2506/2506.00412.md

Title: Concentrating solutions of the fractional (𝑝,𝑞)-Choquard equation with exponential growth

URL Source: https://arxiv.org/html/2506.00412

Markdown Content: Abstract 1Introduction 2Preliminaries 3The autonomous problem ( 𝒬 𝜗 ) 4The auxiliary problem ( 𝒬 𝜀 ) 5Multiplicity and concentration of positive solutions of problem ( ⁢ 𝒬 ⁢ ) References Concentrating solutions of the fractional ( 𝑝 , 𝑞 ) -Choquard equation with exponential growth Yueqiang Song 1 , Xueqi Sun1, Dušan D. Repovš2,3,4 1 College of Mathematics, Changchun Normal University, Changchun, 130032, P.R. China Yueqiang Song: songyq16@mails.jlu.edu.cn Xueqi Sun: sunxueqi1@126.com 2 Faculty of Education, University of Ljubljana, Ljubljana, 1000, Slovenia 3 Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, 1000, Slovenia 4 Institute of Mathematics, Physics and Mechanics, Ljubljana, 1000, Slovenia Dušan D. Repovš: dusan.repovs@guest.arnes.si Abstract.

This article deals with the following fractional ( 𝑝 , 𝑞 ) -Choquard equation with exponential growth of the form:

𝜀 𝑝 ⁢ 𝑠 ⁢ ( − Δ ) 𝑝 𝑠 ⁢ 𝑢 + 𝜀 𝑞 ⁢ 𝑠 ⁢ ( − Δ ) 𝑞 𝑠 ⁢ 𝑢 + 𝑍 ⁢ ( 𝑥 ) ⁢ ( | 𝑢 | 𝑝 − 2 ⁢ 𝑢 + | 𝑢 | 𝑞 − 2 ⁢ 𝑢 )

𝜀 𝜇 − 𝑁 ⁢ [ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑢 ) ] ⁢ 𝑓 ⁢ ( 𝑢 ) in ℝ 𝑁 ,

where 𝑠 ∈ ( 0 , 1 ) , 𝜀 > 0 is a parameter, 2 ≤ 𝑝

𝑁 𝑠 < 𝑞 , and 0 < 𝜇 < 𝑁 . The nonlinear function 𝑓 has an exponential growth at infinity and the continuous potential function 𝑍 satisfies suitable natural conditions. With the help of the Ljusternik-Schnirelmann category theory and variational methods, the multiplicity and concentration of positive solutions are obtained for 𝜀

0 small enough. In a certain sense, we generalize some previously known results.

∗ Corresponding author: Dušan D. Repovš Mathematics Subject Classification (2020): 35A15, 35A23, 35J35, 35J60, 35R11. Keywords: Fractional double phase operator, Critical exponential growth, Mountain Pass Theorem, Trudinger-Moser inequality, Variational method. 1.Introduction

In this paper, we consider the multiplicity and concentration of solutions for the following fractional ( 𝑝 , 𝑞 ) -Choquard problem in ℝ 𝑁 :

𝜀 𝑝 ⁢ 𝑠 ⁢ ( − Δ ) 𝑝 𝑠 ⁢ 𝑢 + 𝜀 𝑞 ⁢ 𝑠 ⁢ ( − Δ ) 𝑞 𝑠 ⁢ 𝑢 + 𝑍 ⁢ ( 𝑥 ) ⁢ ( | 𝑢 | 𝑝 − 2 ⁢ 𝑢 + | 𝑢 | 𝑞 − 2 ⁢ 𝑢 )

𝜀 𝜇 − 𝑁 ⁢ [ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑢 ) ] ⁢ 𝑓 ⁢ ( 𝑢 ) ,

( 𝒬 )

where 𝜀 is small positive parameter, 0 < 𝜇 < 𝑁 , 0 < 𝑠 < 1 , 2 ≤ 𝑝

𝑁 𝑠 < 𝑞 , the continuous potential 𝑍 is bounded from below by 𝑍 0

0 , the nonlinearity 𝑓 has an exponential critical growth at infinity, and ( − Δ ) ℘ 𝑠

( ℘ ∈ { 𝑝 , 𝑞 } ) is the fractional ℘ -Laplace operator defined by

( − Δ ) ℘ 𝑠 ⁢ 𝑢 ⁢ ( 𝑥 )

2 ⁢ lim 𝑟 → 0 + ∫ ℝ 𝑁 ∖ 𝐵 𝑟 ⁢ ( 𝑥 ) | 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) | ℘ − 2 ⁢ ( 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ⁢ 𝑑 𝑦 ⁢  for every  ⁢ 𝑥 ∈ ℝ 𝑁 ,

up to a normalization constant in the integral, where 𝑢 ∈ 𝐶 0 ∞ ⁢ ( ℝ 𝑁 ) and 𝐵 𝑟 ⁢ ( 𝑥 ) denotes the ball with center 𝑥 of radius 𝑟

0 .

Many scholars have studied fractional and nonlocal operators because of their applications in various contexts, for example, in optimization, finance, crystal dislocations, phase transitions, etc. For more on these topics, we refer to Ambrosio [4] and di Nezza et al. [21].

We shall assume that the potential function 𝑍 and the nonlinearity 𝑓 satisfy the following conditions:

( 𝒵 1 )

There exists 𝑍 0

0 such that 𝑍 ⁢ ( 𝑥 ) ≥ 𝑍 0 , for every 𝑥 ∈ ℝ 𝑁 ;

( 𝒵 2 )

There exists an open bounded set Ω ⊂ ℝ 𝑁 such that

𝑍 0

inf 𝑥 ∈ Ω 𝑍 ⁢ ( 𝑥 ) < min 𝑥 ∈ ∂ Ω 𝑍 ⁢ ( 𝑥 ) .

( 𝑓 1 )

𝑓 is a continuous function such that 𝑓 ⁢ ( 𝑡 )

0 and for every 𝑡 ≤ 0 and every 𝑞 1 , 𝑞 2 , such that

𝑞 1 ≥ 𝑞 , 𝑞 2 ≥ 𝑁 𝑠 ,

there exist real numbers 𝑎 1

0 , 𝑎 2

0 , and 𝛽 0 , with 0 < 𝛽 0 < 𝛼 ∗ ⁢ ( 𝑠 , 𝑁 ) , such that

𝑓 ′ ⁢ ( 𝑡 ) ≤ 𝑎 1 ⁢ | 𝑡 | 𝑞 1 − 2 + 𝑎 2 ⁢ ℋ 𝑁 , 𝑠 ⁢ ( 𝛽 0 ⁢ | 𝑡 | 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢ | 𝑡 | 𝑞 2 − 2 ⁢  for every  ⁢ 𝑡 ≥ 0 ,

where (see Parini and Ruf [32] and Zhang [40])

ℋ 𝑁 , 𝑠 ⁢ ( 𝑡 )

𝑒 𝑡 − ∑ 𝑗

0 𝑗 𝑝 − 2 𝑡 𝑗 𝑗 ! , 𝑗 𝑝

min { 𝑗 ∈ ℕ : 𝑗 ≥ 𝑝 } ,  and 

𝛼 ∗

𝛼 ∗ ⁢ ( 𝑠 , 𝑁 )

𝑁 ⁢ [ 2 ⁢ ( 𝑁 ⁢ 𝜔 𝑁 ) 2 ⁢ Γ ⁢ ( 𝑝 + 1 ) 𝑁 ! ⁢ ∑ 𝑘

0 ∞ ( 𝑁 + 𝑘 − 1 ) ! 𝑘 ! ⁢ 1 ( 𝑁 + 2 ⁢ 𝑘 ) 𝑝 ] 𝑠 / ( 𝑁 − 𝑠 ) ,

𝜔 𝑁

𝜋 𝑁 / 2 Γ ⁢ ( 1 + 𝑁 / 2 ) .

(1.1) ( 𝑓 2 )

lim 𝑡 → 0 + 𝑓 ′ ⁢ ( 𝑡 ) 𝑡 𝑞 − 2

0 .

( 𝑓 3 )

There exists 𝜃 > 𝑞 such that 𝑓 ⁢ ( 𝑡 ) ⁢ 𝑡 ≥ 𝜃 ⁢ 𝐹 ⁢ ( 𝑡 ) > 0 , for every 𝑡 > 0 , where 𝐹 ⁢ ( 𝑡 )

∫ 0 𝑡 𝑓 ⁢ ( 𝜏 ) ⁢ 𝑑 𝜏 .

( 𝑓 4 )

There exists 𝛾 1

0 large enough such that 𝐹 ⁢ ( 𝑡 ) ≥ 𝛾 1 ⁢ | 𝑡 | 𝜃 , for every 𝑡 ≥ 0 , where 𝜃 is as given in ( 𝑓 3 ) .

( 𝑓 5 )

The function 𝑡 ↦ 𝑓 ⁢ ( 𝑡 ) ⁢ 𝑡 1 − 𝑞 is strictly increasing on ℝ +

( 0 , + ∞ ) .

Once 𝑠

1 , problem ( 𝒬 ) reduces to a typical ( 𝑝 , 𝑞 ) -elliptic equation:

− Δ 𝑝 ⁢ 𝑢 − Δ 𝑞 ⁢ 𝑢 + 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ ( | 𝑢 | 𝑝 − 2 ⁢ 𝑢 + | 𝑢 | 𝑞 − 2 ⁢ 𝑢 )

𝐻 ⁢ ( 𝑢 ) ⁢  in  ⁢ ℝ 𝑁 ,

(1.2)

where 𝐻 is nonlinear reaction, Δ ℘ ⁢ 𝑢

div ⁢ ( | ∇ 𝑢 | ℘ − 2 ⁢ ∇ 𝑢 ) , and ℘ ∈ { 𝑝 , 𝑞 } . It has been widely studied in physics, biophysics, plasma physics, chemical reaction design and elsewhere. For more physical examples, we refer to Antontsev and Shmarev [8], Benci et al. [9], and Cherfil and Il’yasov [16], and the references therein. The multiple phases equation was proposed in the study of the Born-Infeld equation (see Bonheure et al. [10], Born and Infeld [11], and Brézis and Lieb [12]), which models electromagnetic fields, electrostatics and electrodynamics, and was a model based on the Maxwell-Lagrangian density

− div ⁢ ( ∇ 𝑢 ( 1 − 2 ⁢ | ∇ 𝑢 | 2 ) 1 / 2 )

ℎ ⁢ ( 𝑢 ) ⁢ in ⁢ ℝ 𝑁 .

When 𝑝

𝑞 , problem ( 𝒬 ) becomes the fractional 𝑝 -Laplace Choquard equation of the form:

𝜀 𝑝 ⁢ 𝑠 ⁢ ( − Δ ) 𝑝 𝑠 ⁢ 𝑢 + 𝑍 ⁢ ( 𝑥 ) ⁢ | 𝑢 | 𝑝 − 2 ⁢ 𝑢

𝜀 𝜇 − 𝑁 ⁢ [ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑢 ) ] ⁢ 𝑓 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) ⁢ in  ⁢ ℝ 𝑁 ,

(1.3)

where 𝜀 > 0 is a sufficiently small parameter, typically the Planck constant and 𝐹 ⁢ ( 𝑡 )

∫ 0 𝑡 𝑓 ⁢ ( 𝜏 ) ⁢ 𝑑 𝜏 . We say that a solution of problem (1.3) is semi-classical if 𝜀 → 0 + . From the physics point of view, the semi-classical solution is also a solutions of problem (1.3), when 𝜀 → 0 + . Floer and Weinstein [22] established the existence of semi-classical solutions of problem (1.3). A special form of problem (1.3) is

− 𝜀 2 ⁢ Δ ⁢ 𝑢 + 𝑍 ⁢ ( 𝑥 ) ⁢ 𝑢

𝜀 𝜇 − 𝑁 ⁢ [ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑢 ) ] ⁢ 𝑓 ⁢ ( 𝑢 ) ⁢ in ⁢ ℝ 𝑁 ⁢  for every  ⁢ 0 < 𝜇 < 𝑁 ,

(1.4)

where 𝑓 is a nonlinear reaction. It is worth noting that problem (1.4) was introduced in the theory of the Bose-Einstein condensation and used to describe the finite-range many body interactions between particles. There are already many works on this topic. By variational methods, Alves et al. [1] considered the concentration solutions of problem (1.4) in ℝ 2 , where 𝑓 has exponential critical growth and 𝑍 satisfies some appropriate conditions. Once 𝐹 ⁢ ( 𝑢 )

| 𝑢 | 𝑝 in problem (1.4), one obtains the following Choquard equation

− Δ ⁢ 𝑢 + 𝑍 ⁢ ( 𝑥 ) ⁢ 𝑢

( 𝐼 𝛼 ∗ | 𝑢 | 𝑝 ) ⁢ | 𝑢 | 𝑝 − 2 ⁢ 𝑢 ⁢ in ⁢ ℝ 𝑁 ,

(1.5)

where 𝐼 𝛼 is the Riesz potential, Γ is the Gamma function, and 𝑍 is a potential function.

