Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 42, pp. 1–30.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.42

RADIAL BOUNDED SOLUTIONS FOR MODIFIED

SCHRÖDINGER EQUATIONS

FEDERICA MENNUNI, ADDOLORATA SALVATORE

Abstract. We study the quasilinear elliptic equation

− div(a(x, u,∇u)) +At(x, u,∇u) + |u|p−2u = g(x, u) in RN ,

with N ≥ 2 and p > 1. Here, A : RN×R×RN → R is a given C1-Carathéodory
function that grows as |ξ|p withAt(x, t, ξ) =

∂A
∂t

(x, t, ξ), a(x, t, ξ) = ∇ξA(x, t, ξ)

and g(x, t) is a given Carathéodory function on RN × R which grows as |ξ|q
with 1 < q < p.

Suitable assumptions on A(x, t, ξ) and g(x, t) set off the variational struc-

ture of above problem and its related functional J is C1 on the Banach space
X = W 1,p(RN ) ∩ L∞(RN ). To overcome the lack of compactness, we assume

that the problem has radial symmetry, then we look for critical points of J
restricted to Xr, subspace of the radial functions in X.

Following an approach that exploits the interaction between the intersection

norm in X and the norm in W 1,p(RN ), we prove the existence of at least two

weak bounded radial solutions, one positive and one negative. For this, we
apply a generalized version of the Minimum Principle.

1. Introduction

In this article we look for weak radial bounded solutions for the quasilinear
elliptic equation

−div(a(x, u,∇u)) +At(x, u,∇u) + |u|p−2u = g(x, u) in RN , (1.1)

where p > 1 and N ≥ 2, A : RN × R × RN → R is a C1-Carathéodory function
with partial derivatives

At(x, t, ξ) =
∂A

∂t
(x, t, ξ), a(x, t, ξ) =

( ∂A
∂ξ1

(x, t, ξ), . . . ,
∂A

∂ξN
(x, t, ξ)

)
and g : RN × R → R is a suitable Carathéodory function.

Equation (1.1) generalizes quasilinear equations describing several physical phe-
nomena such as the self-channeling of a high-power ultra short laser, or also some
problems which arise in plasma physics, fluid mechanics, mechanics and in the con-
densed matter theory (see [35] and references therein or also [16] for some model
problems).

2020 Mathematics Subject Classification. 35J20, 35J92, 35Q55, 58E05.
Key words and phrases. Quasilinear elliptic equation; modified Schrödinger equation;

positive radial bounded solution; weak Cerami-Palais-Smale condition; minimum principle.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted April 30, 2024. Published July 31, 2024.

1



2 F. MENNUNI, A. SALVATORE EJDE-2024/42

If A(x, t, ξ) = Ā|ξ|p with Ā real constant, (1.1) turns out to be the p-Laplacian
equation

−∆pu+ |u|p−2u = g(x, u) in RN . (1.2)

In the case p = 2, equation (1.2) reduces to the following Schrödinger equation

−∆u+ u = g(x, u) in RN

which is a central topic in Nonlinear Analysis, see [4, 6, 19, 20, 23, 36, 37]. Many
authors studied also (1.2) in the general case p > 1, see [3, 5, 27, 30].

We note that (1.2) has a variational structure, but there is a lack of compactness
as the problem is settled in the whole Euclidean space RN and classical variational
tools do not work; thus suitable assumptions on the involved functions are required.

On the other hand, even if the function A(x, t, ξ) has the form 1
pA1(x, t)|ξ|p

but the coefficient A1(x, t) is not constant, besides the lack of compactness the
study of equation (1.1) presents another difficulty: the loss of a direct variational
formulation in the space W 1,p(RN ). Let us point out that this problem arises also
if we look for solutions verifying homogeneous Dirichlet conditions in a bounded
domain Ω. Indeed, the natural action functional

J1(u) =
1

p

∫
Ω

A1(x, u)|∇u|p dx+
1

p

∫
Ω

|u|p dx−
∫
Ω

G(x, u) dx,

is not well defined in W 1,p
0 (Ω) if A1(x, t) is unbounded with respect to t. Moreover,

even if A1(x, t) is strictly positive and bounded with respect to t but ∂A1

∂t (x, t) ̸= 0,

then J1 is defined in W 1,p
0 (Ω) but it is Gâteaux differentiable only along directions

of W 1,p
0 (Ω) ∩ L∞(Ω).

Thus, many authors have studied (1.1) by using non-smooth techniques or in-
troducing a suitable change of variable if the term A(x, t, ξ) has a very particular
form or giving a “good” definition of critical point either on bounded domains or
in unbounded ones, see [1, 2, 7, 8, 17, 18, 21, 22, 28, 29, 35].

More recently, Candela and Palmieri in [10]-[12] considered the functional

J (u) =

∫
Ω

A(x, u,∇u) dx+
1

p

∫
Ω

|u|p dx−
∫
Ω

G(x, u) dx,

defined on the Banach spaceW 1,p
0 (Ω)∩L∞(Ω) equipped with the intersection norm.

Introducing a new weak Cerami-Palais-Smale condition (see Definition 2.1) they
state some abstract critical points Theorems. Using this variational approach,
the existence of at least one bounded solution of (1.1) in the case A(x, t, ξ) =
1
pA1(x, t)|ξ|p has been stated when g(x, t) grows as |t|q with q > p but subcritical

and the involved functions are radially symmetric in [14] or the term |u|p−2u is
multiplied by a weight V (x) verifying suitable assumptions in [15] (see also [31] and
[38] where a generalized (p, q)-Laplacian operator in RN is studied).

Always in the presence of a suitable weight V (x), the existence of solutions of
equation like to (1.1) has been investigated in [33] (see also [32]) if A(x, t, ξ) is a
more general function which grows as |ξ|p and g(x, t) has a sub-p-linear growth of
the type

|g(x, t)| ≤ η(x)|t|q−1

with η suitable measurable function and 1 < q < p.
We notice that the results stated in [32, 33] do not cover the case V (x) = 1, so

they do not apply to the equation (1.1). Therefore, in this paper we want to look



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 3

for solutions of (1.1) when A(x, t, ξ) and g(x, t), in addition to hypotheses similar
to those ones required in [33], are radially symmetric in x. To this aim, in Lemma
4.11 we will state a convergence results in RN already proved in bounded domains
by Boccardo, Murat and Puel in [7, Lemma 5] (see also [31, Lemma 4.5]).

This article is organized as follows. In Section 2 we introduce a weak Cerami-
Palais-Smale condition and the related Minimum Principle (see Proposition 2.2).
In Section 3 we give some preliminary assumptions on the functions A(x, t, ξ) and
g(x, t) that ensure a variational formulation for the equation (1.1). In Section 4 we
consider some further assumptions, then we state our main results (see Theorem 4.5)
and we prove some properties of the action functional J and a convergence result
à la Boccardo-Murat-Puel in RN . Finally in Section 5 we prove that J verifies the
weak Cerami-Palais-Smale condition in the subspace Xr of the radial functions of
X = W 1,p(RN ) ∩ L∞(RN ) and then we state the existence of two nontrivial weak
radial bounded solutions, one negative and one positive, thus concluding the proof
of Theorem 4.5.

2. Abstract tools

In this section we denote by (X, ∥ · ∥X) a Banach space with dual space (X ′, ∥ ·
∥X′), (W, ∥ · ∥W ) another Banach space such that X ↪→ W continuously, and by
J : X → R a given C1 functional.

Nevertheless, to avoid any ambiguity, we will henceforth denote by X the space
equipped with its norm ∥ · ∥X , while, if the norm ∥ · ∥W is involved, we will write
it explicitly.

For simplicity, taking β ∈ R, we say that a sequence (un)n ⊂ X is a Cerami-
Palais-Smale sequence at level β, briefly (CPS)β-sequence, if

lim
n→+∞

J(un) = β and lim
n→+∞

∥dJ (un) ∥X′(1 + ∥un∥X) = 0.

Moreover, β is a Cerami-Palais-Smale level, briefly (CPS)-level, if there exists a
(CPS)β-sequence.

The functional J satisfies the classical Cerami-Palais-Smale condition in X at
the level β if every (CPS)β-sequence converges in X up to subsequences. However,
thinking about the setting of our problem, in general a (CPS)β-sequence may also
exist which is unbounded in ∥ · ∥X but converges with respect to ∥ · ∥W . Then, we
can weaken the Cerami-Palais-Smale condition in an appropriate way according to
some ideas developed in previous papers (see, for example, [10]–[12]).

Definition 2.1. The functional J satisfies the weak Cerami-Palais-Smale condition
at level β (β ∈ R), briefly (wCPS)β condition, if for every (CPS)β-sequence (un)n,
a point u ∈ X exists such that

(i) limn→+∞ ∥un − u∥W = 0 (up to subsequences),
(ii) J(u) = β, dJ(u) = 0.

If J satisfies the (wCPS)β condition at each level β ∈ I, I real interval, we say
that J satisfies the (wCPS) condition in I.

Let us point out that, because of the convergence only in the norm of W , the
(wCPS)β condition implies that the set of critical points of J at the β level is
compact with respect to ∥ · ∥W , so that we can state a Deformation Lemma and
some abstract theorems about critical points (see [12]). In particular, the following
Minimum Principle applies (for the proof, see [12, Theorem 1.6]).



4 F. MENNUNI, A. SALVATORE EJDE-2024/42

Proposition 2.2 (Minimum Principle). If J ∈ C1(X,R) is bounded from below in
X and (wCPS)β holds at level β = infX J ∈ R, then J attains its infimum, i.e.,
u0 ∈ X exists such that J(u0) = β.

