arX

iv:2

107.

1328

1v1

[m

ath.

AP]

28

Jul 2

021

Existence of normalized solutions for the planar

Schrödinger-Poisson system with exponential critical

nonlinearity

Claudianor O. Alves1 *†, Eduardo de S. Böer2 ‡ and Olímpio H. Miyagaki2 §

1 Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande

58109-970 Campina Grande-PB-Brazil

2 Department of Mathematics, Federal University of São Carlos,

13565-905 São Carlos, SP - Brazil

July 29, 2021

Abstract: In the present work we are concerned with the existence of normalized solutions to

the following Schrödinger-Poisson System

−∆u+ λu+ µ(ln | · | ∗ |u|2)u = f(u) in R2,∫

R2

|u(x)|2dx = c, c > 0,

for µ ∈ R and a nonlinearity f with exponential critical growth. Here λ ∈ R stands as a

Lagrange multiplier and it is part of the unknown. Our main results extend and/or complement

some results found in [3] and [13].

Mathematics Subject Classification: 35J60, 35J15, 35A15, 35J10.

Key words. Schrödinger equation, exponential critical growth, variational techniques, prescribed norm.

*Corresponding author†E-mail address: coalves@mat.ufcg.edu.br‡E-mail address: eduardoboer04@gmail.com, Tel. +55.51.993673377§E-mail address: ohmiyagaki@gmail.com, Tel.: +55.16.33519178 (UFSCar).

1

2

1 Introduction

In the present paper we are interested with the existence of normalized solutions for the

following Schrödinger-Poisson System,

{

iψt −∆ψ + W (x)ψ + γωψ = 0 in RN × R

∆ω = |ψ|2 in RN ,(1.1)

where ψ : RN × R → C is the time-dependent wave function, W : RN → R is a real external

potential and γ > 0 is a parameter. The function ω stands as an internal potential for a nonlocal

self-interaction of the wave function ψ. The usual ansatz ψ(x, t) = e−iΘtu(x), with Θ ∈ R, for

standing wave solutions of (1.1) leads to

{

−∆u+W (x)u+ γωu = 0 in RN

∆ω = u2 in RN ,(1.2)

with W (x) = W (x) + Θ. From the second equation of (1.2) we observe that ω : RN → R is

determined only up to harmonic functions. In this point of view, it is natural to choose ω as the

Newton potential of u2, i.e., ΓN ∗ u2, where ΓN is the well-known fundamental solution of the

Laplacian

ΓN(x) =

1

N(2 −N)σN|x|2−N if N ≥ 3,

1

2πln |x| if N = 2,

in which σN denotes the volume of the unit ball in RN . From this formal inversion of the second

equation in (1.2), as it is detailed in [11], we obtain the following integro-differential equation

−∆u+W (x)u+ γ(ΓN ∗ |u|2)u = b|u|p−2u, p > 2, b > 0, in RN . (1.3)

Then, we make a quick overview of the literature. To begin with, we note that the case

N = 3 has been extensively studied, due to its relevance in physics. It is curious that,

although this equation is called “Choquard equation”, it has first studied by Fröhlich and Pekar

in [17, 18, 28], to describe the quantum mechanics of a polaron at rest in the particular case

W (x) ≡ a > 0 and γ > 0. Then, it was introduced by Choquard in 1976, to study an electron

trapped in its hole. From the applied point of view, the local nonlinear term on the right side

of equation (1.3) usually appears in Schrödinger equations as a model to the interaction among

particles.

We could discuss a bunch of variations of these kind of equations in the three dimensional

case, but in order to make it concise, we refer the readers to the following papers, and the

references therein, [4, 6, 7, 21, 23, 24, 26].

Once we turn our attention to the case N = 2, we immediately see that the literature is

scarce. In this case, we can cite the recent works of [1, 9, 10, 13, 14, 16]. In [14, 16], the authors

3

have proved the existence of infinitely many geometrically distinct solutions and a ground state

solution, considering W continuous and Z2-periodic and W (x) ≡ a > 0, respectively, and

the particular case f(u) = b|u|p−2u. Then, in [1], the authors have dealt with equation (1.4)

considering a nonlinearity with exponential critical growth. They have proved the existence of

a ground state solution via minimization over Nehari manifold. Moreover, in [9], the authors

have proved existence and multiplicity results for the p−fractional Laplacian operator. Finally,

in [10], the authors deal with a (p,N)-Choquard equation and prove existence and multiplicity

results.

We call attention to [13] in which work the authors have dealt with the existence of stationary

waves with prescribed norm considering λ ∈ R and that consists in a key reference to our work.

Another important reference is [2], since we manage to adapt some techniques from both of

them. We also refer to [2] for a relevant overview about Schrödinger prescribe norm problems.

In the present paper we focus on finding prescribe norm solutions for the planar equation

−∆u+ λu+ µ(ln | · | ∗ |u|2)u = f(u) in R2∫

R2

|u(x)|2dx = c, c > 0 . (1.4)

where λ, µ ∈ R and f : R → R is continuous, with primitive F (t) =t∫

0

f(s)ds. This

approach seems to be particularly meaningful from the physical point of view, because there

is a conservation of mass.

The main difficulties in the proof our main results are associated with the fact that we are

working with critical nonlinearities in the whole R2 and with the logarithm term, which are

unbounded and changes sign.

Our aim is to extend and/or complement the results already obtained in the literature and

cited above, more precisely the results found in [3] and [13], by working with a nonlinearity that

has an exponential critical growth. We recall that a function h has an exponential subcritical

growth at +∞, if

lims→+∞

h(t)

eαt2 − 1

= 0 , for all α > 0,

and we say that h has α0-critical exponential growth at +∞, if

limt→+∞

h(t)

eαt2 − 1

=

{

0, ∀ α > α0,

+∞, ∀ α < α0.

As usual conditions while dealing with this kind of growth, found in works such as [15,25], we

assume that f satisfies

(f1) f ∈ C(R,R), f(0) = 0 and has a critical exponential growth with α0 = 4π.

4

(f2) lim|t|→0

|f(t)||t|τ = 0, for some τ > 3.

From (f1) and (f2), given ε > 0, α > α0, fixed, for all p > 2, we can find two constants

b1 = b1(p, α, ε) > 0 and b2 = b2(p, α, ε) > 0 such that

f(t) ≤ ε|t|τ + b1|t|p−1(eαt2 − 1) , ∀ t ∈ R, (1.5)

and

F (t) ≤ ε|t|τ+1 + b2|t|p(eαt2 − 1) , ∀t ∈ R. (1.6)

In order to verify that (PS) sequences are bounded in H1(R2), we will need the following

conditions:

(f3) there exists θ > 6 such that f(t)t ≥ θF (t) > 0, for all t ∈ R \ {0},

(f4) there exist q > 4 and ν > ν0 such that F (t) ≥ ν|t|q, for all t ∈ R.