When 𝑝

𝛼

2 , 𝑁

3 , and 𝑍 ⁢ ( 𝑥 )

𝜈 , problem (1.5) reduces to the Choquard-Pekar type equation

− Δ ⁢ 𝑢 + 𝜈 ⁢ 𝑢

( 𝐼 2 ∗ 𝑢 2 ) ⁢ 𝑢 ,  for every  ⁢ 𝑥 ∈ ℝ 3 ,

(1.6)

which was proposed in 1976 by Lieb [26], in order to describe an electron trapped in its own hole. Problem (1.6) is called the Schrödinger-Newton equation. Inspired by the work of Lieb [26] and Lions [28], many researchers have studied the Choquard equation by variational methods.

Recently, these methods have become more useful for establishing the existence of weak solutions of the Choquard equations. For example, Chen and Yang [15] studied the following Choquard equation with upper critical exponent on a bounded domain

− Δ ⁢ 𝑢

𝜇 ⁢ 𝑓 ⁢ ( 𝑥 ) ⁢ | 𝑢 | 𝑝 − 2 ⁢ 𝑢 + 𝑔 ⁢ ( 𝑥 ) ⁢ ( 𝐼 𝛼 ∗ ⁢ ( 𝑔 ⁢ | 𝑢 | 2 𝛼 ∗ ) ) ⁢ | 𝑢 | 2 𝛼 ∗ − 2 ⁢ 𝑢 ⁢  for every  ⁢ 𝑥 ∈ Ω ,

where 𝜇

0 is a parameter, 𝑁

4 , 0 < 𝛼 < 𝑁 , 𝐼 𝛼 is the Riesz potential, 𝑁 𝑁 − 2 < 𝑝 < 2 , Ω is a bounded domain with smooth boundary, and 𝑓 and 𝑔 are continuous functions. For 𝜇 small enough, with the help of variational methods, they established the relationship between the number of solutions and the profile of potential 𝑔 .

Yang and Zhao [38] studied the singularly perturbed fractional Choquard equation

𝜀 2 ⁢ 𝑠 ⁢ ( − Δ ) 𝑠 ⁢ 𝑢 + 𝑉 ⁢ ( 𝑥 ) ⁢ 𝑢

𝜀 𝜇 − 3 ⁢ ( ∫ ℝ 3 | 𝑢 ⁢ ( 𝑦 ) | 2 𝜇 , 𝑠 ∗ + 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ) ⁢ ( | 𝑢 | 2 𝜇 , 𝑠 ∗ − 2 ⁢ 𝑢 + 1 2 𝜇 , 𝑠 ∗ ⁢ 𝑓 ⁢ ( 𝑢 ) ) ⁢  in  ⁢ ℝ 𝑁 ,

(1.7)

where 2 𝜇 , 𝑠 ∗

6 − 𝜇 3 − 2 ⁢ 𝑠 is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality and the continuous function 𝑓 satisfies subcritical growth conditions. By variational methods, penalization techniques and the Lyusternik-Schnirelmann theory, the authors established the multiplicity and concentration behaviour of solutions for problem (1.7).

We need to point out some recent results: Zuo et al. [44] developed a variational approach, based on the scaling function method to solve optimization problems. Here, the authors dealt with the mass subcritical case, and referred to the fractional framework setting. Zhang et al. [41] considered a class of fractional parabolic equation with general nonlinearities. The authors established monotone increasing property of the positive solutions in one direction. Based on this, nonexistence of the solutions was demonstrated, via a contradiction argument. For more information, we refer to Chen et al. [13, 14], Cingolani and Tanaka [17], Clemente et al. [18], and Böer and Miyagaki [19], and the references therein.

For fractional ( 𝑝 , 𝑞 ) -Laplace problems, some interesting existence and multiplicity results have emerged in recent years. Zhang et al. [42] studied multiplicity and concentration solution for the double phase equation in ℝ 𝑁 , especially, they assumed the nonlinearity of 𝑓 ∈ 𝐶 1 ⁢ ( ℝ 𝑁 ) and that the continuous potential function satisfies the global condition. Later, using the penalization method, the Ljusternik-Schnirelmann theory, and variational methods, Ambrosio [5] first studied existence of multiple solutions and concentration of the ( 𝑝 , 𝑞 ) -fractional Choquard equation

( − Δ ) 𝑝 𝑠 ⁢ 𝑢 + ( − Δ ) 𝑞 𝑠 ⁢ 𝑢 + 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ ( | 𝑢 | 𝑝 − 2 ⁢ 𝑢 + | 𝑢 | 𝑞 − 2 ⁢ 𝑢 )

( | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑢 ) ) ⁢ 𝑓 ⁢ ( 𝑢 ) in ℝ 𝑁 ,

(1.8)

where 𝜇 ∈ [ 0 , 𝑝 ⁢ 𝑠 ) , 0 < 𝑠 < 1 ,

1 < 𝑝 < 𝑞 < 𝑁 𝑠 ,

𝑓 has subcritical growth and the potential function 𝑍 satisfies the local conditions. Molica Bisci et al.  [29] extended the results of Zhang et al. [42] to the fractional Choquard problem (1.8), where the potential function 𝑍 ⁢ ( 𝑥 ) satisfies the global condition. With the help of the Ljusternik-Schnirelmann category theory and variational methods, Liang et al. [24] explored the multiplicity and concentration behaviors of solutions for the ( 𝑝 , 𝑞 ) fractional Choquard equation with exponential growth.

To the best of our knowledge, there are no known results concerning problem ( 𝒬 ), when the continuous function 𝑓 has the exponential growth behavior at infinity in the sense of Trudinger-Moser. Inspired by the results of Liang et al. [24], we show in this paper the existence and concentration behavior of solutions of problem ( 𝒬 ) involving exponential growth. In comparison to Liang et al. [24], we assume the nonlinearity and function 𝑓 is supposed to be only continuous, which makes the corresponding Nehari manifold possibly nondifferentiable. Therefore, we cannot directly use the differentiability of the Nehari manifold. Furthermore, it is not possible to apply the Ljusternik-Schnirelmann category theory on the Nehari manifold in order to obtain the multiplicity of solutions for problem ( 𝒬 ). Whereas Liang et al. [24] have 𝑓 ∈ 𝐶 1 ⁢ ( ℝ ) , we need to apply some other techniques to overcome this difficulty.

In addition, this is the first time that problem ( 𝒬 ) with the Trudinger-Moser nonlinearities has been studied in both cases: 𝑠 ∈ ( 0 , 1 ) and 𝑠 → 1 − . Comparing with Ambrosio [5], he studied the subcritical growth 𝑝 < 𝑁 𝑠 and the local case. We obtain the Sobolev embedding from 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) into 𝐿 𝑡 ⁢ ( ℝ 𝑁 ) , for every 𝑡 ∈ [ 𝑝 , 𝑝 𝑠 ∗ ] . However, in this paper we consider the case 𝑁

𝑝 ⁢ 𝑠 , so the embedding 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ↪ 𝐿 ∞ ⁢ ( ℝ 𝑁 ) may not exist. To overcome this obstacle, it is essential to apply the fractional Trudinger-Moser inequality, which is the main difference with Ambrosio [5], Molica Bisci et al. [29], and Zhang et al. [42]. The other major challege which we encountered, is the loss of compactness of the Palais-Smale sequences associated with the underlying functionals, corresponding to problems ( 𝒬 ) and ( 𝒬 𝜀 ), so we use certain analytical techniques to overcome this obstacle.

Definition 1.1.

We denote the category of a set 𝐴 with respect to a set 𝐵 by cat 𝐵 ⁢ ( 𝐴 ) as the least integer 𝑘 such that 𝐴 ⊂ 𝐴 1 ∪ ⋯ ∪ 𝐴 𝑘 , where each 𝐴 𝑖 , 𝑖

1 , ⋯ , 𝑘 , is a closed and contractible subset of 𝐵 . We set cat 𝐵 ⁢ ( ∅ )

0 and cat 𝐵 ⁢ ( 𝐴 )

∞ if there is no integer with the above property.

Let

ℳ

{ 𝑥 ∈ ℝ 𝑁 : 𝑍 ⁢ ( 𝑥 )

𝑍 0 }

(1.9)

and for every 𝛿

0 define

ℳ 𝛿

{ 𝑥 ∈ ℝ 𝑁 : dist ⁢ ( 𝑥 , ℳ ) ≤ 𝛿 } .

We are now ready to state the main results of this paper.

Theorem 1.1.

Suppose that conditions ( 𝒵 1 ) , ( 𝒵 2 ) , and ( 𝑓 1 ) − ( 𝑓 5 ) are satisfied. Then for every 𝛿 > 0 , there exists 𝜀 𝛿 > 0 such that problem ( 𝒬 ) has at least cat ℳ 𝛿 ⁢ ( ℳ ) positive (weak) solutions for every 𝜀 > 0 satisfying 𝜀 < 𝜀 𝛿 . Furthermore, let 𝑤 𝜀 be a solution of problem ( 𝒬 ) and 𝜁 𝜀 its global maximum. Then, up to a subsequence, 𝜁 𝜀 → 𝑦 ∈ ℳ and lim 𝜀 → 0 + 𝑍 ⁢ ( 𝜁 𝜀 )

𝑍 0 .

Theorem 1.2.

Suppose that conditions ( 𝒵 1 ) , ( 𝒵 2 ) , and ( 𝑓 1 ) − ( 𝑓 5 ) are satisfied. Let 𝑤 𝜀 be a solution of problem ( 𝒬 ), which exists by Theorem 1.1, and let 𝜁 𝜀 be its global maximum. Then 𝑢 𝜀 ⁢ ( 𝑥 )

𝑤 𝜀 ⁢ ( 𝜀 ⁢ 𝑥 + 𝜁 𝜀 ) converges strongly in 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ∩ 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) to a ground state solution 𝑢 of the following problem

( − Δ ) 𝑝 𝑠 ⁢ 𝑢 + ( − Δ ) 𝑞 𝑠 ⁢ 𝑢 + 𝑍 0 ⁢ ( | 𝑢 | 𝑝 − 2 ⁢ 𝑢 + | 𝑢 | 𝑞 − 2 ⁢ 𝑢 )

[ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑢 ) ] ⁢ 𝑓 ⁢ ( 𝑢 ) ⁢  in  ⁢ ℝ 𝑁 .

Remark 1.1.

As 𝑠 → 1 − 1 , condition ( 𝑓 1 ) reduces to the following condition

( 𝑓 1 ) ′ Continuous function 𝑓 ⁢ ( 𝑡 ) vanishes for every 𝑡 ∈ ( − ∞ , 0 ) and for every 𝑞 1 , 𝑞 2 , with

𝑞 1 ≥ 𝑞 , 𝑞 2

𝑁 ,

there exist constants 𝑎 1

0 , 𝑎 2

0 and 𝛼 0 , with 0 < 𝛽 0 < 𝛼 ∗ , such that

𝑓 ⁢ ( 𝑡 ) ≤ 𝑎 1 ⁢ | 𝑡 | 𝑞 1 − 1 + 𝑎 2 ⁢ ℋ 𝑁 ⁢ ( 𝛽 0 ⁢ | 𝑡 | 𝑁 / ( 𝑁 − 1 ) ) ⁢ | 𝑡 | 𝑞 2 − 1 ⁢  for every  ⁢ 𝑡 ∈ ℝ 0 + ,

where

ℋ 𝑁 ⁢ ( 𝑡 )

𝑒 𝑡 − ∑ 𝑗

0 𝑁 − 2 𝑡 𝑗 𝑗 ! , 0 < 𝛼 ∗ ≤ 𝛼 ∗ ⁢ ( 1 , 𝑁 )

lim 𝑠 → 1 − 𝛼 ∗ ⁢ ( 𝑠 , 𝑁 )

and 𝛼 ∗ ⁢ ( 𝑠 , 𝑁 ) is given in (1.1).

