Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 100, pp. 1–24.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER

EQUATIONS INVOLVING THE FRACTIONAL (p, q)-LAPLACIAN

AND CRITICAL NONLINEARITIES

HUILIN LV, SHENZHOU ZHENG, ZHAOSHENG FENG

Abstract. In this article, we consider the existence of ground state positive

solutions for nonlinear Schrödinger equations of the fractional (p, q)-Laplacian

with Rabinowitz potentials defined in Rn,

(−∆)s1p u+ (−∆)s2q u+ V (εx)(|u|p−2u+ |u|q−2u) = λf(u) + σ|u|q
∗
s2

−2
u.

We prove existence by confining different ranges of the parameter λ under the

subcritical or critical nonlinearities caused by σ = 0 or 1, respectively. In
particular, a delicate calculation for the critical growth is provided so as to

avoid the failure of a global Palais-Smale condition for the energy functional.

1. Introduction

Let 0 < s1 < s2 < 1, 1 < p < q < n/s2, σ ∈ {0, 1}, ε > 0 be small number and
q∗s2 = nq

n−s2q be the fractional critical exponent for s2 and q. We are to consider

the existence results for the following Schrödinger equations of a fractional (p, q)-
Laplacian in Rn:

(−∆)s1p u+ (−∆)s2q u+ V (εx)(|u|p−2u+ |u|q−2u) = λf(u) + σ|u|q
∗
s2
−2u, (1.1)

where λ > 0 is a parameter specified later for the parameter σ = 0 or 1, V : Rn → R
is a continuous function satisfying global Rabinowitz condition, and f : R→ R is a
continuous function with subcritical growth. Here, the fractional t-Laplace operator
(−∆)st , for s ∈ {s1, s2} and t ∈ {p, q}, is defined as

(−∆)stu(x) = 2 lim
ε→0

∫
Rn\Bε(x)

|u(x)− u(y)|t−2(u(x)− u(y))

|x− y|n+st
dy.

Problems of type (1.1) are well known as double-phase equations, appearing
in the case of two different materials, where the fractional operator (−∆)st with
s ∈ {s1, s2} and t ∈ {p, q} described the geometry of a composite of two materials.
Recently, a considerable attention has been devoted to the work on nonlocal prob-
lems driven by fractional operators, particularly on fractional p-Laplacian due to
both its interesting theoretical structure and concrete applications such as finance,

2010 Mathematics Subject Classification. 35R11, 35A15, 58E05.
Key words and phrases. Nonlinear Schrödinger equations; nonlocal (p, q)-Laplacian;

critical growth; Rabinowitz potentials; Nehari manifold.
©2021. This work is licensed under a CC BY 4.0 license.

Submitted May 18, 2021. Published December 20, 2021.

1



2 H. LV, S. ZHENG, Z. FENG EJDE-2021/100

obstacle problems, phase transitions, optimization, anomalous diffusion, conserva-
tion laws, image processing and many others. For more details see [23, 22, 12, 31]
and the references therein.

In the special case s1 = s2 = 1 and V ≡ 1, Problem (1.1) reduces to the following
well-known (p, q)-Laplacian equation in Rn,

−∆pu−∆qu+ |u|p−2u+ |u|q−2u = f(x, u), (1.2)

where ∆pu = div(|∇u|p−2∇u). As explained in [16], the study of Equation (1.2)
can be extended to a more general reaction-diffusion system

ut = div(A(u)∇u) + r(x, u) with A(u) = |∇u|p−2 + |∇u|q−2. (1.3)

This has a great number of applications in plasma physics, solid state physics, bio-
physics and chemical reactions. In such applications, u usually corresponds to the
concentration term, div(A(u)∇u) represents the diffusion with diffusion coefficient
A(u), and the reaction term r(x, u) relates to source and loss processes. In chemical
and biological applications, the reaction term r(x, u) has a polynomial form with
respect to the concentration u with variable coefficients, see [16]. The existence
results for several classes of equations of (p, q)-Laplacian type defined in bounded
domains or in the whole of Rn can be found in [4, 19, 30] and the references therein.

It is a well-known fact that as a special model of the differential operator (1.3)
it is the following so-called double-phase one

Lu := div(|∇u|p−2∇u+ a(x)|∇u|q−2∇u),

which is related to the energy functional

u 7→
∫

Ω

(|∇u|p + a(x)|∇u|q)dx (1.4)

with 1 < p < q and a ∈ L∞(Ω) with a(x) ≥ 0 for almost all x ∈ Ω. Roughly
speaking, the integral functional (1.4) is characterized by the fact that the energy
density changes its growth properties and ellipticity according to the point in the
domain. To be more precise, the modulating potential a(x) dictates the geometry of
a composite material made of two different components, with distinct power hard-
ening exponents p and q. Following Marcellini’s terminology in [26], the integrand
H(x, ξ) = |ξ|p + a(x)|ξ|q for all (x, ξ) ∈ Ω×Rn has different growth near the origin
and at infinity (unbalanced growth), that is,

|ξ|p ≤ H(x, ξ) ≤ |ξ|q + 1 for a.e. x ∈ Ω and for all ξ ∈ Rn.

The study of the double-phase functionals of the form (1.4) has generated con-
siderable interests after the initial work by Zhikov in [40] to describe models of
strongly anisotropic materials, see for instance [7, 8, 17, 18] and the references
therein. While s1 = s2 and σ = 0, the problem (1.1) boils down to the following
fractional (p, q)-Laplacian equations

(−∆)spu+ (−∆)squ+ V (εx)(|u|p−2u+ |u|q−2u) = f(u) in Rn, (1.5)

which has been extensively considered by several authors in recent years. For p =
q 6= 2, there has been a source of inspiration around its existence and multiplicity
results in the last decade due to two phenomena: the nonlocal character of the
operator and its nonlinearity; see for instance [33, 14, 24, 38] and the references
therein. Indeed, we would like to stress that standard arguments used to investigate
the linear case p = q = 2 seem to be inapplicable to the nonlinear case on account of



EJDE-2021/100 EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS 3

the lack of Hilbertian structure of W s,p(Rn) for p 6= 2. For the reader’s convenience,
we refer to [15, 21, 35, 39], where existence and multiplicity results were obtained
in the case p = q = 2. However, for p 6= q in the nonlocal framework only few
recent papers deal with problems like (1.5). For example, Filippis and Palatucci
[20] proved a Hölder regularity result for nonlocal double-phase equations. And Lin
and Zheng [25] established the multiplicity and asymptotic behavior of solutions to
fractional (p, q)-Kirchhoff type problems with critical Sobolev-Hardy exponent. We
refer the readers to some recent papers [5, 6, 13, 27] for other interesting double-
phase problems in both local and nonlocal cases.

While s1 = s2, σ = 1 and p = q 6= 2, this becomes the following fractional
nonlocal Schrödinger equation involving a critical Sobolev exponent

(−∆)spu+ V (εx)|u|p−2u = f(u) + |u|p
∗
s−2u in Rn.

We refer to [3] for this class of p-fractional Schrödinger equation, where the authors
presented the existence and multiplicity results of it. For p 6= q, Ambrosio in [2] es-
tablished an existence result for the fractional (p, q)-Laplacian equation with critical
growth. The multiplicity results for fractional (p, q)-Laplacian equations involving
critical nonlinearities in bounded domains has been obtained in [9]. Motivated by
all these works, we are to consider doubly fractional Schrödinger equations of (p, q)-
Laplacian with critical growth. More precisely, we are interested in the existence
result of positive solutions to Problem (1.1).

It is an important observation that the lack of compactness due to the critical
exponent greatly increases the methodological difficulties. The main contribution
of our work is to deal with the possibility of loss of compactness due to the critical
nonlinearity of the double-phase Schrödinger equations in the fractional setting
suitably. We then apply this to obtain sufficient existence conditions for equations
like (1.1) in all Rn, which generalizes the results of Ambrosio’s paper [4] for the
local case.