Next we provide some important definitions for our work. Once we will use variational

techniques, we consider the associated Euler-Lagrange functional I : H1(R2) → R ∪ {∞}given by

I(u) =1

2A(u) +

µ

4V (u)−

∫

R2

F (u)dx, (1.7)

where

A(u) =

∫

R2

|∇u|2dx and V (u) =

∫

R2

∫

R2

ln(|x− y|)u2(x)u2(y)dxdy.

It is easy to verify, from Moser-Trudinger inequality, Lemma 2.5, and Hardy-Littlewood-

Sobolev inequality (HLS) (found in [22]), that I is well-defined on the slightly smaller Hilbert

space

X =

u ∈ H1(R2); ||u||2∗ =∫

R2

ln(1 + |x|)u2(x)dx <∞

⊂ H1(R2), (1.8)

endowed with the norm || · ||X =√

|| · ||2 + || · ||2∗, where || · || is the usual norm in H1(R2).

Moreover, I is C1 on X (see [1] and [14]) and any critical point u of I∣

∣

S(c)corresponds to a

solution of (1.4), where λ ∈ R appears as a Lagrange multiplier and

S(c) = {u ∈ X ; ||u||22 = c}.

The two first results of the present paper involving the existence of solution are the

following:

Theorem 1.1. Suppose that f satisfies (f1)−(f4). Then, there are µ0, ν0 > 0 such that problem

5

(1.4) has at least one weak solution u ∈ S(c) such that I(u) = mν > 0 for µ ∈ (0, µ0),

c ∈ (0, 1) and ν > ν0, where

mν = infγ∈Γ

maxt∈[0,1]

I(γ(t)), (1.9)

with Γ = {γ ∈ C([0, 1], S(c)) ; γ(0) = u1 and γ(1) = u2}, u1, u2 ∈ S(c).

In a similar way, we get the following result.

Theorem 1.2. Suppose that f satisfies (f1) − (f4) and µ > 0. Then, there are c0, ν0 such that

problem (1.4) has at least one weak solution u ∈ S(c) such that I(u) = mν > 0 for c ∈ (0, c0)

and ν > ν0.

Once the proof of Theorem 1.2 is similar to the proof of Theorem 1.1, where the only change

is the control of the inequalities from parameter µ to the mass c, we will not write it down.

As an immediate consequence from Theorems 1.1 and 1.2 we get the following type of

least energy level. We do not call it a ground state since it cannot be take over all the possible

solutions for (1.4).

Corolary 1.1. Under the hypotheses of Theorem 1.1 (or Theorem 1.2) problem (1.4) has a least

energy level solution u ∈ S(c), in the sense that, there is a function u ∈ S(c) satisfying

I(u) = ml = inf{I(v) ; I∣

∣

′

S(c)(v) = 0 and Q(v) = 0},

where Q is defined in (4.3).

Related to the existence of multiple solutions, we will use a genus approach that is based on

the ideas of [3]. In order to do so, we need the following condition.

(f ′1) f ∈ C(R,R), f(0) = 0, f is odd and has a critical exponential growth with α0 = 4π.

As in the existence case, we have two results which differ by the way that we control some

estimates, either by the parameter µ or by the mass c.

Theorem 1.3. Suppose that f satisfies (f ′1)− (f4), c ∈ (0, 1) and µ ∈ (0, µ1), for µ1 defined in

(5.1). Then, given n ∈ N, there is ν = ν(n) > 0 sufficiently large such that, if ν ≥ ν, (1.4) has

at least n non-trivial weak solutions , uj ∈ S(c), verifying I(uj) < 0, for 1 ≤ j ≤ n.

Theorem 1.4. Suppose that f satisfies (f ′1)− (f4), c ∈ (0,min{c1, 1}), , for c1 defined in (5.1),

and µ ∈ R+. Then, given n ∈ N, there is ν = ν(n) > 0 sufficiently large such that, if ν ≥ ν,

(1.4) has at least n non-trivial weak solutions , uj ∈ S(c), verifying I(uj) < 0, for 1 ≤ j ≤ n.

Before concluding this section, we would like point out that our main results complement the

study made in [13] because in that paper it was not considered the case where the nonlinearity

has a critical exponential growth, while in [3], the authors neither consider the existence of

normalized solutions nor the presence of an unbounded indefinite internal potential.

6

The paper is organized as follows: in Section 2 we present some technical and essential

results, some of them already derived in previous works. Section 3 is devoted to the study of

the geometry of the associated functional and some convergence results. Section 4 consists in

the proof of our existence main results and, finally, Section 5 is concerned with the proof of the

multiplicity results.

Throughout the paper, we will use the following notations:

• We fix the values r1, r2 > 0 such that r1 > 1, r1 ∼ 1 and 1r1+ 1

r2= 1.

• Ls(R2) denotes the usual Lebesgue space with norm || · ||s.

• X ′ denotes the dual space of X .

• Br(x) is the ball centered in x with radius r > 0, simply Br if x = 0.

• Ki, bi, i ∈ N, stand for important constants that appear in the estimates obtained.

• Ci, i ∈ N, will denote different positive constants whose exact values are not essential to

the exposition of arguments.

2 Framework and some Technical Results

In this section, we will focus in presenting additional framework properties and a few technical

results. Some of them can be found in [2, 10, 13, 14] and we will omit their proofs here.

We begin defining three auxiliary symmetric bilinear forms

(u, v) 7→ B1(u, v) =

∫

R2

∫

R2

ln(1 + |x− y|)u(x)v(y)dxdy,

(u, v) 7→ B2(u, v) =

∫

R2

∫

R2

ln

(

1 +1

|x− y|

)

u(x)v(y)dxdy,

(u, v) 7→ B(u, v) = B1(u, v)− B2(u, v) =

∫

R2

∫

R2

ln(|x− y|)u(x)v(y)dxdy.

The above definitions are understood to being over measurable functions u, v : R2 → R, such

that the integrals are defined in the Lebesgue sense. Then, we can define the functionals

V1 : H1(R2) → [0,∞], V2 : L83 (R2) → [0,∞) given by V1(u) = B1(u

2, u2) and

V2(u) = B2(u2, u2), respectively. Moreover, one should observe that V (u) = V1(u)− V2(u).

