRN
u2dx−ZRN
G(u) dx,
(2.3)
since, generally,
Jµ
is not dierentiable at
u
along every
ϕ∈Hs(RN)
(see Lemma 2.3); in
particular, we cannot ensure that a solution satises the Nehari identity. On the other hand, in
some circumstances, we will show that this is the case (cf. Proposition 4.3).
Finally, we will use the following denition.
Denition 2.5
A function
u:RN→R
is called
Schwarz-symmetric
if it is nonnegative, radially
symmetric, and radially nonincreasing.
3 The mass-free problem
The purpose of this section is to discuss the free-mass problem (i.e.,
µ
xed). For any
µ∈R
we introduce a Pohoºaev set
Pµ:= u∈Hs(RN)\ {0} | Pµ(u)=0,P
rad
µ:= Pµ∩Hs
rad
(RN),
(3.1)
15

where the Pohoºaev functional
Pµ:Hs(RN)→R
is dened by
Pµ(u) := ZRN
|(−∆)s/2u|2dx+ 2∗
sZRN
µ
2u2−G(u) dx.
(3.2)
Recalled (2.3), we dene the least action
a(µ) := inf
P
rad
µ
Jµ.
(3.3)
We notice that, generally, the set
Pµ
could be empty, and in this case,
a(µ)=+∞
.
Here we list a series of properties that hold for
µ∈R
under (g0)(g2).
Lemma 3.1
Assume
(g0)

(g2)
and let
µ∈R
. Then, the following statements are equivalent.
(a)
µ < µ0
.
(b) There exists
t=t(µ)= 0
such that
G(t)>1
2µt2
.
(c) There exists
u∈ P
rad
µ.
Proof.
(a)
⇐⇒
(b) is trivial. If (c) holds, then
u= 0
; thus, by the Pohoºaev identity,
ZRN
G(u)−µ
2u2dx=ZRN
|(−∆)s/2u|2dx > 0,
hence there exists
t(µ) := u(xµ)
,
xµ∈RN
, such that (b) is veried. Assume (b) holds, from [8,
Proof of Theorem 2] (see also [20, Lemma 3.1]) there exists
¯u∈H1
rad
(RN)⊂Hs
rad
(RN)
such that
RRNG(¯u)−µ
2¯u2dx > 0
. Then, for
t > 0
, straightforward computations show that
Pµ¯u(·/t)=ZRN
|(−∆)s/2¯u|2dxtN−2s+ 2∗
sZRN
µ
2¯u2−G(¯u) dxtN
is positive when
t
is small and negative when
t
is large, hence there exists
¯
t > 0
such that
Pµ¯u(·/¯
t)= 0
, whence (c).
Remark 3.2
By the previous proof we notice the following fact: for every
u∈Hs(RN)\ {0}
such that
Pµ(u)≤0
, we have
RRNG(u)−µ
2u2dx > 0
, thus there exists
t0>0
such that
u(·/t0)∈ Pµ
. Moreover, using that
Pµ(u)≤0
and that
t∈(0,∞)7→ 1
2tN−2s−1
2∗
stN∈R
is
maximized at
t= 1
, there holds
Jµu(·/t0)=tN−2s
0
2ZRN
|(−∆)su|2dx−tN
0ZRN
G(u)−µ
2u2dx
≤ tN−2s
0
2−tN
0
2∗
s!ZRN
|(−∆)su|2dx≤s
NZRN
|(−∆)su|2dx≤Jµ(u),
that is,
Jµu(·/t0)≤Jµ(u)
. Notice that this fact does not depend on the sign of
µ
.
Lemma 3.3
Let
µ∈R
and assume
(g0)

(g2)
. If
Pµ=∅
and
u∈ Pµ
is such that
Jµ(u) =
infPµJµ
, then
ω:= RRNG(u)−µ
2u2dx > 0
and
1
2|(−∆)s/2u|2
2= min 1
2|(−∆)s/2v|2
2|v∈Hs(RN),ZRN
G(v)−µ
2v2dx=ω.
(3.4)
16

Proof.
First, notice that since
Pµ=∅
, there exists a function
v∈Hs(RN)
such that
RRNG(v)−
µ
2v2dx= 1
, thus the minimization problem
T:= inf 1
2|(−∆)s/2v|2
2|v∈Hs(RN),ZRN
G(v)−µ
2v2dx= 1
is well dened. Moreover, every such function veries
G(v)∈L1(RN)
. We adapt now the proof
of [13, Lemma 1 (ii)]. We notice that we do not need to assume that the Pohoºaev identity holds
for every solution or that
T
is a priori attained. Since
u∈ Pµ
, we have that
Jµ(u)≥2s(N−2s)N
2s−1N−N
2sTN
2s=: e
T .
Now, let
(vn)n
be a minimizing sequence for
T
, and let
tn>0
be such that
vn(·/tn)∈ Pµ
. Then,
since
Jµ(u) = infPµJµ
,
e
T≤Jµ(u)≤Jµvn(·/tn)=e
T+on(1),
and we conclude letting
n→+∞
and via scaling arguments.
Proposition 3.4
Assume
(g0)

(g2)
and let
µ∈R
. Then
a(µ)≥0
and the following relation
holds
a(µ) = inf nJµ(u)|u∈Hs
rad
(RN)\ {0}, Pµ(u)≤0o.
(3.5)
Moreover,
a:R→[0,+∞]
is nondecreasing. Finally, if
µ > 0
and
P
rad
µ=∅
, then
a
is continuous
at
µ
.
Proof.
By denition of
P
rad
µ
, we have
a(µ) = s
Ninf
u∈P
rad
µ
|(−∆)s/2u|2
2≥0.
To show relation (3.5), observe that if
P
rad
µ=∅
, then
u∈Hs
rad
(RN)\ {0} | Pµ(u)≤0=∅
as
well from Remark 3.2, hence both sides of (3.5) equal
+∞
. Next, assume
Pµ(u)≤0
; thus, by
Remark 3.2 there exists
t0>0
such that
Pµu(·/t0)= 0
, and moreover,
Jµu(·/t0)≤Jµ(u)
,
hence
a(µ)≤Jµu(·/t0)≤Jµ(u)
and we can pass to the inmum. The reverse inequality is obvious. We show now the mono-
tonicity. Let
µ1< µ2
. If
P
rad
µ2=∅
, then it clearly holds. Otherwise, let
u∈ P
rad
µ2
. Then
Pµ1(u)< Pµ2(u)=0
and by (3.5) we have
a(µ1)≤Jµ1(u)< Jµ2(u);
(3.6)
passing to the inmum, we have the claim. We show now the continuity. The upper semi-
continuity follows by the fact that
a(µ)
is the inmum of the family of continuous functions
{µ7→ Jµ(u)|u∈Hs
rad
(RN)\ {0}, Pµ(u)=0}
. We focus on the lower semicontinuity and x
µn→µ > 0
. From (g0) and Lemma 3.1,
P
rad
µn=∅
for all suciently large
n
; in addition, we
can assume
µ∗:= infnµn>0
, and by upper semicontinuity,
M:= supna(µn)<∞
. For each
n
, let
(un
k)k⊂Hs
rad
(RN)\ {0}
be a minimizing sequence for
a(µn)
, that is,
Pµn(un
k)=0
and
Jµn(un
k)→a(µn)
as
k→+∞
. Let now
vn:= un
kn
be such that
Jµn(vn)≤a(µn) + 1
n≤M+ 1
.
We show that
(vn)n
is bounded. Indeed, by the Pohoºaev identity we have
s
N|(−∆)s/2vn|2
2=Jµn(vn)≤M+ 1
17

and, again by the Pohoºaev identity and the assumptions on
G+
,
µ∗
2|vn|2
2≤µn
2|vn|2
2≤ZRN
G(vn) dx+C≤δ
2|vn|2
2+Cδ|vn|2∗
s
2∗
sdx+C≤δ
2|vn|2
2+C,
where, by choosing
δ < µ∗
, we obtain the boundedness of
vn
in
Hs(RN)
. Thus,
vn v
in
Hs
rad
(RN)
, which yields
Pµ(v)≤lim infnPµn(vn) = 0
and, in view of (3.5),
a(µ)≤Jµ(v)≤lim inf
nJµn(vn)≤lim inf
na(µn),
which is the claim.
Proposition 3.5
Assume
(g0)