Invoking Theorem 1.1 and Theorem 1.2 for the case when 𝑠 → 1 − 1 , we get the following results for problem ( 𝒬 ), respectively.

Corollary 1.3.

Suppose that conditions ( 𝒵 1 ) , ( 𝒵 2 ) , ( 𝑓 1 ) ′ , and ( 𝑓 2 ) − ( 𝑓 5 ) hold. Then for every 𝛿 > 0 , there exists 𝜀 𝛿 > 0 such that problem ( 𝒬 ) has at least cat ℳ 𝛿 ⁢ ( ℳ ) positive (weak) solutions for every 𝜀 ∈ ( 0 , 𝜀 𝛿 ) . Furthermore, let 𝑤 𝜀 be a solution of problem ( 𝒬 ) and 𝜁 𝜀 its global maximum. Then, up to a subsequence, 𝜁 𝜀 → 𝑦 ∈ ℳ and lim 𝜀 → 0 + 𝑍 ⁢ ( 𝜁 𝜀 )

𝑍 0 .

Corollary 1.4.

Suppose that conditions ( 𝒵 1 ) , ( 𝒵 2 ) , ( 𝑓 1 ) ′ , and ( 𝑓 2 ) − ( 𝑓 5 ) hold. If 𝑤 𝜀 is a solution of problem ( 𝒬 ), which exists by Corollary 1.3, and 𝜁 𝜀 is its global maximum, then 𝑢 𝜀 ⁢ ( 𝑥 )

𝑤 𝜀 ⁢ ( 𝜀 ⁢ 𝑥 + 𝜁 𝜀 ) converges strongly in 𝑊 1 , 𝑁 ⁢ ( ℝ 𝑁 ) ∩ 𝑊 1 , 𝑞 ⁢ ( ℝ 𝑁 ) to a ground state solution 𝑢 of

− Δ 𝑁 ⁢ 𝑢 − Δ 𝑞 ⁢ 𝑢 + 𝑍 0 ⁢ ( | 𝑢 | 𝑁 − 2 ⁢ 𝑢 + | 𝑢 | 𝑞 − 2 ⁢ 𝑢 )

[ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑢 ) ] ⁢ 𝑓 ⁢ ( 𝑢 ) ⁢  in  ⁢ ℝ 𝑁

and there exist 𝑐

0 , 𝐶

0 such that | 𝑤 𝜀 ⁢ ( 𝑥 ) | ≤ 𝐶 ⁢ 𝑒 − 𝑐 ⁢ | 𝑥 − 𝜁 𝜀 | / 𝜀 , for every 𝑥 ∈ ℝ 𝑁 .

The organization of this paper is as follows. In Section 2, we introduce some notations and recall certain technical results which will be needed in the paper. In Section 3, we study the autonomous problem ( 𝒬 𝜗 ) associated to problem ( 𝒬 ). In Section 4, we deal with the auxiliary problem ( 𝒬 𝜀 ). In addition, we also verify that the Palais-Smale condition holds for its energy functional and apply some new tools to obtain a multiplicity result. In Section 5, we establish the multiplicity of solutions for the modified problem and complete the proof of the main results.

2.Preliminaries

In this section, we state some results and notions which will be used later. For all other background material we refer to the comprehensive monograph by Papageorgiou et al. [31].

Let us recall that 𝑝

𝑁 𝑠 in problem ( 𝒬 ). For 1 < 𝑝 < ∞ , we define the fractional Sobolev space 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) as

𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 )

{ 𝑢 ∈ 𝐿 𝑝 ⁢ ( ℝ 𝑁 ) : [ 𝑢 ] 𝑠 , 𝑝

( ∬ ℝ 2 ⁢ 𝑁 | 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) | 𝑝 | 𝑥 − 𝑦 | 2 ⁢ 𝑁 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦 ) 1 / 𝑝 < ∞ }

and we endow it with the norm

‖ 𝑢 ‖ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 )

( ‖ 𝑢 ‖ 𝐿 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + [ 𝑢 ] 𝑠 , 𝑝 𝑝 ) 1 / 𝑝 .

By Pucci et al. [33, Lemma 10], the fractional Sobolev space 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) is a uniformly convex Banach space.

Now, fix 𝜗

0 and endow 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) with the norm

‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 )

( 𝜗 ⁢ ‖ 𝑢 ‖ 𝐿 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + [ 𝑢 ] 𝑠 , 𝑝 𝑝 ) 1 / 𝑝 .

Obviously, the norm ∥ ⋅ ∥ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) is equivalent to ∥ ⋅ ∥ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) on 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) .

Let conditions ( 𝒵 1 ) and ( 𝒵 2 ) be satisfied. We denote by 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) the completion of 𝐶 0 ∞ ⁢ ( ℝ 𝑁 ) , with norm

‖ 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 )

( [ 𝑢 ] 𝑠 , 𝑝 𝑝 + ‖ 𝑢 ‖ 𝑝 , 𝑍 , 𝜀 𝑝 ) 1 / 𝑝 , ‖ 𝑢 ‖ 𝑝 , 𝑍 , 𝜀 𝑝

∫ ℝ 𝑁 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ | 𝑢 ⁢ ( 𝑥 ) | 𝑝 ⁢ 𝑑 𝑥 ⁢  for every  ⁢ 𝜀

0 .

Pucci et al.  [33, Lemma 10] showed that 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) is also a uniformly convex Banach space for 1 < 𝑝 < ∞ . Moreover, 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) is a reflexive Banach space. Invoking conditions ( 𝒵 1 ) − ( 𝒵 2 ) , and Di Nezza et al. [21, Theorem 6.9], we obtain the continuous embedding 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ↪ 𝐿 𝜈 ⁢ ( ℝ 𝑁 ) for arbitrary 𝜈 ∈ [ 𝑁 / 𝑠 , ∞ ) .

Let 𝑠 ∈ ( 0 , 1 ) and 𝑝 , 𝑞 ∈ ( 1 , ∞ ) . The natural solution space of problem ( 𝒬 ) is defined as

𝒲 𝜀

𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ∩ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 )

and is equipped with the norm

‖ 𝑢 ‖ 𝒲 𝜀

‖ 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) + ‖ 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) .

With the aid of above definitions, assumptions ( 𝒵 1 ) − 𝒵 2 ) , and the fact that 𝑝

𝑁 𝑠 , it is easy to get the continuous embeddings

𝒲 𝜀 ↪ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ↪ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ↪ 𝐿 𝜈 ⁢ ( ℝ 𝑁 ) ⁢  for every  ⁢ 𝜈 ∈ [ 𝑁 / 𝑠 , ∞ ) .

Therefore, there exist the best constants

𝑆 𝜈 , 𝜀

inf 𝑢 ∈ 𝒲 𝜀

𝑢 ≠ 0 ‖ 𝑢 ‖ 𝒲 𝜀 ‖ 𝑢 ‖ 𝐿 𝜈 ⁢ ( ℝ 𝑁 ) ⁢  for every  ⁢ 𝜈 ∈ [ 𝑁 / 𝑠 , ∞ ) .

By a change of variable 𝑥 ↦ 𝜀 ⁢ 𝑥 , problem ( 𝒬 ) becomes equivalent to the following equation

( − Δ ) 𝑁 / 𝑠 𝑠 ⁢ 𝑢 + ( − Δ ) 𝑞 𝑠 ⁢ 𝑢 + 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ ( | 𝑢 | 𝑁 𝑠 − 2 ⁢ 𝑢 + | 𝑢 | 𝑞 − 2 ⁢ 𝑢 )

[ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑢 ) ] ⁢ 𝑓 ⁢ ( 𝑢 ) ⁢  in  ⁢ ℝ 𝑁

( 𝒬 𝜀 )

which is variational and the (weak) solutions of problem ( 𝒬 𝜀 ) satisfy the following definition.

Definition 2.1.

Let 𝑢 ∈ 𝒲 𝜀 . If for every 𝜑 ∈ 𝒲 𝜀 , we have

∑ ℘ ∈ { 𝑝 , 𝑞 }
∬ ℝ 2 ⁢ 𝑁 | 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) | ℘ − 2 ⁢ ( 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) ) ⁢ ( 𝜑 ⁢ ( 𝑥 ) − 𝜑 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦

+ ∫ ℝ 𝑁 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ ( | 𝑢 | 𝑝 − 2 ⁢ 𝑢 + | 𝑢 | 𝑞 − 2 ⁢ 𝑢 ) ⁢ 𝜑 ⁢ 𝑑 𝑥

∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑓 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝜑 ⁢ ( 𝑥 ) ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦 ,

then 𝑢 is called a weak solution of problem ( 𝒬 𝜀 ).

3.The autonomous problem ( 𝒬 𝜗 )

Fix 𝜗

0 . In this section, we shall consider the autonomous problem ( 𝒬 𝜗 ) , associated with problem ( 𝒬 ), that is

( − Δ ) 𝑁 / 𝑠 𝑠 ⁢ 𝑢 + ( − Δ ) 𝑞 𝑠 ⁢ 𝑢 + 𝜗 ⁢ ( | 𝑢 | 𝑁 𝑠 − 2 ⁢ 𝑢 + | 𝑢 | 𝑞 − 2 ⁢ 𝑢 )

[ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑢 ) ] ⁢ 𝑓 ⁢ ( 𝑢 ) ⁢ in ⁢ ℝ 𝑁 .

( 𝒬 𝜗 )

We consider the Euler-Lagrange functional ℰ 𝜗 : 𝑊 𝑠 , 𝑁 / 𝑠 ⁢ ( ℝ 𝑁 ) ∩ 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) → ℝ corresponding to problem ( 𝒬 𝜗 ) as follows

ℰ 𝜗 ⁢ ( 𝑢 )

1 𝑝 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 1 𝑞 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − 1 2 ⁢ ∫ ℝ 𝑁 ℒ 𝜇 ⁢ ( 𝑥 ) ⁢ 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝑑 𝑥 ,

(3.1)

where

ℒ 𝜇 ⁢ ( 𝑢 ) ⁢ ( 𝑥 )

∫ ℝ 𝑁 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 .

Here, 𝒲

𝑊 𝑠 , 𝑁 / 𝑠 ⁢ ( ℝ 𝑁 ) ∩ 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) is the Banach space, with the norm

‖ 𝑢 ‖

‖ 𝑢 ‖ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) + ‖ 𝑢 ‖ 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) .

We also endow 𝒲 with the equivalent norm

‖ 𝑢 ‖ 𝜗 , 𝒲

‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) + ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) .

Thus, 𝒲 is a uniformly convex Banach space and so 𝒲 is also a reflexive Banach space. By Di Nezza et al. [21, Theorem 6.9 ], we obtain the continuous embeddings

𝒲 ↪ 𝑊 𝑠 , 𝑁 / 𝑠 ⁢ ( ℝ 𝑁 ) ↪ 𝐿 𝜈 ⁢ ( ℝ 𝑁 ) ⁢  for every  ⁢ 𝜈 ∈ [ 𝑁 / 𝑠 , ∞ ) .

Hence, there exists the best constant 𝐴 𝜈 , 𝜂

0 given by

𝐴 𝜈 , 𝜗

inf 𝑢 ∈ 𝒲

𝑢 ≠ 0 ‖ 𝑢 ‖ 𝜗 , 𝒲 ‖ 𝑢 ‖ 𝐿 𝜈 ⁢ ( ℝ 𝑁 ) ⁢  for every  ⁢ 𝜈 ∈ [ 𝑁 / 𝑠 , ∞ ) .

Lemma 3.1.