Remark 2.1. (i) From 0 ≤ ln(1 + r) ≤ r, for r > 0, and (HLS), for u, v ∈ L43 (R2),

|B2(u, v)| ≤ K0||u|| 43||v|| 4

3, where K0 > 0 is the (HLS) best constant. Hence,

|V2(u)| ≤ K0||u||483, ∀ u ∈ L

83 (R2), (2.1)

7

(ii) We recall the so-called Gagliardo-Nirenberg inequality

||u||s ≤ K1s

GN ||∇u||σ2 ||u||1−σ2 , for t > 2 and σ = 2

(

1

2− 1

s

)

. (2.2)

(iii) From equations (2.1) and (2.2), we obtain a positive constant K1 > 0 such that

|V2(u)| ≤ K1c32

√

A(u), ∀ u ∈ H1(R2). (2.3)

(iv) Observing that

ln(1+ |x− y|) ≤ ln(1+ |x|+ |y|) ≤ ln(1+ |x|)+ ln(1+ |y|), for all x, y ∈ R2, (2.4)

we obtain the estimate

B1(uv, wz) ≤ ||u||∗||v||∗||w||2||z||2 + ||u||2||v||2||w||∗||z||∗, (2.5)

for all u, v, w, z ∈ L2(R2).

We are going to need the following results from [14].

Lemma 2.1. ( [14, Lemma 2.2]) (i) The space X is compactly embedded in Ls(R2), for all

s ∈ [2,∞).

(ii) The functionals V0, V1, V2 and I are of class C1 on X . Moreover, V ′i (u)(v) = 4Bi(u

2, uv),

for u, v ∈ X and i = 0, 1, 2.

(iii) V2 is continuous (in fact continuously differentiable) on L83 (R2) .

Lemma 2.2. ( [14, Lemma 2.1]) Let (un) be a sequence in L2(R2) and u ∈ L2(R2) \ {0} such

that un → u pointwise a.e. on R2. Moreover, let (vn) be a bounded sequence in L2(R2) such

that

supn∈N

B1(u2n, v

2n) <∞.

Then, there exist n0 ∈ N and C > 0 such that ||un||∗ < C, for n ≥ n0. If, moreover,

B1(u2n, v

2n) → 0 and ||vn||2 → 0, as n→ ∞,

then

||vn||∗ → 0 , as n→ ∞.

Lemma 2.3. ( [14, Lemma 2.6]) Let (un), (vn) and (wn) be bounded sequences in X such that

un ⇀ u in X . Then, for every z ∈ X , we have B1(vnwn , z(un − u)) → 0, as n→ +∞.

Also, we borrow the following result from [13].

8

Lemma 2.4. ( [13, Lemma 2.5]) Let (un) ⊂ S(c) and assume the existence of ε ∈ (0, c) such

that for all R > 0, we have

lim infn

supy∈Z2

∫

BR(y)

|un|2 dx ≤ c− ε.

Then,

lim supn

V1(un) = +∞.

Corolary 2.1. Let (un) ⊂ S(c) and assume that (V1(un)) is bounded. Then, there existsR0 ≥ 2

such that

lim infn

supy∈Z2

∫

BR0(y)

|un|2 dx >c

2.

Proof. From Lemma 2.4, for ε = c2, there exists R > 0 satisfying

lim infn

supy∈Z2

∫

BR(y)

|un|2 dx >c

2.

Consequently, the corollary follows considering R0 = R + 2.

Now, we turn our attention to the term with exponential critical growth. First of all, we

remember the well-known Moser-Trudinger inequality.

Lemma 2.5. [12] If α > 0 and u ∈ H1(R2), then

∫

R2

(

eα|u|2 − 1

)

dx < +∞.

Moreover, if ||∇u||22 ≤ 1, ||u||22 ≤ M < ∞ and α < 4π, then there exists Kα,M = C(M,α),

such that∫

R2

(

eα|u|2 − 1

)

dx < Kα,M .

Then, inspired by [2], we prove the following corollary.

Corolary 2.2. Let (un) ⊂ S(c) satisfying lim supA(un) < 1 − c. Then, for all p ≥ 1, there

exist values β > 1, β ∼ 1, p0 = p0(p) > 2 and a constant K2 = K2(β, c) > 0 such that

∫

R2

|un|p(e4βπ|un(x)|2 − 1)dx ≤ K2||un||pp0, ∀ n ∈ N.

Proof. Since lim supn

A(un) < 1 − c, there exist d > 0 and n0 ∈ N such that ||un||2 < d < 1,

for all n ≥ n0. Thus, there exists β > 1, β ∼ 1 with 1 < β < 1d

and s1 > 1, s1 ∼ 1 with

9

1 < s1 <1dβ

. Let s2 > 2 such that 1s1+ 1

s2= 1. Consequently,

∫

R2

|un|p(e4βπ|un(x)|2 − 1)dx ≤ ||un||pps2

∫

R2

(e4βs1dπ(un(x)

||un|| )2

− 1)dx

1s1

≤ C1||un||pps2,

for all n ≥ n0. Now, consider

C2 = max

∫

R2

(e4βs1π|u1(x)|2 − 1)dx

1s1

, ... ,

∫

R2

(e4βs1π|un0 (x)|2 − 1)dx

1s1

.

Then, for each 1 ≤ n ≤ n0, we have

∫

R2

|un|p(e4βπ|un(x)|2 − 1)dx ≤ C2||un||pps2.

Therefore, the lemma follows setting p0 = ps2 > 2 and K2 = max{C1, C2} > 0.

From now on, unless we say otherwise, β > 0 will stand as that given by Corollary 2.2.

The next lemma plays an important role in our work but since its proof is very similar to that

in [9, Lemma 3.7], we omit it here.

Lemma 2.6. Let (un) ⊂ X be bounded in H1(R2) and R > 0 satisfying

lim infn

supy∈Z2

∫

BR(y)

u2n(x)dx > 0. (2.6)

Then, there exists u ∈ H1(R2) \ {0} and a sequence (yn) ⊂ Z2 such that, up to a subsequence,

un(x) = un(· − yn)⇀ u in H1(R2).

In the last remark of this section we discuss the geometry of a real function that appears in

some further arguments.

Remark 2.2. Consider the real function h : (0,+∞) → (0,+∞) given by h(t) = at−bt12 +d,

for a, b, d > 0. Note that h′(t) = a − b

2t12

and h′′(t) = b

4t32

. One can see that h′(t) = 0 for

t =(

b

2a

)2and h′′(t) > 0 for all t ∈ (0,+∞). Consequently, h is convex and t =

(

b

2a

)2is a

global minimum for h. Moreover, h(t) = −b2

4a+ d. Thus, we conclude that

h(t) ≥ 0 if and only ifb2

4a≤ d.

3 Geometry of I and key auxiliary results

The present section will be devoted to derive some geometrical properties of the functional I and

some very important auxiliary results, such as those involving boundedness and convergence of

10

sequences on X . We start defining the map H : H1(R2) → R by H(u, t) = etu(etx), for all

x ∈ R2 and t ∈ R \ {0}, and, for each u ∈ H1(R2) fixed, the function ϕu(t) : R → R given by

ϕu(t) = I(H(u, t)).

Remark 3.1. For any u ∈ S(c), one should observe that H(u, t) ∈ S(c), for all t ∈ R \ {0}.