Before stating main result, let us introduce main assumptions imposed on the
potential V and the nonlinearity f . Throughout this paper, we assume that V :
Rn → R is a continuous function satisfying the following condition by Rabinowitz
as in [34]:

0 < V0 = inf
x∈RN

V (x) < lim inf
|x|→∞

V (x) = V∞ ∈ (0,∞], (1.6)

and we shall consider it in the two cases of V∞ <∞ and V∞ =∞ in the following.
For the nonlinearity f : R→ R, it is assumed that:

(A1) f ∈ C0(R,R) and f(t) = 0 for all t ≤ 0;
(A2) lim|t|→0 |f(t)|/|t|p−1 = 0;
(A3) there exists r ∈ (q, q∗s2) with q∗s2 = nq/(n− s2q), such that

lim
|t|→∞

|f(t)|
|t|r−1

= 0;

(A4) there exists θ ∈ (q, q∗s2) such that 0 < θF (t) = θ
∫ t

0
f(τ)dτ ≤ tf(t) for all

t > 0;
(A5) the map t 7→ f(t)/tq−1 is increasing in (0,∞).

Regarding the existence result, we prove that there exists at least one non-
negative non-trivial solution to Problem (1.1) in the subcritical for all λ > 0 and
for small ε. For the critical case, we prove the existence of at least one non-negative



4 H. LV, S. ZHENG, Z. FENG EJDE-2021/100

non-trivial solution to Problem (1.1) provided small ε and λ large enough. To be
more precise, our first result of the paper is the following.

Theorem 1.1. Let σ = 0. Assume that (1.6) and (A1)–(A5) hold. Then there
exists ε0 > 0 such that, for all ε ∈ (0, ε0), Problem (1.1) with subcritical growth
admits at least one positive ground state solution for any λ > 0.

The proof of this theorem relies heavily on existence of the corresponding au-
tonomous problem motivated from [3]. As usual, the presence of the fractional
(p, q)-Laplacian operator leads to more intriguing analysis. Indeed, the arguments
used in the study of ((1.5) seem not to be trivially adaptable to handling two totally
different components. Therefore, some appropriate technical lemmas (see Lemma
2.4 and Lemma 2.5) and much more delicate estimates will be needed.

Further, we consider the existence result of Problem (1.1) for the critical case
with σ = 1.

Theorem 1.2. Let σ = 1. Assume that (1.6) and (A1)–(A5) hold. Then there
exists ε0 > 0 such that, for all ε ∈ (0, ε0), Problem (1.1) with critical growth admits
at least one positive ground state solution for any λ ≥ λ∗, where λ∗ is a positive
constant.

The idea for proving this theorem is also based on suitable variational tech-
niques. Owing to the combination of two nonhomogeneous fractional involved op-
erators with different scaling properties, it is a rather delicate situation, and more
estimates will be needed to achieve our result. Particularly, we would like to point
out that the calculations performed for the setting σ = 1 to recover compactness
are much more complicated with respect to the case σ = 0 due to the presence
of critical exponent. More precisely, the main difficulty in the critical case is that
the energy functional fails to satisfy the Palais-Smale condition globally. This is
different from the calculations performed as in [29] and the optimal asymptotic
behavior of p-minimizers established as in [10]. Instead, we provide some technical
results which allow us to avoid unnecessary calculations and prove the Palais-Smale
condition for the critical case (see Lemma 4.4 and Lemma 4.5 ). To the best of our
knowledge, these are new contributions to show the existence results of Problem
(1.1) for double-phase nonlocal Schrödinger equations involved in both subcritical
and critical growth.

The remainder of this paper is organized as follows. In Section 2, we give some
related notations, recall basic facts about the involved fractional Sobolev spaces
and provide various useful lemmas. We devoted Section 3 to the proof of existence
result for the subcritical case when σ = 0. In Section 4, we deal with the critical
case when σ = 1, and finally give the proof of Theorem 1.2.

2. Preliminaries

This section is devoted to some well-known facts about the fractional Sobolev
spaces and some technical lemmas we will use later. Throughout this paper,
C(n, ν, L, · · · ) stands for a universal constant depending only on prescribed quanti-
ties and possibly varying from line to line. However, the ones we need to emphasize
will be denoted with special symbols, such as C1, C2, C∗, Cξ.

For p ∈ [1,∞] and A ⊂ Rn, we denote by |u|Lp(A) the Lp(A)-norm of a function
u : Rn → R belonging to Lp(A), and by |u|p its Lp(Rn)-norm. We define Ds,p(Rn)



EJDE-2021/100 EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS 5

as the closure of C∞c (Rn) with respect to the following semi-norm

[u]ps,p =

∫
R2n

|u(x)− u(y)|p

|x− y|n+sp
dx dy

for s ∈ (0, 1) and p ∈ (1,∞). The fractional Sobolev space W s,p(Rn) is defined as
the set of all functions u ∈ Lp(Rn) such that [u]s,p <∞, equipped with the norm

‖u‖ps,p = [u]ps,p + |u|pp.

Furthermore, for u, v ∈W s,p(Rn), we put

〈u, v〉s,p =

∫
R2n

|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|n+sp
(v(x)− v(y)) dx dy.

In the following, we denote by BR(x) the ball of radius R with the center at x.
While x = 0, we briefly write BR = BR(0). First let us begin with the following
Sobolev embedding relation.

Lemma 2.1 ([31]). Let s ∈ (0, 1) and p ∈ [1,∞) such that n > sp. Then there
exists a constant S∗ > 0 such that, for any u ∈ Ds,p(Rn) it holds

|u|pp∗s ≤ S
−1
∗ [u]ps,p.

Moreover, W s,p(Rn) is continuously embedded in Lq(Rn) for any q ∈ [p, p∗s] and
compactly embedded in Lqloc(Rn) for any q ∈ [1, p∗s).

The following compactness result of Lions-type is recalled which will be used in
the main proof later.

Lemma 2.2 ([3]). Let n > sp and r ∈ [p, p∗s). If {un} is a bounded sequence in
W s,p(Rn) with

lim
n→∞

sup
y∈Rn

∫
BR(y)

|un|rdx = 0,

where R > 0, then un → 0 in Lt(Rn) for all t ∈ (p, p∗s).

Lemma 2.3 ([3]). Let u ∈ W s1,p(Rn) ∩ W s2,q(Rn) and φ ∈ C∞c (Rn) such that
0 ≤ φ ≤ 1, φ = 1 in B1 and φ = 0 in Rn\B2. Set φr(x) = φ(xr ). Then

lim
r→∞

‖uφr − u‖1 = 0 and lim
r→∞

‖uφr − u‖2 = 0 .

For any ε > 0, we define the space related to the potential V (εx),

Xε =
{
u ∈W s1,p(Rn) ∩W s2,q(Rn) :

∫
Rn
V (εx)(|u|p + |u|q)dx <∞

}
equipped with the norm

‖u‖ε = ‖u‖1 + ‖u‖2,
where

‖u‖ti = [u]tsi,t +

∫
Rn
V (εx)|u|tdx for all t > 1 and i = {1, 2}.

Thanks to the assumption (1.6) and Lemma 2.1, it is easy to check that it holds
the following result.

Lemma 2.4. The space Xε is continuously embedded into W s1,p(Rn)∩W s2,q(Rn).
Moreover, Xε is continuously embedded in Lt(Rn) for any t ∈

[
p, q∗s2

]
, and com-

pactly embedded in Ltloc

(
Rn
)

for any t ∈
[
1, q∗s2

)
.



6 H. LV, S. ZHENG, Z. FENG EJDE-2021/100

Proof. For any u ∈ Xε, by the assumption (1.6) and the definition of ‖ · ‖i for
i = {1, 2} we have

min{1, V0}‖u‖ps1,p ≤ ‖u‖
p
1,

min{1, V0}‖u‖ps2,q ≤ ‖u‖
p
2.