3. Variational setting and first properties

Here and in the following, let N = {1, 2, . . . } be the set of the strictly positive
integers and we denote by x · y the inner product in RN and | · | the standard norm
on any Euclidean space as the dimension of the considered vector is clear and no
ambiguity arises. Furthermore, we denote by:

• BR(x) = {y ∈ RN : |y−x| < R} the open ball in RN with center in x ∈ RN

and radius R > 0;
• Bc

R = RN \BR(0) the complement of the open ball BR(0) in RN ;
• meas(Ω) the usual Lebesgue measure of a measurable set Ω in RN ;

• Ll(RN ) the Lebesgue space with norm |u|l =
(∫

RN |u|l dx
)1/l

if 1 ≤ l < +∞;

• L∞(RN ) the space of Lebesgue-measurable and essentially bounded func-
tions u : RN → R with norm

|u|∞ = ess supRN |u|;

• W 1,p(RN ) the classical Sobolev space with norm ∥u∥p = (|∇u|pp + |u|pp)
1
p if

1 ≤ p < +∞;
• W 1,p

r (RN ) = {u ∈ W 1,p(RN ) : u(x) = u(|x|) a.e. x ∈ RN} the subspace of
the radial functions of W 1,p(RN ) equipped with the norm ∥ · ∥p with dual
space (W 1,p

r (RN ))′.

From the Sobolev Embedding Theorems, for any l ∈ [p, p∗] with p∗ = pN
N−p if

N > p, or any l ∈ [p,+∞[ if p = N , the Sobolev space W 1,p(RN ) is continuously
embedded in Ll(RN ), i.e., a constant σl > 0 exists such that

|u|l ≤ σl∥u∥p for all u ∈W 1,p(RN ) (3.1)

(see, e.g., [9, Corollaries 9.10 and 9.11]). Clearly, it is σp = 1. On the other hand, if
p > N then W 1,p(RN ) is continuously imbedded in L∞(RN ) (see, e.g., [9, Theorem
9.12]). Thus, we define

X :=W 1,p(RN ) ∩ L∞(RN ), ∥u∥X = ∥u∥p + |u|∞. (3.2)

From now on, we assume 1 < p ≤ N as, otherwise, it is X =W 1,p(RN ) and the
proofs can be simplified.

Lemma 3.1. For any l ≥ p the Banach space X is continuously embedded in
Ll(RN ), i.e., a constant σl > 0 exists such that

|u|l ≤ σl∥u∥X for all u ∈ X. (3.3)

Proof. If p = N or if p ≤ l ≤ p∗ the embedding (3.3) follows from (3.1) and (3.2).
On the other hand, if l > p∗ then, taking any u ∈ X, again (3.2) implies∫

RN

|u|l dx ≤ |u|l−p
∞

∫
RN

|u|p dx ≤ |u|l−p
∞ ∥u∥pp ≤ ∥u∥lX ,

thus (3.3) holds with σl = 1. □

From Lemma 3.1 it follows that if (un)n ⊂ X, u ∈ X are such that un → u in X,
then un → u also in Ll(RN ) for any l ≥ p. This result can be weakened as follows.



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 5

Lemma 3.2. If (un)n ⊂ X, u ∈ X, M > 0 are such that

∥un − u∥p → 0 as n→ +∞, (3.4)

|un|∞ ≤M for all n ∈ N, (3.5)

then un → u also in Ll(RN ) for all l ≥ p.

Proof. Let 1 ≤ p < N and l > p∗ (otherwise, it is a direct consequence of (3.1)).
Then, from (3.2), (3.5) and (3.1) we have that∫

RN

|un − u|l dx ≤ |un − u|l−p
∞

∫
RN

|un − u|p dx ≤ (M + |u|∞)l−p∥un − u∥pp,

then (3.4) implies the result. □

From now on, we consider A : RN × R× RN → R and g : RN × R → R be such
that:

(A1) A is a C1-Carathéodory function, i.e., A(·, t, ξ) is measurable for all (t, ξ) ∈
R× RN and A(x, ·, ·) is C1 for a.e. x ∈ RN ;

(A2) some positive continuous functions Φi, ϕi : R → R, i ∈ {0, 1, 2}, exist such
that:

|A(x, t, ξ)| ≤ Φ0(t)|t|p + ϕ0(t)|ξ|p a.e. in RN , for all (t, ξ) ∈ R× RN ,

|At(x, t, ξ)| ≤ Φ1(t)|t|p−1
+ ϕ1(t)|ξ|p a.e. in RN , for all (t, ξ) ∈ R× RN ,

|a(x, t, ξ)| ≤ Φ2(t)|t|p−1
+ ϕ2(t)|ξ|p−1

a.e. in RN , for all (t, ξ) ∈ R× RN ;

(A3) g(x, t) is a Carathéodory function;

(A4) a function η ∈ L
p

p−q (RN ) exists, with 1 < q < p, such that

0 ≤ g(x, t)t ≤ η(x)|t|q a.e. in RN , for all t ∈ R.

Remark 3.3. From (A4) it results that

|g(x, t)| ≤ η(x)|t|q−1
a.e. in RN , for all t ∈ R.

Moreover, (A3) and (A4) imply that G(x, t) =
∫ t

0
g(x, s)ds is a well defined C1-

Carathéodory function in RN × R and

0 ≤ G(x, t) ≤ 1

q
η(x)|t|q a.e. in RN , for all t ∈ R. (3.6)

Remark 3.4. From (A2) it follows that

A(x, 0, 0) = At(x, 0, 0) = 0 and a(x, 0, 0) = 0 for a.e. x ∈ RN .

Moreover, from (A3), (A4) and Remark 3.3 we have that

G(x, 0) = g(x, 0) = 0 for a.e. x ∈ RN .

Hence, u = 0 is a trivial solution of (1.1).

Proposition 3.5. Assumptions (A3) and (A4) imply that∫
RN

G(x, u) dx ∈ R for all u ∈ X (or better for all u ∈W 1,p(RN )),∫
RN

g(x, u)v dx ∈ R for all u, v ∈ X (or better for all u, v ∈W 1,p(RN )).



6 F. MENNUNI, A. SALVATORE EJDE-2024/42

Proof. Let u ∈W 1,p(RN ). As η ∈ L
p

p−q (RN ) and |u|q ∈ L
p
q (RN ), Hölder’s inequal-

ity with p
p−q and p

q conjugate exponents and (3.6) imply that

0 ≤
∫
RN

G(x, u)dx ≤ 1

q

∫
RN

η(x)|u|q dx ≤ 1

q
|η| p

p−q
|u|qp. (3.7)

Moreover, by applying again Hölder’s inequality with p
p−q ,

p
q−1 and p conjugate

exponents, we have∣∣ ∫
RN

g(x, u)v dx
∣∣ ≤ ∫

RN

|η(x)|u|q−1
v| dx ≤ |η| p

p−q
|u|q−1

p |v|p (3.8)

for all u, v ∈W 1,p(RN ). □

Remark 3.6. From (A3) and (A4) we have that

g(x, u) ∈ L
p

p−1
(
RN

)
for all u ∈W 1,p(RN ).

Indeed, Hölder’s inequality with p−1
p−q and p−1

q−1 conjugate exponents implies that∫
RN

|g(x, u)|
p

p−1 dx ≤ |η|
p

p−1
p

p−q
|u|

p(q−1)
p−1

p .

Let us point out that assumptions (A1) and (A2) imply that A(x, u,∇u) ∈
L1(RN ) for any u ∈ X. Therefore, from (3.7) it follows that the functional

J (u) =

∫
RN

A(x, u,∇u) dx+
1

p

∫
RN

|u|p dx−
∫
RN

G(x, u) dx (3.9)

is well defined for all u ∈ X. Moreover, taking v ∈ X, from (3.8), the Gâteaux
differential of functional J in u along the direction v is given by

⟨dJ (u), v⟩ =
∫
RN

a(x, u,∇u) · ∇v dx+

∫
RN

At(x, u,∇u)v dx

+

∫
RN

|u|p−2uv dx−
∫
RN

g(x, u)v dx.

(3.10)

Now, we are ready to state the following regularity result.

Proposition 3.7. Taking p > 1, assume that (A1)—-(A4) hold. If (un)n ⊂ X,
u ∈ X, M > 0 are such that (3.4), (3.5) hold and

un → u a.e. in RN as n→ +∞,

then

J (un) → J (u) and ∥dJ (un)− dJ (u)∥X′ → 0 as n→ +∞.

Hence, J is a C1 functional on X with Fréchet differential defined as in (3.10).

Proof. It is sufficient to simplify the proof of [33, Prop. 3.10] by observing that the
functional u ∈ X 7→ 1

p

∫
RN |u|p dx ∈ R is of class C1. □

4. Statement of main results

From now on, we assume that in addition to (A1)–(A4), functions A(x, t, ξ) and
g(x, t) satisfy the following further conditions:

(A5) there exists a constant α0 > 0 such that

A(x, t, ξ) ≥ α0|ξ|p a.e. in RN , for all (t, ξ) ∈ R× RN ;



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 7

(A6) there exists a constant η0 such that

A(x, t, ξ) ≤ η0 a(x, t, ξ) · ξ a.e. in RN , for all (t, ξ) ∈ R× RN ;

(A7) there exists a constant α1 > 0 such that

a(x, t, ξ) · ξ +At(x, t, ξ)t ≥ α1a(x, t, ξ) · ξ a.e. in RN , for all (t, ξ) ∈ R× RN ;

(A8) there exist constants µ > p and α2 > 0 such that

µA(x, t, ξ)− a(x, t, ξ) · ξ −At(x, t, ξ)t ≥ α2A(x, t, ξ) a.e. in RN ,

for all (t, ξ) ∈ R× RN ;
(A9) for all ξ, ξ∗ ∈ RN , ξ ̸= ξ∗, we have

[a(x, t, ξ)− a(x, t, ξ∗)] · [ξ − ξ∗] > 0 a.e. in RN , for all t ∈ R;

(A10) A(x, t, ξ) = A(|x|, t, ξ) a.e. in RN , for all t ∈ R;
(A11) there exist real constants l1, l2, η1, η2 such that

lim
t→0

Φ1(t)

|t|η1
= l1, lim

t→0

Φ2(t)

|t|η2
= l2

with Φ1,Φ2 as in (A2) and

η1 >
p

N − 1
, η2 >

p− 1

N − 1
; (4.1)

(A12) g(x, t) = g(|x|, t) a.e. in RN , for all t ∈ R;
(A13) the function η introduced in (A4) is such that

ess sup|x|≤1 η(x) < +∞;

(A14) limt→0+
g(x,t)
tp−1 = +∞ uniformly for a.e. x ∈ RN , |x| ≤ 1.