(see Zhang [40, Theorem 1.1]) Let 𝑠 ∈ ( 0 , 1 ) and 𝑁

𝑠 ⁢ 𝑝 . Then for every 𝛼 , with 0 < 𝛼 < 𝛼 ∗ ≤ 𝛼 ∗ ⁢ ( 𝑠 , 𝑁 ) ,

sup 𝑣 ∈ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 )

‖ 𝑣 ‖ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ≤ 1 ∫ ℝ 𝑁 ℋ 𝑁 , 𝑠 ⁢ ( 𝛼 ⁢ | 𝑣 | 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢ 𝑑 𝑥 < ∞ ,

where  ⁢ ℋ 𝑁 , 𝑠 ⁢ ( 𝑡 )

𝑒 𝑡 − ∑ 𝑗

0 𝑗 𝑝 − 2 𝑡 𝑗 𝑗 ! , 𝑗 𝑝

min { 𝑗 ∈ ℕ : 𝑗 ≥ 𝑝 } .

Moreover, for 𝛼

𝛼 ∗ ⁢ ( 𝑠 , 𝑁 ) ,

sup 𝑣 ∈ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 )

‖ 𝑣 ‖ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ≤ 1 ∫ ℝ 𝑁 ℋ 𝑁 , 𝑠 ⁢ ( 𝛼 ⁢ | 𝑣 | 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢ 𝑑 𝑥

∞ ,

where

𝛼 ∗ ⁢ ( 𝑠 , 𝑁 )

𝑁 ⁢ ( 2 ⁢ ( 𝑁 ⁢ 𝜔 𝑁 ) 2 ⁢ Γ ⁢ ( 𝑝 + 1 ) 𝑁 ! ⁢ ∑ 𝑘

0 + ∞ ( 𝑁 + 𝑘 − 1 ) ! 𝑘 ! ⁢ 1 ( 𝑁 + 2 ⁢ 𝑘 ) 𝑝 ) 𝑠 / ( 𝑁 − 𝑠 )

𝑁 ⁢ ( 𝛾 𝑠 , 𝑁 ) 𝑠 / ( 𝑁 − 𝑠 ) .

Remark 3.1.

In Lemma 3.1, if we take the norm ∥ . ∥ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) in 𝑊 𝑠 , 𝑁 / 𝑠 ⁢ ( ℝ 𝑁 ) , then

( max ⁡ { 1 , 𝜗 } ) − 1 / 𝑝 ⁢ ‖ 𝑣 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ≤ ‖ 𝑣 ‖ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ≤ ( min { 1 , 𝜗 } ) − 1 / 𝑝 ⁢ ‖ 𝑣 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) , 𝑣 ∈ 𝑊 𝑠 , 𝑁 / 𝑠 ⁢ ( ℝ 𝑁 ) .

Moreover,

sup 𝑣 ∈ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 )

‖ 𝑣 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ≤ ( min { 1 , 𝜗 } ) 𝑠 / 𝑁 ∫ ℝ 𝑁 ℋ 𝑁 , 𝑠 ⁢ ( 𝛼 ⁢ | 𝑣 | 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢ 𝑑 𝑥 < ∞ ⁢  for every  ⁢ 𝛼 , 0 < 𝛼 < 𝛼 ∗ ≤ 𝛼 ∗ ⁢ ( 𝑠 , 𝑁 ) .

Lemma 3.2 (The Hardy-Littlewood-Sobolev inequality, see Lieb [26, 27]).

Let 𝑟 , 𝑡

1 , and 0 < 𝜇 < 𝑁 such that

1 𝑟 + 𝜇 𝑁 + 1 𝑡

2 .

Then there exists a sharp constant 𝐶 ⁢ ( 𝑟 , 𝑁 , 𝜇 , 𝑡 )

0 such that

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝑢 ⁢ ( 𝑥 ) ⁢ 𝑣 ⁢ ( 𝑦 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦 ≤ 𝐶 ⁢ ( 𝑟 , 𝑁 , 𝜇 , 𝑡 ) ⁢ ‖ 𝑢 ‖ 𝐿 𝑟 ⁢ ( ℝ 𝑁 ) ⁢ ‖ 𝑣 ‖ 𝐿 𝑡 ⁢ ( ℝ 𝑁 ) ⁢  for every  ⁢ 𝑢 ∈ 𝐿 𝑟 ⁢ ( ℝ 𝑁 ) , 𝑣 ∈ 𝐿 𝑡 ⁢ ( ℝ 𝑁 ) .

Since 𝑟

𝑡 , we can use Lemma 3.2 and the following equality

2 𝑡 + 𝜇 𝑁

2 ,

that is, when 𝑡

2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 , then the integral

∫ ℝ 𝑁 [ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑢 ) ] ⁢ 𝐹 ⁢ ( 𝑢 ) ⁢ 𝑑 𝑥 for every ⁢ 𝐹 ⁢ ( 𝑢 )

| 𝑢 | 𝑞

is well-defined on 𝐿 𝑡 ⁢ ( ℝ 𝑁 ) , with 𝑡

2 ⁢ 𝑁 / ( 2 ⁢ 𝑁 − 𝜇 ) , along every 𝑢 ∈ 𝑊 𝑠 , 𝑁 / 𝑠 ⁢ ( ℝ 𝑁 ) , provided that 𝑞 ⁢ 𝑡 ≥ 𝑁 𝑠 , due to the continuous embedding 𝑊 𝑠 , 𝑁 / 𝑠 ⁢ ( ℝ 𝑁 ) ↪ 𝐿 𝜈 ⁢ ( ℝ 𝑁 ) , for every 𝜈 ∈ [ 𝑁 / 𝑠 , ∞ ) . Hence,

𝑞 ≥ 𝑁 𝑠 ⁢ 𝑡

2 ⁢ 𝑁 − 𝜇 2 ⁢ 𝑠 .

Consequently, Lemma 3.1, assumption ( 𝑓 1 ) and the fact that 𝐶 0 ∞ ⁢ ( ℝ 𝑁 ) is dense in 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) , imply that  ℰ 𝜗 is well-defined on 𝒲 and of class 𝐶 1 ⁢ ( 𝒲 ) . Furthermore, for every 𝑢 ∈ 𝒲 ,

⟨ ℰ 𝜗 ′ ⁢ ( 𝑢 ) , 𝜑 ⟩

∑ ℘ ∈ { 𝑝 , 𝑞 } ∫ ℝ 2 ⁢ 𝑁 | 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) | ℘ − 2 ⁢ ( 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) ) ⁢ ( 𝜑 ⁢ ( 𝑥 ) − 𝜑 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦

• 𝜗 ⁢ ∫ ℝ 𝑁 ( | 𝑢 | 𝑁 𝑠 − 2 ⁢ 𝑢
• | 𝑢 | 𝑞 − 2 ⁢ 𝑢 ) ⁢ 𝜑 ⁢ 𝑑 𝑥 − ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝜑 ⁢ ( 𝑥 ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦 ⁢  for every  ⁢ 𝜑 ∈ 𝒲 .

Here ⟨ ⋅ , ⋅ ⟩ is the dual pairing between 𝒲 and its dual space 𝒲 ′ . Consequently, the (weak) solutions of problem (3.1) are also the critical points of 𝒥 𝜗 in  𝒲 .

Lemma 3.3.

Suppose that conditions ( 𝒵 1 ) − ( 𝒵 2 ) , ( 𝑓 1 ) , and ( 𝑓 5 ) hold. Then there exist constants 𝑡 0 , 𝜌 0 > 0 such that ℰ 𝜗 ⁢ ( 𝑢 ) ≥ 𝜌 0 , for every 𝑢 ∈ 𝒲 , with ‖ 𝑢 ‖ 𝜗 , 𝒲

𝑡 0 .

Proof.

By condition ( 𝑓 1 ) , with

𝑞 1 ≥ 𝑞

𝑁 𝑠

2 ⁢ 𝑁 − 𝜇 2 ⁢ 𝑠 , 𝑞 2 ≥ 𝑁 𝑠 ,

there exist 𝑎 1

0 , 𝑎 2

0 such that

𝑓 ′ ⁢ ( 𝑡 ) ≤ 𝑎 1 ⁢ | 𝑡 | 𝑞 1 − 2 + 𝑎 2 ⁢ ℋ 𝑁 , 𝑠 ⁢ ( 𝛽 0 ⁢ | 𝑡 | 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢ | 𝑡 | 𝑞 2 − 2 ⁢  for every  ⁢ 𝑡 ∈ ℝ .

This implies that

𝑓 ⁢ ( 𝑡 ) ≤ 𝑎 1 ⁢ | 𝑡 | 𝑞 1 − 1 + 𝑎 2 ⁢ ℋ 𝑁 , 𝑠 ⁢ ( 𝛽 0 ⁢ | 𝑡 | 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢ | 𝑡 | 𝑞 2 − 1 ⁢  for every  ⁢ 𝑡 ∈ ℝ .

Therefore, we get

| 𝐹 ⁢ ( 𝑡 ) | ≤ 𝑎 1 ⁢ | 𝑡 | 𝑞 1 + 𝑎 2 ⁢ | 𝑡 | 𝑞 2 ⁢ ℋ 𝑁 , 𝑠 ⁢ ( 𝛽 0 ⁢ | 𝑡 | 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢  for every  ⁢ 𝑡 ∈ ℝ .

(3.2)

By Lemma 3.2, we obtain that

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦 ≤ 𝐶 ⁢ ( 𝑟 , 𝑁 , 𝜇 ) ⁢ ‖ 𝐹 ⁢ ( 𝑢 ) ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ⁢ ( ℝ 𝑁 ) 2 ⁢  for every  ⁢ 𝑢 ∈ 𝒲 .

(3.3)

Moreover, (3.2) yields that

‖ 𝐹 ⁢ ( 𝑢 ) ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ⁢ ( ℝ 𝑁 ) ≤ 𝑎 1 ⁢ ‖ 𝑢 𝑞 1 ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ⁢ ( ℝ 𝑁 ) + 𝑎 2 ⁢ ‖ | 𝑢 | 𝑞 2 ⁢ ℋ 𝑁 , 𝑠 ⁢ ( 𝛽 0 ⁢ | 𝑢 | 𝑁 / ( 𝑁 − 𝑠 ) ) ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ⁢ ( ℝ 𝑁 ) .

(3.4)

Applying the Hölder inequality with 𝜅 > 1 and 𝜅 ′ > 1 close to 1 , 1 / 𝜅 + 1 / 𝜅 ′

1 , and using the arguments from the proof of Li and Yang [23, Lemma 2.3 ], we can show that for every

𝔩

2 ⁢ 𝑁 ⁢ 𝜅 ′ 2 ⁢ 𝑁 − 𝜇 ,

there exists a constant 𝐶 ⁢ ( 𝔩 )

0 such that

( ℋ 𝑁 , 𝑠 ⁢ ( 𝛽 0 ⁢ | 𝑡 | 𝑁 / ( 𝑁 − 𝑠 ) ) ) 2 ⁢ 𝑁 ⁢ 𝜅 ′ 2 ⁢ 𝑁 − 𝜇 ≤ 𝐶 ⁢ ( 𝔩 ) ⁢ ℋ 𝑁 , 𝑠 ⁢ ( 𝔩 ⁢ 𝛽 0 ⁢ | 𝑡 | 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢  for every  ⁢ 𝑡 ∈ ℝ .

(3.5)

Hence, (3.5) implies that for every 𝑢 ∈ 𝒲 ,

‖ | 𝑢 | 𝑞 2 ⁢ ℋ 𝑁 , 𝑠 ⁢ ( 𝛽 0 ⁢ | 𝑢 | 𝑁 / ( 𝑁 − 𝑠 ) ) ‖ 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ⁢ ( ℝ 𝑁 )

( ∫ ℝ 𝑁 ( | 𝑢 | 𝑞 2 ⁢ ℋ 𝑁 , 𝑠 ⁢ ( 𝛽 0 ⁢ | 𝑢 | 𝑁 / ( 𝑁 − 𝑠 ) ) ) 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ⁢ 𝑑 𝑥 ) 2 ⁢ 𝑁 − 𝜇 2 ⁢ 𝑁

≤ ‖ 𝑢 ‖ 𝐿 2 ⁢ 𝑁 ⁢ 𝜅 ⁢ 𝑞 2 2 ⁢ 𝑁 − 𝜇 𝑞 2 ⁢ ( ∫ ℝ 𝑁 𝐶 ⁢ ( 𝔩 ) ⁢ ℋ 𝑁 , 𝑠 ⁢ ( 𝔩 ⁢ 𝛽 0 ⁢ | 𝑢 | 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢ 𝑑 𝑥 ) 2 ⁢ 𝑁 − 𝜇 2 ⁢ 𝑁 .