(g2)
and let
µ∈R
.
(i) All minimizers of
Jµ
over
P
rad
µ
(resp.
Pµ
) are solutions to
(−∆)su+µu =g(u)
in
RN.
(3.7)
(ii) All minimizers of
Jµ
over
Pµ
are radially symmetric about a point.
(iii) All nonnegative minimizers of
Jµ
over
Pµ
or
P
rad
µ
are Schwarz-symmetric up to a transla-
tion.
(iv) If
g(t)−µt ≥0
for all
t≤0
, then every solution to
(3.7)
such that
RRNg−(u)udx < +∞
is nonnegative.
The proof of Proposition 3.5 is analogous to those of Propositions 1.7 and 1.12 (see Subsections
4.2,5.2 and 6.2), hence it is omitted. We now state the main theorem of this section.
Theorem 3.6
Let
µ > 0
and
g∈ C(R)
satisfy

|g(t)|≲|t|+|t|2∗
s−1
for every
t∈R
;

lim sup
t→0
G(t)
t2≤0
and
lim sup
|t|→∞
G(t)
|t|2∗
s≤0
;

there exists
t0= 0
such that
G(t0)>1
2µt2
0
.
Then the following statements hold true.
(i) There exists
v0∈ P
rad
µ
such that
Jµ(v0) = a(µ)
and thus
a(µ)>0
. Moreover,
a: (0,+∞)→(0,+∞)
is increasing.
(ii) If
g|(−∞,0) = 0
or
g
is odd, then there exists a Schwarz-symmetric minimizer of
Jµ
over
Pµ
; in particular,
inf
Pµ
Jµ= inf
P
rad
µ
Jµ.
Proof.
(i) From the assumptions, we see that for every
δ > 0
there exists
Cδ>0
such that for every
t∈R
G(t)≤δt2+Cδ|t|2∗
s,
18

whence, for every
u∈Hs(RN)
,
1
2∗
s
Pµ(u)≥1
2∗
s
|(−∆)s/2u|2
2+µ
2|u|2
2−δ|u|2
2−Cδ|u|2∗
s
2∗
s≥min n1
2∗
s,µ
2o−δ∥u∥2
Hs−Cδ∥u∥2∗
s
Hs.
Taking
0< δ < min{1/2∗
s, µ/2}
, we see that
Pµ(u)>0
if
∥u∥Hs
is suciently small.
Let now
(un)n⊂ P
rad
µ
such that
limnJµ(un) = infP
rad
µJµ
. Since
Pµ(un)=0
, we see that
inf
Pµ
Jµ+o(1) = Jµ(un)−1
2∗
s
Pµ(un) = s
N|(−∆)s/2un|2
2,
which yields that
|(−∆)s/2un|2
is bounded, and, using the same
δ
-related argument as before,
that
inf
Pµ
Jµ+o(1) = Jµ(un)−1
2Pµ(un) = −µ
22∗
s
2−1|un|2
2+2∗
s
2−1ZRN
G(un) dx
≤ −µ
22∗
s
2−1|un|2
2+2∗
s
2−1δ|un|2
2+Cδ|un|2∗
s
2∗
s.
Since
|un|2∗
s
is bounded from the previous step and the fractional Sobolev embedding, taking
δ < µ/2
we obtain that
|un|2
is bounded as well, i.e.,
∥un∥Hs
is bounded. We can then consider
a subsequence, still denoted by
un
, and
u0∈Hs
rad
(RN)
such that
un u0
in
Hs(RN)
,
un→u0
a.e. in
RN
, and
un→u0
in
Lq(RN)
for every
q∈(2,2∗
s)
. In particular, using the weak lower
semi-continuity of the norm and Fatou's lemma, we get that
Jµ(u0)≤infP
rad
µJµ
and
Pµ(u0)≤0
.
In addition, recalling
Pµ(un)=0
, we have by the strong convergence in
Lq(RN)
,
|(−∆)s/2u0|2
2+2∗
sµ
2|u0|2
2+ 2∗
sZRN
G−(u0) dx
≤lim sup
n|(−∆)s/2un|2
2+2∗
sµ
2|un|2
2+ 2∗
sZRN
G−(un) dx
= lim sup
nZRN
G+(un) dx=ZRN
G+(u0) dx;
if
u0≡0
, the above relation implies
un→0
in
Hs(RN)
, which is impossible because
Pµ(u)>0
for small
u
. Since
u0= 0
, by Remark 3.2 we can nd
t0>0
such that
Pµu0(·/t0)= 0
and
Jµu(·/t0)≤Jµ(u0)
, so the sought minimizer is
v0:= u0(·/t0)
. By the Pohoºaev identity we
also have
a(µ) = Jµ(v0) = s
N∥(−∆)s/2v0∥2
2>0
.
To show the strict monotonicity, arguing as in (3.6), in the case
µ2>0
we choose
u
as the
minimum corresponding to
µ2
. Hence, we have
a(µ1)≤Jµ1(u)< Jµ2(u) = a(µ2).
(ii) We start with the case when
g
is odd. Let
(un)n⊂ Pµ
be such that
limnJµ(un) = infPµJµ
.
From Lemma 2.1,
Jµ(|un|)≤Jµ(un)
and
Pµ(|un|)≤Pµ(un) = 0
. If
tn>0
is such that
Pµ|un|(·/tn)= 0
, then we obtain
infPµJµ≤Jµ|un|(·/tn)≤Jµ(|un|)≤Jµ(un)
by Remark
3.2. This proves that, up to replacing
un
with
|un|(·/tn)
, we can assume that
un≥0
for every
n
.
In a very similar way, using the Schwarz rearrangement and the fractional PolyáSzegö inequality
[46], we can assume that every
un
is Schwarz symmetric. We can now conclude in the very same
way as in point (i). If
g|(−∞,0) = 0
, we argue as above, except we use
u+
n
instead of
|un|
.
Remark 3.7
We observe the following facts.
19

(i) Theorem 3.6 (i) is a renement of [14, Theorem 1.2 (i)], where the main additional as-
sumption is the existence of
q∈(2,2∗
s)
such that
limt→0g(t)/|t|q−1= 0
. At the same time,
Proposition 3.5 shows that [14, Theorem 1.2 (ii)(iii)] holds under considerably weaker
assumptions (in particular,
(g1)
).
(ii) In Proposition 3.5 (i), we are not claiming that, for general
g
, there exists a minimizer
over
Pµ
; on the other hand, from Proposition 3.5 (ii), we deduce that
P
rad
µ
is a natural
subset where to search for such a minimizer.
4 Some general properties
For
ρ∈[−∞,+∞)
, recall
P(ρ)
from (1.22) and
µ0
from (1.18). We observe that, by Lemma
3.1, if
ρ∈[−∞, µ0)
, then
P(ρ)⊃ P
rad
(ρ)=∅
; in particular,
P =∅
always holds.
Remark 4.1
We point out that if
(µ, u)
minimizes
Im
over
P(ρ)
for some
ρ∈[−∞,+∞)
, then
u
minimizes
Jµ
over
Pµ
.
4.1 Connection among inma
We prove now Propositions 1.5 and 1.6.
Proof of Proposition 1.5.
Let
(µn, un)n⊂ P
be a minimizing sequence for
infPIm
, that
is,
Im(µn, un) = infPIm+o(1)
; recall that
un= 0
by the denition of
P
. Consider
tn:=
(m/|un|2
2)1/N
such that
vn:= un(·/tn)∈ Sm
. We notice, by the Pohoºaev identity,
Im(µn, un) = 1
2|(−∆)s/2un|2
2+µn|un|2
2
21−tN
n−ZRN
G(un) dx
=1
2|(−∆)s/2un|2
2+1−tN
nZRN
G(un) dx−1
2∗
s
|(−∆)s/2un|2
2−ZRN
G(un) dx
=1
2−1
2∗
s1−tN
n|(−∆)s/2un|2
2−tN
nZRN
G(un) dx
=1
2−1
2∗
s1−tN
n1
tN−2s
n
|(−∆)s/2vn|2
2−ZRN
G(vn) dx
≥1
2|(−∆)s/2vn|2
2−ZRN
G(vn) dx=K(vn),
where we used that
1
2−1
2∗
s1−tN≥1
2tN−2s
for all
t > 0
, which is true by a straightforward
computation. Thus, we have shown
inf
Sm
K≤K(vn)≤Im(µn, un) = inf
PIm+o(1).
Passing to the limit in
n
, we obtain
inf
Sm
K≤inf
PIm.
(4.1)
Assume now
ρ∈[0,+∞)
and
infDmK < −ρ
2m
. We can thus choose a minimizing sequence
(un)n⊂ Dm
such that
limnK(un) = infDmK
and
K(un)<−ρ
2m≤0
for every
n
(which implies
un= 0
) and dene
µn:= 2
N|un|2
2s|(−∆)s/2un|2
2−NK(un)≥ − 2
|un|2
2
K(un)> ρ
20

so that
(µn, un)∈ P(ρ)
. This, together with
µn>0
and
|un|2
2≤m
, yields
inf
Dm
K+on(1) = K(un)≥Im(µn, un)≥inf
P(ρ)
Im,
and passing to the limit in
n
we get
infDmK≥infP(ρ)Im
. Arguing similarly for
ρ∈R
and
infSmK < −ρ
2m
we obtain
infSmK≥infP(ρ)Im.
Choosing
ρ=−∞
and recalling (4.1), we get
(1.23). Gathering the previous inequalities we conclude the proof.
Finally, we show the last relation arguing as in the proof of (3.5). Indeed, considered
(µ, u)∈
(ρ, +∞)×Hs(RN)
with
P(µ, u)≤0
, there exists
t0>0
such that
P(µ, u(·/t0)) = 0
and
Im(µ, u(·/t0)) ≤Im(µ, u)
, thus
inf
P(ρ)
Im≤Im(µ, u(·/t0)) ≤Im(µ, u)
and hence the claim passing to the inmum (the other inequality is obvious).
Proof of Proposition 1.6.
Let
µ∈(ρ, µ0)
. From Lemma 3.1,
Pµ=∅
(thus
infPµJµ
is well
dened). Then, for any
u∈ Pµ
, we have
inf
P(ρ)
Im≤Im(µ, u) = Jµ(u)−µ
2m.
Passing to the inmum over
u
, we have
infP(ρ)Im≤infPµJµ−µ
2m
. Passing to the inmum over
µ
, we have
infP(ρ)Im≤infρ<µ<µ0inf PµJµ−µ
2m
.
Let now
(ν, u)∈ Pρ
, which implies
ν∈(ρ, µ0)
. As a consequence
Im(ν, u) = Jν(u)−ν
2≥inf
Pν
Jν−ν
2≥inf
ρ<µ<µ0inf
Pµ
Jµ−µ
2m.
Passing to the inmum over
(ν, u)
we obtain
infP(ρ)Im≥infρ<µ<µ0inf PµJµ−µ
2m
, whence
the claim.
4.2
L2
-minima, Pohoºaev identity, and critical points
Proposition 4.2
Assume
(g0)