Example 4.1. The function

A(x, t, ξ) =
1

p

(
A1(x) +A2(x)|t|θ

)
|ξ|p a.e. in RN , for all (t, ξ) ∈ R× RN

with p > 1 and θ > 1, satisfies (A1), (A2), (A5)–(A11) if A1 and A2 are two radial
functions and there exists a constant ᾱ0 > 0 such that

A1, A2 ∈ L∞(RN ), A1(x) ≥ ᾱ0, A1(x) ≥ 0 a.e. in RN .

We point out some direct consequences of the previous hypotheses.

Remark 4.2. In assumption (A5) we always suppose α0 ≤ 1 while from (A5) and
(A6) we suppose α1 ≤ 1 in (A7).

Remark 4.3. From (A7) and (A8) it follows that

(µ− α2)A(x, t, ξ) ≥ α1 a(x, t, ξ) · ξ a.e. in RN , for all (t, ξ) ∈ R× RN ;

hence, if also (A5) and (A6) hold, we have α2 < µ. So,

A(x, t, ξ) ≥ α3a(x, t, ξ) · ξ a.e. in RN for all (t, ξ) ∈ R× RN , (4.2)

with α3 = α1

µ−α2
> 0. Moreover, from (4.2) and (A8) we have that

µA(x, t, ξ)− a(x, t, ξ) · ξ −At(x, t, ξ)t ≥ α2α3 a(x, t, ξ) · ξ a.e. in RN ,

for all (t, ξ) ∈ R× RN .



8 F. MENNUNI, A. SALVATORE EJDE-2024/42

Remark 4.4. We note that from (A5)–(A8) it follows that

−(1− α1)a(x, t, ξ) · ξ ≤ At(x, t, ξ)t ≤ (µ− α2)A(x, t, ξ) ≤ (µ− α2)η0a(x, t, ξ)ξ

which implies that
|At(x, t, ξ)t| ≤ ca(x, t, ξ)ξ (4.3)

with c = max{(µ− α2)η0, (1− α1)}.

Now, we are able to state our main existence result.

Theorem 4.5. Assume that (A1)–(A14) hold, then problem (1.1) admits at least
two weak nontrivial radial bounded solutions, one negative and one positive.

We will prove Theorem 4.5 by applying Proposition 2.2 to a suitable restriction
of the functional J introduced in (3.9). To this aim, the following results will be
useful.

Proposition 4.6. Assume that conditions (A1)–(A5) hold. Then, there exists
positive constants b1, b2 such that

J (u) ≥ b1∥u∥pp − b2∥u∥qp for each u ∈ X.

Hence, functional J is bounded from below, i.e., there exists a constant α ∈ R such
that

J (u) ≥ α for any u ∈ X, with α = min
s≥0

(b1s
p − b2s

q).

Proof. From (A5) and (3.7) we have

J (u) =

∫
RN

A(x, u,∇u)dx+
1

p

∫
RN

|u|p dx−
∫
RN

G(x, u) dx

≥ α0

∫
RN

|∇u|p dx+
1

p

∫
RN

|u|p dx− 1

q
|η| p

p−q
|u|qp

≥ b1∥u∥pp − b2∥u∥qp
where b1 = min{α0,

1
p} and b2 = 1

q |η| p
p−q

. □

Lemma 4.7. Assume that g(x, t) satisfies conditions (A3) and (A4), with 1 < q <
p, and consider (wn)n, (vn)n ⊂ X and v, w ∈ X such that

∥wn∥p ≤M1 for all n ∈ N, wn → w a.e. in RN , (4.4)

∥vn∥p ≤M2 for all n ∈ N, vn → 0 a.e. in RN , (4.5)

for some constants M1, M2 > 0. Then

lim
n→+∞

∫
RN

g(x,wn)vn dx = 0.

Proof. From (4.4), (4.5) and (A3) we have

g(x,wn)vn → 0 a.e. in RN . (4.6)

Moreover, from (3.8) and by applying again (4.4) and (4.5), it follows that∫
RN

|g(x,wn)vn| dx ≤ |η| p
p−q

|wn|q−1
p |vn|p ≤ |η| p

p−q
∥wn∥q−1

p ∥vn∥p ≤Mq−1
1 M2|η| p

p−q
.

As η ∈ L
p

p−q (RN ), for each ϵ > 0 there exists R > 0 such that∫
RN\BR(0)

|g(x,wn)vn| dx < ϵ (4.7)



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 9

for all n ∈ N. On the other hand, from the absolute continuity of the Lebesgue’s

integral taking ϵ′ =
(

ϵ

Mq−1
1 M2

) p
p−q

there exists δϵ > 0 such that∫
A

|η|
p

p−q dx ≤ ϵ′

for all measurable set A ⊂ BR(0) with meas(A) < δϵ. Thus, it follows that∫
A

|g(x,wn)vn| dx ≤ ϵ

for all n ∈ N and for all measurable set A with meas(A) < δϵ. Hence, by Vitali’s
Convergence Theorem

g(x,wn)vn → 0 in L1(BR(0)). (4.8)

The conclusion follow from (4.7) and (4.8). □

From now on, to overcome the lack of compactness of the problem we reduce to
work in the space of radial functions which is a natural constraint if the problem is
radially invariant (see [34]). Thus, in our setting, we consider the space

Xr :=W 1,p
r (RN ) ∩ L∞(RN )

endowed with norm ∥ · ∥X and we denote by (X ′
r, ∥ · ∥X′

r
) its dual space.

Lemma 4.8 (Radial Lemma). If N ≥ 2 and p > 1, for all u ∈W 1,p
r (RN ) it holds

|u(x)| ≤ C
∥u∥p
|x|

N−1
p

a.e. in RN , (4.9)

for a suitable constant C depending only on N and p.

For a proof of the above lemma, see [26, Lemma II.1].

Lemma 4.9. If p > 1 then the following compact embeddings hold:

W 1,p
r (RN ) ↪→↪→ Ll(RN ) for any p < l < p∗.

The proof of the above lemma is essentially contained in [13, Theorem 3.2] (see
also [14, Lemma 4.8]).

Remark 4.10. By assumptions (A10) and (A12), we can be reduced to looking
for critical points of the restriction of J in (3.9) to Xr, which we still denote as J
for simplicity (see [34]).

We recall that Proposition 3.7 implies that functional J is C1 on the Banach
space Xr, too, if also (A1)–(A4) hold.

Now, we want to extend to RN a result stated by Boccardo–Murat–Puel in
bounded domains (see [7, Lemma 5]).

Lemma 4.11. Assume that (A1), (A2), (A5), (A6), (A9)–(A11) hold. Let (un)n ⊂
Xr, u ∈ Xr be such that

un ⇀ u weakly in W 1,p
r (RN ), (4.10)

un → u a.e. in RN , (4.11)

|un|∞ ≤M for all n ∈ N, (4.12)∫
RN

[a(x, un,∇un)− a(x, un,∇u)] · ∇(un − u)dx→ 0. (4.13)



10 F. MENNUNI, A. SALVATORE EJDE-2024/42

Then ∫
RN

|∇un|p dx→
∫
RN

|∇u|p dx as n→ +∞. (4.14)

Proof. We will use arguments similar to those ones used in bounded domains in
[31, Lemma 4.5] (see also [7, Lemma 5]). We will prove that any subsequence of
(un)n admits a subsequence satisfying (4.14) and then (4.14) holds for all sequence
(un)n.

Let fn be defined by

fn = [a(x, un,∇un)− a(x, un,∇u)] · ∇(un − u).

From (A9) it follows that fn ≥ 0 a.e. in RN and from (4.13) we have fn → 0 in
L1(RN ).

Thus, from [9, Theorem 4.9] a function h̄ ∈ L1(RN ) and a subset Z of RN exist
such that meas(Z) = 0 and, up to a subsequence,

fn(x) → 0 and fn(x) ≤ h̄(x) <∞ for all x ∈ RN \ Z, for all n ∈ N. (4.15)

Moreover, since u ∈ X and (4.11)–(4.12) hold, we can assume that

un(x) → u(x), |u(x)| < +∞ and |∇u(x)| < +∞, for all x ∈ RN \ Z.
(4.16)

From (A2) and (A6) we also have

fn(x) ≥
α0

η0
[|∇un|p + |∇u|p]− Φ2(un)|un|p−1|∇u| − ϕ2(un)|∇un|p−1|∇u|

− Φ2(u)|u|p−1|∇un| − ϕ2(u)|∇u|p−1|∇un|.

Since Φ2, ϕ2 are continuous functions, by (4.12), (4.15) and (4.16) we find that

(∇un(x))n is bounded for all x ∈ RN \ Z.

Let ξ∗(x) be a cluster point of (∇un(x))n. We have |ξ∗(x)| <∞ and, since fn(x) →
0 and a is a Carathéodory function, it follows that

[a(x, u, ξ∗)− a(x, u,∇u)] · (ξ∗ −∇u) = 0,

hence (A9) implies that ∇u(x) = ξ∗(x) for all x ∈ RN \ Z. From this, we deduce
that ∇un(x) converges to ∇u(x) without passing to subsequence. Hence,

∇un(x) → ∇u(x) for all x ∈ RN \ Z. (4.17)

Thus, from (A1), (4.16) and (4.17) we have that

a(x, un(x),∇un(x)) → a(x, u(x),∇u(x)) for all x ∈ RN \ Z

and then

a(x, un,∇un) · ∇un → a(x, u,∇u) · ∇u a.e. in RN . (4.18)

Now, from (A5) and (A6) it follows that

a(x, un,∇un) · ∇un ≥ 0 a.e. in RN . (4.19)

From (4.12) and (A2) we obtain that

|a(x, un,∇un)| ≤ c
(
|∇un|p−1

+ |un|p−1)
. (4.20)



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 11

Since (4.10) holds, un is bounded in W 1,p(RN ), thus from (4.20) the sequence

(a(x, un,∇un))n is bounded in (L
p

p−1 (RN ))N , hence, up to subsequences, it weakly

converges to a(x, u,∇u) in (L
p

p−1 (RN ))N . It follows that∫
RN

a(x, un,∇un) · ∇u dx→
∫
RN

a(x, u,∇u) · ∇u dx.