Therefore, it follows that the embedding Xε ↪→W s1,p(Rn)∩W s2,q(Rn) is continu-
ous. By Lemma 2.1, it is evident that the embedding Xε ↪→ Lt(Rn) is continuous
for any t ∈ [p, q∗s2 ] and the embedding Xε ↪→ Ltloc(Rn) is obviously compact for any
t ∈ [1, q∗s2). �

In addition, by considering V being coercive, we obtain the following compact-
ness lemma.

Lemma 2.5. Let V∞ = ∞. Then Xε is compactly embedded in Lt(Rn) for any
t ∈ [p, q∗s2).

Proof. For t = p, let {un} be a sequence such that un ⇀ 0 in Xε. By Lemma 2.4
we know that Xε ⊂ Lp

(
Rn
)
. Then un ⇀ 0 in W s1,p(Rn) ∩W s2,q(Rn) and un → 0

in Lp
(
BR
)
. Therefore, for any ε > 0 there exists n0 > 0 such that∫

BR

|u|pdx ≤ ε for any n ≥ n0. (2.1)

Since V (x) is coercive, there exists R = Rε > 0 such that

1

V (εx)
< ε for any |x| > R. (2.2)

Let us set
T := sup

n∈N
‖un‖ε <∞. (2.3)

Hence, for any n ≥ n0, by using (2.1), (2.2) and (2.3) we conclude that∫
Rn
|u|pdx =

∫
BR

|u|pdx+

∫
Rn\BR

|u|pdx

≤ ε+ ε

∫
Rn\BR

V (εx)|u|pdx ≤ ε(1 + T p).

Then we have un → 0 in Lp
(
Rn
)
. As for p < t < q∗s2 , by using the interpolation

inequality and Lemma 2.1 we see that

|u|t ≤ C[u]αs2,q|u|
1−α
p ,

where 1
t = α

p + 1−α
q∗s2

, which yields the required result. �

Finally, we recall the following splitting lemma which will be very useful in
our main proof, which is proved by Ambrosio and Repovš (cf. [4]) following the
arguments developed by Brezis and Lieb [11].

Lemma 2.6. Let {un} be a sequence such that un ⇀ u in Xε, and vn = un − u.
Then we have

(i) [vn]ps1,p + [vn]qs2,q =
(
[un]ps1,p + [un]qs2,q

)
−
(
[u]ps1,p + [u]qs2,q

)
+ on(1);

(ii)
∫
Rn V (εx)(|vn|p+|vn|q)dx =

∫
Rn V (εx)

(
(|un|p+|un|q)dx−(|u|p+|u|q)

)
dx+

on(1);
(iii)

∫
Rn(F (vn)− F (un) + F (u))dx = on(1);



EJDE-2021/100 EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS 7

(iv) sup
‖w‖ε≤1

∫
Rn |(f(vn)− f(un) + f(u))w|dx = on(1).

3. Subcritical case while σ = 0

3.1. Functional setting in the subcritical case. In this section we consider the
problem

(−∆)s1p u+ (−∆)s2q u+ V (εx)
(
|u|p−2u+ |u|q−2u

)
= λf(u) in Rn,

u ∈W s1,p(Rn) ∩W s2,q(Rn), u > 0 in Rn.
(3.1)

Let us define the energy functional associated with (3.1),

Iε(u) =
1

p
‖u‖p1 +

1

q
‖u‖q2 − λ

∫
Rn
F (u)dx,

which is well-defined for all u : Rn → R belonging to the fractional space Xε.
Therefore, from the assumptions on f it is easy to check that Iε ∈ C1(Xε,R) and
its differential is given by

〈I ′ε(u), v〉

= 〈u, v〉s1,p + 〈u, v〉s2,q +

∫
Rn
V (εx)(|u|p−2u+ |u|q−2u)v dx− λ

∫
Rn
f(u)v dx

for any u, v ∈ Xε.
Now we check that Iε possesses a mountain pass geometry (cf. [1]).

Lemma 3.1. The functional has a mountain pass geometry shown as follows:

(i) there exist α, ρ > 0 such that Iε(u) ≥ α with ‖u‖ε = ρ;
(ii) there exists e ∈ Xε with ‖e‖ε > ρ such that Iε(e) < 0.

Proof. (i) By growth assumptions (A2) and (A3) on f , we readily see that for any
ξ > 0 there exists a constant Cξ > 0 such that

|f(t)| ≤ ξ|t|p−1 + Cξ|t|r−1 ∀t ∈ R, (3.2)

|F (t)| ≤ ξ

p
|t|p +

Cξ
r
|t|r ∀t ∈ R. (3.3)

Therefore, using V0 ≤ V (εx) and taking ξ ∈ (0, V0

λ ), we have

Iε(u) ≥ 1

p
‖u‖p1 +

1

q
‖u‖q2 − λ

ξ

p
|t|pp − λ

Cξ
r
|t|rr

≥ C1‖u‖p1 +
1

q
‖u‖q2 − λ

Cξ
r
|t|rr.

By choosing ‖u‖ε = ρ ∈ (0, 1) and using 1 < p < q, we have ‖u‖p1 < 1 which
leads to ‖u‖p1 ≥ ‖u‖

q
1. This fact combined with an elementary convex inequality

at + bt ≥ 2−t+1(a+ b)t, ∀a, b ≥ 0 and t > 1, and Lemma 2.4 yield

Iε(u) ≥ C2‖u‖qε − λ
Cξ
r
|t|rr ≥ C2‖u‖qε − C3‖u‖rε.

Therefore, we can find α > 0 such that Iε(u) ≥ α for ‖u‖ε = ρ due to r > q.
(ii) By assumption (A4) we can infer that for some C1, C2 > 0 it holds

F (t) ≥ C1t
θ − C2 for any t > 0.



8 H. LV, S. ZHENG, Z. FENG EJDE-2021/100

By taking u ∈ C∞c (Rn) such that u ≥ 0 and u 6= 0, then we know that

Iε(tu) ≤ tp

p
‖u‖p1 +

tq

q
‖u‖q2 − λtθC1

∫
Rn
uθdx+ C3 → −∞ as t→∞,

where we have used θ > q > p. This completes the proof. �

Therefore, invoking a variant of the mountain pass theorem without Palais-Smale
condition (cf. [37]), we can see that there exists a sequence {un} ⊂ Xε such that

Iε(un)→ cε and I ′ε(un)→ 0,

where

cε = inf
γ∈Γ

max
t∈[0,1]

Iε(γ(t)) with Γ = {γ ∈ C0([0, 1], Xε) : γ(0) = 0, Iε(γ(1)) < 0}.

Further, as in [37] we can use the equivalent characterization of cε, which is more
appropriate to our aim, given by

cε = inf
u∈Xε\{0}

max
t≥0

Iε(γ(t)).

We also introduce the Nehari manifold associated with Iε, which is defined by

Nε = {u ∈ Xε\{0} : 〈I ′ε(u), u〉 = 0}.
Next, we prove that any Palais-Smale sequence of Iε is bounded.

Lemma 3.2. Let {un} be a Palais-Smale sequence of Iε at level c. Then {un} is
bounded in Xε.

Proof. Let {un} ⊂ Xε be a Palais-Smale sequence at the level c, that is

Iε(un) = c+ on(1) and I ′ε(un) = on(1).

Using (A4) and considering θ > q > p we can deduce that

c(1 + ‖un‖ε) ≥ Iε(un)− 1

θ
〈I ′ε(un), un〉

=
(1

p
− 1

θ

)
‖un‖p1 +

(1

q
− 1

θ

)
‖un‖q2 + λ

∫
Rn

(1

θ
f(un)un − F (un)

)
dx

≥
(1

q
− 1

θ

)
(‖un‖p1 + ‖un‖q2).

Next we prove it by contradiction. Assume that ‖un‖ε = ‖u‖1 + ‖u‖2 → ∞, we
distinguish it in the following three cases:

Case 1. If ‖un‖1 → ∞ or ‖un‖2 → ∞, then for n sufficiently large we have

‖un‖q−p2 ≥ 1, that is‖un‖q2 ≥ ‖un‖
p
2. Thus

c(1 + ‖un‖ε) ≥
(1

q
− 1

θ

)
(‖un‖p1 + ‖un‖p2) ≥ C(‖un‖1 + ‖un‖2)p = C‖un‖pε ,

which gives a contradiction because p > 1.