Moreover, we have the following:

(i) V (H(u, t)) = V (u)− t||u||42,

(ii) A(H(u, t)) = e2tA(u),

(iii) ||H(u, t)||pp = e(p−2)t||u||pp, for all p ≥ 1.

Lemma 3.1. Assume (f1)− (f4) and u ∈ S(c). Then,

(1) A(H(u, t)) → +∞ and ϕu(t) → −∞, as t→ +∞ .

(2) A(H(u, t)) → 0 and

{

ϕu(t) → +∞ , if µ > 0

ϕu(t) → −∞ , if µ < 0, as t→ −∞.

(3) ϕu(t) → I(u), as t→ 0.

Proof. (1) From Remark 3.1-(ii), A(H(u, t)) → +∞, as t → +∞. Moreover, from condition

(f4), Remark 3.1 and q > 4, we have

ϕu(t) ≤e2t

2A(u) +

µ

4V (u)− µ

4c2t− νe(q−2)t||u||qq → −∞ , as t→ +∞.

(2) Once again, from Remark 3.1, A(H(u, t)) → 0 and ||H(u, t)||pp → 0, as t → −∞, for all

p > 2. Thus, there are t0 < 0 and d ∈ (0, 1) such that βdr1 < 1 and ||H(u, t)||2 ≥ d, for all

t ∈ (−∞, t0], with r1 > 1, r1 ∼ 1 and 1r1+ 1

r2= 1, β > 1, β ∼ 1.

Then, from (1.6), we have

|F (H(u, t))| ≤ ε|H(u, t)|τ+1 + b2|H(u, t)|q(e4πβ|H(u,t)|2 − 1), ∀ t ≤ t0,

and applying Hölder inequality and Lemma 2.5, we obtain a constant C1 > 0 such that

∫

R2

F (H(u, t))dx ≤ ε||H(u, t)||τ+1τ+1 + C1||H(u, t)||qqr2 → 0 , as t→ −∞.

Consequently, from this fact and from Remark 3.1, we get item (2).

For the next result, let us consider the subsets of S(c):

A− = {u ∈ S(c) ; V (u) < 0},A+ = {u ∈ S(c) ; V (u) ≥ 0},Ar = {u ∈ S(c) ; A(u) ≤ r},

A+r = A+ ∩ Ar and A−

r = A− ∩ Ar, for r > 0.

11

Lemma 3.2. The sets A−, A+ and A+r are non-empty, for all r > 0.

Proof. Let u ∈ S(c). From Remark 3.1, we can choose t1 > 0 sufficiently large such that

u1 = H(u, t1) ∈ S(c) and V (u1) < 0 and, in a similar way, we can find t2 < 0 such that

u2 = H(u, t2) ∈ S(c) and V (u2) ≥ 0. Moreover, from Remark 3.1, there exists t3 < 0

sufficiently large satisfying A(u3) ≤ r and V (u3) ≥ 0, for u3 = H(u, t3).

Lemma 3.3. Let µ ∈ R and (un) ⊂ S(c) verifying lim supn

A(un) ≤ 1 − c and I(un) ≤ d, for

some d ∈ R and for all n ∈ N. Then, there exists a sequence (yn) ⊂ Z2 and u ∈ X \ {0} such

that un = un(· − yn)⇀ u in X .

Proof. Since (A(un)) is bounded, from equation (2.1), (V2(un)) is also bounded. Moreover,

from (1.6), (2.2) and Corollary 2.2, we see that

∫

R2

F (un)dx

is bounded. Thus, once

I(un) ≤ d, for all n ∈ N, we have

V1(un) =4

µI(un)−

2

µA(un) + V2(un) +

4

µ

∫

R2

F (un)dx ≤ C1, ∀ n ∈ N.

From Lemma 2.6 and Corollary 2.1, there exists a sequence (yn) ⊂ Z2 such that, up to a

subsequence, un ⇀ u in H1(R2) with u 6= 0. We can assume, without loss of generality that

un(x) → u(x) a.e. in R2. Thus, once V1(un) = V1(un), from Lemma 2.2, (||un||∗) is bounded.

Therefore, up to a subsequence, we conclude that un ⇀ u in X with u 6= 0.

The next lemma is one of the key results to obtain our main theorems.

Lemma 3.4. Let µ ∈ R and (un) ⊂ S(c) satisfying lim supn

A(un) ≤ 1 − c and I(un) ≤ d, for

some d ∈ R and for all n ∈ N. Then, up to a subsequence, (un) is bounded in X .

Proof. First of all, since (un) is bounded in H1(R2) and from Lemma 3.3, passing to a

subsequence if necessary, un = un(· − yn) ⇀ u in X , u 6= 0 and un(x) → u(x) pointwise a.e.

in R2. Moreover, un → u in L2(R2).

Claim: (yn) is a bounded sequence.

Indeed, since u 6= 0 in L2(R2), there are R1 > 0, n1 ∈ N and C1 > 0 such that

||un||22,BR1≥ C1 > 0, for all n ≥ n1. If |yn| ≤ 2R1 for all n ∈ N the claim is proved.

From this, assume that there is n ∈ N such that |yn| > 2R1. Recalling that

1 + |x+ y| ≥ 1 +|y|2

≥√

1 + |y| ∀x ∈ BR1 and y ∈ Bc2R1

,

we have that

||un||2∗ =∫

R2

ln(1 + |x|)u2n(x− yn)dx =

∫

R2

ln(1 + |x+ yn|)u2n(x)dx

≥ C2||un||22,BR1ln(1 + |yn|) = C3 ln(1 + |yn|).

12

As (||un||∗) is bounded, the above inequality implies that (yn) is bounded, showing the claim.

The boundedness of (yn) yields that (un) is bounded in X , because (un) is bounded in

H1(R2) and, for all n ∈ N,

‖un‖2∗ =∫

R2

ln(1 + |x|)|un|2 dx =

∫

R2

ln(1 + |x− yn|)|un|2 dx ≤ ‖un‖2∗ + ln(1 + |yn|)‖un‖22.

The above inequality together with the boundedness of (un) in X implies that (un) is bounded

in X .

The last result of this section is the other key to obtain our main theorems.

Lemma 3.5. Let µ ∈ R and (un)be a (PS) sequence for I restricted to S(c) bounded in X

and satisfying lim supn

A(un) ≤ 1 − c. Then, up to a subsequence, un → u in X with u 6= 0.

Particularly, u is a critical point for I restricted to S(c).

Proof. First of all, once (un) is bounded in X , passing to a subsequence if necessary, we obtain

that un ⇀ u in X and un → u in Ls(R2), for all s ≥ 2. In particular, u ∈ S(c). Thus, from

Corollary 2.2, we get that

∫

R2

F (un)dx

and

∫

R2

f(un)undx

are bounded. Moreover, from

(2.1) and (2.5), we obtain that (A(un)), (V1(un)) and (V2(un)) are also bounded.