In a similar way, we prove that∫
RN

a(x, un,∇u) · ∇u dx→
∫
RN

a(x, u,∇u) · ∇u dx.

Now, we prove that∫
RN

a(x, un,∇u) · ∇un dx→
∫
RN

a(x, u,∇u) · ∇u dx. (4.21)

Clearly, from (A1), (4.11), and (4.17) it follows that

a(x, un,∇u) · ∇un → a(x, u,∇u) · ∇u a.e. in RN . (4.22)

Moreover, ∣∣ ∫
RN

[a(x, un,∇u) · ∇un − a(x, u,∇u) · ∇u] dx
∣∣

≤
∫
RN

|a(x, un,∇u)||∇un| dx+

∫
RN

a(x, u,∇u) · ∇u dx
(4.23)

where a(x, u,∇u) · ∇u ∈ L1(RN ) while from (A2), Hölder inequality, (4.10) and
(4.12)∣∣ ∫

RN

a(x, un,∇u) · ∇un dx
∣∣ ≤ c(|∇u|p−1

p + |(Φ2(un))
1

p−1 |un||p−1
p . (4.24)

We notice that from (A11) we have

lim
t→0

Φ2(t)

|t|η2
= l2 ≥ 0

hence, there exists δ̄ > 0 such that

Φ2(t) < (l2 + 1)|t|η2 for all t ∈ R, |t| < δ̄.

Therefore, taking M̄ = supn ∥un∥p and R̄ such that CM̄

R̄
N−1

p

< δ̄, using (4.9) in

Radial Lemma it holds

|un(x)| ≤
CM̄

|x|
N−1

p

≤ CM̄

R̄
N−1

p

< δ̄ for all x ∈ RN , |x| > R̄

and therefore, using again Radial Lemma a constant C̄ > 0 exists such that for
|x| > R̄,

(Φ2(un))
p

p−1 |un|p ≤ (l2 + 1)
p

p−1 |un|
η2p
p−1 |un|p ≤ C̄

|x|(N−1)(
η2
p−1+1)

∈ L1(Bc
R̄) (4.25)

since from (4.1) and simple calculations it follows that (N−1)( η2

p−1+1) > N . Thus,

from (4.23)–(4.25) for each ϵ > 0 there exists R > R̄ such that∣∣ ∫
Bc

R

[a(x, un,∇u) · ∇un − a(x, u,∇u) · ∇u] dx
∣∣ ≤ ϵ. (4.26)



12 F. MENNUNI, A. SALVATORE EJDE-2024/42

On the other hand, from (4.24) and (4.12), since un → u in Lp(BR(0)) for each
ϵ > 0, ∣∣ ∫

BR(0)

[a(x, un,∇u) · ∇un − a(x, u,∇u) · ∇u] dx
∣∣

≤ c(|∇u|p−1
p,BR(0) + |u|p−1

p,BR(0)).

(4.27)

From the absolute continuity of the Lebesgue integral, there exists δϵ > 0 such
that ∣∣ ∫

A

[a(x, un,∇u) · ∇un − a(x, u,∇u) · ∇u] dx
∣∣ < ϵ (4.28)

for all measurable set A ⊂ BR(0) with meas(A) < δϵ. Hence, from (4.22) Vitali’s
Theorem holds and∫

BR(0)

a(x, un,∇u) · ∇un dx→
∫
BR(0)

a(x, u,∇u) · ∇u dx. (4.29)

Finally, (4.21) follows from (4.26) and (4.29). Hence, from (4.13) we finally find
that ∫

RN

a(x, un,∇un) · ∇un dx→
∫
RN

a(x, u,∇u) · ∇u dx. (4.30)

Now, we set

yn = a(x, un,∇un) · ∇un and y = a(x, u,∇u) · ∇u.

So, from (4.19), (4.18), (A2) and (4.30) we obtain that

yn ≥ 0, yn → y a.e. in RN , y ∈ L1(RN ),

∫
RN

yn dx→
∫
RN

y dx.

From Brezis-Lieb’s Lemma [9] it results

a(x, un,∇un) · ∇un → a(x, u,∇u) · ∇u in L1(RN ),

hence, using again [9, Theorem 4.9] a function H ∈ L1(RN ) exists such that

a(x, un,∇un) · ∇un ≤ H(x) a.e. in RN . (4.31)

Moreover, from (A5), (A6) and (4.31) we have that

α0

η0
(|∇un|p) ≤ a(x, un,∇un) · ∇un ≤ H(x),

thus, (4.14) follows from (4.17) and Lebesgue’s Convergence Theorem. □

5. Proof of the main result

The aim of this section is to prove that J satisfies the (wCPS)β-condition in Xr

and then to apply Proposition 2.2 to the functional J on Xr. To prove the weak
Cerami-Palais-Smale condition, we need some preliminary lemmas.

Firstly, let us point out that, while if p > N the two norms ∥ · ∥X and ∥ · ∥p
are equivalent, if p ≤ N sufficient conditions are required for the boundedness of
a W 1,p-function. Even if we are working in W 1,p

r (RN ), we need a condition for
functions u in W 1,p(Ω), Ω bounded, as in the following result.



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 13

Lemma 5.1. Let Ω be an open bounded domain in RN with boundary ∂Ω, consider
p, r so that 1 < p ≤ r < p∗, p ≤ N , and take v ∈ W 1,p(Ω). If γ > 0 and k0 ∈ N
exist such that

k0 ≥ ess sup∂Ω v(x),∫
Ω+

k

|∇v|p dx ≤ γ
(
kr meas(Ω+

k ) +

∫
Ω+

k

|v|r dx
)

for all k ≥ k0,

with Ω+
k = {x ∈ Ω : v(x) > k}, then ess supΩ v is bounded from above by a positive

constant which can be chosen so that it depends only on meas(Ω), N , p, r, γ, k0,
|v|p∗ (|v|l for some l > r if p∗ = +∞). Vice versa, if

−k0 ≤ ess inf∂Ω v(x)

and ∫
Ω−

k

|∇v|p dx ≤ γ
(
kr meas(Ω−

k ) +

∫
Ω−

k

|v|r dx
)

for all k ≥ k0

holds with Ω−
k = {x ∈ Ω : v(x) < −k}, then ess supΩ(−v) is bounded from above by

a positive constant which can be chosen so that it depends only on meas(Ω), N , p,
r, γ, k0, |v|p∗ (|v|l for some l > r if p∗ = +∞).

The proof follows from [24, Theorem II.5.1] but reasoning as in [11, Lemma 4.5].
By applying Lemma 5.1, we will prove that the weak limit in W 1,p

r (RN ) of a
(CPS)β-sequence has to be bounded in RN . For simplicity, in the following proofs,
when a sequence (un)n is involved, we use the notation (εn)n for any infinitesimal
sequence depending only on (un)n while (εk,n)n for any infinitesimal sequence de-
pending not only on (un)n but also on some fixed integer k. Moreover, c denotes
any strictly positive constant independent of n which can change from line to line.

Proposition 5.2. Let 1 < q < p and assume that (A1)–(A7), (A10), (A12), (A13)
hold. Then, taking any β ∈ R and a (CPS)β-sequence (un)n ⊂ Xr, it follows that
(un)n is bounded in W 1,p

r (RN ) and a constant β0 > 0 exists such that

|un(x)| ≤ β0 for a.e. x ∈ RN with |x| ≥ 1 and for all n ∈ N. (5.1)

Moreover, there exists u ∈ Xr such that, up to subsequences,

un ⇀ u weakly in W 1,p
r (RN ), (5.2)

un → u strongly in Ll(RN ) for each l ∈]p, p∗[, (5.3)

un → u a.e. in RN , (5.4)

as n→ +∞.

Proof. Let β ∈ R be fixed and consider a sequence (un)n ⊂ Xr such that

J (un) → β and ∥dJ (un)∥X′
r
(1 + ∥un∥Xr

) → 0 as n→ +∞. (5.5)

From Proposition 4.6, as q < p, (un)n is bounded in W 1,p
r (RN ) and therefore

Lemma 4.8 implies the uniform estimate (5.1). Furthermore, u ∈ W 1,p
r (RN ) exists

such that (5.2)–(5.4) hold, up to subsequences.
Now, we have just to prove that u ∈ L∞(RN ). Clearly, (5.1) and (5.4) imply

ess sup|x|≥1 |u(x)| ≤ β0 < +∞. (5.6)

Then, it is sufficient to prove that

ess sup|x|≤1 |u(x)| < +∞. (5.7)



14 F. MENNUNI, A. SALVATORE EJDE-2024/42

Arguing by contradiction, let us assume that either

ess sup|x|≤1 u(x) = +∞ (5.8)

or

ess sup|x|≤1(−u(x)) = +∞. (5.9)

If, for example, (5.8) holds then, for any fixed k ∈ N, k > β0 we have that

meas(B+
k ) > 0 with B+

k = {x ∈ B1(0) : u(x) > k}. (5.10)

We note that the choice of k and (5.6) imply that

B+
k = {x ∈ RN : u(x) > k}. (5.11)

Moreover, if we set

B+
k,n = {x ∈ B1(0) : un(x) > k}, n ∈ N,

the choice of k and (5.1) imply that

B+
k,n = {x ∈ RN : un(x) > k} for all n ∈ N. (5.12)

Now, consider the new function R+
k : t ∈ R → R+

k t ∈ R such that

R+
k t =

{
0 if t ≤ k

t− k if t > k .