Case 2. If ‖un‖1 →∞ and ‖un‖2 is bounded, then we have

c(1 + ‖un‖1 + ‖un‖2) = c(1 + ‖un‖ε) ≥
(1

q
− 1

θ

)
‖un‖p1,

which implies that

c
( 1

‖un‖p1
+

1

‖un‖p−1
1

+
‖un‖2
‖un‖p1

)
≥
(1

q
− 1

θ

)
.



EJDE-2021/100 EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS 9

Passing to the limit as n → ∞, we obtain 0 < 1
q −

1
θ ≤ 0, which leads to a

contradiction.

Case 3. If ‖un‖2 → ∞ and ‖un‖1 is bounded, we can do it by a similar way as
Case 2. Putting the three cases together completes the proof. �

3.2. Autonomous subcritical problem. We devote this subsection to the fol-
lowing autonomous problem associated with Problem (3.1) for any µ > 0:

(−∆)s1p u+ (−∆)s2q u+ µ
(
|u|p−2u+ |u|q−2u

)
= λf(u) in Rn,

u ∈W s1,p(Rn) ∩W s2,q(Rn), u > 0 in Rn.
(3.4)

The corresponding energy functional is

Iµ(u) =
1

p
‖u‖pµ,1 +

1

q
‖u‖qµ,2 − λ

∫
Rn
F (u)dx,

which is well-defined on the space Yµ = W s1,p(Rn) ∩W s2,q(Rn) equipped with the
norm

‖u‖µ = ‖u‖µ,1 + ‖u‖µ,2,
where

‖u‖tµ,i = [u]tsi,t + µ|u|tt for all t > 1 and i = {1, 2}.

It is easy to check that Iµ ∈ C1(Yµ,R) and its differential is given by

〈I ′µ(u), v〉 = 〈u, v〉s1,p + 〈u, v〉s2,q + µ

∫
Rn

(|u|p−2u+ |u|q−2u)v dx− λ
∫
Rn
f(u)v dx

for any u, v ∈ Yµ. Let us define the Nehari manifold associated with Iµ as follows:

Nµ = {u ∈ Yµ\{0} : 〈I ′µ(u), u〉 = 0}.

It follows from (A4) that

Iµ(u) = Iµ(u)− 1

q
〈I ′µ(u), u〉

=
(1

p
− 1

q

)
‖u‖pµ,1 − λ

∫
Rn

(
F (u)− 1

q
f(u)v

)
dx

≥
(1

p
− 1

q

)
‖u‖pµ,1 for all u ∈ Nµ.

(3.5)

We easily check that Iµ has a mountain pass geometry, and we denote by cµ its
mountain pass level. Moreover, by the standard arguments from [37] and (3.5) we
can show that

0 < cµ = inf
u∈Nµ

Iµ(u) = inf
u∈Yµ\{0}

max
t≥0

Iµ(tu).

With these facts in hand, we arrive to the following lemma.

Lemma 3.3. Let t ∈
[
p, q∗s2

)
, and {un} ⊂ Nµ be a minimizing sequence for Iµ.

Then, {un} is bounded in Yµ; moreover, there exist a sequence {yn} ⊂ Rn and
constants R, β > 0 with

lim inf
n→∞

∫
BR(yn)

|un|tdx ≥ β > 0.



10 H. LV, S. ZHENG, Z. FENG EJDE-2021/100

Proof. Arguing as in the proof of Lemma 3.2, we immediately see that {un} is
bounded in Yµ. To prove the latter conclusion of this Lemma, by contradiction we
assume that for any R > 0 it holds

lim
n→∞

sup
y∈Rn

∫
BR(y)

|un|tdx = 0.

Then, it follows from Lemma 2.2 that

un → 0 in Lt(Rn) for all t ∈
(
p, q∗s2

)
. (3.6)

We fix ξ ∈ (0, µλ ), and take into account {un} ∈ Nµ and (3.2) to deduce that

0 = 〈I ′µ(un), un〉
≥ ‖un‖pµ,1 + ‖un‖qµ,2 − λξ|t|pp − λCξ|t|rr
≥ C1‖un‖pµ,1 + ‖un‖qµ,2 − λCξ|t|rr,

which combined with (3.6) implies that ‖un‖µ → 0 as n → ∞. This leads to a
contradiction because of Iµ(un)→ cµ > 0, which completes the proof. �

Next we prove a useful compactness result for the autonomous problem (3.4).

Theorem 3.4. Under assumptions (A1)–(A5), Problem (3.4) admits a positive
ground state solution.

Proof. We see from a variant of the mountain pass theorem without Palais-Smale
condition (cf. [37]) that there exists a Palais-Smale sequence {un} ⊂ Yµ for Iµ at
the level cµ. Then, using Lemma 3.3 we know that {un} is bounded in Yµ, and we
may assume that

un ⇀ u in Yµ,

un → u in Ltloc(Rn) for all t ∈
[
1, q∗s2

)
.

Now show that the weak limit u is a critical point of Iµ. To this end, we consider
the sequence

hn(x, y) =
|un(x)− un(x)|p−2(un(x)− un(x))

|x− y|
n+s1p

p′
,

and let

h(x, y) =
|u(x)− u(x)|p−2(u(x)− u(x))

|x− y|
n+s1p

p′

with p′ = p
p−1 . We easily check that {hn} is a bounded sequence in the reflexive

Banach space Lp
′(R2n

)
with hn → h a.e. in R2n. Then there exists a subsequence,

still denoted by {hn}, such that hn ⇀ h in Lp
′(R2n

)
, that is to say,∫

R2n

hn(x, y)g(x, y) dx dy →
∫
R2n

h(x, y)g(x, y) dx dy

for all g ∈ Lp
(
R2n

)
. For any v ∈ C∞c (Rn), by taking

g(x, y) =
v(x)− v(x)

|x− y|
n+s1p
p

∈ Lp
(
R2n

)
,

we can see that∫
R2n

|un(x)− un(y)|p−2(un(x)− un(y))

|x− y|n+s1p
(v(x)− v(y)) dx dy



EJDE-2021/100 EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS 11

→
∫
R2n

|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|n+s1p
(v(x)− v(y)) dx dy.

Similarly, we also prove that∫
R2n

|un(x)− un(y)|q−2(un(x)− un(y))

|x− y|n+s2q
(v(x)− v(y)) dx dy

→
∫
R2n

|u(x)− u(y)|q−2(u(x)− u(y))

|x− y|n+s2q
(v(x)− v(y)) dx dy.

Note that ∫
Rn
|un|p−2unv dx→

∫
Rn
|u|p−2uv dx,∫

Rn
|un|q−2unv dx→

∫
Rn
|u|q−2uv dx,∫

Rn
f(un)v dx→

∫
Rn
f(u)v dx

and the fact that 〈I ′µ(un), v〉 = on(1), we can deduce that 〈I ′µ(u), v〉 = 0 for all
v ∈ C∞c (Rn). By the density of C∞c (Rn) in Yµ, we obtain that u is a critical point
of Iµ, which implies that 〈I ′µ(u), u〉 = 0.

Next, we prove that Iµ(u) = cµ, which is divided into two cases:

Case 1. For u 6= 0, it suffices only to show that

‖un‖pµ,1 → ‖u‖
p
µ,1, (3.7)

and then similarly, we can see that ‖un‖qµ,2 → ‖u‖
q
µ,2. Lemma 2.6 leads to that

un → u in Yµ. This together with the fact that Iµ(un) → cµ deduces the desired
result.

To prove (3.7), we observe that Fatou’s lemma yields

‖u‖pµ,1 ≤ lim inf
n→∞

‖un‖pµ,1.