Claim: There exists a value λ ∈ R such that (un) is a (PS) sequence for the functional

I(u) := I(u) + λ2||u||22.

Indeed, once (un) is bounded inX , from [8, Lemma 3], adapted from the unit sphere to S(c),

we know that ||I∣

∣

′

S(c)(un)||X′ = o(1) is equivalent to ||I ′(un)− 1

cI ′(un)(un)un||X′ = o(1).

Set

λn = −1

cI ′(un)(un) = −1

c

A(un) +µ

4V (un)−

∫

R2

f(un)undx

, ∀ n ∈ N.

Then, (λn) ⊂ R is bounded and, up to a subsequence, λn → λ in R. Hence, observing that

I ′(u) = I ′(u) + λu and that, for v ∈ X \ {0},

|I ′(un)(v)| ≤ |I ′(un)(v) + λn〈un, v〉|+ |λ− λn|||v||c12 , ∀ n ∈ N,

we conclude that (un) is a (PS) sequence for I, proving the claim.

Now, since un ⇀ u in X and un → u in Ls(R2), for all s ≥ 2, we have

(i) 0 ≤ |I ′(un)(un − u)| ≤ ||I ′(un)||X′||un − u||X → 0.

(ii)

∣

∣

∣

∣

∣

∣

∫

R2

f(un)undx

∣

∣

∣

∣

∣

∣

≤ ε||un||τ2τ ||un − u||2 + C1||un||q−12(q−1)r2

||un − u||2r2 → 0.

(iii) |V ′2(un)(un − u)| ≤ C2||un||38

3

||un − u|| 83→ 0.

(iv) V ′1(un)(un − u) = B1(u

2n, u(un − u)) = B1(u

2n, (un − u)2) +B1(u

2n, u(un − u)) .

13

From Lemma 2.3, B1(u2n, u(un − u)) → 0. Thus, since B1(u

2n, (un − u)2) ≥ 0 and from

(i)-(iv), we obtain

o(1) = I ′(un)(un − u) = o(1) + A(un)−A(u) +µ

4V ′(un)(un − u)−

∫

R2

f(un)(un − u)dx

≥ o(1) + A(un)− A(u) ≥ o(1), ∀n ∈ N.

Consequently, A(un) → A(u) and, in particular, un → u in H1(R2). Then, going back

to the above inequality, we conclude that B1(u2n, (un − u)2) → 0 and, from Lemma 2.2,

||un − u||∗ → 0. Therefore, un → u in X .

Finally, for v ∈ X , we have

|I∣

∣

′

S(c)(u)(v)| = lim |I

∣

∣

′

S(c)(un)v| ≤ ||v||X lim ||I

∣

∣

′

S(c)(un)||X′ = 0,

which implies that u is a critical point to I restricted to S(c).

4 Proof of Theorems 1.1 and 1.2

Inspired by [2], we will construct a suitable mountain pass level. From Lemma 3.1, for each

u ∈ S(c) there are a value tu ≤ 0 such that ϕu(t) > 0 for all t ≤ tu and a value tu,A < 0

verifying A(H(u, t)) < ρc2

for all t ≤ tu,A. Thus, we can define the following real values

−∞ < t := sup{tu < 0 ; u ∈ A−} ≤ 0 and −∞ < t := sup{

tu,A < 0 ; u ∈ A−}

≤ 0.

Consider t0 = min{t, t}. Thus, we can choose u0 ∈ A− such that there exists t1 < 0, with

t0−1 < t1 < t0, satisfying ϕu0(t1) > 0 and A(H(u0, t1)) <ρc2

. Set u1 = H(u0, t1). Moreover,

from Lemma 3.1, there exists t2 > 0 such that u2 = H(u0, t2) satisfies A(u2) > 2ρc and

I(u2) < 0. Therefore, we have the mountain pass level mν as defined in (1.9).

The above construction makes possible to find a bound to µ that does not depend on the fixed

function u0. As one can observe in the proof of Lemma 4.2, if we simply consider a function

u0 fixed and a value t1 < 0 given by Lemma 3.1, the constant that appears will depend on u0.

Lemma 4.1. Assume (f1) − (f3) and µ > 0 sufficiently small. Then, for u1 ∈ X with

A(u1) ≤ ρc, there exists a value ρc = ρ(c) > 0 such that

0 < I(u1) < infu∈B

I(u),

where

B = {u ∈ S(c) ; A(u) = 2ρc}.

Proof. Set ρc < 1−c2

and v ∈ S(c) with A(v) = 2ρc. Then, from equations (1.6), (2.2), Lemma

14

2.5 and v ∈ S(c), we have

∫

R2

F (v)dx ≤ C2A(v)τ−12 + C3A(v)

qr2−22r2 .

Thus, since F (u1), V1(v), V2(u1) ≥ 0 and from (2.1), we have

I(v)− I(u1) =1

2(A(v)− A(u1)) +

µ

4(V (v)− V (u1)) +

∫

R2

(F (u1)− F (v))dx

≥ A(v)− A(u1)−µ

4V1(u1)−

µ

4V2(v)−

∫

R2

F (v)dx

≥ 1

2ρc −

µ

2‖u1‖2∗c−

µ

4K1c

32

√2ρ

12c − C4ρ

τ−12

c − C5ρqr2−22r2

c .

Moreover, once c32 < c and ρ

12c < 1, for ‖u1‖2∗ = C1 we get that

I(v)− I(u1) ≥(

1

4ρc −

µ

2C1c−

µ

4K1

√2c

)

+

(

1

4ρc − C4ρ

τ−12

c − C5ρqr2−22r2

c

)

.

Therefore, fixing ρc even smaller of such way that

1

4ρc − C4ρ

τ−12

c − C5ρqr2−22r2

c ≥ 1

8ρc

and µc = µc(ρc) > 0 such that

1

4ρc −

µ

2C1c−

µ

4K1

√2c ≥ 1

8ρc, ∀µ ∈ (0, µc),

we get

I(v)− I(u1) ≥1

4ρc > 0,

and so,

infv∈B

I(v) > I(u1).

Moreover, we also have that

I(u1) ≥1

2A(u1)−

µ

4K1c

32A(u1)

12 − C1A(u1)

τ−12 − C2A(u1)

qr2−22r2 > 0,

for ρc > 0 sufficiently small. This proves the desired result.

Lemma 4.2. We have maxt∈[0,1]

I(γ(t)) > max{I(u1), I(u2)}, for all γ ∈ Γ.

Proof. Let γ ∈ Γ. Then, ||γ(t)||22 = c for all t ∈ [0, 1]. Moreover, A(γ(0)) < ρc2

and

A(γ(1)) > 2ρc. Thus, ||γ(0)||2 < c + 2ρc < ||γ(1)||2 and, from the Intermediate Value

Theorem, there exists t ∈ [0, 1] such that ||γ(t)||2 = c+ 2ρc.