(3.6)

We apply Lemma 3.1, taking ‖ 𝑢 ‖ 𝜗 , 𝒲 small enough, and get

𝔩 ⁢ 𝛽 0 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / ( 𝑁 − 𝑠 ) ≤ 𝔩 ⁢ 𝛽 0 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝒲 𝑁 / ( 𝑁 − 𝑠 ) < 𝛼 ∗ ,

(3.7)

hence

∫ ℝ 𝑁 ℋ 𝑁 , 𝑠 ⁢ ( 𝔩 ⁢ 𝛽 0 ⁢ | 𝑢 | 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢ 𝑑 𝑥

∫ ℝ 𝑁 ℋ 𝑁 , 𝑠 ⁢ ( 𝔩 ⁢ 𝛽 0 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / ( 𝑁 − 𝑠 ) ⁢ ( | 𝑢 | ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ) 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢ 𝑑 𝑥 < ∞ .

(3.8)

Together with (3.3)-(3.8), assuming ‖ 𝑢 ‖ 𝜗 , 𝒲 to be small enough, we conclude that there exist appropriate constants 𝔥 1

0 and 𝔥 2

0 such that

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦 ≤ 𝔥 1 ⁢ ‖ 𝑢 ‖ 𝐿 2 ⁢ 𝑁 ⁢ 𝑞 1 2 ⁢ 𝑁 − 𝜇 ⁢ ( ℝ 𝑁 ) 2 ⁢ 𝑞 1 + 𝔥 2 ⁢ ‖ 𝑢 ‖ 𝐿 2 ⁢ 𝑁 ⁢ 𝜅 ⁢ 𝑞 2 2 ⁢ 𝑁 − 𝜇 2 ⁢ 𝑞 2 .

(3.9)

Thus, by (3.1) and the continuity of the embeddings 𝒲 ↪ 𝑊 𝑠 , 𝑁 / 𝑠 ⁢ ( ℝ 𝑁 ) ↪ 𝐿 𝑡 ⁢ ( ℝ 𝑁 ) , for every 𝑡 ≥ 𝑁 / 𝑠 , we obtain that

ℰ 𝜗 ⁢ ( 𝑢 )

≥ 𝑠 𝑁 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / 𝑠 + 1 𝑞 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − 𝔥 1 ⁢ 𝐴 2 ⁢ 𝑁 ⁢ 𝑞 1 2 ⁢ 𝑁 − 𝜇 , 𝜗 − 2 ⁢ 𝑞 1 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝒲 2 ⁢ 𝑞 1 − 𝔥 2 ⁢ 𝐴 2 ⁢ 𝑁 ⁢ 𝜅 ⁢ 𝑞 2 2 ⁢ 𝑁 − 𝜇 , 𝜗 − 2 ⁢ 𝑞 2 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝒲 2 ⁢ 𝑞 2

≥ 2 1 − 𝑞 𝑞 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝒲 𝑞 − 𝔥 1 ⁢ 𝐴 2 ⁢ 𝑁 ⁢ 𝑞 1 2 ⁢ 𝑁 − 𝜇 , 𝜗 − 2 ⁢ 𝑞 1 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝒲 2 ⁢ 𝑞 1 − 𝔥 2 ⁢ 𝐴 2 ⁢ 𝑁 ⁢ 𝜅 ⁢ 𝑞 2 2 ⁢ 𝑁 − 𝜇 , 𝜗 − 2 ⁢ 𝑞 2 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝒲 2 ⁢ 𝑞 2

(3.10)

for ‖ 𝑢 ‖ 𝜗 , 𝒲 small enough. Let

𝒞 ⁢ ( 𝑡 )

2 1 − 𝑞 𝑞 − 𝔞 1 ⁢ 𝐴 2 ⁢ 𝑁 ⁢ 𝑞 1 2 ⁢ 𝑁 − 𝜇 , 𝜗 − 2 ⁢ 𝑞 1 ⁢ 𝑡 2 ⁢ 𝑞 1 − 𝑁 𝑠 − 𝔥 2 ⁢ 𝐴 2 ⁢ 𝑁 ⁢ 𝜅 ⁢ 𝑞 2 2 ⁢ 𝑁 − 𝜇 , 𝜗 − 2 ⁢ 𝑞 2 ⁢ 𝑡 2 ⁢ 𝑞 2 − 𝑁 𝑠 , 𝑡 ≥ 0 .

We claim that there exists 𝑡 0

0 so small that

𝒞 ⁢ ( 𝑡 0 ) ≥ 2 1 − 𝑞 2 ⁢ 𝑞

𝒞 0 .

Clearly, 𝒞 is continuous in ℝ 0 + and lim 𝑡 → 0 + 𝒞 ⁢ ( 𝑡 )

2 ⁢ 𝒞 0 , so there exists 𝑡 0 such that 𝒞 ⁢ ( 𝑡 ) ≥ 𝒞 0 , for every 𝑡 ∈ [ 0 , 𝑡 0 ] . We take 𝑡 0 even smaller, if necessary, so that ‖ 𝑢 ‖ 𝜗 , 𝒲

𝑡 0 satisfies (3.7). This proves the claim. Hence ℰ 𝜗 ⁢ ( 𝑢 ) ≥ 𝒞 0 ⁢ 𝑡 0 𝑞

𝜌 0 , for every 𝑢 ∈ 𝒲 , with ‖ 𝑢 ‖ 𝜗 , 𝒲

𝑡 0 . This completes the proof of Lemma 3.3. ∎

In the sequel, 𝒜 𝜇 : 𝒲 → ℝ will denote the functional

𝒜 𝜇 ⁢ ( 𝑢 )

1 2 ⁢ ∫ ℝ 𝑁 ℒ 𝜇 ⁢ ( 𝑢 ) ⁢ ( 𝑥 ) ⁢ 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝑑 𝑥 ,

(3.11)

where ℒ 𝜇 is given in (3.1).

Lemma 3.4.

Suppose that conditions ( 𝒵 1 ) , ( 𝒵 2 ) , and ( 𝑓 4 ) hold. Then there exists a nonnegative function 𝑣 ∈ 𝐶 0 ∞ ⁢ ( ℝ 𝑁 ) , with ‖ 𝑣 ‖ 𝜗 , 𝒲

𝑡 0 , such that ℰ 𝜗 ⁢ ( 𝑣 ) < 0 , where 𝑡 0

0 is the number given by Lemma 3.3.

Proof.

Fix 𝑢 0 ∈ 𝐶 0 ∞ ⁢ ( ℝ 𝑁 ) ∖ { 0 } , with 𝑢 0 ≥ 0 in ℝ 𝑁 . Set

ℋ 𝜇 ⁢ ( 𝑡 )

𝒜 𝜇 ⁢ ( 𝑡 ⁢ 𝑢 0 / ‖ 𝑢 0 ‖ 𝜗 , 𝒲 ) for every 𝑡

0 ,

where 𝒜 𝜇 is defined in (3.11). Condition ( 𝑓 4 ) gives

ℋ 𝜇 ′ ⁢ ( 𝑡 )

ℋ 𝜇 ′ ⁢ ( 𝑡 ⁢ 𝑢 0 / ‖ 𝑢 ‖ 𝜗 , 𝒲 ) ⁢ 𝑢 0 ‖ 𝑢 0 ‖ 𝜗 , 𝒲

∫ ℝ 𝑁 [ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑡 ⁢ 𝑢 0 / ‖ 𝑢 0 ‖ 𝜗 , 𝒲 ) ] ⁢ 𝑓 ⁢ ( 𝑡 ⁢ 𝑢 0 / ‖ 𝑢 0 ‖ 𝜗 , 𝒲 ) ⁢ 𝑢 0 ‖ 𝑢 0 ‖ 𝜗 , 𝒲 ⁢ 𝑑 𝑥

𝜃 𝑡 ⁢ ℋ 𝜇 ⁢ ( 𝑡 ) .

Integrating the above inequality on [ 1 , 𝑡 ⁢ ‖ 𝑢 0 ‖ 𝜗 , 𝒲 ] , with 𝑡

1 / ‖ 𝑢 0 ‖ 𝜗 , 𝒲 , we get

ℋ 𝜇 ⁢ ( 𝑡 ⁢ ‖ 𝑢 0 ‖ 𝜗 , 𝒲 ) ≥ ℋ 𝜇 ⁢ ( 1 ) ⁢ ( 𝑡 ⁢ ‖ 𝑢 0 ‖ 𝜗 , 𝒲 ) 𝜃

which implies that

𝒜 𝜇 ⁢ ( 𝑡 ⁢ 𝑢 0 ) ≥ 𝒜 𝜇 ⁢ ( 𝑢 0 ‖ 𝑢 0 ‖ 𝜗 , 𝒲 ) ⁢ ‖ 𝑢 0 ‖ 𝜗 , 𝒲 𝜃 ⁢ 𝑡 𝜃 .

Therefore, we have

ℰ 𝜗 ⁢ ( 𝑡 ⁢ 𝑢 0 )

𝑠 ⁢ 𝑡 𝑁 / 𝑠 𝑁 ⁢ ‖ 𝑢 0 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / 𝑠 + 𝑡 𝑞 𝑞 ⁢ ‖ 𝑢 0 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − ∫ ℝ 𝑁 ℒ 𝜇 ⁢ ( 𝑡 ⁢ 𝑢 0 ) ⁢ ( 𝑥 ) ⁢ 𝐹 ⁢ ( 𝑡 ⁢ 𝑢 0 ) ⁢ 𝑑 𝑥

≤ 𝑠 ⁢ 𝑡 𝑁 / 𝑠 𝑁 ⁢ ‖ 𝑢 0 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / 𝑠 + 𝑡 𝑞 𝑞 ⁢ ‖ 𝑢 0 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − ℒ 𝜇 ⁢ ( 𝑢 0 ‖ 𝑢 0 ‖ 𝜗 , 𝒲 ) ⁢ ‖ 𝑢 0 ‖ 𝜗 𝜃 ⁢ 𝑡 𝜃

for every 𝑡 > 1 / ‖ 𝑢 0 ‖ 𝜗 , 𝒲 , choosing 𝑣

𝑡 ⁢ 𝑢 0 and 𝑡 large enough. This completes the proof of Lemma 3.4. ∎

Lemmas 3.3 and 3.4 show that ℰ 𝜗 satisfies the geometric conditions of the Mountain Pass Theorem, therefore there exists a Palais-Smale sequence { 𝔲 𝑛 } 𝑛 ⊂ 𝒲 for ℰ 𝜗 at level 𝑐 𝜗 , briefly ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜗 , that is,

ℰ 𝜗 ⁢ ( 𝔲 𝑛 ) → 𝑐 𝜗 and ℰ 𝜗 ′ ⁢ ( 𝔲 𝑛 ) → 0 ⁢  in  𝒲 ′ ,  as  ⁢ 𝑛 → ∞ ,

𝑐 𝜗

inf 𝜁 ∈ Π max 𝑡 ∈ [ 0 , 1 ] ⁡ ℰ 𝜗 ⁢ ( 𝜁 ⁢ ( 𝑡 ) ) ,

(3.12)

where Π

{ 𝜁 ∈ 𝐶 ⁢ ( [ 0 , 1 ] , 𝒲 ) : 𝜁 ⁢ ( 0 )

0 , ℰ 𝜗 ⁢ ( 𝜁 ⁢ ( 1 ) ) < 0 } . We denote Nehari manifold ℳ 𝜗 related to ℰ 𝜗 by

ℳ 𝜗

{ 𝑢 ∈ 𝒲
{ 0 } : ⟨ ℰ 𝜗 ′ ⁢ ( 𝑢 ) , 𝑢 ⟩

0 } .

Let us define

𝒯 𝜗 + := { 𝑢 ∈ 𝒲 : | supp ⁢ ( 𝑢 + ) |

0 }

(3.13)

and 𝕊 𝑍 0 +

𝕊 𝑍 0 ∩ 𝒯 𝜗 + , where 𝕊 𝜗 is the unit sphere in 𝒲 . We know that 𝒯 𝜗 + is an open subset of 𝒲 .