(g2)
. Let
u∈Hs(RN)
.
(i) If
u∈ Sm
is an
L2
-minimum for
K
, i.e.,
K(u) = infSmK
, then
u
is a solution of
(1.1)
for some Lagrange multiplier
µ∈R
, and
(µ, u)
satises the Pohoºaev identity.
(ii) If
u∈ Dm
is such that
K(u) = infDmK
and
|u|2
2< m
, then
u
is a solution of
(1.1)
with
Lagrange multiplier
µ= 0
, and
(µ, u)
satises the Pohoºaev identity.
(iii) If
(µ, u)∈ P
and
K(u)≤0
, then
µ > 0
.
(iv) If
u∈ Dm
is such that
K(u) = infDmK < 0
, then
u∈ Sm
; in particular,
K(u) = infSmK
.
Proof.
(i) First, we show that
u
is a solution. Indeed, let
ϕ∈Hs(RN)∩L1(RN)
; being
u= 0
,
for
t∈R
with
|t|
small we can assume
u+tϕ = 0
. Thus, for each
t
we choose
θ(t)>0
such
that
|(u+tϕ)(·/θ(t))|2
2=m
, that is,
θ(t) := m|u+tϕ|−2
21/N .
Let
γ(t) := K(u+tϕ)(·/θ(t))
;
due to the dierentiability of
θ
and the arguments of the proof of Lemma 2.3, we have that
γ
is
dierentiable. Being
t= 0
a point of minimum for
γ
, we have
γ′(0) = 0
, that is (notice
θ(0) = 1
),
N−2s
2θ′(0)|(−∆)s/2u|2
2+ZRN
(−∆)s/2u(−∆)s/2ϕdx−Nθ′(0) ZRN
G(u) dx−ZRN
g(u)ϕdx= 0
21

and hence (being
θ′(t) = −2
Nm1/N |u+tϕ|−2/N −2
2RRN(u+tϕ)ϕdx
)
ZRN
(−∆)s/2u(−∆)s/2ϕdx−2
m1
2∗
s
|(−∆)s/2u|2
2−ZRN
G(u) dxZRN
uϕ dx−ZRN
g(u)ϕdx= 0.
Set
µ:= −2
m1
2∗
s
|(−∆)s/2u|2
2−ZRN
G(u)∈R,
(4.2)
we have the claim. We see that the Pohoºaev identity directly comes from the denition of
µ
in
(4.2).
(ii) The statement is obvious for
u≡0
. Arguing as in (i), let
ϕ∈Hs(RN)∩L1(RN)
; being
u= 0
, for
t∈R
with
|t|
small we can assume
u+tϕ = 0
and
|u+tϕ|2
2< m
. By the minimality
of
t= 0
for
t7→ K(u+tϕ)
we obtain
ZRN
(−∆)s/2u(−∆)s/2ϕdx−ZRN
g(u)ϕdx= 0
which implies that
u
is a solution with zero Lagrange multiplier. To show the Pohoºaev identity,
consider the map
t7→ K(u(·/t))
, where
|u(·/t)|2
2< m
for
t
close to
1
. Being
t= 1
a point of
minimum, we have
N−2s
2|(−∆)s/2u|2
2−NZRN
G(u) dx= 0
which is the claim.
The proof of (iii) is straightforward. Point (iv) follows by (ii) and (iii).
We prove now Proposition 1.7. We underline that, unlike [14], we employ a direct proof
without relying on a deformation lemma.
Proof of Proposition 1.7.
Let
r, t ∈R
and
ϕ∈Hs(RN)∩L1(RN)
, and dene
σ=σ(r, t)>0
such that
Pµ+r, u(·/σ) + tϕ(·/σ)= 0
, that is,
σ(r, t) := 1
2∗
s
|(−∆)s/2(u+tϕ)|2
2ZRN
G(u+tϕ)−µ+r
2(u+tϕ)2dx−1!1
2s
.
Indeed, since
P(µ, u) = 0
, we know that
RRNG(u)−µ
2u2dx > 0
, thus for
|r|,|t|
small the
above expression is well dened. Moreover we can assume
|r|
small such that
µ+r > ρ
, thus
(µ+r, u(·/σ) + tϕ(·/σ)) ∈ P(ρ)
.
Being
P(µ, u)=0
, we have
σ(0,0) = 1
. Next, we compute
Imµ+r, u(·/σ) + tϕ(·/σ)
=σN−2s
2|(−∆)s/2(u+tϕ)|2
2+µ+r
2σN|u+tϕ|2
2−m−σNZRN
G(u+tϕ) dx.
By Lemma 2.3, the map
t7→ RRNG(u+tϕ) dx
, and so
σ
as well, is dierentiable at
0
. Since
(µ, u)
is a minimizer, the gradient of
Imµ+r, u(·/σ) + tϕ(·/σ)
at
(r, t) = (0,0)
is zero. At the
same time, recalling that
σ(0,0) = 1
,
∂rImµ+r, u(·/σ)|(t,r)=(0,0) =N−2s
2∂rσ(0,0)P(µ, u) + 1
2(|u|2
2−m) = 1
2(|u|2
2−m)
and
∂tImµ, u(·/σ) + tϕ(·/σ)|(t,r)=(0,0) =
22

=N−2s
2∂tσ(0,0)P(µ, u) + ZRN
(−∆)s/2u(−∆)s/2ϕ+µuϕ −g(u)ϕdx
=ZRN
(−∆)s/2u(−∆)s/2ϕ+µuϕ −g(u)ϕdx.
Thus, we obtain the claim. We conclude by applying Proposition 1.5.
Proof of Proposition 1.8.
It follows by Proposition 4.2 and Proposition 1.5.
Proof of Proposition 1.10.
(i) From Proposition 1.8 we have that
Im(µ, u) = infPIm
, hence, by Remark 4.1,
J(µ, u) =
infPµJµ
, and (1.24) follows from Proposition 1.5.
(ii) By (1.24) and Proposition 1.5 we have
J(µ, u) = inf
Pµ
Jµ= inf
Sm
K+µ
2m= inf
PIm+µ
2m,
that is,
Im(µ, u) = infPIm
. From Proposition 1.7 we obtain that
u∈ Sm
and we conclude.
4.3 Nehari identity
Proposition 4.3 (Nehari identity)
Assume
(g0)

(g2)
and let
µ∈R
. If
u∈Hs(RN)
is a
solution to
(1.1)
such that
RRNg−(u)udx < ∞
, then
u
satises the Nehari identity
|(−∆)s/2u|2
2+µ|u|2
2=ZRN
g(u)udx.
Proof.
With the notation of Lemma 2.2, for every
R > 0
there holds
ZRN
(−∆)s/2u(−∆)s/2(ϕRu) dx+µZRN
u2ϕRdx=ZRN
g(u)ϕRudx.
(4.3)
Since
|ϕR| ≤ 1
and
limR→+∞ϕR= 1
a.e. in
RN
, from the dominated convergence theorem we
obtain
lim
R→+∞ZRN
u2ϕRdx=ZRN
u2dx
and
lim
R→+∞ZRN
g(u)ϕRudx=ZRN
g(u)udx;
using also Lemma 2.2, we can pass to the limit in (4.3) and conclude.
5 The superlinear (perturbed) problem
In this section, where we always assume (g0)(g2), we study the following perturbed problem