By contradiction, let us assume that

‖u‖pµ,1 < lim sup
n→∞

‖un‖pµ,1. (3.8)

We notice that

cµ + on(1) = Iµ(un)− 1

q
〈I ′µ(un), un〉

=
(1

p
− 1

q

)
‖un‖pµ,1 + λ

∫
Rn

(1

q
f(un)un − F (un)

)
dx.

(3.9)

Recalling that
lim sup
n→∞

(an + bn) ≥ lim sup an+
n→∞

lim inf
n→∞

bn

and p < q, using (3.8), (3.9), Fatou’s lemma again and the fact that 〈I ′µ(u), u〉 = 0,
it yields that

cµ ≥
(1

p
− 1

q

)
lim sup
n→∞

‖un‖pµ,1 + λ lim inf
n→∞

∫
Rn

(1

q
f(un)un − F (un)

)
dx

>
(1

p
− 1

q

)
‖u‖pµ,1 + λ

∫
Rn

(1

q
f(u)u− F (u)

)
dx

= Iµ(u)− 1

q
〈I ′µ(u), u〉 = Iµ(u) ≥ cµ,



12 H. LV, S. ZHENG, Z. FENG EJDE-2021/100

which gives a contradiction.

Case 2. For u = 0, we argue it as in the proof of Lemma 3.3, we can find a
sequence {yn} ⊂ Rn and constants R, β > 0 such that

lim inf
n→∞

∫
BR(yn)

|un|qdx ≥ β > 0.

Set vn(x) = un(x + yn). From the invariance by translations of Rn, it is clear to
show that {vn} ⊂ Yµ is also a bounded Palais–Smale sequence for Iµ at the level
cµ, and vn ⇀ v 6= 0 in Yµ. Hence, we can proceed as before to check that {vn}
converges strongly in Yµ.

Finally, we prove that the ground state obtained above is positive. In fact, thanks
to 〈I ′µ(u), u−〉 = 0, assumption (A1) and the inequality

|x− y|t−2(x− y)(x− − y−) ≥ |x− − y−|t for all t ≥ 1,

for u− = min{u, 0} we have that

‖u−‖pµ,1 + ‖u−‖qµ,2 ≤ 0,

which implies that u− = 0, which meads that u ≥ 0 in Rn. By employing a variant
of the maximum principle (cf. [32]) we conclude that u > 0 in Rn. This completes
the proof. �

3.3. Proof of Theorem 1.1. In this subsection, we concentrate on the existence
of the solution to Problem (3.1) and then give the proof of Theorem 1.1 provided
that ε is sufficiently small. We shall start with the following inevitable lemma.

Lemma 3.5. Let t ∈
[
p, q∗s2

)
, and {un} ⊂ Nε be a sequence such that Iε(un)→ cε

and un ⇀ 0 in Xε. Then, one of the following alternatives occurs:

(a) un → 0 in Xε;
(b) there exist a sequence {yn} ⊂ Rn and constants R, β > 0 such that

lim inf
n→∞

∫
BR(yn)

|un|tdx ≥ β > 0.

Proof. It is natural that (b) fails while (a) happens. Conversely, we assume that
(b) does not hold. Then for any R > 0 it holds

lim
n→∞

sup
y∈Rn

∫
BR(y)

|un|tdx = 0.

By using Lemma 3.2 and Lemma 2.2, it follows that

un → 0 in Lt(Rn) for all t ∈
(
p, q∗s2

)
.

We argue it as in the proof of Lemma 3.3, and we conclude that ‖un‖ε → 0 as
n→∞. Thus we complete the proof. �

Before establishing a compactness result for Iε, it is necessary to prove the fol-
lowing auxiliary lemma.

Lemma 3.6. For V∞ <∞, let {vn} ⊂ Nε be a sequence such that Iε(vn)→ c and
vn ⇀ 0 in Xε. If vn 6→ 0 in Xε, then we have c ≥ c∞, where c∞ is the infimum of
IV∞ over NV∞ .



EJDE-2021/100 EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS 13

Proof. It follows from Lemma 3.2 that {vn} is bounded in Xε. Let {tn} ⊂ (0,∞)
be such that {tnvn} ⊂ NV∞ . Hence, it suffices to prove that

lim sup
n→∞

tn ≤ 1.

In fact, by contradiction we suppose that there exist δ > 0 such that

tn ≥ 1 + δ for all n ∈ N. (3.10)

Note that {vn} ⊂ Xε is a bounded Palais-Smale sequence for Iε, we easily see that
〈I ′ε(vn), vn〉 = 0, which means that

[vn]ps1,p + [vn]qs2,q +

∫
Rn
V (εx)(|vn|p + |vn|q)dx− λ

∫
Rn
f(vn)vndx = 0.

This combined with the fact that tnvn ∈ NV∞ yields

tp−qn [vn]ps1,p+[vn]qs2,q+t
p−q
n V∞

∫
Rn
|vn|pdx+V∞

∫
Rn
|vn|qdx−λ

∫
Rn

f(tnvn)vqn
(tnvn)q−1

dx = 0,

which implies that

λ

∫
Rn

( f(tnvn)

(tnvn)q−1
− f(vn)

vq−1
n

)
vqndx

≤
∫
Rn

(V∞ − V (εx))|vn|pdx+

∫
Rn

(V∞ − V (εx))|vn|qdx
(3.11)

for any required tn and p < q. By using assumption (1.6), for any ζ > 0, there
exists a constant R > 0 such that

V (εx) ≥ V∞ − ζ for all |x| ≥ R. (3.12)

In addition, using the boundedness of {vn} in Xε together with the fact that
vn → 0 in Lp(BR), we can deduce that∫

Rn
(V∞ − V (εx))|vn|pdx

=

∫
BR

(V∞ − V (εx))|vn|pdx+

∫
Rn\BR

(V∞ − V (εx))|vn|pdx

≤ V∞
∫
BR

|vn|pdx+ ζ

∫
Rn\BR

|vn|pdx

≤ on(1) + ζC.

(3.13)

Similarly, we also find that∫
Rn

(V∞ − V (εx))|vn|qdx ≤ on(1) + ζC. (3.14)

Combining (3.11), (3.13) and (3.14) we have∫
Rn

( f(tnvn)

(tnvn)q−1
− f(vn)

vq−1
n

)
vqndx ≤ on(1) + ζC. (3.15)

With the help of Lemma 3.5, we can infer that there exist a sequence {yn} ⊂ Rn,
such that for the constants R, β > 0 it holds

lim inf
n→∞

∫
BR(yn)

|vn|tdx ≥ β > 0 (3.16)



14 H. LV, S. ZHENG, Z. FENG EJDE-2021/100

with t ∈
[
p, q∗s2

)
. By considering v̂n = vn(x + yn), then, up to a subsequence

we can assume that v̂n ⇀ v̂ in Xε. By formula (3.16) there exists Ω ⊂ Rn with
positive measure such that v̂ > 0 in Ω, which combined (3.10) and (3.15) yields the
inequality ∫

Ω

( f((1 + δ)v̂n)

((1 + δ)v̂n)q−1
− f(v̂n)

v̂q−1
n

)
v̂qndx ≤ on(1) + ζC.

Therefore, passing to the limit as n→∞, using Fatou’s lemma and (A5), yields

0 <

∫
Ω

( f((1 + δ)v̂)

((1 + δ)v̂)q−1
− f(v̂)

v̂q−1

)
v̂qdx ≤ ζC for all ζ > 0,

which leads to a contradiction. For the remainder we consider the following two
cases.

Case 1. For lim supn→∞ tn = 1, up to a subsequence, there exists {tn} such that
tn → 1. Considering that Iε(vn)→ c, we have

c+ on(1) = Iε(vn)

= Iε(vn)− IV∞(tnvn) + IV∞(tnvn)

≥ Iε(vn)− IV∞(tnvn) + c∞.