15

On the other side, ||γ(t)||2 = A(γ(t)) + c. Consequently, A(γ(t)) = 2ρc. Therefore, from

Lemma 4.1, I(u1) < I(γ(t)) and the result follows.

As a consequence of Lemma 4.2, mν > 0. Next we seek for an useful upper bound for mν .

Lemma 4.3. There are µ0, ν0 > 0 such that mν ≤ (1− c)(θ − 6)

8θ, for ν > ν0 and µ ∈ (0, µ0).

Proof. Consider the path γ0(t) = H(u0, (1− t)t1 + tt2) ∈ Γ. Then,

maxt∈[0,1]

I(γ0(t)) ≤ maxr≥0

{r

2A(u0)−

µ

4c2(t0 − 1)− νr

q−22 ||u0||qq

}

= maxr≥0

{r

2A(u0)− νr

q−22 ||u0||qq

}

+µ

4c2K4,

where K4 = −(t0 − 1) > 0. Hence, we obtain a constant K5 = K5(q, u0) > 0 such that

mν ≤ K5

(

1

ν

)2

q−4

+K4µ

4c2,

and the result follows for

µ <(1− c)(θ − 6)

4c2θK4= µ0

and

ν ≥(

16θK5

(1− c)(θ − 6)

)q−42

= ν0.

As an immediate consequence of the last lemma is the corollary below

Corolary 4.1. The Lemma 4.3 is true letting µ to be any positive real number and controlling

the mass c. In this case, it is enough to consider ν ≥ ν0 and

c < min

{

1,(6− θ) +

√

(θ − 6)2 + 16K4µθ(6− θ)

8K4µθ

}

= c0. (4.1)

In the sequel, let (un) ⊂ S(c) be the sequence constructed in [19] satisfying

I(un) → mν , ||I∣

∣

′

S(c)(un)||X′ → 0 and Q(un) → 0, as n→ +∞, (4.2)

where Q : H1(R2) → R is given by

Q(u) = A(u)− µ

4||u||42 + 2

∫

R2

F (u)dx−∫

R2

f(u)udx. (4.3)

16

Lemma 4.4. Let (un) be the sequence given in (4.2). Then, decreasing if necessary µ0 given in

Lemma 4.3, we have

lim supn

∫

R2

f(un)undx ≤ 4θ

θ − 6mν , ∀µ ∈ (0, µ0).

A similar results holds by fixing µ and decreasing if necessary the number c0 given in (4.1).

Proof. From (f3), V1(u) ≥ 0 and (2.1), we have

mν + o(1) ≥ I(un)−1

4Q(un) ≥

1

4A(un)−

µ

4K1c

32A(un)

12 +

µ

16c2 +

(θ − 6)

4θ

∫

R2

f(un)undx.

(4.4)

Since c ∈ (0, 1), it follows that

mν + o(1) ≥ I(un)−1

4Q(un) ≥

1

4A(un)−

µ

4K1cA(un)

12 +

µ

16c2 +

(θ − 6)

4θ

∫

R2

f(un)undx.

Using Remark 2.2, with a = 14, b = µ

4K1c and d = µ

16c2, decreasing if necessary µ0, we obtain

mν + o(1) ≥ (θ − 6)

4θ

∫

R2

f(un)undx, ∀ n ∈ N.

Now, if we fix µ > 0, we apply Remark 2.2, with a = 14, b = µ

4K1c

3/2 and d = µ16c2 in (4.4) to

get the inequality above, decreasing if necessary c0 given in (4.1).

Corolary 4.2. Let (un) be the sequence given in (4.2). Then, for ν ≥ ν0 and µ > 0,

lim supn

A(un) ≤(1− c)

2+µ

4c2.

Proof. From (4.2),

A(un) = Q(un) +µ

4c2 − 2

∫

R2

F (un)dx+

∫

R2

f(un)undx ≤ o(1) +µ

4c2 +

∫

R2

f(un)undx.

Now, the result follows employing Lemma 4.4.

As an immediate consequence of the last corollary we have

Corolary 4.3. Let (un) be the sequence given in (4.2). Then, decreasing if necessary µ0 > 0

given in Lemma 4.3 we get lim supn

A(un) ≤ 1 − c. Moreover, a similar estimate holds fixing

µ > 0 and considering

0 < c <−1 +

√1 + 2µ

µ.

Lemma 4.5. Let (un) be the sequence given in (4.2). Then, (un) is bounded in H1(R2).

17

Proof. From Corollary 4.3, there exists n0 ∈ N such that A(un) ≤ 2− c, for all n ≥ n0. Thus,

A(un) ≤ C2, for all n ∈ N, where C2 = max{A(u1), A(u2), ...A(un0), 2 − c}. Therefore,

||un|| ≤ (C2 + c)12 , for all n ∈ N.

We will only write the proof of Theorem 1.1, because the same argument works to prove

Theorem 1.2.

Proof of Theorem 1.1: Let (un) be the sequence given in (4.2). Then, from Corollary 4.3,

lim supn

A(un) ≤ 1 − c. Thus, from Lemma 3.5, without loss of generality, we can assume that

un → u in X , u 6= 0 and u is a critical point for I restricted to S(c). Moreover, I(u) = mν .

5 Proof of Theorems 1.3 and 1.4

This section is devoted to the proof of the multiplicity results. We start recalling the reader the

definition of the Krasnoselski’s genus. Consider

Λ = {K ⊂ X ; K is symmetric and closed} (with respect to topology in X)

and

ΣK = {k ∈ N ; there exists φ ∈ C0(K,Rk \ {0}) such that φ(−u) = φ(u)}.

Then, we define the Krasnoselski’s genus of K, denoted by γ(K), as follows

γ(K) =

{

inf ΣK , if ΣK 6= ∅+∞ , if ΣK = ∅ and γ(∅) = 0.

Basic properties of genus can be found in [27, Chapter II.5]. In the sequence, we consider Z

as a real Banach space with norm || · ||Z , H as a Hilbert space with inner product 〈·, ·〉H and

M = {u ∈ Z ; 〈u, u〉H = m}, for m > 0. We assume that Z is continuously embedded in H.

Moreover, define

Υk = {K ⊂ M ; K is simmetric, closed and γ(K) ≥ k} , for n ∈ N.

Then, we are ready to enunciate a crucial result, adapted from [20, Theorem 2.1].

Theorem 5.1. ( [3, Theorem 2.1]) Let I : Z → R an even functional of C1 class. Suppose that

I∣

∣

Mis bounded from below and satisfies the (PS)d condition for all d < 0, and Υk 6= ∅ for

each k = 1, ..., n. Then, the minimax values −∞ < d1 ≤ d2 ≤ · · · ≤ dn can be defined as

dk = infK∈Υk

supu∈K

I(u) , for 1 ≤ k ≤ n.