Invoking the fact above and the definition of 𝕊 𝜗 + , we obtain that 𝕊 𝜗 + is an incomplete 𝐶 1 , 1 -manifold of codimension 1 modelled on 𝒲 and contained in 𝒯 𝜗 + . Therefore, 𝒲

𝑇 𝑢 ⁢ 𝕊 𝑍 0 + ⊕ ℝ ⁢ 𝑢 for every 𝑢 ∈ 𝕊 𝑍 0 + , where

𝑇 𝑢 ⁢ 𝕊 𝜗 +

{ 𝑣 ∈ 𝑊 : ∑ ℘ ∈ { 𝑝 , 𝑞 } ∬ ℝ 2 ⁢ 𝑁 | 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) | ℘ − 2 ⁢ ( 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) ) ⁢ ( 𝜑 ⁢ ( 𝑥 ) − 𝜑 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 𝑑 𝑥 𝑑 𝑦

+ ∫ ℝ 𝑁 𝜗 ( | 𝑢 | 𝑝 − 2 𝑢 + | 𝑢 | 𝑞 − 2 𝑢 ) 𝜑 𝑑 𝑥

0 } .

Since 𝑓 is only continuous, the following result plays an important role in overcoming the nondifferentiability of ℳ 𝜗 and the incompleteness of 𝕊 𝜗 .

Lemma 3.5.

Suppose that conditions ( 𝒵 1 ) − ( 𝒵 2 ) and ( 𝑓 1 ) − ( 𝑓 5 ) hold. Then

( 𝑖 )

For every 𝑢 ∈ 𝒯 𝜗 + and ℬ 𝑢 : [ 0 , ∞ ) → ℝ , defined as ℬ 𝑢 ⁢ ( 𝑡 ) := ℰ 𝜗 ⁢ ( 𝑡 ⁢ 𝑢 ) , there exists a unique 𝑡 𝑢

0 such that ℬ 𝑢 ′ ⁢ ( 𝑡 )

0 in ( 0 , 𝑡 𝑢 ) and ℬ 𝑢 ′ ⁢ ( 𝑡 ) < 0 on ( 𝑡 𝑢 , + ∞ ) .

( 𝑖 ⁢ 𝑖 )

ℳ 𝜗 is bounded away from 0 and ℳ 𝜗 is closed in 𝒲 . There exists 𝜏

0 independent on 𝑢 , such that 𝑡 𝑢 ≥ 𝜏 , for every 𝑢 ∈ 𝕊 𝜗 . Moreover, for each compact set 𝒦 ⊂ 𝕊 𝜗 , there exists 𝐶 𝒦 such that 𝑡 𝑢 ≤ 𝐶 𝒦 , for every 𝑢 ∈ 𝒦 .

( 𝑖 ⁢ 𝑖 ⁢ 𝑖 )

The map 𝑚 ~ 𝜗 : 𝒯 𝜗 + → ℳ 𝜗 , given by 𝑚 ~ 𝜗 ⁢ ( 𝑢 )

𝑡 𝑢 ⁢ 𝑢 , is continuous and 𝑚 𝜗 := 𝑚 ~ 𝜗 | 𝕊 𝜗 + is a homeomorphism between 𝕊 𝜗 + and ℳ 𝜗 , and 𝑚 𝜗 − 1 ⁢ ( 𝑢 )

𝑢 ‖ 𝑢 ‖ 𝜗 , 𝒲 .

Proof.

For every 𝑢 ∈ 𝒯 𝜗 + and 𝑡 ≥ 0 ,

ℬ 𝑢 ⁢ ( 𝑡 )

ℰ 𝜗 ⁢ ( 𝑡 ⁢ 𝑢 )

𝑡 𝑝 𝑝 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 𝑞 𝑞 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − ∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝑡 ⁢ 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝐹 ⁢ ( 𝑡 ⁢ 𝑢 ⁢ ( 𝑥 ) ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦 .

By (3.9), (3) and the continuity of the embeddings 𝒲 ↪ 𝑊 𝑠 , 𝑁 / 𝑠 ⁢ ( ℝ 𝑁 ) ↪ 𝐿 𝑡 ⁢ ( ℝ 𝑁 ) for every 𝑡 ∈ [ 𝑁 𝑠 , + ∞ ) , we get

ℬ 𝑢 ⁢ ( 𝑡 )

ℰ 𝜗 ⁢ ( 𝑡 ⁢ 𝑢 ) ≥ 𝑡 𝑝 𝑝 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 𝑞 𝑞 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − 𝔞 1 ⁢ ‖ 𝑢 ‖ 𝐿 2 ⁢ 𝑁 ⁢ 𝑞 1 2 ⁢ 𝑁 − 𝜇 ⁢ ( ℝ 𝑁 ) 2 ⁢ 𝑞 1 − 𝐷 ⁢ ‖ 𝑢 ‖ 𝐿 2 ⁢ 𝑁 ⁢ 𝜅 ⁢ 𝑞 2 2 ⁢ 𝑁 − 𝜇 2 ⁢ 𝑞 2

≥ 𝑡 𝑝 𝑝 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 𝑞 𝑞 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − 𝑡 2 ⁢ 𝑞 1 ⁢ 𝔞 1 ⁢ 𝐴 2 ⁢ 𝑁 ⁢ 𝑞 1 2 ⁢ 𝑁 − 𝜇 , 𝜗 − 2 ⁢ 𝑞 1 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝒲 2 ⁢ 𝑞 1 − 𝑡 2 ⁢ 𝑞 2 ⁢ 𝐷 ⁢ 𝐴 2 ⁢ 𝑁 ⁢ 𝜅 ⁢ 𝑞 2 2 ⁢ 𝑁 − 𝜇 , 𝜗 − 2 ⁢ 𝑞 2 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝒲 2 ⁢ 𝑞 2

(3.14)

which yields ℬ 𝑢 ⁢ ( 𝑡 ) → 0 + , as 𝑡 → 0 + . Moreover, we obtain that ℬ 𝑢 ⁢ ( 𝑡 ) → − ∞ , as 𝑡 → ∞ . Therefore, there exists 𝑡 𝑢 ∈ ( 0 , ∞ ) such that ℬ 𝑢 ⁢ ( 𝑡 𝑢 )

max 𝑡 ≥ 0 ⁡ ℬ 𝑢 ⁢ ( 𝑡 ) . Furthermore, ℬ 𝑢 ′ ⁢ ( 𝑡 𝑢 )

0 . Now we shall verify that 𝑡 𝑢 is a unique critical point of ℬ 𝑢 in ( 0 , ∞ ) . Arguing by contradiction, suppose that there exist 0 < 𝑡 1 < 𝑡 2 < ∞ such that ℬ 𝑢 ′ ⁢ ( 𝑡 1 )

ℬ 𝑢 ′ ⁢ ( 𝑡 2 )

0 . Consequently, we have

𝑡 1 𝑝 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 1 𝑞 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝑡 1 ⁢ 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝑡 1 ⁢ 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝑡 1 ⁢ 𝑢 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦

and

𝑡 2 𝑝 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 2 𝑞 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝑡 2 ⁢ 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝑡 2 ⁢ 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝑡 2 ⁢ 𝑢 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦 .

By two equalities above and ( 𝑓 4 ) , we obtain

0
< ( 1 𝑡 1 𝑞 − 𝑝 − 1 𝑡 2 𝑞 − 𝑝 ) ∥ ⁢ 𝑢 ∥ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝

∫ ℝ 𝑁 ∫ ℝ 𝑁 1 | 𝑥 − 𝑦 | 𝜇 ⁢ | 𝐹 ⁢ ( 𝑡 1 ⁢ 𝑢 ⁢ ( 𝑦 ) ) ( 𝑡 1 ⁢ 𝑢 ⁢ ( 𝑦 ) ) 𝑞 2 ⁢ 𝑓 ⁢ ( 𝑡 1 ⁢ 𝑢 ⁢ ( 𝑥 ) ) ( 𝑡 1 ⁢ 𝑢 ⁢ ( 𝑥 ) ) 𝑞 2 − 1 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) 𝑞 2 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) 𝑞 2 | ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

− ∫ ℝ 𝑁 ∫ ℝ 𝑁 1 | 𝑥 − 𝑦 | 𝜇 ⁢ | 𝐹 ⁢ ( 𝑡 2 ⁢ 𝑢 ⁢ ( 𝑦 ) ) ( 𝑡 2 ⁢ 𝑢 ⁢ ( 𝑦 ) ) 𝑞 2 ⁢ 𝑓 ⁢ ( 𝑡 2 ⁢ 𝑢 ⁢ ( 𝑥 ) ) ( 𝑡 2 ⁢ 𝑢 ⁢ ( 𝑥 ) ) 𝑞 2 − 1 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) 𝑞 2 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) 𝑞 2 | ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

< 0

which is impossible. Therefore, we have completed the proof of ( 𝑖 ) .

( 𝑖 ⁢ 𝑖 ) Let 𝑢 ∈ ℳ 𝜗 . We shall prove that the first part of conclusion is true in the following two cases.

Case 1. ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / ( 𝑁 − 𝑠 )

𝛼 ∗ 𝔩 ⁢ 𝛽 0 ⁢ 𝜎 𝑠 / ( 𝑁 − 𝑠 ) .

In this case we are done.

Case 2. ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / ( 𝑁 − 𝑠 ) < 𝛼 ∗ 𝔩 ⁢ 𝛽 0 ⁢ 𝜎 𝑠 / ( 𝑁 − 𝑠 ) .

Applying the Trudinger-Moser inequality, we obtain

sup 𝑢 ∈ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ∫ ℝ 𝑁 Φ 𝑁 , 𝑠 ⁢ ( 𝛼 0 ⁢ | 𝑢 | 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢ 𝑑 𝑥 < + ∞ ⁢  for every  ⁢ 0 ≤ 𝛼 < 𝛼 ∗ .

(3.15)

Using the Hardy-Littlewood-Sobolev inequality again and ( 𝑓 3 ) , it follows that

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝑢 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 ≤ 𝐶 ⁢ ‖ 𝐹 ⁢ ( 𝑢 ) ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ⁢ ‖ 𝑓 ⁢ ( 𝑢 ) ⁢ 𝑢 ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ≤ 𝐶 ⁢ ‖ 𝑓 ⁢ ( 𝑢 ) ⁢ 𝑢 ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 2 .

By ( 𝑓 1 ) and ( 𝑓 2 ) , for every 𝜀 ∗

0 and 𝑞 1 ≥ 𝑞

𝑁 𝑠

2 ⁢ 𝑁 − 𝜇 2 ⁢ 𝑠 and 𝑞 2 ≥ 𝑁 𝑠 , there exists 𝐶 𝑞 , 𝜀 ∗

0 such that

‖ 𝑓 ⁢ ( 𝑢 ) ⁢ 𝑢 ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ≤ 𝜀 ∗ ⁢ ‖ | 𝑢 | 𝑞 1 ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 + 𝐶 𝑞 , 𝜀 ∗ ⁢ ‖ | 𝑢 | 𝑞 2 ⁢ ℋ 𝑁 , 𝑠 ⁢ ( 𝛼 0 ⁢ | 𝑡 | 𝑁 / ( 𝑁 − 𝑠 ) ) ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇

(3.16)

for every 𝑡 ≥ 0 . Using inequality (3.16) and the definition of 𝐴 𝜈 , 𝜗 , there exists a constant 𝐶 ⁢ ( 𝜀 ∗ ) such that

‖ 𝑓 ⁢ ( 𝑢 ) ⁢ 𝑢 ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 2 ≤ 𝜀 ∗ ⁢ 𝐴 2 ⁢ 𝑁 ⁢ 𝑞 1 2 ⁢ 𝑁 − 𝜇 , 𝜗 − 2 ⁢ 𝑞 1 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝒲 2 ⁢ 𝑞 1 + 𝐶 ⁢ ( 𝑞 , 𝜀 ∗ ) ⁢ ‖ 𝑢 ‖ 𝜗 , 𝒲 2 ⁢ 𝑞 2 ⁢  for some  ⁢ 𝑞 1 ≥ 𝑞 , 𝑞 2 ≥ 𝑁 𝑠 .