(3.17)

Note that

Iε(vn)− IV∞(tnvn)

=
1− tpn
p

[vn]ps1,p +
1− tqn
q

[vn]qs2,q +
1

p

∫
Rn

(V (εx)− tpnV∞)|vn|pdx

+
1

q

∫
Rn

(V (εx)− tqnV∞)|vn|qdx+ λ

∫
Rn

(F (tnvn)− F (vn))dx.

(3.18)

Taking into account that vn → 0 in Lp(BR) as tn → 1, Inequality (3.12) and
assumption (1.6) imply that

V (εx)− tpnV∞ = (V (εx)− V∞) + (1− tpn)V∞ ≥ −ζ + (1− tpn)V∞ for all |x| ≥ R.

Therefore, we conclude that∫
Rn

(V (εx)− tpnV∞)|vn|pdx

=

∫
BR

(V (εx)− tpnV∞)|vn|pdx+

∫
Rn\BR

(V (εx)− tpnV∞)|vn|pdx

≥ (V0 − tpnV∞)

∫
BR

|vn|pdx− ζ
∫
Rn\BR

|vn|pdx+ (1− tpn)V∞

∫
Rn\BR

|vn|pdx

≥ on(1)− ζC.
(3.19)

Similarly, we can prove that∫
Rn

(V (εx)− tqnV∞)|vn|qdx ≥ on(1)− ζC. (3.20)

Using the boundedness of {vn} in Xε, we obtain

1− tpn
p

[vn]ps1,p = on(1) and
1− tqn
q

[vn]qs2,q = on(1). (3.21)



EJDE-2021/100 EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS 15

Combining (3.18), (3.19), (3.20) and (3.21) we have

Iε(vn)− IV∞(tnvn) ≥ λ
∫
Rn

(F (tnvn)− F (vn))dx+ on(1)− ζC. (3.22)

Then it suffices to show that∫
Rn

(F (tnvn)− F (vn))dx = on(1). (3.23)

Employing Lemma 2.6, (3.3) and the boundedness of {vn} in Xε, we can infer that∫
Rn

(
F (tnvn)− F (vn)

)
dx

=

∫
Rn
F ((tn − 1)vn)dx+ on(1)

≤ ξ

p
|tn − 1|p

∫
Rn
|vn|pdx+

Cξ
r
|tn − 1|r

∫
Rn
|vn|rdx+ on(1) ≤ on(1).

Thus, putting (3.17), (3.22) and (3.23) together we obtain

c+ on(1) ≥ on(1)− ζC + c∞,

and then passing to the limit as ζ → 0 yields c ≥ c∞.

Case 2. For lim supn→∞ tn = t0 < 1, there exists a subsequence, still denoted by
{tn} such that tn → t0 and tn < 1 for any n ∈ N. Considering that I ′ε(vn)→ 0, we
have

c+ on(1) = Iε(vn)− 1

q
〈I ′ε(vn), vn〉

=
(1

p
− 1

q

)
‖vn‖p1 + λ

∫
Rn

(1

q
f(vn)vn − F (vn)

)
dx.

(3.24)

Moreover, recalling the fact that tnvn ∈ NV∞ , Inequality (3.13) and assumption
(A5) yields

c∞ ≤ IV∞(tnvn)

= IV∞(tnvn)− 1

q
〈I ′V∞(tnvn), tnvn〉

=
(1

p
− 1

q

)
‖tnvn‖pV∞,1 + λ

∫
Rn

(1

q
f(tnvn)tnvn − F (tnvn)

)
dx

≤
(1

p
− 1

q

)
(‖vn‖p1 + on(1) + ζC) + λ

∫
Rn

(1

q
f(vn)vn − F (vn)

)
dx

= c+ on(1) + ζC,

where we used (3.24) in the last inequality. Therefore, by passing to the limit as
ζ → 0 and n→∞ it implies that c ≥ c∞, which completes the proof. �

We are now in a position to prove the compactness result as follows.

Lemma 3.7. Let {un} ⊂ Nε be such that Iε(un)→ cε, where cε < c∞ for V∞ <∞,
and any cε ∈ R for V∞ =∞. Then {un} has a convergent subsequence in Xε.

Proof. We argue as before, we immediately see that {un} is bounded in Xε, which
means that we can take a sequence {un} such that

un ⇀ u in Xε,

un → u in Ltloc(Rn) for all t ∈
[
1, q∗s2

)
.



16 H. LV, S. ZHENG, Z. FENG EJDE-2021/100

Similar to the proof of Theorem 3.4, it is easy to check that I ′ε(u) = 0. Let
vn = un − u and Iε(vn) → c. Considering that Iε(un) → cε and Lemma 2.6, we
have

c+ on(1) = Iε(vn)

= Iε(un)− Iε(u) + on(1)

= cε − Iε(u) + on(1).

(3.25)

Arguing it as in the proof of [3, Proposition 3.1], we can see that I ′ε(u) = 0. Note
that

Iε(u) = Iε(u)− 1

q
〈I ′ε(u), u〉 ≥ 0. (3.26)

By using (3.25) and (3.26), and considering V∞ <∞, we can see that

c ≤ cε < c∞,

which together with Lemma 3.6 yields vn → 0 in Xε, which means that un → u in
Xε.

On the other hand, by considering V∞ =∞, and using Lemma 2.5 we can infer
that vn → 0 in Lt(Rn) for any t ∈

[
p, q∗s2

)
. This combined with assumptions (A2)

and (A3) yields ∫
Rn
f(vn)vndx = on(1).

In addition, in accordance with 〈I ′ε(vn), vn〉 = on(1) we deduce that

‖vn‖p1 + ‖vn‖q2 = on(1),

which leads to ‖un − u‖ε = on(1) as n→∞. This completes the proof. �

Proof of Theorem 1.1. According to Lemma 3.1, we know that there exists a se-
quence {un} ⊂ Xε such that Iε(un)→ cε and I ′ε(un)→ 0, where

cε = inf
u∈Xε\{0}

max
t≥0

Iε(tu).

By standard arguments, we obtain that {un} is bounded in Xε, which yields that
there exists a subsequence {un} such that un ⇀ u in Xε, where u is denoted by its
weak limit. Moreover, I ′ε(u) = 0. With the help of Lemma 3.7, it is clear to check
that un → u in Xε while V∞ = ∞. Then applying the mountain pass theorem
yields the existence result. It is rather clear to check that cε ≤ Iε(u). On the other
hand, it follows from Fatou’s lemma that

Iε(u) = Iε(u)− 1

q
〈I ′ε(u), u〉 ≤ lim inf

n→∞

(
Iε(un)− 1

q
〈I ′ε(un), un〉

)
= cε,

which implies that the solution obtained above is a ground state solution, that is,
cε = Iε(u).

To complete the proof, it suffices to show that cε < c∞ for small ε while V∞ <∞.
Without loss of generality, let us suppose that

V (0) = V0 = inf
x∈Rn

V (x)

and µ ∈ (V0, V∞). It is clear that c0 < cµ < c∞, where c0 is the infimum of IV0 over
NV0

. By Theorem 3.4, there exists w ∈W s1,p(Rn)∩W s2,q(Rn) as a positive ground
state of the autonomous problem (3.4). Let φ ∈ C∞c (Rn) be a cut-off function such



EJDE-2021/100 EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS 17

that 0 ≤ φ ≤ 1, φ = 1 in B1, and φ = 0 in Rn\B2. We set φr(x) = φ(xr ) and
consider the function wr(x) = φr(x)w(x). Using Lemma 2.3 we see that

lim
r→∞

‖wr − w‖ε = 0. (3.27)

Take tr > 0 such that

Iµ(trwr) = max
t≥0

Iµ(twr),

which leads to that trwr ∈ Nµ. Now we prove that there exists r large enough such
that Iµ(trwr) < c∞. By contradiction we assume Iµ(trwr) ≥ c∞ for any r > 0.
By using the assumption (A5), (3.27), trwr ∈ Nµ and w ∈ Nµ, we can infer that
tr → 1. Therefore, we conclude that

c∞ ≤ lim inf
r→∞

Iµ(trwr) = Iµ(wr) = cµ < c∞,

which leads to a contradiction. In addition, by assumption (1.6) we obtain that for
some ε0 > 0,

V (εx) ≤ µ for all ε ∈ (0, ε0).