(−∆)su+µu =gε(u),
ZRN
u2dx=m,
(µ, u)∈(0,+∞)×Hs(RN),
(5.1)
where, for
ε∈(0,1]
,
gε
is dened as follows: let
ϕε∈ C(R,[0,1])
given by
ϕε(t) := (|t|/ε
for
|t|< ε,
1
for
|t| ≥ ε,
(5.2)
23

and set
gε
−(t) := ϕε(t)g−(t), Gε
−(t) := Zt
0
gε
−(τ) dτ.
Then, we dene
gε(t) := g+(t)−gε
−(t), Gε(t) := Zt
0
gε(τ) dτ=G+(t)−Gε
−(t).
(5.3)
Notice that
ϕε→1
pointwise, thus
gε(t)→g(t), Gε(t)→G(t)
for every
t∈R
as
ε→0+
. Hence, (5.1) can be considered as an approximating family of problems; the advantage
is given by the modication in zero, which suppresses the singularity in the sublinear case:
lim
t→0
gε(t)
t= 0.
(5.4)
We will make large use of the following monotonicity property: if
0< ε′< ε ≤1
, then
Gε
−≤Gε′
−≤G−,
which implies, moreover,
Gε≥Gε′≥G, |Gε| ≤ |Gε′| ≤ |G|.
The purpose of this section is to obtain the existence of a Pohoºaev minimum for the problem
(5.1), furnishing, in addition, a slight generalization of [20] by exploiting the rened Theorem
3.6; such a generalization holds for more general nonlinearities not related to the perturbation
process, see Remark 5.3 below.
5.1
ε
xed: some renement for the superlinear problem
We start by considering
ε∈(0,1]
xed. Assuming
µ > 0
, to avoid technical issues related to
the boundary of
(0,+∞)
, we exploit the identication
µ≡eλ.
(5.5)
A similar approach that does not make use of (5.5) is developed in [30, Section 4.2].
Remark 5.1
The condition
µ > 0
is a classical requirement for nonlinearities with superlinear
growth at the origin  cf.
(5.4)
 because it leads to (see [8,18])
lim
t→0
gε(t)−µt
t<0.
Making use of (5.5), we can thus reformulate the search for solutions to (5.1) as critical points
of the Lagrangian functional
Im
ε:R×Hs(RN)→R
dened as
Im
ε(λ, u) := 1
2|(−∆)s/2u|2
2+eλ
2(|u|2
2−m)−ZRN
Gε(u) dx.
Let us also dene
Pε:R×Hs(RN)→R
as
Pε(λ, u) := |(−∆)s/2u|2
2+ 2∗
seλ
2|u|2
2−ZRN
Gε(u) dx,
24

Pε,+:= (λ, u)∈R×Hs(RN)|Pε(λ, u)=0, u = 0,P
rad
ε,+:= Pε,+∩R×Hs
rad
(RN),
and
dm
ε,+:= inf
P
rad
ε,+
Im
ε.
With an abuse of notation, we redene consequently on
R×Hs(RN)
also
Im(λ, u), P (λ, u)
dened in (1.13) and (1.16). We also set
P+:= (λ, u)∈R×Hs(RN)|P(λ, u)=0, u = 0,P
rad
+:= P+∩R×Hs
rad
(RN).
(5.6)
Notice that, with the identication
µ≡eλ>0
, we have
dm
+:= inf
(λ,u)∈P
rad
+
Im(λ, u) = inf
(µ,u)∈P
rad
(0)
Im(µ, u).
(5.7)
Moreover, we introduce
Kε(u) := 1
2|(−∆)s/2u|2
2−ZRN
Gε(u) dx, κm
ε:= inf
S
rad
m
Kε, m
ε:= inf
D
rad
m
Kε.
The following is the main theorem of this section.
Theorem 5.2
Assume that
(g0)

(g4)
hold, then there exists
mε≥0
such that for every
m>mε
there exists a solution
(µε, uε)∈ P
rad
ε,+
to
(5.1)
verifying
Im(λε, uε) = dm
ε,+<0
.
Moreover, if
g|(−∞,0) = 0
or
g
is odd, then for every
m>mε
there exists a Schwarz-symmetric
minimizer of
Im
ε
over
Pε,+
; in particular,
inf
Pε,+
Im
ε= inf
P
rad
ε,+
Im
ε.
Remark 5.3
We highlight that the only additional property of
gε
with respect to
(g0)

(g4)
that
is used in Theorem 5.2 is the superlinearity at zero of
gε
, i.e.,
(5.4)
(or equivalently, from
(g1)
,
limt→0gε
−(t)/t = 0
). Consequently, Theorem 5.2 and all the other properties presented in this
section still hold true with
g
instead of
gε
under the additional assumption
lim
t→0
g−(t)
t= 0.
This furnishes a small generalization of the results in [20].
Remark 5.4
We underline that the solution found in Theorem 5.2 has also a minimax charac-
terization. Introduced
Ωε:= (λ, u)∈R×Hs(RN)|Pε(λ, u)>0∪R× {0},
we have
∂Ωε=Pε,+
(see [30, Lemma 2.3.1]), and this is treated as a mountain in the minimax
approach: set
Γm
ε:= ξ∈ C[0,1],R×Hs
rad
(RN)ξ(0) ∈R× {0}, ξ(1) ∈ Ωε,max
i=0,1Im
εξ(i)≤dm
ε,+−1,
one can prove that [20, Section 6]
dm
ε,+= inf
ξ∈Γm
ε
max
t∈[0,1] Im
εξ(t).
Moreover, by arguing as in [20], we also obtain (write
g0:= g
and
G0:= G
for convenience)
25


if
g
is odd, then for each
ε∈(0,1]
and
k∈N
there exists
mε,k ≥0
such that, for
m>mε,k
,
there exist
k
(couple of) solutions to
(5.1)
;

if
lim
t→0
G(t)
|t|¯p+1 = +∞
, then for each
ε∈[0,1]
, we have
mε= 0
(and
mε,k = 0
if
g
is odd).
Notice that, in this case,
g=g+
in a neighborhood of the origin.
In [20], the existence of a normalized solution is pursued by a Mountain Pass approach,
involving a suitable augmented functional and a deformation lemma; the found solution is shown
a posteriori to be a Pohoºaev minimum. Here we provide a dierent approach, by minimizing
directly the functional on the Pohoºaev product set: this proof is not trivial, since a precise
sequence of minimizers has to be selected; on the other hand, it requires fewer tools.
To prove Theorem 5.2, we need preliminary estimates on
dm
ε,+
. This will be given by the
interplay of the value of the Pohoºaev minimum on the product space and the one with xed
µ
.
5.1.1 The Pohoºaev asymptotic geometry
To estimate
dm
ε,+
, we introduce now the Pohoºaev geometry related to a xed frequency
λ
,
in the spirit of Section 3. For
λ∈R
we dene
J(λ, u) := 1
2|(−∆)s/2u|2
2+eλ
2|u|2
2−ZRN
G(u) dx,
Jε(λ, u) := 1
2|(−∆)s/2u|2
2+eλ
2|u|2
2−ZRN
Gε(u) dx,
P(λ) := u∈Hs(RN)|(λ, u)∈ P+,P
rad
(λ) := P(λ)∩Hs
rad
(RN),
Pε(λ) := u∈Hs(RN)|(λ, u)∈ Pε,+,P
rad
ε(λ) := Pε(λ)∩Hs
rad
(RN),
a(λ) := inf
P
rad
(λ)J(λ, ·), aε(λ) := inf
P
rad
ε(λ)Jε(λ, ·).
(5.8)
Notice that
J(λ, u) = Jeλ(u)
and
P(λ) = Peλ
with the notations of Section 3. Moreover, we
make again an abuse of notation by writing
a(λ)≡a(eλ)
 see (3.3). We observe that
Jε≤J.
To compare
a(λ)
with
aε(λ)
, we make use of (3.5): as a matter of fact, by
Pε≤P
we have
{P= 0}⊂{Pε≤0}
and thus
aε(λ)≤a(λ).
(5.9)
We highlight that, arguing as in [14], we could give a Mountain Pass characterization of
a(λ)
;
nonetheless, we will not make use of this characterization for the limit problem. Moreover, we
introduce
µε:= sup
t=0
Gε(t)
t2/2
≥sup
t=0
G(t)
t2/2=µ0∈(0,+∞]
λε:= log(µε)
≥log(µ0) =: λ0∈(−∞,+∞]
(5.10)
with the convention that
log(+∞) = +∞
. Observe that
µ0>0
follows from (g4). We highlight
that, in what follows, we will sometimes need to distinguish the cases
λε∈R
and
λε= +∞
.
We dene, for
m > 0
, the values
bm
ε:= inf
λ<λεaε(λ)−eλ
2m
and
bm:= inf
λ<λ0a(λ)−eλ
2m.
(5.11)
26

Moreover, by Lemma 3.1 the following thresholds are well dened
mε:= inf
λ<λε
aε(λ)
eλ/2
and
m0:= inf
λ<λ0
a(λ)
eλ/2.
(5.12)
Observe that, from Lemma 3.1 and since
λε≥λ0
and
a0≥aε≥0
 cf. Proposition 3.4, we have
bm≥bm
ε
and
m0≥mε≥0.
(5.13)
As a straightforward consequence of Proposition 1.6, Theorem 3.6, and the denitions of
λε
and
mε
, we obtain the following result.
Proposition 5.5
Assume
(g0)

(g2)
and
(g4)
. Then

For every
λ∈R
there exists a solution
uλ,ε ∈ P
rad
ε(λ)
to
(−∆)su+eλu=gε(u)
in
RN.
Additionally,
Jε(λ, uλ,ε) = inf
P
rad
ε(λ)Jε(λ, u) = aε(λ)>0.
As a consequence,
a(λ)>0
.

dm
ε,+=bm
ε
; if
m>mε
, then
dm
ε,+<0
.