By definition and (5.11), respectively (5.12), it results

R+
k u(x) =

{
0 if x ̸∈ B+

k

u(x)− k if x ∈ B+
k ,

R+
k un(x) =

{
0 if x ̸∈ B+

k,n

un(x)− k if x ∈ B+
k,n .

(5.13)
Clearly, (5.1), (5.6) and k > β0 imply

R+
k u ∈W 1,p

0 (B1(0)) and R+
k un ∈W 1,p

0 (B1(0)) for all n ∈ N . (5.14)

From (5.2) it follows that R+
k un ⇀ R+

k u weakly in W 1,p
r (RN ), then, from (5.14), in

W 1,p
0 (B1(0)). As W 1,p

0 (B1(0)) ↪→↪→ Ll(B1(0)) for any 1 ≤ l < p∗, then

lim
n→+∞

∫
B1(0)

|R+
k un|

l dx =

∫
B1(0)

|R+
k u|

l dx for 1 ≤ l < p∗. (5.15)

Moreover, from (5.3) we have un → u strongly in Ll(B1(0)) for any l ∈]p, p∗[ and
then

lim
n→+∞

∫
B1(0)

|un|l dx =

∫
B1(0)

|u|l dx for 1 ≤ l < p∗. (5.16)

Thus, by the weak lower semi-continuity of the norm ∥ · ∥p, we have that∫
RN

|∇R+
k u|

p dx+

∫
RN

|R+
k u|

p dx ≤ lim inf
n→+∞

(∫
RN

|∇R+
k un|

p dx+

∫
RN

|R+
k un|

p dx
)
,

i.e., from (5.13)–(5.15) we have∫
B+

k

|∇u|p dx+

∫
B1(0)

|R+
k u|

p dx ≤ lim inf
n→+∞

(∫
B+

k,n

|∇un|p dx+

∫
B1(0)

|R+
k un|

p dx
)

= lim inf
n→+∞

∫
B+

k,n

|∇un|p dx+

∫
B1(0)

|R+
k u|

p dx.



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 15

Hence, ∫
B+

k

|∇u|p dx ≤ lim inf
n→+∞

∫
B+

k,n

|∇un|p dx. (5.17)

On the other hand, since ∥R+
k un∥X ≤ ∥un∥X holds, it follows that

|⟨dJ (un), R
+
k un⟩| ≤ ∥dJ (un)∥X′

r
∥un∥X .

Then (5.5) and (5.10) imply that nk ∈ N exists such that

⟨dJ (un), R
+
k un⟩ < meas(B+

k ) for all n ≥ nk. (5.18)

Let us point out that, since α1 ≤ 1, assumptions (A5)–(A7) imply that

⟨dJ (un), R
+
k un⟩ =

∫
B+

k,n

a(x, un,∇un) · ∇un dx+

∫
B+

k,n

At(x, un,∇un)(un − k) dx

+

∫
B+

k,n

|un|p−2un(un − k) dx−
∫
B+

k,n

g(x, un)R
+
k un dx

=

∫
B+

k,n

(
1− k

un

)
[a(x, un,∇un) · ∇un +At(x, un,∇un)un] dx

+

∫
B+

k,n

k

un
a(x, un,∇un) · ∇un dx+

∫
B+

k,n

|un|p−2un(un − k) dx

−
∫
B+

k,n

g(x, un)R
+
k un dx

≥ α1

∫
B+

k,n

a(x, un,∇un) · ∇un dx−
∫
B+

k,n

g(x, un)R
+
k un dx.

Hence, from the previous inequalities, (A5) and (A6) it follows that

α0α1

η0

∫
B+

k,n

|∇un|p dx ≤ ⟨dJ (un), R
+
k un⟩+

∫
B+

k,n

g(x, un)R
+
k un dx. (5.19)

Now, from (5.14), (5.15) and (A4) we obtain

lim
n→+∞

∫
RN

g(x, un)R
+
k un dx =

∫
RN

g(x, u)R+
k u dx. (5.20)

Thus, from (5.17)–(5.20) and (A13) we obtain that∫
B+

k

|∇u|p dx ≤ c
(
meas(B+

k ) +

∫
B+

k

g(x, u)R+
k u dx

)
≤ c meas(B+

k ) + c

∫
B+

k

η(x)|u|q dx

≤ c̄
(
meas(B+

k ) +

∫
B+

k

|u|p
)

with c̄ = max{c, ess sup|x|≤1 η(x)} since∫
B+

k

η(x)|u|q dx ≤
∫
B+

k

η(x)|u|p dx ≤ ess sup|x|≤1 η(x)

∫
B+

k

|u|p dx

as q < p and u(x) > 1 for all x ∈ B+
k .



16 F. MENNUNI, A. SALVATORE EJDE-2024/42

Thus, we obtain ∫
B+

k

|∇u|p dx ≤ c̄
(
meas(B+

k ) +

∫
B+

k

|u|p
)
.

As this inequality holds for all k > β0, Lemma 5.1 implies that (5.8) is not true.
Thus, (5.9) must hold. In this case, fixing any k ∈ N, k > β0, we have

meas(B−
k ) > 0, with B−

k = {x ∈ B1(0) : u(x) < −k},
and we can consider R−

k : t ∈ R → R−
k t ∈ R such that

R−
k t =

{
0 if t ≥ −k
t+ k if t < −k .

Thus, reasoning as above, but replacing R+
k with R−

k , and applying again Lemma
5.1 we prove that (5.9) cannot hold. Hence, (5.7) has to be true. □

We are ready to prove the (wCPS) condition in R by adapting the arguments
developed in [10, Proposition 3.4], also in [11, Proposition 4.6], to our setting in
the whole space RN .

Proposition 5.3. If 1 < q < p and (A1)–(A13) hold, then functional J satisfies
the weak Cerami-Palais-Smale condition in Xr at each level β ∈ R.

Proof. Let β ∈ R be fixed and consider a sequence (un)n ⊂ Xr verifying (5.5). By
Proposition 5.2, the uniform estimate (5.1) holds and there exists u ∈ Xr such that,
up to subsequences, (5.2)–(5.4) are satisfied.

We need to prove the following three steps:

(1) Define Tk : R → R such that

Tkt =

{
t if |t| ≤ k

k t
|t| if |t| > k,

(5.21)

with k ≥ max{|u|∞, β0}. Then, as n→ +∞, we have

J (Tkun) → β, (5.22)

∥dJ (Tkun)∥X′
r
→ 0; (5.23)

(2) ∥un − u∥p → 0 if n→ +∞, as

∥Tkun − u∥p → 0 as n→ +∞; (5.24)

(3) J (u) = β and dJ (u) = 0.

Step 1. Taking any k > max{|u|∞, β0}, if we set

Bk,n = {x ∈ B1(0) : |un(x)| > k}, n ∈ N, (5.25)

the choice of k and (5.1) imply that

Bk,n = {x ∈ RN : |un(x)| > k} for all n ∈ N. (5.26)

Then, from (5.21) and (5.26) we have that

Tkun(x) =

{
un(x) for a.e. x ̸∈ Bk,n

k un(x)
|un(x)| for x ∈ Bk,n

(5.27)

and
|Tkun|∞ ≤ k, ∥Tkun∥p ≤ ∥un∥p for each n ∈ N.



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 17

Defining Rk : R → R such that

Rkt = t− Tkt =

{
0 if |t| ≤ k

t− k t
|t| if |t| > k,

from (5.26) it results that

Rkun(x) =

{
0 for a.e. x ̸∈ Bk,n

un(x)− k un(x)
|un(x)| for x ∈ Bk,n;

(5.28)

hence, (5.25) and (5.28) imply that

Rkun ∈W 1,p
0 (B1(0)) for all n ∈ N . (5.29)

Since k > |u|∞, we deduce that

Tku(x) = u(x) and Rku(x) = 0 for a.e. x ∈ RN ;

thus, from (5.2) it follows that Rkun ⇀ 0 weakly in W 1,p
r (RN ), and, from (5.29),

in W 1,p
0 (B1(0)). From the compact embedding of W 1,p

0 (B1(0)) in Ll(B1(0)) for
1 ≤ l < p∗, we have that

lim
n→+∞

∫
RN

|Rkun|l dx = 0 for 1 ≤ l < p∗. (5.30)

Now, arguing as in the proof of (5.19) but replacing R+
k un with Rkun we obtain

α0α1

η0

∫
Bk,n

|∇un|p dx ≤ α1

∫
Bk,n

a(x, un,∇un) · ∇un dx

≤ ⟨dJ (un), Rkun⟩+
∫
Bk,n

g(x, un)Rkun dx.

(5.31)

We note that (5.5) and ∥Rkun∥X ≤ ∥un∥X imply that

lim
n→+∞

|⟨dJ (un), Rkun⟩| = 0; (5.32)

while the boundedness of the sequences (∥un∥p)n and (∥Rkun∥p)n, (5.4), (5.6),
(5.28), and Lemma 4.7 imply that

lim
n→+∞

∫
Bk,n

g(x, un)Rkun dx = 0. (5.33)

From (5.31)–(5.33) we obtain that

lim
n→+∞

∫
Bk,n

|∇un|p dx = 0, (5.34)

lim
n→+∞

∫
Bk,n

a(x, un,∇un) · ∇un dx = 0. (5.35)

Hence, from (5.28), (5.30), and (5.34) it follows that

lim
n→+∞

∥Rkun∥p = 0. (5.36)

Moreover, from (5.4), (5.25), and k > |u|∞ we obtain

lim
n→+∞

meas(Bk,n) = 0, (5.37)



18 F. MENNUNI, A. SALVATORE EJDE-2024/42

which together (5.16) implies

lim
n→+∞

∫
Bk,n

|un|l dx = 0 for 1 ≤ l < p∗. (5.38)

From (3.9) and (5.27) we have

J (Tkun)

=

∫
RN\Bk,n

A(x, un,∇un) dx+

∫
Bk,n

A

(
x, k

un
|un|

, 0

)
dx

+
1

p

∫
RN\Bk,n

|un|p dx+
1

p

∫
Bk,n

kp dx−
∫
RN

G(x, Tkun) dx

= J (un)−
∫
Bk,n

A(x, un,∇un) dx+

∫
Bk,n

A

(
x, k

un
|un|

, 0

)
dx

− 1

p

∫
Bk,n

|un|p dx+
1

p

∫
Bk,n

kp dx−
∫
RN

(G(x, Tkun)−G(x, un)) dx.