(3.17)

In view of ⟨ ℰ 𝜗 ′ ⁢ ( 𝑢 ) , 𝑢 ⟩

0 , we get

‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝑢 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

≤ 𝜀 ∗ ⁢ 𝐴 2 ⁢ 𝑁 ⁢ 𝑞 1 2 ⁢ 𝑁 − 𝜇 , 𝜗 − 2 ⁢ 𝑞 1 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝒲 2 ⁢ 𝑞 1 + 𝐶 ⁢ ( 𝑞 , 𝜀 ∗ ) ⁢ ‖ 𝑢 ‖ 𝜗 , 𝒲 2 ⁢ 𝑞 2 .

Therefore

‖ 𝑢 ‖ 𝜗 , 𝒲 ≥ 𝛼 for every  ⁢ 𝑢 ∈ ℳ 𝜗 .

(3.18)

For any sequence { 𝔲 𝑛 } 𝑛 ⊂ ℳ 𝜗 , such that 𝔲 𝑛 → 𝑢 in 𝒲 , we have to prove that 𝑢 ∈ ℳ 𝜗 . Indeed, by the fact that 𝔲 𝑛 → 𝑢 in 𝒲 , we obtain ‖ 𝔲 𝑛 − 𝑢 ‖ 𝜗 , 𝒲 → 0 , as 𝑛 → ∞ . Using the discussion as in Willem [37, Lemma A.1], there exists a subsequence { 𝔲 𝑛 } 𝑛 of { 𝑢 𝑛 } 𝑛 satisfying ‖ 𝔲 𝑖 + 1 − 𝔲 𝑖 ‖ 𝜗 , 𝒲 ≤ 2 − 𝑖 for every 𝑖 ≥ 1 . We denote 𝒰 ⁢ ( 𝑥 ) := | 𝔲 1 ⁢ ( 𝑥 ) | + ∑ 𝑖

1 ∞ | 𝔲 𝑖 + 1 ⁢ ( 𝑥 ) − 𝔲 𝑖 ⁢ ( 𝑥 ) | . Together with the fact that 𝔲 𝑛 → 𝑢 in 𝒲 , we obtain | 𝔲 𝑛 ⁢ ( 𝑥 ) | ≤ 𝒰 ⁢ ( 𝑥 ) for every 𝑥 ∈ ℝ 𝑁 and | 𝑢 ⁢ ( 𝑥 ) | ≤ 𝒰 ⁢ ( 𝑥 ) in ℝ 𝑁 . Clearly, ‖ 𝒰 ‖ 𝜗 , 𝒲 ≤ ‖ 𝔲 1 ‖ 𝜗 , 𝒲 + ∑ 𝑖

1 ∞ 2 − 𝑖 < + ∞ , hence 𝒰 ∈ 𝒲 .

Since { 𝔲 𝑛 } 𝑛 is a subsequence of { 𝔲 𝑛 } 𝑛 , we have

‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

lim 𝑛 → ∞ ∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝔲 𝑛 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 .

(3.19)

We shall now prove

lim 𝑛 → ∞ ∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝔲 𝑛 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝔲 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝔲 ⁢ ( 𝑥 ) ) ⁢ 𝔲 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 .

(3.20)

We have

| | 𝐹 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝔲 𝑛 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 − | 𝐹 ⁢ ( 𝔲 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝔲 ⁢ ( 𝑥 ) ) ⁢ 𝔲 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 | ≤ 2 ⁢ | 𝐹 ⁢ ( 𝒰 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝒰 ⁢ ( 𝑥 ) ) ⁢ 𝔲 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 .

Now we shall show that

| 𝐹 ⁢ ( 𝒰 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝒰 ⁢ ( 𝑥 ) ) ⁢ 𝒰 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ∈ 𝐿 1 ⁢ ( ℝ 𝑁 ) .

(3.21)

By Zhang et al. [39, Lemma 2.4], we have

∫ ℝ 𝑁 ℋ 𝑁 , 𝑠 ⁢ ( 𝛼 0 ⁢ | 𝒰 | 𝑁 / ( 𝑁 − 𝑠 ) ) ⁢ 𝑑 𝑥 < + ∞ .

(3.22)

Together with (3.16)-(3.17), we deduce (3.21). Therefore,

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝔲 𝑛 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 − ∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝑢 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 ∈ 𝐿 1 ⁢ ( ℝ 𝑁 ) .

Furthermore,

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝔲 𝑛 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 − ∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝑢 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 → 0 ,

pointwisely on ℝ 𝑁 outside a set of measure zero. By the Dominated Convergence Theorem, we obtain that (3.20) is true. By 𝔲 𝑛 → 𝑢 in 𝒲 ⁢ ( ℝ 𝑁 ) , we have

‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝑢 ⁢ ( 𝑥 ) | | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

which yields 𝑢 ∈ ℳ 𝜗 .

In the sequel, we shall verify that the second part of the conclusion is also true.

Applying ( 𝑖 ) , there exists 𝑡 𝑢

0 such that 𝑡 𝑢 ⁢ 𝑢 ∈ ℳ 𝜗 for every 𝑢 ∈ 𝕊 𝜗 . Therefore, it follows from (3.18) that 𝑡 𝑢 ≥ 𝛼 . We shall argue by contradiction that 𝔲 𝑛 ∈ 𝒦 satisfies 𝑡 𝑛 := 𝑡 𝔲 𝑛 → ∞ . Due to the compactness of 𝒦 , we may suppose that 𝔲 𝑛 → 𝑢 in 𝒲 . Then 𝑢 ∈ 𝒦 ⊂ 𝕊 𝜗 . By ( 𝑓 4 ) , we obtain

ℰ 𝜗 ⁢ ( 𝑡 𝑛 ⁢ 𝔲 𝑛 )

1 𝑝 ⁢ 𝑡 𝑛 𝑝 ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 1 𝑞 ⁢ 𝑡 𝑛 𝑝 ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 − 1 2 ⁢ ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝑡 𝑛 ⁢ 𝔲 𝑛 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝐹 ⁢ ( 𝑡 𝑛 ⁢ 𝔲 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦

≥ 1 𝑝 ⁢ 𝑡 𝑛 𝑝 ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 1 𝑞 ⁢ 𝑡 𝑛 𝑝 ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − 𝛾 1 2 ⁢ 𝑡 𝑛 2 ⁢ 𝜃 ⁢ ‖ | 𝔲 𝑛 | 𝜃 ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ⁢ ( ℝ 𝑁 ) 2 → − ∞

due to 𝜃

𝑞 . However, since 𝑡 𝑛 ⁢ 𝔲 𝑛 ∈ ℳ 𝜗 , we have

ℰ 𝜗 | ℳ 𝜗 ⁢ ( 𝑡 𝑛 ⁢ 𝔲 𝑛 )

1 𝑝 ⁢ 𝑡 𝑛 𝑝 ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 1 𝑞 ⁢ 𝑡 𝑛 𝑝 ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − 1 2 ⁢ ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ 𝐹 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

≤ ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝑡 𝑛 ⁢ 𝔲 𝑛 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ [ 1 𝑝 ⁢ 𝑓 ⁢ ( 𝑡 𝑛 ⁢ 𝔲 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝑡 𝑛 ⁢ 𝔲 𝑛 ⁢ ( 𝑥 ) − 1 2 ⁢ 𝐹 ⁢ ( 𝑡 𝑛 ⁢ 𝔲 𝑛 ⁢ ( 𝑥 ) ) ] ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 ≥ 0

which is impossible.

By ( 𝑖 ) − ( 𝑖 ⁢ 𝑖 ) and the arguments from the proof of Szulkin and Weth [34, Proposition 3.1], we obtain ( 𝑖 ⁢ 𝑖 ⁢ 𝑖 ) . This completes the proof of Lemma 3.5. ∎

Remark 3.2.

By Lemma 3.5, the least energy 𝑐 𝑍 0 satisfies the following equality:

𝑐 𝜗

inf 𝑢 ∈ ℳ 𝜗 ℰ 𝜗 ⁢ ( 𝑢 )

inf 𝑢 ∈ 𝒲
{ 0 } max 𝑡 > 0 ⁡ ℰ 𝜗 ⁢ ( 𝑡 ⁢ 𝑢 )

inf 𝑢 ∈ 𝕊 𝜗 max 𝑡

0 ⁡ ℰ 𝜗 ⁢ ( 𝑡 ⁢ 𝑢 ) .

(3.23)

Considering the functional Φ 𝜗 : 𝒮 𝜗 → ℝ given by

Φ 𝜗 ⁢ ( 𝜔 ) := ℰ 𝜗 ⁢ ( 𝑚 𝜗 ⁢ ( 𝜔 ) )

(3.24)

similarly to Szulkin and Weth [34, Corollary 3.3], we have:

Lemma 3.6.

Suppose that conditions ( 𝒵 1 ) − ( 𝒵 2 ) and ( 𝑓 1 ) − ( 𝑓 5 ) hold. Then the following statements are true:

( 𝑖 )

If { 𝔲 𝑛 } 𝑛 is a ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜗 sequence for Φ 𝜗 , then { 𝑚 𝜗 ⁢ ( 𝔲 𝑛 ) } is a ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜗 sequence for ℰ 𝜗 . If { 𝔲 𝑛 } 𝑛 ⊂ ℳ 𝜗 is a bounded ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜗 sequence for ℰ 𝜗 , then { 𝑚 𝜗 − 1 ⁢ ( 𝔲 𝑛 ) } 𝑛 is a ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜗 sequence for Φ 𝜗 .

( 𝑖 ⁢ 𝑖 )

𝑢 is a critical point of Φ 𝜗 if and only if 𝑚 𝜗 ⁢ ( 𝑢 ) is a nontrivial critical point of ℰ 𝜗 . Moreover, inf ℳ 𝜗 ℰ 𝜗

inf 𝒮 𝜗 Φ 𝜗 .

Lemma 3.7.

(see Liang et al. [25, Lemma 5]) Let ( 𝑓 1 ) be satisfied. We suppose that { 𝔲 𝑛 } 𝑛 is a sequence in 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) such that

lim sup 𝑛 → ∞ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / ( 𝑁 − 𝑠 ) < 𝛼 ∗ 𝔩 ⁢ 𝛽 0 ⁢ 𝜎 𝑠 / ( 𝑁 − 𝑠 ) for some 𝔩

1 ,

where 𝜎

min { 1 , 𝜗 } . Then there exists 𝐶 0

0 such that

| | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝔲 𝑛 ) | ≤ 𝐶 0 ⁢  for every  ⁢ 𝑛 .

Lemma 3.8.

(see Molica Bisci et al. [30, Lemma 4]) Let 𝜍 ∈ [ 𝑁 / 𝑠 , ∞ ) . If { 𝔲 𝑛 } 𝑛 is a bounded sequence in 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) and

lim 𝑛 → ∞ sup 𝑦 ∈ ℝ 𝑁 ∫ 𝐵 𝑅 ⁢ ( 𝑦 ) | 𝔲 𝑛 ⁢ ( 𝑥 ) | 𝜍 ⁢ 𝑑 𝑥

0

for some 𝑅

0 , then 𝔲 𝑛 → 0 in 𝐿 𝜈 ⁢ ( ℝ 𝑁 ) , for every 𝜈 ∈ ( 𝜍 , ∞ ) .

As in the proof of Molica Bisci et al. [30, Theorem 7], we get

Lemma 3.9.

Suppose that conditions ( 𝒵 1 ) − ( 𝒵 2 ) and ( 𝑓 1 ) hold. Then

lim sup 𝑡 → 0 + 𝑓 ⁢ ( 𝑡 ) ⁢ 𝑡 1 − 𝜍

0

for some 𝜍 ≥ 𝑁 / 𝑠 . Let { 𝔲 𝑛 } 𝑛 ⊂ 𝒲 be weakly convergent to 0 and such that

lim sup 𝑛 → ∞ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / ( 𝑁 − 𝑠 ) < 𝛼 ∗ 𝔩 ⁢ 𝛽 0 ⁢ 𝜎 𝑠 / ( 𝑁 − 𝑠 ) ,

where 𝛽 0 , 𝛼 ∗ are given in ( 𝑓 1 ) and (1.1), 𝜎

min { 1 , 𝜗 } , and 𝔩

1 is a suitable constant. If there exists 𝑅

0 such that

lim inf 𝑛 → ∞ sup 𝑦 ∈ ℝ 𝑁 ∫ 𝐵 𝑅 ⁢ ( 𝑦 ) | 𝔲 𝑛 | 𝜍 ⁢ 𝑑 𝑥

0 ,

then

∫ ℝ 𝑁 [ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝔲 𝑛 ) ] ⁢ 𝑓 ⁢ ( 𝔲 𝑛 ) ⁢ 𝔲 𝑛 → 0 and ∫ ℝ 𝑁 [ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝔲 𝑛 ) ] ⁢ 𝐹 ⁢ ( 𝔲 𝑛 ) → 0 ⁢  as  ⁢ 𝑛 → ∞ .