If
λε∈R
, then
min{Pε(λ, u), Jε(λ, u)}>0
for all
λ≥λε
and
u∈Hs(RN)\ {0}
.
We focus now on properties strictly related to the
L2
-subcritical setting (g3). To start, we
observe that for every
δ > 0
there exist
Cδ, cδ>0
such that for every
t∈R
Gε(t)≤G+(t)≤δ
2t2+Cδ
¯p+ 1|t|¯p+1
(5.14)
and
Gε(t)≤G+(t)≤cδ
2t2+δ
¯p+ 1|t|¯p+1.
(5.15)
Lemma 5.6 (Asymptotic geometry)
Assume
(g0)

(g4)
. The following statements hold.

limλ→λ−
εaε(λ)=+∞
. If
λε= +∞
, then
limλ→+∞aε(λ)
eλ= +∞
.

dm
ε,+>−∞
.
Proof.
Recalled (5.14)-(5.15), concerning the rst point we have that the rst limit can be
obtained with the same reasoning as in [20, Proposition 3.3] in light of the monotonicity in
Proposition 3.4, while for the second one we can argue as in [20, Proposition 3.4] once the
following property is proved:
aε(λ)≥(eλ−cδ) inf
P∗(δ)J∗
δ
if
eλ> cδ
and
lim
δ→0+inf
P∗(δ)J∗
δ= +∞,
(5.16)
where
J∗
δ(u) := 1
2|(−∆)s/2u|2
2+1
2|u|2
2−δ
¯p+ 1|u|¯p+1
¯p+1 ,
P∗
δ:= u∈Hs
rad
(RN)|P∗
δ(u)=0\ {0},
27

P∗
δ(u) := |(−∆)s/2u|2
2+ 2∗
s1
2|u|2
2−δ
¯p+ 1|u|¯p+1
¯p+1 
(i.e.,
P∗
δ
and
P∗
δ
are, respectively, the Pohoºaev functional and manifold related to
J∗
δ
). For
every
θ > 0
and
u∈ P(λ)
, we dene
uθ:= θ−N/(4s)u(θ−1/(2s)·)
. Then, using (5.15) and taking
θ=eλ−cδ
(which is positive if
λ > log(cδ)
), it is straightforward to verify that
Jε(λ, u)≥θJ∗
δ(uθ)
and
P∗
δ(uθ)≤0,
therefore the rst property in (5.16) holds thanks to (3.5). As for the second one, by scaling,
one readily checks that
P∗
δ(δ−1/(¯p−1)·) = δ−2/( ¯p−1)P∗
1
, which implies that
P∗
δ=δ−1/(¯p−1)P∗
1
, and
J∗
δ(δ−1/(¯p−1)·) = δ−2/( ¯p−1)J∗
1
. From this, one obtains
infP∗
δJ∗
δ=δ−2/(¯p−1) infP∗
1J∗
1→+∞
as
δ→0+
.
Finally, recalled that
dm
ε,+=bm
ε
by Proposition 5.5, the estimate
bm
ε>−∞
is obtained as in
[20, Proposition 3.7].
5.1.2 Direct minimization on Pohoºaev product
We are now ready to prove the existence of a Pohoºaev minimum as stated in Theorem 5.2.
Proof of Theorem 5.2.
By Proposition 5.5 and Lemma 5.6 we know that
dm
ε,+∈(−∞,0)
. Let
(λn, un)n⊂ P
rad
ε,+
be a minimizing sequence, i.e.,
Im
ε(λn, un)→dm
ε,+
and
Pε(λn, un)=0
. Notice
that, being
un∈ P
rad
ε(λn)
, by Lemma 3.1 we have
λn< λε
for each
n
.
Step 1.
Choice of a proper minimizing sequence
.
We build now a minimizing sequence with additional information on the energy levels and on
the dierential (with respect to
u
) of
Im
ε
. For each
n
, by Proposition 5.5 there exists
vn∈ P
rad
ε(λn)
such that
Im
ε(λn, vn) = aε(λn)−eλn
2m
(5.17)
and
∂uIm
ε(λn, vn)=0.
(5.18)
From (5.17) we obtain
dm
ε,+≤Im
ε(λn, vn) = aε(λn)−eλn
2m≤Jε(λn, un)−eλn
2m=Im
ε(λn, un) = dm
ε,++o(1).
As a consequence, also
(λn, vn)n
is a minimizing sequence.
Step 2.
(λn)n
is bounded below
.
We have
m
2eλn=1
2∗
s
Pε(λn, vn)−Im
ε(λn, vn) + s
N|(−∆)s/2vn|2
2≥ −Im
ε(λn, vn),
whence
lim inf
n
m
2eλn≥ −dm
ε,+>0.
Step 3.
(λn)n
is bounded above
.
In this Step we use the
L2
-subcritical nature of the problem. If
λε<+∞
, then we are done.
Otherwise, if by contradiction
λn→+∞
, then by Lemma 5.6 and (5.17) we have
dm
ε,++o(1) = Im
ε(λn, vn) = eλnaε(λn)
eλn−1
2m→+∞,
28

which is impossible.
Step 4.
(|(−∆)s/2vn|2)n
is bounded
.
Indeed,
s
N|(−∆)s/2vn|2
2=Im
ε(λn, vn)−1
2∗
s
Pε(λn, vn) + m
2eλn=dm
ε,++m
2eλn+o(1)
and thus
|(−∆)s/2vn|2
2
is bounded thanks to Step 3.
Step 5.
(|vn|2)n
is bounded
.
By (5.14) we have
dm
ε,++m
2eλn+o(1) = Im
ε(λn, vn)−1
2Pε(λn, vn) + m
2eλn
=−eλn
22∗
s
2−1|vn|2
2+2∗
s
2−1ZRN
Gε(vn) dx
≤2∗
s
2−1−1
2eλn−δ|vn|2
2+Cδ
¯p+ 1|vn|2∗
s
2∗
s;
by choosing
δ < einf {λn}
(possible thanks to Step 2) and exploiting Step 4 (which implies that
|vn|2∗
s
is bounded), we obtain that
|vn|2
is bounded as well.
Step 6.
(vn)n
converges strongly in
Hs(RN)
.
In this step, we rely on the superlinearity of the perturbed problem. From Steps 25, there
exist
λ∈R
and
v∈Hs
rad
(RN)
such that, up to a subsequence,
λn→λ
,
vn v
in
Hs
rad
(RN)
,
vn→v
in
Lq(RN)
for
q∈(2,2∗
s)
, and
vn(x)→v(x)
for a.e.
x∈RN
. In particular, from
(g0)(g3),
lim
nZRN
gε(vn)vdx=ZRN
gε(v)vdx
and
lim
nZRN
g+(vn)vndx=ZRN
g+(v)vdx;
moreover, from (5.18) we have
∂uIm
ε(λn, vn)[v] = 0 = ∂uIm
ε(λn, vn)[vn].
Thus, exploiting the weak convergence, Fatou's lemma (recall
gε
−(t)t≥0
for all
t∈R
) and that
λn→λ
, we obtain
|(−∆)s/2v|2
2+eλ|v|2
2−ZRN
gε(v)vdx
= lim
nZRN
(−∆)s/2vn(−∆)s/2vdx+eλnZRN
vnvdx+ZRN
gε(vn)vdx= 0
= lim
nZRN
|(−∆)s/2vn|2dx+eλnZRN
v2
ndx+ZRN
gε(vn)vndx
= lim
nZRN
|(−∆)s/2vn|2dx+eλnZRN
v2
ndx+ZRN
gε
−(vn)vndx−ZRN
g+(v)vdx
≥lim inf
nZRN
|(−∆)s/2vn|2dx+eλnZRN
v2
ndx−ZRN
gε(v)vdx
≥ |(−∆)s/2v|2
2+eλ|v|2
2−ZRN
gε(v)vdx,
thus
lim inf
nZRN
|(−∆)s/2vn|2dx+eλnZRN
v2
ndx=|(−∆)s/2v|2
2+eλ|v|2
2;
29

this means that, up to a subsequence,
vn→v
strongly in
Hs(RN)
.
Step 7.
Conclusions
.
By the strong convergence, we have
Im
ε(λ, v) = dm
ε,+
, i.e.,
(λ, v)
is a Pohoºaev minimum in
the product space. The fact that
(µε, uε) := (eλ, v)
is a solution to (5.1) follows from Proposition
1.7. Finally, assume
g|(−∞,0) = 0
or
g
odd, which imply the same properties on
gε
. Then, in
Step 1, we may assume that each
vn
is Schwarz-symmetric in light of Theorem 3.6 (ii), hence so
is
v
.
5.2 Symmetry results
Exploiting the perturbed
gε
, we prove here Proposition 1.12 (i)(iii). We recall that
G
does
not map
Hs(RN)
to
L1(RN)
; with the intention of gaining a symmetry result arguing as in [40,
Proof of Theorem 4.1], we show the following lemma.
Lemma 5.7
Assume
(g0)