Moreover, the following statement are valid:

(i) dk is a critical value for I∣

∣

M, provided dk < 0 ;

18

(ii) If dk = dk+1 = · · · = dk+l−1 = d < 0, for some k, l ≥ 1, then γ(Kd) ≥ l, where Kd is the

set of critical points of I∣

∣

Min the level d ∈ R. Hence, I

∣

∣

Mhas at least n critical points.

Our approach is based in [3] (see also [5]), which we refer the reader for more details in the

arguments. Precisely, in what follows we will verify the necessary conditions to apply Theorem

5.1 in order to obtain at least n solutions for (1.4).

First of all, observe that

I(u) ≥ 1

2A(u)− µ

4K1c

32A(u)

12 − C1A(u)

τ−12 − C2A(u)

qr2−22r2 ≥ h(A(u)

12 ),

for C1, C2 > 0 and h : R → R given by

h(t) =1

2t2 − µ

4K1c

32 t− C1t

τ−1 − C2tqr2−2

r2 .

Since τ − 1 > 2 and qr2−2r2

> 2, there exists a value a > 0 such that, if µc32 < a, then there are

R0, R1 > 0 satisfying

h(t) ≤ 0 , for t ∈ [0, R0],

h(t) ≥ 0 , for t ∈ [R0, R1],

h(t) < 0 , for t ∈ (R1,+∞).

Moreover, we can define the values

µ1 =a

c32

and c1 =

(

a

µ

)32

. (5.1)

Now, forR0, R1 given above, define T : R+ → [0, 1] as a non-decreasing function such that

T ∈ C∞ and

T (t) =

{

1 , for t ∈ [0, R0],

0 , for t ∈ [R1,+∞).

Thus, we consider the truncated functional IT : X → R given by

IT (u) =1

2A(u) +

µ

4V (u)− T (A(u)

12 )

∫

R2

F (u)dx.

Similarly as above, we have

I(u) ≥ h(A(u)12 ), (5.2)

where h : R → R is defined as

h(t) =1

2t2 − µ

4K1c

32 t− T (t)[C1t

τ−1 + C2tqr2−2

r2 ].

Without loss of generality, we assume that

1

2t2 − C1t

τ−1 − C2tqr2−2

r2 ≥ 0 , ∀t ∈ [0, R0], and R0 <√1− c.

19

Lemma 5.1. (1) IT ∈ C1(X,R).

(2) If IT (u) ≤ 0, then A(u)12 < R0 and I(v) = IT (v) for all v in a small neighbourhood of

u in X .

(3) If µ ∈ (0, µ1), then IT restricted to S(c) verifies the (PS)d condition in every level d < 0.

Proof. Items (1) and (2) can be proved by standard arguments, so we omit it here. Let us prove

item (3). Let (un) ⊂ S(c) be a (PS) sequence for IT restricted to S(c) in the level d < 0.

Then, up to a subsequence, I(un) ≤ 0, for all n ∈ N. Thus, from item (2), we conclude that

lim supn

A(un) ≤ 1− c. Consequently, the result follows from Lemmas 3.4 and 3.5.

In what follows we will need the level sets

IdT = {u ∈ S(c) ; IT (u) ≤ d}.

Lemma 5.2. For each n ∈ N and µ ∈ R, there are ǫn = ǫ(n) > 0 and νn = ν(n) > 0 such that

γ(I−ǫT ) ≥ n, for all ǫ ∈ (0, ǫn) and ν ≥ νn.

Proof. For each n ∈ N, as in [3], consider the n-dimensional spaceEn ⊂ X with the orthogonal

base B = {u1, ..., un}, that is,

∫

R2

∇uj∇ukdx =

∫

R2

ujukdx =

∫

R2

ln(1 + |x|)ujukdx = 0,

if j 6= k, A(uj) = ρ2 < 1− c, ||uj||22 = c and ||uj|| =√

ρ2 + c, for each 1 ≤ j ≤ n. Define

Zn = {t1u1+· · · tnun ; t21+· · · t2n = 1} and Sρ2+c = {(y1, ..., yn) ∈ Rn ; y21+· · · y2n = ρ2+c}.

Considering the map Φ : Zn → Sρ2+c given by Φ(u) = (t1√

ρ2 + c, ..., tn√

ρ2 + c), for

u =n∑

j=1

tjuj , one can easily see that Zn and Sρ2+c are homeomorphic (with respect to X).

Thus, by genus properties, we get that γ(Zn) = n.

Now, since dimEn < +∞, all the norms are equivalent. Then, the value

an = inf

{

||u||qq ; u ∈ S

(

c

ρ2

)

∩ En and A(u) = 1

}

is well-defined and positive. Moreover, there exists a constant Pn = P (n) > 0 such that

||u||2∗ ≤ PnA(u), for all u ∈ En.

Observe that, once B is orthogonal, A(v) = ρ2 for all v ∈ Zn and, considering 0 < ρ < R0,

20

we have I(v) = IT (v). Hence, from condition (f4), we have

IT (v) = I(v) ≤ 1

2ρ2A

(

v

ρ

)

+µ

4ρ4

∥

∥

∥

∥

v

ρ

∥

∥

∥

∥

2

2

∥

∥

∥

∥

v

ρ

∥

∥

∥

∥

2

∗

− νρq∥

∥

∥

∥

v

ρ

∥

∥

∥

∥

q

q

≤ 1

2ρ2 +

µ

4Pncρ

2 − νanρq, ∀ v ∈ Zn.

Therefore, choosing ρ ∈ (0, R0) there is ǫn > 0 and a value νn > 0 sufficiently large such that

IT (v) ≤ −ǫn, for all v ∈ Zn and ν ≥ νn. Consequently, Zn ⊂ I−ǫT and, from a genus property,

γ(I−ǫT ) ≥ n.

Lemma 5.3. Let Γk = {D ⊂ S(c) ; D is symmetric, closed and γ(D) ≥ k},

dk = infD∈Γk

supu∈D

IT (u)

and Kd = {u ∈ S(c) ; I ′T (u) = 0 and IT (u) = d}. Assume that µ ∈ (0, µ1), where µ1

is given by (5.1). If dk < 0 then dk is a critical value of IT restricted to S(c). Moreover, if

dk = · · ·dk+r := d < 0, for some k, r ≥ 1, then Kd 6= ∅ and γ(Kd) ≥ r + 1. Particularly, IT

restricted to S(c) has at least k non-trivial critical points.

Proof. From Lemma 5.2, for each k ∈ N, there exists ǫk > 0 such that γ(I−ǫT ) ≥ k, for all

ǫ ∈ (0, ǫk). Since IT is continuous and even, I−ǫT ∈ Γk. Consequently, dk ≤ −ǫk < 0, for all

k ∈ N. On the other hand, from (5.2), IT is bounded from bellow over S(c), which implies

dk > −∞, for all k ∈ N. Hence, from Lemma 5.1-(3) and Theorem 5.1, dk is a critical value of

IT restricted to S(c).