Hence, we deduce that

cε ≤ max
t≥0

Iε(twr) ≤ max
t≥0

Iµ(twr) = Iµ(trwr) < c∞

for all ε ∈ (0, ε0), which completes the proof. �

4. Critical case while σ = 1

4.1. Functional setting in the critical case. In this section we focus on the
critical case while σ = 1

(−∆)s1p u+ (−∆)s2q u+ V (εx)(|u|p−2u+ |u|q−2u) = λf(u) + |u|q
∗
s2
−2u in Rn,

u ∈W s1,p(Rn) ∩W s2,q(Rn), u > 0 in Rn.
(4.1)

It is clear that the energy functional associated with (4.1) is

Jε(u) =
1

p
‖u‖p1 +

1

q
‖u‖q2 − λ

∫
Rn
F (u)dx− 1

q∗s2
|u|

q∗s2
q∗s2

and its differential is given by

〈J ′ε(u), v〉 = 〈u, v〉s1,p + 〈u, v〉s2,q +

∫
Rn
V (εx)(|u|p−2u+ |u|q−2u)v dx

− λ
∫
Rn
f(u)v dx−

∫
Rn
|u|q

∗
s2
−2uv dx

for any u, v ∈ Xε. We also introduce the Nehari manifold associated with Jε,

Mε = {u ∈ Xε\{0} : 〈J ′ε(u), u〉 = 0}.

It is easy to check that Jε possesses a mountain pass geometry shown as follows
(cf. [1]). For simplicity, we here omit the proof because of its similarity to Lemma
3.1.

Lemma 4.1. The functional Jε satisfies the following conditions:

(i) there exists α, ρ > 0 such that Jε(u) ≥ α with ‖u‖ε = ρ;
(ii) there exists e ∈ Xε with ‖e‖ε > ρ such that Jε(e) < 0.



18 H. LV, S. ZHENG, Z. FENG EJDE-2021/100

In view of Lemma 4.1 we define the mountain pass level

dε = inf
ξ∈Λ

max
t∈[0,1]

Jε(ξ(t)) = inf
u∈Xε\{0}

max
t≥0

Jε(ξ(t)),

where

Λ = {ξ ∈ C0([0, 1], Xε) : ξ(0) = 0, Jε(ξ(1)) < 0}.
Next we prove that any Palais-Smale sequence of Jε is bounded.

Lemma 4.2. If {un} is a Palais-Smale sequence of Jε at level c, then {un} is
bounded in Xε.

Proof. Using assumption (A4) and q∗s2 > θ > q > p, we conclude that

c(1 + ‖un‖ε) ≥ Jε(un)− 1

θ
〈J ′ε(un), un〉

=
(1

p
− 1

θ

)
‖un‖p1 +

(1

q
− 1

θ

)
‖un‖q2

+ λ

∫
Rn

(1

θ
f(un)un − F (u)

)
dx+

(1

θ
− 1

q∗s2

)
|u|

q∗s2
q∗s2

≥
(1

q
− 1

θ

)
(‖un‖p1 + ‖un‖q2).

Then we argue it as in the proof of Lemma 3.2, and we easily get the desired
result. �

4.2. Autonomous critical problem. First, let us consider the autonomous prob-
lem associated with Problem (4.1) as follows:

(−∆)s1p u+ (−∆)s2q u+ µ
(
|u|p−2u+ |u|q−2u

)
= λf(u) + |u|q

∗
s2
−2u in Rn,

µ > 0, u ∈W s1,p(Rn) ∩W s2,q(Rn), u > 0 in Rn.
(4.2)

Therefore, the corresponding energy functional is defined as

Jµ(u) =
1

p
‖u‖pµ,1 +

1

q
‖u‖qµ,2 − λ

∫
Rn
F (u)dx− 1

q∗s2
|u|

q∗s2
q∗s2
,

and its differential is given by

〈J ′µ(u), v〉 = 〈u, v〉s1,p + 〈u, v〉s2,q + µ

∫
Rn

(|u|p−2u+ |u|q−2u)v dx

− λ
∫
Rn
f(u)v dx−

∫
Rn
|u|q

∗
s2
−2uv dx

for any u, v ∈ Yµ. Moreover, the Nehari manifold associated with Jµ is

Mµ = {u ∈ Yµ\{0} : 〈J ′µ(u), u〉 = 0}.
Arguing it as before, it is standard to check that Jµ has a mountain pass geometry,
and we denote by dµ its mountain pass level.

Remark 4.3. As in [36] we have the following variational characterization of the
infimum of Jµ over Mµ:

0 < dµ = inf
u∈Mµ

Jµ(u) = inf
ξ∈Λ

max
t∈[0,1]

Jε(ξ(t)) = inf
u∈Yµ\{0}

max
t≥0

Jµ(tu).

To prove the existence of a nontrivial solution to Problem (4.2), we firstly need
to prove the following fundamental result.



EJDE-2021/100 EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS 19

Lemma 4.4. There exists λ∗ > 0 such that dµ ∈
(
0, s2n S

n
s2q

∗
)

for all λ ≥ λ∗.

Proof. Let v ∈ C∞c (Rn) be a non-zero function such that v ≥ 0 in Rn. Then there
exists tλ > 0 such that Jµ(tλv) = maxt≥0 Jµ(tv), which yields 〈J ′µ(tλv), tλv〉 = 0.
Therefore,

tpλ‖v‖
p
µ,1 + tqλ‖v‖

q
µ,2 = λ

∫
Rn
f(tλv)tλv dx+ t

q∗s2
λ |v|

q∗s2
q∗s2
. (4.3)

Using assumption (A4) we find that

tpλ‖v‖
p
µ,1 + tqλ‖v‖

q
µ,2 ≥ t

q∗s2
λ |v|

q∗s2
q∗s2
,

which combined with p < q < q∗s2 yields that tλ is bounded. Then there exists a
subsequence {tλn} such that tλn → t0 ≥ 0 as λn → ∞. Now we prove t0 = 0 by
contradiction. Let us suppose that t0 > 0 such that

tpλn‖v‖
p
µ,1 + tqλn‖v‖

q
µ,2 → T ∈ (0,∞),

λn

∫
Rn
f(tλnv)tλnv dx+ t

q∗s2
λn
|v|

q∗s2
q∗s2
→∞,

which certainly goes against (4.3). Thus, tλn → 0 as λn →∞.
Next, we write h(t) = tv for t ∈ [0, 1]. Then h ∈ Λ, and we obtain

0 < dµ ≤ max
t∈[0,1]

Jµ(h(t)) ≤ max
t≥0

Jµ(tv) = Jµ(tλv) ≤ tpλ‖v‖
p
µ,1 + tqλ‖v‖

q
µ,2.

Taking λ sufficiently large we obtain

tpλ‖v‖
p
µ,1 + tqλ‖v‖

q
µ,2 <

s2

n
S

n
s2q

∗ ,

which completes the proof. �

Lemma 4.5. Let t ∈
[
p, q∗s2

)
, and {un} ⊂ Mµ be a minimizing sequence for Jµ.

Then, {un} is bounded in Yµ, and there exist a sequence {yn} ⊂ Rn and constants
R, β > 0 such that

lim inf
n→∞

∫
BR(yn)

|un|tdx ≥ β > 0.

Proof. It is easy to see that {un} is bounded in Yµ. We assume by contradiction
that for any R > 0 it holds

lim
n→∞

sup
y∈Rn

∫
BR(y)

|un|tdx = 0.

Then, it follows from Lemma 2.2 that

un → 0 in Lt(Rn) for all t ∈
(
p, q∗s2

)
. (4.4)

Employing (3.2), (3.3) and (4.4) we can infer that

0 ≤
∫
Rn
f(un)undx ≤ ξ

∫
Rn
|un|p + on(1),

0 ≤
∫
Rn
F (un)dx ≤ C ξ

∫
Rn
|un|p + on(1).