(g2)
. For
v∈Hs(RN)
and
x= (x1, x′)∈R×RN−1
, dene
v1(x) := (v(x1, x′)
if
x1<0
v(−x1, x′)
if
x1≥0
and
v2(x) := (v(−x1, x′)
if
x1<0
v(x1, x′)
if
x1≥0.
Then
RRNG(v1) dx+RRNG(v2) dx= 2 RRNG(v) dx
.
Proof.
It is clear that
RRNG+(v1) dx+RRNG+(v2) dx= 2 RRNG+(v) dx
. Additionally, from
the monotone convergence theorem,
ZRN
G−(v1) dx+ZRN
G−(v2) dx= lim
ε→0+ZRN
Gε
−(v1) dx+ZRN
Gε
−(v2) dx
= 2 lim
ε→0+ZRN
Gε
−(v) dx= 2 ZRN
G−(v) dx,
whence the statement.
Proof of Proposition 1.12 (i)(iii).
(i) Let
(µ, u)∈ P(ρ)
be a minimizer as in the statement. In view of Remark 4.1 and Lemma
3.3,
u
solves the minimization problem (3.4). Then, exploiting Lemma 5.7, we can argue in a
similar way to [40, Proof of Theorem 4.1]
5
. Likewise if
u
minimizes
K
over
Sm
.
(ii) Let
(µ, u)∈ P(ρ)
be a minimizer as in the statement, and let us denote by
u∗
the Schwarz
rearrangement of
u
. We rst observe that
ZRN
G−(u) dx=ZRN
G−(u∗) dx;
(5.19)
indeed, (5.19) holds with
Gε
−
instead of
G−
(see [14, Proposition B.1]), thus from the monotone
convergence theorem
ZRN
G−(u) dx= lim
ε→0+ZRN
Gε
−(u) dx= lim
ε→0+ZRN
Gε
−(u∗) dx=ZRN
G−(u∗) dx,
and we get the claim. Now, assume by contradiction that
u
does not equal its Schwarz rear-
rangement
u∗= 0
. Then, from [14, Propositions B.1 and B.3],
|(−∆)s/2u∗|2
2<|(−∆)s/2u|2
2
and
|u∗|2=|u|2,
5
We observe that we formally deal with
Case A
therein but minimize over
Hs(RN)
.
30

which imply
Jµ(u∗)< Jµ(u)
and
Pµ(u∗)< Pµ(u)=0.
From Remark 3.2, we nd
t∗>0
such that
Pµu∗(·/t∗)= 0
and
inf
Pµ
Jµ≤Jµu∗(·/t∗)≤Jµ(u∗)< Jµ(u) = inf
Pµ
Jµ,
a contradiction. Similarly for
L2
-minima.
(iii) We claim that
RRN|g−(u)u−|dx < +∞
, where, we recall,
u±= max{±u, 0}
. As a matter
of fact, since
g−(u)u−=g−(−u−)u−≤0
, there holds
|g−(u)u−|=−g−(−u−)u−≤g−(u+)u+−g−(−u−)u−=g−(u)u+−g−(u)u−=g−(u)u∈L1(RN),
which holds by the assumptions. As a consequence, with the notation of Lemma 2.2, for every
R > 0
there holds
ZRN
(−∆)s/2u(−∆)s/2(ϕRu−) dx+µZRN
uϕRu−dx=ZRN
g(u)ϕRu−dx,
and passing to the limit we obtain
ZRN
(−∆)su(−∆)su−dx+µZRN
uu−dx=ZRN
g(u)u−dx.
(5.20)
Finally, using (5.20) and Lemma 2.1,
0≤ZRNg(−u−)−µ(−u−)u−dx=ZRN
g(u)u−−µuu−dx=ZRN
(−∆)s/2u(−∆)s/2u−dx
≤ −|(−∆)s/2u−|2
2≤0,
and we conclude.
6 The sublinear problem
6.1
ε
-independent estimates
Let
µε=eλε
,
uε
,
mε
, and
dm
ε,+
be the quantities found in Theorem 5.2 and
m0
be as in
(5.12). From now on, we are interested in the dependence on
ε
; in particular, we will ensure that
some objects related to problem (5.1) enjoy some uniformity in
ε > 0
. We start by showing that
(dm
ε,+)ε∈(0,1]
is bounded and far from zero.
Lemma 6.1
Let
m>m0
. There exist
ζm, βm∈R
such that
−∞ < ζm≤dm
ε,+≤βm<0
for every
ε∈(0,1].
Proof.
By Lemma 5.6 and the monotonicity of
ε7→ aε(λ)
and
ε7→ λε
 cf. (5.9) and (5.10), we
have
dm
ε,+=bm
ε≥bm
1=: ζm
. Again by Lemma 5.6 we have, for every
λ < λ0≤λε
,
dm
ε,+≤eλ
2aε(λ)
eλ/2−m≤eλ
2a(λ)
eλ/2−m.
Let now
δ:= m−m0
2>0
. By the denition of inmum
m0= infλ<λ0
a(λ)
eλ/2
there exists
λδ< λ0
such that
dm
ε,+≤eλδ
2a(λδ)
eλδ/2−m≤eλδ
2(m0+δ−m) = −eλδ
4(m−m0) =: βm<0.
31

We show now some estimates on the solutions
(λε, uε)
. Here we use the
L2
-subcritical growth
(g3).
Lemma 6.2
Let
m>m0
. Then
(λε, uε)ε∈(0,1]
is bounded in
R×Hs
rad
(RN)
.
Proof.
Observe rst that, since
(λε, uε)
satises the Pohoºaev identity and by Lemma 6.1, we
have
m
2eλε=1
2∗
s
Pε(λε, uε)−Im
ε(λε, uε) + s
N|(−∆)s/2uε|2
2≥ −dm
ε,+≥ −βm
which gives a bound below on
(λε)ε∈(0,1]
. Next, from (g0),(g1), and (g3), in a similar way to
(5.15), for every
δ > 0
there exists
cδ>0
such that for all
t∈R
g+(t)t≤cδt2+δ|t|¯p+1.
Then, from (2.2) and the fact that
(λε, uε)
is a critical point of
Im
ε
, we have
0 = |(−∆)s/2uε|2
2+eλε|uε|2
2+ZRN
gε
−(uε)uεdx−ZRN
g+(uε)uεdx
≥ |(−∆)s/2uε|2
2+eλε−cδ|uε|2
2+ZRN
gε
−(uε)uεdx−δ|uε|¯p+1
¯p+1
≥ |(−∆)s/2uε|2
2+eλε−cδ|uε|2
2−δC
GN
|(−∆)s/2uε|2
2|uε|4s/N
2
=1−δC
GN
m2s/N |(−∆)s/2uε|2
2+eλε−cδm,
and we conclude by taking
δ < (C
GN
m2s/N )−1
.
6.2 Passage to the limit
We want to pass to the limit with the solutions
(λε, uε)
found in Theorem 5.2. Recall that,
by Lemma 6.1 and Proposition 1.5, we have
−∞ < ζm≤dm
ε,+=Im
ε(λε, uε) = κm
ε=ε
m=Kε(uε)≤βm<0;
moreover, by Lemma 6.2,
(λε, uε)ε∈(0,1]
is bounded in
R×Hs
rad
(RN)
. Therefore, up to a subse-
quence,
(uε u0
in
Hs
rad
(RN),
λε→λ0
in
R.
(6.1)
In addition, we have
|u0|2
2≤m
by the weak convergence. We write
µε:= eλε
,
µ0:= eλ0
.
We want to show that
(µ0, u0)∈(0,+∞)×Hs
rad
(RN)
is the desired solution of Theorem 1.1.
To this aim, the rst step is to prove the pointwise convergence of
Gε(uε)
to
G(u0)
.
Lemma 6.3
Let
u0∈Hs
rad
(RN)
as in
(6.1)
. Then,
gε(uε(x)) →g(u0(x))
and
Gε(uε(x)) →
G(u0(x))
as
ε→0+
for a.e.
x∈RN
.
Proof of Lemma 6.3.
Let
x∈RN
such that
uε(x)→u0(x)∈R
. We prove rst the claim on
gε
. We write
(0,1] = {ε∈(0,1] | |uε(x)|> ε}∪{ε∈(0,1] | |uε(x)| ≤ ε}=: Ix
1∪Ix
2.
32