In the sequence, suppose that dk = · · · dk+r := d < 0, for some k, r ≥ 1. From the first

part, Kd 6= ∅. Observe, now, that a sequence (un) ⊂ Kd is a (PS)d sequence. Hence, from

Lemma 5.1-(1), R0 <√1− c and Lemma 3.5, Kd is compact. Finally, from a deformation

lemma, genus properties and Theorem 5.1, the result follows.

Proof of Theorems 1.3 and 1.4. One should observe that, from Lemma 5.1-(2), critical points

to IT restricted to S(c) are precisely critical points of I restricted to S(c). Therefore, the proof

follows from Lemma 5.3.

Acknowledgements: C.O. Alves was supported by CNPq/Brazil 304804/2017- ; E.S. Böer was

supported by Coordination of Superior Level Staff Improvement-(CAPES)-Finance Code 001

and São Paulo Research Foundation-(FAPESP), grant ♯ 2019/22531-4, and O.H. Miyagaki was

supported by National Council for Scientific and Technological Development-(CNPq), grant ♯

307061/2018-3 and FAPESP grant ♯ 2019/24901-3.

Declarations

Conflict of Interest. On behalf of all authors, the corresponding author states that there is no

conflict of interest.

Data Availability Statement. This article has no additional data.

21

References

[1] Alves, C.O. and Figueiredo, G.M. (2019) Existence of positive solution for a planar

Schrödinger-Poisson system with exponential growth, Journal of Mathematical Physics.

60, 011503.

[2] Alves, C.O., Ji, C. and Miyagaki, O.H. (2021) Normalized solutions for a Schrödinger

equation with critical growth in RN , ArXiv:2102.03001 [Math].

[3] Alves, C.O., Ji C. and Miyagaki, O.H. (2021) Multiplicity of normalized solutions for a

Schrödinger equation with critical growth in RN , arXiv:2103.07940v2 [Math].

[4] Ambrosetti, A. and Ruiz, D. (2008) Multiple bound states for the Schrödinger–Poisson

Problem. Communications in Contemporary Mathematics, 10, 391–404.

[5] Azorero, J.G. and Alonso, I.P. (1991) Multiplicity of Solutions for Elliptic Problems

with Critical Exponent or with a Nonsymmetric Term, Transactions of the American

Mathematical Society. 323, 877-895.

[6] Azzollini, A. and Pomponio, A. (2008) Ground state solutions for the nonlinear

Schrödinger– Maxwell equations. Journal of Mathematical Analysis and Applications, 345,

90–108.

[7] Bellazzini, J., Jeanjean, L., and Luo, T. (2013) Existence and instability of standing waves

with prescribed norm for a class of Schrödinger-Poisson equations. Proceedings of the

London Mathematical Society, 107, 303–339, arXiv: 1111.4668.

[8] Berestycki, H. and Lions, P.-L.(1983) Nonlinear scalar field equations, II existence of

infinitely many solutions, Arch. Rational Mech. Anal. 82, 347–375.

[9] Böer, E. de S. and Miyagaki, O. H. (2021) Existence and multiplicity of solutions for

the fractional p-Laplacian Choquard logarithmic equation involving a nonlinearity with

exponential critical and subcritical growth, J. Math. Phys. 62, 051507.

[10] Böer, E. de S. and Miyagaki, O.H. (2021) (p,N)-Choquard logarithmic equation

involving a nonlinearity with exponential critical growth: existence and multiplicity,

ArXiv:2105.11442 [Math], submitted.

[11] Bonheure, D., Cingolani, S., and Van Schaftingen, J. (2017) The logarithmic Choquard

equation: Sharp asymptotics and nondegeneracy of the groundstate. Journal of Functional

Analysis, 272, 5255–5281.

[12] Cao, D. M. (1992) Nontrivial solution of semilinear elliptic equations with critical

exponent in R2. Communications in Partial Differential Equations, 17, 407–435.

22

[13] Cingolani, S. and Jeanjean, L. (2019) Stationary waves with prescribed L2-norm for the

planar Schrödinger-Poisson system, SIAM J. Math. Anal. 51, 3533–3568.

[14] Cingolani, S. and Weth, T. (2016) On the planar Schrödinger–Poisson system. Annales de

l’Institut Henri Poincare (C) Non Linear Analysis, 33, 169–197.

[15] do Ó, J. M., Miyagaki, O. H. and Squassina, M. (2015) Nonautonomous fractional

problems with exponential growth. Nonlinear Differential Equations and Applications No-

DEA, 22, 1395–1410.

[16] Du, M. and Weth, T. (2017) Ground states and high energy solutions of the planar

Schrödinger–Poisson system. Nonlinearity, 30, 3492–3515.

[17] Fröhlich, H. (1937) Theory of electrical breakdown in Ionic crystals. Proceedings of the

Royal Society of London. Series A, Mathematical and Physical Sciences, 160, 230–241.

[18] Fröhlich, H. (1954) Electrons in lattice fields. Advances in Physics, 3, 325–361.

[19] Jeanjean. L. (1997) Existence of solutions with prescribed norm for semilinear elliptic

equations. Nonlinear Anal. 28, 1633-1659.

[20] Jeanjean, L. and Lu, S.-S. (2019) Nonradial normalized solutions for nonlinear scalar field

equations, Nonlinearity. 32, 4942–4966.

[21] Lieb, E. H. (1977) Existence and uniqueness of the minimizing solution of Choquard’s

nonlinear equation. Studies in Applied Mathematics, 57, 93–105.

[22] Lieb, E. H. (1983) Sharp constants in the Hardy-Littlewood-Sobolev and related

inequalities. The Annals of Mathematics, 118, 349.

[23] Lions, P.-L. (1987) Solutions of Hartree-Fock equations for Coulomb systems.

Communications in Mathematical Physics, 109, 33–97.

[24] Penrose, R. (1996) On gravity’s role in quantum state reduction. General Relativity and

Gravitation, 28, 581–600.

[25] Ruf, B. and Sani, F. (2013) Ground states for elliptic equations in R2 with exponential

critical growth. Magnanini, R., Sakaguchi, S., and Alvino, A. (eds.), Geometric properties

for parabolic and elliptic PDE’s, vol. 2, pp. 251–267, Springer, Milan.

[26] Ruiz, D. (2006) The Schrödinger–Poisson equation under the effect of a nonlinear local

term. Journal of Functional Analysis, 237, 655–674.

[27] Struwe, M. Variational Methods: applications to nonlinear partial differential equations

and Hamiltonian systems, 3rd ed, Springer, Berlin; New York, 2000.

[28] Wilson, A. J. C. (1955) Untersuchungen über die Elektronentheorie der Kristalle by S. I.

Pekar. Acta Crystallographica, 8, 70–70.