(5.39)

From (A5), (A6) and (5.35) we have∫
Bk,n

A(x, un,∇un) dx ≤ η0

∫
Bk,n

a(x, un,∇un) · ∇un dx→ 0, (5.40)

while (A2), (5.4), (5.37), and (5.38) imply∫
Bk,n

A
(
x, k

un
|un|

, 0
)
dx ≤

∫
Bk,n

Φ0

(
k
un
|un|

)
kp dx

≤
(
max
|t|≤k

Φ0(t)
)
kp measBk,n → 0

(5.41)

and

−1

p

∫
Bk,n

|un|p dx+
1

p

∫
Bk,n

kp dx→ 0. (5.42)

Furthermore, from (5.27), we have∫
RN

(G(x, Tkun)−G(x, un)) dx =

∫
Bk,n

(
G
(
x, k

un
|un|

)
−G(x, un)

)
dx→ 0 (5.43)

since (3.7), (5.37), and (5.38) imply that∫
Bk,n

G
(
x, k

un
|un|

)
dx ≤ 1

q
|η| p

p−q
kq(meas(Bk,n))

q
p → 0

and ∫
Bk,n

G(x, un) dx ≤ 1

q
|η| p

p−q

(∫
Bk,n

|un|q dx
) q

p → 0.

Then, (5.22) follows from (5.5) and (5.39)–(5.43).
To prove (5.23), we take v ∈ Xr such that ∥v∥X = 1; hence, |v|∞ ≤ 1, ∥v∥W ≤ 1.

From (3.10) and (5.27) we have

⟨dJ (Tkun), v⟩

=

∫
RN

a(x, Tkun,∇Tkun) · ∇v dx+

∫
RN

At(x, Tkun,∇Tkun)v dx

+

∫
RN

|Tkun|p−2Tkunv dx−
∫
RN

g(x, Tkun)v dx



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 19

=

∫
RN\Bk,n

a(x, un,∇un) · ∇v dx+

∫
Bk,n

a
(
x, k

un
|un|

, 0
)
· ∇v

+

∫
RN\Bk,n

At(x, un,∇un)v dx+

∫
Bk,n

At

(
x, k

un
|un|

, 0
)
v dx

+

∫
RN\Bk,n

|un|p−2unv dx+

∫
Bk,n

kp−1 un
|un|

v dx−
∫
RN

g(x, Tkun)v dx

= ⟨dJ (un), v⟩ −
∫
Bk,n

a(x, un,∇un) · ∇v dx−
∫
Bk,n

At(x, un,∇un)v dx

−
∫
Bk,n

|un|p−2unv dx+

∫
Bk,n

(g(x, un)− g(x, Tkun))v dx+ ϵn,

since (A2), (5.37), Hölder inequality and |∇v|p ≤ 1, |v|∞ ≤ 1 imply that∣∣ ∫
Bk,n

a
(
x, k

un
|un|

, 0
)
· ∇v dx

∣∣ ≤ ∫
Bk,n

Φ2

(
k
un
|un|

)
kp−1|∇v|dx

≤
(
max
|t|≤k

Φ2(t)
)(∫

Bk,n

kp dx
) p−1

p → 0,

(5.44)

∣∣ ∫
Bk,n

At

(
x, k

un
|un|

, 0
)
v dx

∣∣ ≤ ∫
Bk,n

Φ1

(
k
un
|un|

)
kp−1dx

≤
(
max
|t|≤k

Φ1(t)
)
kp−1 meas(Bk,n) → 0,

(5.45)

∣∣ ∫
Bk,n

kp−1 un
|un|

v dx
∣∣ ≤ kp−1 meas(Bk,n) → 0, (5.46)

where all the limits hold uniformly with respect to v.
Furthermore, from (4.3) and (5.35) we have that

lim
n→+∞

∫
Bk,n

|At(x, un,∇un)un|dx = 0,

and then, since 1 ≤ k ≤ |un| on Bk,n and |v|∞ ≤ 1, we obtain∣∣ ∫
Bk,n

At(x, un,∇un)v dx
∣∣ ≤ ∫

Bk,n

|At(x, un,∇un)|dx

≤
∫
Bk,n

|At(x, un,∇un)||un| dx→ 0

(5.47)

uniformly with respect to v, while from (5.38), Hölder inequality and |v|p ≤ 1 we
have ∣∣ ∫

Bk,n

|un|p−2unv dx
∣∣ ≤ (∫

Bk,n

|un|p dx
) p−1

p → 0.

Moreover, from (3.8), (5.37), (5.38), and |v|p ≤ 1 it results∣∣ ∫
Bk,n

g(x, un)v dx
∣∣ ≤ |η| p

p−q

(∫
Bk,n

|un|p dx
) q−1

p → 0

uniformly with respect to v, and∣∣ ∫
Bk,n

g(x, Tkun)v dx
∣∣ ≤ |η| p

p−q

(∫
Bk,n

|Tkun|p dx
) q−1

p → 0



20 F. MENNUNI, A. SALVATORE EJDE-2024/42

uniformly with respect to v. Thus, summing, from (5.5) we obtain

|⟨dJ (Tkun), v⟩| ≤ εk,n +
∣∣ ∫

Bk,n

a(x, un,∇un) · ∇v dx
∣∣. (5.48)

Now, to estimate the last integral in (5.48), following the notation introduced in
the proof of Proposition 5.2, let us consider the set B+

k,n and the test function

φ+
k,n = vR+

k un.

By definition, we have ∥φ+
k,n∥X ≤ 2∥un∥X ; thus, (5.5) implies

∥dJ (un)∥X′
r
∥φ+

k,n∥X ≤ εn.

From definition (5.13) and direct computations we note that

⟨dJ (un), φ
+
k,n⟩ =

∫
B+

k,n

a(x, un,∇un)R+
k un · ∇v dx+

∫
B+

k,n

a(x, un,∇un) · v∇un dx

+

∫
B+

k,n

At(x, un,∇un)vR+
k un dx+

∫
B+

k,n

|un|p−2unvR
+
k un dx

−
∫
B+

k,n

g(x, un)vR
+
k un dx ,

where, since B+
k,n ⊂ Bk,n, from (5.37) we have

lim
n→+∞

meas(B+
k,n) = 0,

while |v|∞ ≤ 1, (5.35), (5.47), (5.38), and (3.8) imply∣∣ ∫
B+

k,n

a(x, un,∇un) · v∇un dx
∣∣ ≤ ∫

B+
k,n

a(x, un,∇un) · ∇un dx→ 0,

∣∣ ∫
B+

k,n

At(x, un,∇un)vR+
k undx

∣∣ ≤ ∫
B+

k,n

|At(x, un,∇un)|(un − k)dx

≤
∫
B+

k,n

|At(x, un,∇un)|un dx→ 0,

∣∣ ∫
B+

k,n

|un|p−2unvR
+
k un dx

∣∣ ≤ ∫
B+

k,n

|un|p dx→ 0,

∣∣ ∫
B+

k,n

g(x, un)vR
+
k un dx

∣∣ ≤ ∫
B+

k,n

|g(x, un)||un| dx

≤ |η| p
p−q

(∫
B+

k,n

|un|p
) q−1

p → 0

uniformly with respect to v. From the previous estimates it follows that

lim
n→+∞

∫
B+

k,n

a(x, un,∇un)R+
k un · ∇v dx = 0 (5.49)

Now, if we fix k > max{|u|∞, β0} + 1, all the previous computations hold also
for k − 1 and then in particular, (5.34), (5.38), and (5.49) become

lim
n→+∞

∫
Bk−1,n

|∇un|p dx = 0, lim
n→+∞

∫
Bk−1,n

|un|p dx = 0, (5.50)



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 21

lim
n→+∞

∫
B+

k−1,n

a(x, un,∇un)R+
k−1un · ∇v dx = 0. (5.51)

From (5.51) since B+
k,n ⊂ B+

k−1,n, we have

ϵk,n =

∫
B+

k−1,n

a(x, un,∇un)R+
k−1un · ∇v dx

=

∫
B+

k,n

a(x, un,∇un)R+
k−1un · ∇v dx

+

∫
B+

k−1,n\B
+
k,n

a(x, un,∇un)R+
k−1un · ∇v dx

=

∫
B+

k,n

a(x, un,∇un)R+
k un · ∇v dx+

∫
B+

k,n

a(x, un,∇un) · ∇v dx

+

∫
B+

k−1,n\B
+
k,n

a(x, un,∇un)R+
k−1un · ∇v dx

where (A2), (5.13), the properties of B+
k−1,n \ B+

k,n, Hölder inequality, |∇v|p ≤ 1,

and (5.50) imply∣∣ ∫
B+

k−1,n\B
+
k,n

a(x, un,∇un)R+
k−1un · ∇v dx

∣∣
≤ k

∫
B+

k−1,n\B
+
k,n

|a(x, un,∇un)||∇v| dx

≤ kmax
|t|≤k

Φ2(t)

∫
B+

k−1,n\B
+
k,n

|un|p−1|∇v| dx

+ kmax
|t|≤k

ϕ2(t)

∫
B+

k−1,n\B
+
k,n

|∇un|p−1|∇v| dx

≤ kmax
|t|≤k

Φ2(t)
(∫

B+
k−1,n\B

+
k,n

|un|p dx
) p−1

p

+ kmax
|t|≤k

ϕ2(t)
(∫

B+
k−1,n\B

+
k,n

|∇un|p dx
) p−1

p → 0.