Clearly,
0∈Ix
1∪Ix
2
. If
0∈Ix
1
then, observed that
gε(uε(x)) = g(uε(x))
for every
ε∈Ix
1
, we get
lim
ε→0+, ε∈Ix
1
gε(uε(x)) = lim
ε→0+, ε∈Ix
1
g(uε(x)) = g(u0(x)).
If
0∈Ix
2
, let
δ > 0
. By the assumptions, we know that there exists
ωδ>0
such that
|g(t)| ≤ δ
for every
|t| ≤ ωδ
. Observed that
|gε(uε(x))| ≤ |g+(uε(x))|+|gε
−(uε(x))| ≤ |g+(uε(x))|+|g−(uε(x))|
=|g+(uε(x)) + g−(uε(x))|=|g(uε(x))| ≤ δ
for every
ε∈Ix
2∩(0, ωδ]
, we obtain
lim sup
ε→0+, ε∈Ix
2
|gε(uε(x))| ≤ δ;
by the arbitrariness of
δ
, we achieve
lim
ε→0+, ε∈Ix
2
gε(uε(x)) = 0;
on the other hand, by denition of
Ix
2
, we have
lim
ε→0+, ε∈Ix
2
|uε(x)| ≤ lim
ε→0+, ε∈Ix
2
ε= 0
and thus
u0(x) = limε→0+, ε∈Ix
2uε(x) = 0
; since
g(u0(x)) = g(0) = 0
, we have reached
lim
ε→0+, ε∈Ix
2
gε(uε(x)) = g(u0(x)),
which is the claim. We prove now the claim on
Gε
. Obviously,
G+(uε(x)) →G+(u0(x))
; noticed
that, in a similar way as before, we can show
gε
−(u0(x)t)→gε
−(u0(x)t)
for each
t∈[0,1]
, and
observed that
|gε
−(uε(x)t)|≤|g−(uε(x)t)| ≤ 1 + |uε(x)|2∗
s≤2 + |u0(x)|2∗
s
for
0< ε ≪1
and any
t∈[0,1]
, we can use the dominated convergence theorem and obtain
Gε
−(uε(x)) = uε(x)Z1
0
gε
−(uε(x)t) dt→u0(x)Z1
0
g−(u0(x)t) dt=G−(u0(x)).
We observe that the limit function obtained in (6.1), at the moment, could be trivial. In what
follows, we show that this is not the case and, moreover,
u0
is a solution enjoying minimality
properties.
Proposition 6.4
Let
(µ0, u0)∈(0,+∞)×Hs
rad
(RN)
as in
(6.1)
. Then
u0∈ Sm
,
(µ0, u0)∈ P
,
it is a solution to
(1.8)
, and
K(u0) = Im(µ0, u0) = m=κm=dm
+=dm<0.
Moreover,
uε→u0
in
Hs(RN)
,
RRNGε
−(uε) dx→RRNG−(u0) dx < ∞
, and
dm
ε,+→dm
+
as
ε→0+
.
33

Proof.
We show rst that
u0
minimizes
K
over the
L2
-ball. Since
Gε≥G
, we have
m
ε= inf
D
rad
m
Kε≤inf
D
rad
m
K≤K(u0).
On the other hand, by exploiting that
(−∆)s/2uε(−∆)s/2u0
in
L2(RN)
, the strong conver-
gence in
L1(RN)
of
G+(uε)
, and Fatou's lemma for
Gε
−(uε)
(here we use Lemma 6.3),
K(u0) = 1
2|(−∆)s/2u0|2
2−ZRN
G+(u0) dx+ZRN
G−(u0) dx
≤lim inf
ε→0+1
2|(−∆)s/2uε|2
2−lim
ε→0+ZRN
G+(uε) dx+ lim inf
ε→0+ZRN
Gε
−(uε) dx
≤lim inf
ε→0+Kε(uε) = lim inf
ε→0+m
ε≤lim sup
ε→0+
m
ε≤inf
D
rad
m
K≤K(u0).
This yields that
u0
is a minimum over the
L2
-ball and the following holds as
ε→0+
:
m
ε→m
,
RRNGε
−(uε) dx→RRNG−(u0) dx < ∞
, and
(−∆)s/2uε→(−∆)s/2u0
in
L2(RN)
. By the fact
that
m
ε≤βm<0
, we obtain
K(u0)≤βm<0
; in particular,
u0= 0
. Moreover, by Proposition
4.2 and Proposition 1.5 we know that
(µ0, u0)
is a solution of (1.1), it satises the Pohoºaev
identity,
u0∈ Sm
(whence
Im(µ, u0) = K(u0)
), and
κm=m=dm
+=dm
, thus
uε→u0
in
L2(RN)
and hence in
Hs(RN)
.
To show the Nehari identity, we need the following lemma.
Lemma 6.5
The solution
u0
given in
(6.1)
satises
RRNg−(u0)u0dx < ∞
.
Proof.
Let us recall that, as
ε→0+
,
µε→µ0
,
uε→u0
in
Hs(RN)
and a.e. in
RN
, and
gε(uε)→g(u0)
a.e. in
RN
. Additionally,
(µε, uε)
satises the Nehari identity
|(−∆)s/2uε|2
2+µε|u0|2
2=ZRN
g+(uε)−gε
−(uε) dx.
Since
g−(t)t≥0
and
ϕε(t)≥0
for all
t∈R
, we also have that
gε
−(t)t≥0
for all
t∈R
, hence
u0
satises the following Nehari inequality
|(−∆)s/2u0|2
2+µ0|u0|2
2+ZRN
g−(u0)u0dx≤lim
ε→0+|(−∆)s/2uε|2
2+µε|uε|2
2+ZRN
gε
−(uε)uεdx
= lim
ε→0+ZRN
g+(uε)uεdx=ZRN
g+(u0)u0dx
thanks to (g1),(g3), and Fatou's lemma. In particular, this gives the claim.
Proof of Theorem 1.1.
The theorem is a consequence of Lemma 6.5, Proposition 4.3, Propo-
sition 6.4, Proposition 1.6 and Proposition 1.10.
Proof of Proposition 1.12 (iv).
From Theorem 5.2, for every
ε∈(0,1]
there exists
(λε, uε)∈
Pε,+
Schwarz-symmetric such that
Im
ε(λε, uε) = infPε,+Im
ε= infP
rad
ε,+Im
ε
. Then, repeating all the
steps in Section 6, not only do we re-obtain Theorem 1.1, but we also get the same outcome where
κm
,
m
,
a(µ)
, and
dm
are replaced, respectively, with
infSmK
,
infDmK
,
infPµJµ
, and
infPIm
,
but the pair
(µ0, u0)
is the same in both cases. In particular,
κm= infSmK, m= infDmK
,
a(µ) = infPµJµ
, and
dm= infPIm
.
34

7 Further existence results
7.1 Existence for all masses
In Theorem 1.1, we obtained the existence of a solution for large masses. When we rule out
the possibility of
g
to be superlinear at the origin (or positive nearby), which clearly mimics
the logarithmic case, we can achieve existence also for all masses. Obviously, we already have
existence for all masses if
m0= 0
, which is why we assume
m0>0
in Theorem 7.2 below.
Lemma 7.1
Under assumptions
(g0)

(g4)
, for every
ω≥0
there exists
eµω∈(0,+∞)
such that
inf
0<µ<µ0
a(µ)
µ+ω′= inf
0<µ<eµω
a(µ)
µ+ω′
(7.1)
for every
ω′∈[0, ω]
. In particular,
m0= inf0<µ<eµ0
a(µ)
µ/2.
Moreover,
eµω
ω→0
as
ω→+∞.
Proof.
If
µ0<+∞
, we take
eµω:= µ0
. Otherwise, we set
M:= infµ∈(0,+∞)a(µ)
µ
and
eµω:= inf eµ > 0
a(µ)
µ+ω≥M+ 1
for each
µ≥eµ.
We observe that
eµω<+∞
from Lemma 5.6, while
eµω>0
from its denition and that of
M
.
Additionally, from the continuity of
a: (0,+∞)→[0,+∞)
(see Proposition 3.4), we have
a(eµω)
eµω+ω=M+ 1.
(7.2)
Consequently, if
ν≥eµω
and
ω′∈[0, ω]
, then
a(ν)
ν+ω′≥M+ 1 >inf
0<µ<+∞
a(µ)
µ+ω′,
whence (7.1). We move to the last claim; we can assume
eµω→+∞
as
ω→+∞
(otherwise, the
claim is obvious). Since (7.2) can be rewritten as
ω
eµω
=1
M+ 1
a(eµω)
eµω
−1,
the claim follows from Lemma 5.6.
Theorem 7.2
Assume
(g0)

(g4)
,
m0>0
, and
lim sup
t→0
g(t)
t=: −ω∈[−∞,0).
Then, there exists
m1=m1(ω)∈[0, m0)
such that, for every
m > m1
, there exists a solution
(µ0, u0)∈R× S
rad
m
to
(1.8)
, which satises the Nehari and Pohoºaev identities, and such that
Im(µ, u0) = K(u0) = a(µ0)−µ0
2m=κm=dm= inf
−∞<ν<µ0a(ν)−ν
2m;
if
κm≤0
then
µ0>0
. Finally, if
ω=∞
, then
m1= 0
.
35