Passing to the limit as ξ → 0, we obtain∫
Rn
f(un)undx = on(1) and

∫
Rn
F (un)dx = on(1), (4.5)



20 H. LV, S. ZHENG, Z. FENG EJDE-2021/100

which combined with 〈J ′µ(un), un〉 = 0 yields

‖un‖pµ,1 + ‖un‖qµ,2 − |un|
q∗s2
q∗s2

= on(1).

Moreover, from the boundedness of {un} in Yµ we may assume that

‖un‖pµ,1 + ‖un‖qµ,2 → L ≥ 0 and |un|
q∗s2
q∗s2
→ L ≥ 0. (4.6)

If L = 0, then ‖un‖µ → 0 as n→∞, which is a contradiction because of Jµ(un)→
dµ > 0. Therefore, in the following we assume that L > 0. By taking into account
(4.5) and (4.6), we conclude that

dµ = Jµ(un) + on(1)

=
1

p
‖un‖pµ,1 +

1

q
‖un‖qµ,2 − λ

∫
Rn
F (un)dx− 1

q∗s2
|un|

q∗s2
q∗s2

≥ 1

q
L− 1

q∗s2
L+ on(1)

=
s2

n
L+ on(1).

(4.7)

On the other hand, by using Lemma 2.1 it follows that

|un|qq∗s2 ≤ S
−1
∗ ([un]qs2,q + µ|un|qq) = S−1

∗ ‖u‖
q
µ,2 ≤ S−1

∗ (‖un‖pµ,1 + ‖un‖qµ,2).

Passing to the limit as n→∞, we find that

L
q
q∗s2 ≤ S−1

∗ L,

which together with (4.7) yields dµ ≥ s2
n S

n
s2q

∗ . This yields a contradiction in view
of Lemma 4.4. Thus we completes the proof. �

Let us now prove the existence result for the autonomous critical case.

Theorem 4.6. Under assumptions (A1)–(A5), Problem (4.2) admits a positive
ground state solution.

Proof. This proof follows the argument developed as in Theorem 3.4. We here need
to replace (3.9) by

dµ + on(1)

= Jµ(un)− 1

q
〈J ′µ(un), un〉

=
(1

p
− 1

q

)
‖un‖pµ,1 + λ

∫
Rn

(1

q
f(un)un − F (un)

)
dx+

(1

q
− 1

q∗s2

)
|u|

q∗s2
q∗s2
.

Recalling that

lim sup
n→∞

(an + bn + cn) ≥ lim sup
n→∞

an + lim inf
n→∞

(bn + cn)

≥ lim sup
n→∞

an + lim inf
n→∞

bn + lim inf
n→∞

cn,

which implies that

dµ ≥
(1

p
− 1

q

)
lim sup
n→∞

‖un‖pµ,1 + λ lim inf
n→∞

∫
Rn

(1

q
f(un)un − F (un)

)
dx

+
(1

q
− 1

q∗s2

)
lim inf
n→∞

|u|
q∗s2
q∗s2
.



EJDE-2021/100 EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS 21

We complete the proof by using Lemma 4.5 instead of Lemma 3.3. �

4.3. Proof of Theorem 1.2. By a similar argument as Lemma 4.5, the critical
version of Lemma 3.5 is presented here.

Lemma 4.7. Assume that dε <
s2
n S

n
s2q

∗ . Let t ∈
[
p, q∗s2

)
, and {un} ⊂ Mε be a

sequence such that Jε(un) → dε and un ⇀ 0 in Xε. Then, one of the following
alternatives occurs:

(i) un → 0 in Xε;
(ii) there exist a sequence {yn} ⊂ Rn and constants R, β > 0 such that

lim inf
n→∞

∫
BR(yn)

|un|tdx ≥ β > 0.

For the case of V∞ < ∞, we immediately obtain the following result along the
lines of the proof of Lemma 3.6.

Lemma 4.8. Assume that V∞ < ∞, and let {vn} ⊂ Mε be a sequence such that
Jε(vn) → d and vn ⇀ 0 in Xε. If vn 6→ 0 in Xε, then d ≥ d∞, where d∞ is the
infimum of JV∞ over MV∞ .

Next, we can give the following compactness result in the critical case.

Lemma 4.9. Let {un} ⊂ Mε be such that Jε(un) → dε, where dε < d∞ for

V∞ <∞, and dε <
s2
n S

n
s2q

∗ for V∞ =∞. Then {un} has a convergent subsequence
in Xε.

Proof. We first argue it as in the proof of Lemma 3.7 so that we know that {un}
is bounded in Xε, which yields that we may assume that un ⇀ u in Xε. It is clear
that J ′ε(u) = 0. Let vn = un − u and Jε(vn) → d. By Brezis-Lieb Lemma in [11]
and [28, Lemma 3.3] we find that

|vn|
q∗s2
q∗s2

= |un|
q∗s2
q∗s2
− |u|

q∗s2
q∗s2

+ on(1).

We replace (3.25) by

d+ on(1) = Jε(vn)

= Jε(un)− Jε(u) + on(1)

= dε − Jε(u) + on(1)

and notice that

Jε(u) = Jε(u)− 1

q
〈J ′ε(u), u〉

=
1

p
‖u‖p1 + λ

∫
Rn

(1

q
f(u)u− F (u)

)
dx+

(1

q
− 1

q∗s2

)
|u|

q∗s2
q∗s2
≥ 0,

where we used assumption (A4) in the last inequality. Therefore, for the case of
V∞ <∞ we deduce that

d ≤ dε < d∞,

which together with Lemma 4.8 yields vn → 0 in Xε, that is to say, un → u in Xε.
For the case of V∞ =∞, by using Lemma 2.5 we can infer that vn → 0 in Lt(Rn)

for any t ∈
[
p, q∗s2

)
. This combined with assumptions (A2) and (A3) implies that∫

Rn
f(vn)vn dx = on(1),



22 H. LV, S. ZHENG, Z. FENG EJDE-2021/100

which together with 〈J ′ε(vn), vn〉 = 0 yields

‖vn‖p1 + ‖vn‖q2 − |vn|
q∗s2
q∗s2

= on(1).

Considering the boundedness of {vn}, we may assume that

‖vn‖p1 + ‖vn‖q2 → L ≥ 0 and |vn|
q∗s2
q∗s2
→ L ≥ 0.

If L > 0, we follow the lines of the proof of Lemma 4.5 to get d ≥ s2
n S

n
s2q

∗ . Note

that d ≤ dε <
s2
n S

n
s2q

∗ , which yields a contradiction. This leads to that L = 0,
which completes the proof. �

We are finally ready to give the proof of Theorem 1.2.

Proof of Theorem 1.2. The proof follows the same lines as in the proof of Theorem
1.1, once one replace Lemma 3.1, Lemma 3.2, Theorem 3.4 and Lemma 3.7 by
Lemma 4.1, Lemma 4.2, Theorem 4.6 and Lemma 4.9, respectively. �

Acknowledgements. This research was supported by National Natural Science
Foundation of China 12071021.

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Huilin Lv

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China
Email address: huilinlv@bjtu.edu.cn



24 H. LV, S. ZHENG, Z. FENG EJDE-2021/100

Shenzhou Zheng (corresponding author)

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

Email address: shzhzheng@bjtu.edu.cn

Zhaosheng Feng

School of Mathematical and Statistical Sciences, University of Texas Rio Grande Val-
ley, Edinburg, TX 78539, USA

Email address: zhaosheng.feng@utrgv.edu


	1. Introduction
	2. Preliminaries
	3. Subcritical case while =0
	3.1. Functional setting in the subcritical case
	3.2. Autonomous subcritical problem
	3.3. Proof of Theorem ??

	4. Critical case while =1
	4.1. Functional setting in the critical case
	4.2. Autonomous critical problem
	4.3. Proof of Theorem ??
	Acknowledgements

	References