The above arguments imply∣∣ ∫
B+

k,n

a(x, un,∇un) · ∇v dx
∣∣ ≤ εk,n. (5.52)

Similar arguments apply also if we consider B−
k,n and the test functions

φ−
k,n = vR−

k un, φ−
k−1,n = vR−

k−1un;

hence, we have ∣∣ ∫
B−

k,n

a(x, un,∇un) · ∇v dx
∣∣ ≤ εk,n. (5.53)

Thus, (5.23) follows from (5.48), (5.52) and (5.53) as all εk,n are independent of v.



22 F. MENNUNI, A. SALVATORE EJDE-2024/42

Step 2. We note that (5.2)–(5.4) imply that, if n→ +∞,

Tkun ⇀ u weakly in W 1,p
r (RN ),

Tkun → u strongly in Ll(RN ) for each l ∈]p, p∗[,
Tkun → u a.e. in RN .

Now, arguing as in [1], let us consider the real map

ψ : t ∈ R 7→ ψ(t) = teη̄t
2

∈ R,

where η̄ > ( β
2α )

2 will be fixed once α, β > 0 are chosen in a suitable way later. By
definition,

αψ′(t)− β|ψ(t)| > α

2
for all t ∈ R. (5.54)

If we define vk,n = Tkun − u, since k > |u|∞, we have that |vk,n|∞ ≤ 2k for all
n ∈ N. Therefore,

|ψ(vk,n)| ≤ ψ(2k), 0 < ψ′(vk,n) ≤ ψ′(2k) a.e. in RN for all n ∈ N, (5.55)

ψ(vk,n) → 0, ψ′(vk,n) → 1 a.e. in RN as n→ +∞. (5.56)

Furthermore, we note that

|ψ(vk,n)| ≤ |vk,n|e4k
2η̄ a.e. in RN for all n ∈ N,

thus, direct computations imply that (∥ψ(vk,n)∥X)n is bounded, and so from (5.56),
up to subsequences, we have

ψ(vk,n)⇀ 0 weakly in W 1,p
r (RN ), (5.57)

while from (5.23) it follows that

⟨dJ (Tkun), ψ(vk,n)⟩ → 0 as n→ +∞,

where

⟨dJ (Tkun), ψ(vk,n)⟩

=

∫
RN\Bk,n

a(x, un,∇un) · ∇ψ(vk,n) dx+

∫
Bk,n

a
(
x, k

un
|un|

, 0
)
· ∇ψ(vk,n) dx

+

∫
RN\Bk,n

At(x, un,∇un)ψ(vk,n) dx+

∫
Bk,n

At

(
x, k

un
|un|

, 0
)
ψ(vk,n) dx

+

∫
RN\Bk,n

|un|p−2unψ(vk,n) dx+

∫
Bk,n

kp−1 un
|un|

ψ(vk,n) dx

−
∫
RN

g(x, Tkun)ψ(vk,n) dx.

Since (∥ψ(vk,n)∥X)n is bounded, arguing as in (5.44)–(5.46) it follows that

lim
n→+∞

∫
Bk,n

a
(
x, k

un
|un|

, 0
)
· ∇ψ(vk,n) dx = 0,

lim
n→+∞

∫
Bk,n

At

(
x, k

un
|un|

, 0
)
ψ(vk,n) dx = 0,

lim
n→+∞

∫
Bk,n

kp−1 un
|un|

ψ(vk,n) dx = 0.



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 23

Furthermore, from Lemma 4.7 with wn = Tkun and vn = ψ(vk,n), we have

lim
n→+∞

∫
RN

g(x, Tkun)ψ(vk,n) dx = 0.

Hence, summing, the previous relations imply

εk,n =

∫
RN\Bk,n

a(x, un,∇un)ψ′(vk,n) · ∇vk,n dx

+

∫
RN\Bk,n

At(x, un,∇un)ψ(vk,n) dx

+

∫
RN\Bk,n

|un|p−2unψ(vk,n) dx.

(5.58)

We note that from (A2),∣∣ ∫
RN\Bk,n

At(x, un,∇un)ψ(vk,n) dx
∣∣

≤
∫
RN\Bk,n

(
Φ1(un)|un|p−1 +max

|t|≤k
ϕ1(t)|∇un|p

)
|ψ(vk,n)|dx.

(5.59)

We prove that

lim
n→+∞

∫
RN\Bk,n

Φ1(un)|un|p−1|ψ(vk,n)| dx = 0. (5.60)

In fact, since the sequence (un)n is bounded in W 1,p
r (RN ), there exists a constant

M̃ > 0 such that

∥un∥p ≤ M̃, ∥un − u∥p ≤ M̃ for all n ∈ N.

Moreover, from assumption (A11),

lim
t→0

Φ1(t)

|t|η1
= l1 with l1 ≥ 0,

hence, there exists δ1 > 0 such that

Φ1(t) < (l1 + 1)|t|η1 for all t ∈ R, |t| < δ1. (5.61)

Now, fixing ϵ > 0, as from (4.1) it follows that (η1 + p)N−1
p > N , then there exists

Rϵ such that

CM̃

R
N−1

p
ϵ

< δ1, (5.62)

(l1 + 1)(CM̃)p+η1e
η̄ C2M̃2

Rϵ
2N−1

p

∫
Bc

Rϵ

1

|x|(η1+p)N−1
p

dx < ϵ (5.63)

where C is the constant introduced in (4.9). From (4.9) and (5.62), it follows that

|un(x)| ≤ C
M̃

|x|
N−1

p

≤ C
M̃

R
N−1

p
ϵ

< δ1 a.e. x ∈ RN with |x| > Rϵ;

hence, (5.61), (4.9), and (5.63) imply∫
(RN\Bk,n)∩Bc

Rϵ

Φ1(un)|un|p−1|ψ(vk,n)| dx



24 F. MENNUNI, A. SALVATORE EJDE-2024/42

≤
∫
(RN\Bk,n)∩Bc

Rϵ

(l1 + 1)|un|η1+p−1|un − u|eη̄∥un−u∥2
W dx

≤ (l1 + 1)(CM̃)η1+pe
η̄ C2M̃2

Rϵ
2N−1

p

∫
Bc

Rϵ

1

|x|(η1+p)N−1
p

dx < ϵ

while from Hölder’s inequality∫
(RN\Bk,n)∩BRϵ

Φ1(un)|un|p−1|ψ(vk,n)| dx

≤
(
max
|t|≤k

Φ1(t)
)
|un|p−1

p

(∫
BRϵ

|ψ(vk,n)|p dx
)1/p

→ 0

since (5.57) implies that ψ(vk,n) → 0 in Lp
loc(RN ). Then, (5.60) holds and from

(A5) and (A6) it follows that∫
RN\Bk,n

|∇un|p|ψ(vk,n)| dx

≤ η0
α0

∫
RN\Bk,n

a(x, un,∇un) · ∇un|ψ(vk,n)| dx

=
η0
α0

∫
RN\Bk,n

a(x, un,∇un) · ∇vk,n|ψ(vk,n)| dx

+
η0
α0

∫
RN\Bk,n

a(x, un,∇un) · ∇u|ψ(vk,n)|dx,

(5.64)

where the boundedness of (un)n in W 1,p
r (RN ), (A2), Hölder’s inequality, (5.56) and

the Lebesgue Dominated Convergence Theorem imply that∣∣ ∫
RN\Bk,n

a(x, un,∇un) · ∇u|ψ(vk,n)|dx
∣∣

≤
∫
RN\Bk,n

Φ2(un)|un|p−1|∇u||ψ(vk,n)|dx

+

∫
RN\Bk,n

ϕ2(un)|∇un|p−1|∇u||ψ(vk,n)|dx

≤
(
max
|t|≤k

Φ2(t)
)
|un|p−1

p

(∫
RN\Bk,n

|∇u|p|ψ(vk,n)|pdx
)1/p

+
(
max
|t|≤k

ϕ2(t)
)
|∇un|p−1

p

(∫
RN\Bk,n

|∇u|p|ψ(vk,n|p) dx
)1/p

→ 0.

(5.65)

From (5.58)–(5.60), (5.64), (5.65), (A5) and (A6) we obtain

ϵk,n ≥
∫
RN\Bk,n

a(x, un,∇un)ψ′(vk,n) · ∇vk,ndx

− η0
α0

max
|t|≤k

ϕ1(t)

∫
RN\Bk,n

a(x, un,∇un) · ∇vk,n|ψ(vk,n)| dx

+

∫
RN\Bk,n

|un|p−2unψ(vk,n) dx.



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 25

Thus, setting

hk,n(x) = ψ′(vk,n)−
η0
α0

max
|t|≤k

ϕ1(t)|ψ(vk,n)|,

and choosing, in the definition of ψ, constants α = 1 and β = η0

α0
max|t|≤k ϕ1(t),

from (5.54) it results

hk,n(x) >
1

2
a.e. in RN . (5.66)

Therefore,

εk,n ≥
∫
RN\Bk,n

hk,na(x, un,∇un) · ∇vk,n dx

+

∫
RN\Bk,n

|un|p−2unψ(vk,n) dx

=

∫
RN\Bk,n

a(x, u,∇u) · ∇vk,n dx

+

∫
RN\Bk,n

hk,n (a(x, un,∇un)− a(x, un,∇u)) · ∇vk,n dx

+

∫
RN\Bk,n

(hk,na(x, un,∇u)− a(x, u,∇u)) · ∇vk,n dx

+

∫
RN\Bk,n

(|un|p−2un − |u|p−2u)ψ(vk,n) dx

+

∫
RN\Bk,n

|u|p−2uψ(vk,n) dx,

(5.67)

where (5.2), respectively (5.57) imply that

lim
n→+∞

∫
RN\Bk,n

a(x, u,∇u) · ∇vk,n dx = 0, lim
n→+∞

∫
RN\Bk,n

|u|p−2uψ(vk,n) dx = 0.