Proof.
Let
ω∈(0, ω)
and dene, for
t∈R
,
hω(t) := g(t) + ωt, Hω(t) := G(t) + ω
2t2.
We observe that

hω
is continuous and
hω(0) = 0
,

lim sup|t|→+∞
|hω(t)|
|t|2∗
s−1<∞
,

there exists
t0= 0
such that
Hω(t0)>0
.
Moreover, it is straightforward to check that
lim supt→0
hω(t)
t<0
and
lim sup|t|→+∞
hω(t)
|t|¯p−1t≤0
,
thus
hω
satises the assumptions of Theorem 1.1. In particular, for
m>mω:= inf
0<µ<µω
aω(µ)
µ/2
there exists an
L2
-minimum
uω∈ S
rad
m
with Lagrange multiplier
µω>0
, i.e.,
Kω(uω) = inf
D
rad
m
Kω= inf
S
rad
m
Kω=: κm
ω<0;
(7.3)
here
µω:= sup
t=0
Hω(t)
t2/2=µ0+ω, Kω(u) := 1
2|(−∆)s/2u|2
2−ZRN
Hω(u) dx,
aω(µ) := inf
P
rad
µ,ω
Jµ,ω, Jµ,ω(u) := 1
2|(−∆)s/2u|2
2+µ
2|u|2
2−ZRN
Hω(u) dx,
Pµ,ω(u) := ZRN
|(−∆)s/2u|2dx+ 2∗
sZRN
µ
2u2−Hω(u) dx,
Pµ,ω := u∈Hs(RN)\ {0} | Pµ,ω(u)=0,P
rad
µ,ω := Pµ,ω ∩Hs
rad
(RN).
We notice that
(µω, uω)
satises the Nehari and Pohoºaev identities with respect to
Hω
. Since
Kω(u) = K(u)−ω
2|u|2
2
, we obtain
K(uω)−ω
2m=K(uω)−ω
2|uω|2
2=Kω(uω) = inf
S
rad
m
Kω=κm−ω
2m,
whence
K(uω) = κm
, which means that
uω
is an
L2
-minimum for the original problem. Moreover,
κm=κm
ω+ω
2m.
(7.4)
In addition,
uω
is a solution to the EulerLagrange equation corresponding to
Kω
, i.e., for some
Lagrange multiplier
µω>0
,
(−∆)s/2uω+µωuω=hω(uω)
in
RN,
which means that
u0:= uω
is a solution to (1.8) with Lagrange multiplier
µ0:= µω−ω
(7.5)
36

and
(µ0, u0)
satises the Nehari and Pohoºaev identities with respect to
G
. Additionally, for any
ν > ω
, we have
Jν,ω(u) = Jν−ω(u)
and
Pν,ω(u) = Pν−ω(u)
, therefore
1
2mω≤inf
ω<ν<µ0+ω
aω(ν)
ν= inf
0<ν−ω<µ0
a(ν−ω)
ν= inf
0<ν<µ0
a(ν)
ν+ω= inf
0<ν<µ0
a(ν)
ν
ν
ν+ω.
(7.6)
Let
eµ=eµω∈(0,+∞)
be the value given in Lemma 7.1. Since the function
ν∈(0,+∞)7→
ν
ν+ω∈(0,+∞)
is increasing, from (7.6) we obtain
1
2mω≤inf
0<ν<eµ
a(ν)
ν
ν
ν+ω≤eµ
eµ+ωinf
0<ν<eµ
a(ν)
ν=eµ
eµ+ωinf
0<ν<µ0
a(ν)
ν=eµ
eµ+ω
m0
2<1
2m0.
In addition, if
ω=∞
, taking
ω
arbitrarily large we obtain
mω→0
as
ω→+∞
again from Lemma 7.1. As a consequence, for any
m > 0
, there exists
mω∈[0, m)
and thus an
L2
-minimum
uω
. We conclude setting
m1:= inf
0<ω<ω mω;
noticed that
ω7→ mω
is nonincreasing (this is a direct consequence of
Jµ,ω ≤Jµ
,
Pµ,ω ≤Pµ
,
µω≥µ0
, and an argument similar to (5.9)(5.13)), we further observe that
m1= limω→ω−mω
.
The sign of
µ0
is a simple consequence of the Pohoºaev identity. Finally, the fact that
K(u0) = Im(µ0, u0) = dm= inf
−∞<ν<µ0a(ν)−ν
2m=a(µ0)−µ0
2m
follows from Propositions 1.5,1.6, and 1.10.
Remark 7.3
By denition,
m1
is nonincreasing with respect to
ω
. Moreover, in the case of
small masses, Theorem 7.2 gives no information on the sign of the Lagrange multiplier and of
the minimal energy  see indeed
(7.4)
and
(7.5)
.
Proof of Theorem 1.3.
It is a consequence of Theorem 7.2.
7.2 Existence under relaxed assumptions
We deal now with Proposition 1.14. We thus search for a normalized solution which is
nonnegative, possibly considering also the case
lim sup
|t|→+∞
g−(t)
|t|2∗
s−1= +∞.
The idea is to modify the nonlinearity
g
. We begin with the following result.
Lemma 7.4
Let
(g0)

(g2)
hold. Assume
g(t) = 0
for
|t| ≥ t1
, and let
u
be a nonnegative
solution of
(1.1)
for some
µ≥0
. Then
u∈L∞(RN)
and
|u|∞≤t1
.
Proof.
Let
v:= (u−t1)+
. It is easy to check, via explicit computations, that
v∈Hs(RN)
. Let
ϕR
be as in Lemma 2.2 and set
vR:= vϕR
. Then
vR∈Hs(RN)∩L1(RN)
and hence
1
2ZRN
(−∆)s/2u(−∆)s/2vRdx+µ
2ZRN
uvRdx=ZRN
g(u)vRdx.
37

We notice that
g(u)vR≡0
by the assumptions. Since
u≥0
and
µ≥0
, we have
µuvR≥0
and
thus
ZRN
(−∆)s/2u(−∆)s/2vRdx≤0.
By Lemma 2.2 we have
ZRN
(−∆)s/2u(−∆)s/2(u−t1)+dx≤0.
On the other hand,
ZRN
(−∆)s/2u(−∆)s/2(u−t1)+dx
=CN,s ZRNZRN(u(x)−t1)−(u(y)−t1)(u(x)−t1)+−(u(y)−t1)+
|x−y|N+2s
≥CN,s Z{u(x)≥t1}Z{u(y)≥t1}(u(x)−t1)+−(u(y)−t1)+2
|x−y|N+2s
=CN,s ZRNZRN(u(x)−t1)+−(u(y)−t1)+2
|x−y|N+2s=|(−∆)s/2(u−t1)+|2
2.
Joining the two relations found we obtain
(u−t1)+= 0
, i.e.,
u≤t1
. Finally, using
vR:=
(u+t1)−ϕR
and
g(t)=0
for
t≤ −t1
, we similarly obtain
u≥ −t1
.
Proof of Proposition 1.14.
Let us assume that
t0
, dened in (g4), is positive (if it is negative,
then we argue likewise). This implies that
g
is positive somewhere on
(0,+∞)
. Let
t1
be the
rst positive zero of
g
such that
g
is positive somewhere on
(0, t1)
(if
g(t)>0
for all
t > 0
, we
set
t1= +∞
). Then, we set
¯g(t) := g(t)χ(0,t1)(t).
Clearly,
¯g
satises (g0)(g4), thus Proposition 1.12 applies (or Remark 1.13 if (1.21) holds),
and we have a Schwarz-symmetric solution
(µ0, u0)
, which satises the Nehari and Pohoºaev
identities, to
(−∆)su+µu = ¯g(u).
If
t1= +∞
, then
¯g(u0) = g(u0)
, so
(µ0, u0)
is a solution to (1.8); otherwise, from Lemma 7.4,
|u0|∞≤t1
, hence
¯g(u0) = g(u0)
once again, and we conclude as before.
We end this section with the examples mentioned in the introduction.
Proof of Corollary 1.15.
As in [45, Proof of Theorem 1.6], we prove that when
β < 0
and
q∈(1,2∗
s−1]
,
max G⋚0
if and only if
β⋚−α(q+1)
q−1e−(q+1)/2
, hence the second part (existence)
follows from Theorem 1.3 and Proposition 1.12 (iv) (see also Remark 1.13). If
m>m0
, then
the claim follows by Theorem 1.1. The rst part (nonexistence) is a direct consequence of the
Pohoºaev identity, together with Proposition 1.8.
Proof of Corollary 1.16.
We argue as in Corollary 1.15.
Acknowledgments.
This work was supported by the Thematic Research Programme Vari-
ational and geometrical methods in partial dierential equations, University of Warsaw, Excel-
lence Initiative - Research University. The rst author is supported by INdAM-GNAMPA Project
Metodi variazionali per problemi dipendenti da operatori frazionari isotropi e anisotropi, codice
CUP #E5324001950001#. The second author is supported by INdAM-GNAMPA Project Prob-
lemi di ottimizzazione in PDEs da modelli biologici, codice CUP #E5324001950001#. The rst
author expresses his gratitude to the University of Warsaw for hosting him in November 2024
and February 2025.
38

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