Now, we want to prove that

lim
n→+∞

∫
RN\Bk,n

(
hk,na(x, un,∇u)− a(x, u,∇u)

)
· ∇vk,n dx = 0. (5.68)

Indeed, recalling that (∇vk,n)n is bounded in Lp(RN ), arguing as in the proof of
(4.26), from (A11) for all ϵ > 0 there exists Rϵ > 0 such that∫

(RN\Bk,n)∩Bc
Rϵ

|hk,na(x, un,∇u)− a(x, u,∇u)|
p

p−1 dx < ϵ (5.69)

where (RN \ Bk,n) ∩ Bc
Rϵ

= Bc
Rϵ

(0). On the other hand, we note that (A1), (5.4)
and (5.56) infer that

hk,na(x, un,∇u)− a(x, u,∇u) → 0 a.e. inRN ,

while from Hölder’s inequality it follows that∣∣ ∫
RN\Bk,n

(
hk,na(x, un,∇u)− a(x, u,∇u)

)
· ∇vk,n dx

∣∣
≤

(∫
RN\Bk,n

|hk,na(x, un,∇u)− a(x, u,∇u)|
p

p−1 dx
) p−1

p |∇vk,n|p.
(5.70)



26 F. MENNUNI, A. SALVATORE EJDE-2024/42

From (5.55) and (A2) we have that for each x ∈ (RN \Bk,n),

|hk,na(x, un,∇u)− a(x, u,∇u)|
p

p−1

≤
(
ψ′(2k)

(
Φ2(un)|un|p−1 + (max

|t|≤k
ϕ2(t))|∇u|p−1

)
+ |a(x, u,∇u)|

) p
p−1

≤ c(1 + |∇u|p),

(5.71)

hence, the Lebesgue Dominated Convergence Theorem implies that

lim
n→+∞

∫
(RN\Bk,n)∩BRϵ (0)

|hk,na(x, un,∇u)− a(x, u,∇u)|
p

p−1 dx = 0. (5.72)

Thus, from (5.66) and (5.67), by using the previous estimate, the strong convexity

of the power function with exponent p > 1, (A9) and eη̄v
2
k,n ≥ 1 we obtain

εk,n ≥ 1

2

∫
RN\Bk,n

(
a(x, un,∇un)− a(x, un,∇u)

)
· ∇(un − u) dx

+

∫
RN\Bk,n

(|un|p−2un − |u|p−2u)(un − u) dx.

Using again (A9) and the strong convexity of the power function with exponent
p > 1 we have

lim
n→+∞

∫
RN\Bk,n

(
a(x, un,∇un)− a(x, un,∇u)

)
· ∇(un − u) dx = 0, (5.73)

lim
n→+∞

∫
RN\Bk,n

(|un|p−2un − |u|p−2u)(un − u) dx = 0 (5.74)

Next we prove that

lim
n→+∞

∫
RN\Bk,n

|un − u|p dx = 0. (5.75)

In fact, if p ≥ 2,

|un − u|p ≤ (|un|p−2un − |u|p−2u)(un − u) a.e. x ∈ RN , for all n ∈ N; (5.76)

thus, (5.76) implies (5.75). On the other hand, if p ∈ (1, 2), it is p
p−1 > p; thus, as

(Tkun)n is bounded in W 1,p(RN ) and |Tkun| ≤ k a.e. x ∈ RN \Bk,n, for all n ∈ N,
it follows that (Tkun)n is bounded in Lℓ(RN ) for any ℓ ≥ p, and in particular is

bounded in L
p

p−1 (RN ). Passing to a subsequence, Tkun ⇀ u in L
p

p−1 (RN ), hence

lim
n→+∞

∫
RN

|Tkun|p−2Tkunu dx =

∫
RN

|u|p dx,

which implies, together (5.37), that

lim
n→+∞

∫
RN\Bk,n

|un|p−2unu dx =

∫
RN

|u|p dx. (5.77)

Moreover, since (Tkun)n is bounded in Lp(RN ) and u ∈ L
p

p−1 (RN ), up to sub-
sequences, we have that

lim
n→+∞

∫
RN

|u|p−2uTkun dx =

∫
RN

|u|p dx,



EJDE-2024/42 MODIFIED SCHRÖDINGER EQUATIONS 27

i.e., using again (5.37),

lim
n→+∞

∫
RN\Bk,n

|u|p−2uun dx =

∫
RN

|u|p dx. (5.78)

Hence, from (5.74), (5.77), (5.78), (5.37), and (5.38) we obtain

0 = lim
n→+∞

∫
RN\Bk,n

(
|un|p + |u|p − |un|p−2unu− |u|p−2uun

)
dx

= lim
n→+∞

∫
RN\Bk,n

|un|p dx+

∫
RN

|u|p dx

− lim
n→+∞

∫
RN\Bk,n

|un|p−2unu dx− lim
n→+∞

∫
RN\Bk,n

|u|p−2uun dx

= lim
n→+∞

∫
RN

|un|p dx−
∫
RN

|u|p dx,

i.e.,

lim
n→+∞

∫
RN

|un|p dx =

∫
RN

|u|p dx.

Thus, by applying Brezis-Lieb’s Lemma (see [9]), condition (5.75) follows, also in
the case 1 < p < 2.

In each case, Tkun → u in Lp(RN ). Finally, as Tkun → u a.e. in RN and
|Tkun|∞ ≤ k for all n ∈ N, from (5.73), we can apply Lemma 4.11 to the sequence
(Tkun)n obtaining that ∇Tkun → ∇u in Lp(RN ). Thus, (5.24) follows.

Step 3. The proof follows from (5.24), Proposition 3.7, (5.22) and (5.23). □

Proof of Theorem 4.5. The functional J is bounded from below in X (see Propo-
sition 4.6) and satisfies condition (wCPS) in R (see Proposition 5.3), thus, from
Proposition 2.2, J admits a minimum point u∗ in X. Clearly, it is

J (u∗) = min
u∈X

J (u) ≤ J (0) = 0.

Now, we prove that u∗ is not trivial since J (u∗) < 0. To this aim, we consider

φ1 ∈ W 1,p
0 (B1(0)) the unique eigenfunction associated to the first eigenvalue λ1 of

−∆p in B1(0) (see [25]). It results

φ1 > 0 a.e. in B1(0), φ1 ∈ L∞(B1(0)),∫
B1(0)

|φ1|p dx = 1,

∫
B1(0)

|∇φ1|p dx = λ1.

We denote again by φ1 its null extension to RN \B1(0).
Let us remark that φ1 is radial since by the Pólya-Szegö inequality we have

λ1 = |∇φ1|pp ≥ |∇φ⋆
1|pp,

where φ⋆
1 is the Schwartz rearrangement of φ1.

Taking τ ∈ (0, 1), from (A2) we have

J (τφ1) =

∫
RN

A(x, τφ1,∇(τφ1))dx+
1

p

∫
RN

|τφ1|p dx−
∫
RN

G(x, τφ1)dx

≤
∫
B1(0)

(Φ0(τφ1(x))|τφ1(x)|p + ϕ0(τφ1(x))|∇(τφ1(x))|p) dx



28 F. MENNUNI, A. SALVATORE EJDE-2024/42

+
τp

p

∫
B1(0)

|φ1|p dx−
∫
Ω

G(x, τφ1)dx

≤ c1τ
p −

∫
B1(0)

G(x, τφ1)dx,

where c1 = max0≤t≤|φ1|∞ Φ0(t) + λ1max0≤t≤|φ1|∞ ϕ0(t) +
1
p .

Now, from (A14) there exists a constant δ > 0 such that for each s ∈ [0, δ] and
for a.e. x ∈ B1(0) it is G(x, s) > 2c1s

p. Then, for any τ > 0 small sufficient, in
particular 0 < τ < δ

|φ1|∞ , it results

J (τφ1) ≤ c1τ
p − 2c1τ

p < 0.

Finally, let us prove that J has at least two solutions, one negative and one
positive. For this, let us denote by u+ = max{0, u} and u− = max{0,−u}, the
positive and the negative part of u, respectively, so that u = u+ − u−.

If we replace g(x, u) by g+(x, u) := g(x, u+), all the previous statements still
hold true for the functional J+ obtained by replacing G with G+, defined as

G+(x, t) =
∫ t

0
g+(x, s)ds. In particular, J+ has a nontrivial critical point u. Hence,

from (A5) and (A6) we find that

0 = ⟨dJ+(u),−u−⟩ =
∫
RN

a(x,−u−,∇(−u−))∇(−u−) dx

+

∫
RN

At(x,−u−,∇(−u−))(−u−)dx+

∫
RN

|u−|pdx−
∫
RN

g+(x, u)u−dx

≥ α0α1

η0

∫
RN

|∇u−|p dx+

∫
RN

|u−|p dx

≥ α0α1

η0
∥u−∥W .

Hence, u− = 0 a.e. in RN , and u is a positive critical point of J .
Similarly, replacing g(x, u) with g(x,−u−), we find a negative solution of (1.1).

□

Acknowledgments. This research was supported by Horizon Europe Seeds STEPS:
STEerability and controllability of PDEs in Agricolture and Physical Models, CUP:
H91I21001640006. The authors are members of Research Group INdAM-GNAMPA.

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Federica Mennuni (corresponding author)
Dipartimento di Matematica, Università di Bologna, Via Zamboni, 33, 40126 Bologna,

Italy
Email address: federica.mennuni@unibo.it

Addolorata Salvatore

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona
4, 70125 Bari, Italy

Email address: addolorata.salvatore@uniba.it


	1. Introduction
	2. Abstract tools
	3. Variational setting and first properties
	4. Statement of main results
	5. Proof of the main result
	Acknowledgments.

	References