Lemma 3.10.

(see Liang et al. [24, Lemma 2.6]) Suppose that conditions ( 𝑓 3 ) and ( 𝑓 4 ) hold. Then there exists a constant 𝐶 𝛾 1 such that 𝜌 0 ≤ 𝑐 𝜗 ≤ 𝐶 𝛾 1 , where 𝜌 0 is the number determined by Lemma  3.3 ,

𝛾 1 is the constant given in ( 𝑓 4 ) ,

𝐶 𝛾 1

𝑎 ⁢ ( 1 − 𝑁 2 ⁢ 𝜃 ⁢ 𝑠 ) ⁢ ( 𝑎 ⁢ 𝑁 2 ⁢ 𝜃 ⁢ 𝑠 ⁢ 𝑏 ) 𝑁 / ( 2 ⁢ 𝜃 ⁢ 𝑠 − 𝑁 ) ,

and

𝑎

𝑠 𝑁 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 𝑠 + 1 𝑞 ⁢ ‖ 𝑢 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 .

Proposition 3.1.

Suppose that conditions ( 𝒵 1 ) − ( 𝒵 2 ) and ( 𝑓 1 ) − ( 𝑓 5 ) hold. Then problem ( 𝒬 𝜗 ) has a nontrivial nonnegative (weak) solution.

Proof.

By Lemmas 3.3 and 3.4, there exists a ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜗 sequence { 𝔲 𝑛 } 𝑛 ⊂ 𝒲 , satisfying (3.12). We shall divide the proof into two steps.

Step 1. { 𝔲 𝑛 } 𝑛 is a bounded sequence in 𝒲 .

Up to a subsequence, we may suppose that { 𝔲 𝑛 } 𝑛 is strongly convergent in 𝒲 . From (3.12), we have

ℰ 𝜗 ⁢ ( 𝔲 𝑛 ) − 1 𝜃 ⁢ ⟨ ℰ 𝜗 ′ ⁢ ( 𝔲 𝑛 ) , 𝔲 𝑛 ⟩

𝑐 𝜗 + 𝑜 𝑛 ⁢ ( 1 ) + 𝑜 𝑛 ⁢ ( 1 ) ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝒲 ⁢  as  ⁢ 𝑛 → ∞ ,

(3.25)

where 𝜃 is given in ( 𝑓 3 ) . Moreover,

ℰ 𝜗 ⁢ ( 𝔲 𝑛 ) − 1 𝜃 ⁢ ⟨ ℰ 𝜗 ′ ⁢ ( 𝔲 𝑛 ) , 𝔲 𝑛 ⟩

( 𝑠 𝑁 − 1 𝜃 ) ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / 𝑠 + ( 1 𝑞 − 1 𝜃 ) ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

• ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝔲 𝑛 ) | 𝑥 − 𝑦 | 𝜇 ⁢ [ 1 𝜃 ⁢ 𝑓 ⁢ ( 𝔲 𝑛 ) ⁢ 𝔲 𝑛 − 1 2 ⁢ 𝐹 ⁢ ( 𝔲 𝑛 ) ] ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦 .

Condition ( 𝑓 2 ) implies that

ℰ 𝜗 ⁢ ( 𝔲 𝑛 ) − 1 𝜃

⟨ ℰ 𝜗 ′ ⁢ ( 𝔲 𝑛 ) , 𝔲 𝑛 ⟩ ≥ ( 𝑠 𝑁 − 1 𝜃 ) ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / 𝑠 + ( 1 𝑞 − 1 𝜃 ) ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 .

(3.26)

Combining (3.25) and (3.26), we get as 𝑛 → ∞ ,

( 𝑠 𝑁 − 1 𝜃 ) ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / 𝑠 + ( 1 𝑞 − 1 𝜃 ) ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑞 ≤ 𝑐 𝜗 + 𝑜 𝑛 ⁢ ( 1 ) + 𝑜 𝑛 ⁢ ( 1 ) ⁢ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝒲 .

(3.27)

Observing that lim 𝑥 → ∞ , 𝑦 → ∞ 𝔠 ⁢ 𝑥 𝑁 / 𝑠 + 𝔨 ⁢ 𝑦 𝑞 𝑥 + 𝑦

∞ , for fixed numbers 𝔠

0 , 𝔨

0 , we conclude that { 𝔲 𝑛 } 𝑛 is a bounded sequence in 𝒲 . Since ℰ 𝜗 ⁢ ( 𝔲 𝑛 ) − 1 𝜃 ⁢ ⟨ ℰ 𝜗 ′ ⁢ ( 𝔲 𝑛 ) , 𝔲 𝑛 ⟩ → 𝑐 𝜗 as 𝑛 → ∞ , it follows that

lim sup 𝑛 → ∞ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / 𝑠 ≤ 𝑐 𝜗 1 𝑝 − 1 𝜃 ≤ 𝐶 𝛾 1 1 𝑝 − 1 𝜃

(3.28)

and

lim sup 𝑛 → ∞ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 ≤ 𝑐 𝜗 1 𝑞 − 1 𝜃 ≤ 𝐶 𝛾 1 1 𝑞 − 1 𝜃 .

(3.29)

From (3.28), we have

lim sup 𝑛 → ∞ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / ( 𝑁 − 𝑠 ) < 𝛼 ∗ 𝔩 ⁢ 𝛽 0 ⁢ 𝜎 𝑠 / ( 𝑁 − 𝑠 )

(3.30)

for some 𝔩

0 and 𝛾 1 large enough such that

𝐶 𝛾 1 1 𝑝 − 1 𝜃 < ( 𝛼 ∗ 𝔩 ⁢ 𝛽 0 ) ( 𝑁 − 𝑠 ) / 𝑁 ⁢ 𝜎 ,

when 𝛾 1 ≥ 𝛾 ∗ , where

𝛾 ∗

1 | 𝑁 ⁢ 𝐵 1 ⁢ ( 0 ) | ⁢ 2 ⁢ 𝑎 ⁢ ( 𝑁 − 𝜇 ) 𝔅 ⁢ ( 𝑁 , 𝑁 − 𝜇 + 1 )

(3.31)

and 𝔅

∫ 0 1 𝑡 𝑥 − 1 ⁢ ( 1 − 𝑡 ) 𝑦 − 1 ⁢ 𝑑 𝑡 , 𝑥

0 , 𝑦

0 is Beta function. This implies that

𝛾 1 ≥ 1 | 𝑁 ⁢ 𝐵 1 ⁢ ( 0 ) | ⁢ 𝑎 ⁢ 𝑁 ⁢ ( 𝑁 − 𝜇 ) 𝜃 ⁢ 𝑎 ⁢ 𝔅 ⁢ ( 𝑁 , 𝑁 − 𝜇 + 1 ) ⁢ [ 𝑎 ⁢ ( 1 − 𝑁 2 ⁢ 𝜃 ⁢ 𝑠 ) ( 𝑠 𝑁 − 1 𝜃 ) ⁢ 𝜎 ⁢ ( 𝛼 ∗ 𝔟 ⁢ 𝛽 0 ) ( 𝑁 − 𝑠 ) / 𝑁 ] 2 ⁢ 𝜃 ⁢ 𝑠 − 𝑁 2 ⁢ 𝑁

𝛾 ∗ ∗ .

Hence, (3.30) holds for every 𝛾 1 ≥ max ⁡ { 𝛾 ∗ , 𝛾 ∗ ∗ } .

Step 2.   We shall show that there exist 𝑅

0 , 𝛿

0 and a sequence { 𝑦 𝑛 } 𝑛 ⊂ ℝ 𝑁 such that

lim inf 𝑛 → ∞ ∫ 𝐵 𝑅 ⁢ ( 𝑦 𝑛 ) | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ 𝑑 𝑥 ≥ 𝛿

0 , ℘ ∈ { 𝑝 , 𝑞 } .

(3.32)

Arguing by contradiction, we assume that for some 𝑅

0 , we have

lim 𝑛 → ∞ sup 𝑦 ∈ ℝ 𝑁 ∫ 𝐵 𝑅 ⁢ ( 𝑦 ) | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ 𝑑 𝑥

0 , ℘ ∈ { 𝑝 , 𝑞 } .

(3.33)

Then Lemma 3.8 yields that 𝔲 𝑛 → 0 in 𝐿 𝜈 ⁢ ( ℝ 𝑁 ) , for every 𝜈

℘ . Condition (3.33), Lemma 3.9 and the Trudinger-Moser inequality imply that

∫ ℝ 𝑁 [ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝔲 𝑛 ) ] ⁢ 𝑓 ⁢ ( 𝔲 𝑛 ) ⁢ 𝔲 𝑛 ⁢ 𝑑 𝑥 → 0 ⁢ as ⁢ 𝑛 → ∞ .

Therefore,

𝑜 𝑛 ⁢ ( 1 )

⟨ ℰ 𝜗 ′ ⁢ ( 𝔲 𝑛 ) , 𝔲 𝑛 ⟩

‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − ∫ ℝ 𝑁 [ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝔲 𝑛 ) ] ⁢ 𝑓 ⁢ ( 𝔲 𝑛 ) ⁢ 𝔲 𝑛 ⁢ 𝑑 𝑥

‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 + 𝑜 𝑛 ⁢ ( 1 ) ⁢  as  ⁢ 𝑛 → ∞ .

So 𝔲 𝑛 → 0 in 𝑊 𝑠 , 𝑁 / 𝑠 ⁢ ( ℝ 𝑁 ) ∩ 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) . Passing to the limit as 𝑛 → ∞ , we have

ℰ 𝜗 ⁢ ( 𝔲 𝑛 )

‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 𝑝 + ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 𝑞 − ∫ ℝ 𝑁 [ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝔲 𝑛 ) ] ⁢ 𝐹 ⁢ ( 𝔲 𝑛 ) ⁢ 𝑑 𝑥 → 0

which contradicts with the fact that ℰ 𝜗 ⁢ ( 𝔲 𝑛 ) → 𝑐 𝜗

0 , as 𝑛 → ∞ . Thus (3.32) holds.

Put 𝔳 𝑛

𝔲 𝑛 ( ⋅ + 𝑦 𝑛 ) , then from (3.32) we have

∫ 𝐵 𝑅 ⁢ ( 0 ) | 𝔳 𝑛 | ℘ ⁢ 𝑑 𝑥 ≥ 𝛿 / 2

0 , ℘ ∈ { 𝑝 , 𝑞 } ⁢  for some  ⁢ 𝛿

0 .

(3.34)

Because ℰ 𝜗 and ℰ 𝜗 ′ are invariant under translation, we have

ℰ 𝜗 ⁢ ( 𝔳 𝑛 ) → 𝑐 𝜗 and ℰ 𝜗 ′ ⁢ ( 𝔳 𝑛 ) → 0 in ⁢ 𝒲 ′ .

Since ‖ 𝔳 𝑛 ‖ 𝜗 , 𝒲

‖ 𝔲 𝑛 ‖ 𝜗 , 𝒲 for every 𝑛 , then { 𝔳 𝑛 } 𝑛 is also bounded in 𝒲 and

lim sup 𝑛 → ∞ ‖ 𝔳 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / ( 𝑁 − 𝑠 )

lim sup 𝑛 → ∞ ‖ 𝔲 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / ( 𝑁 − 𝑠 ) < 𝛼 ∗ 𝔩 ⁢ 𝛽 0 ⁢ 𝜎 𝑠 / ( 𝑁 − 𝑠 ) .

(3.35)

Hence, choosing a subsequence if necessary, we may assume that there exists 𝔳 ∈ 𝒲 such that 𝔳 𝑛 ⇀ 𝔳 in 𝑊 ,

𝔳 𝑛 → 𝔳 in 𝐿 𝜗 ⁢ ( 𝐵 𝑅 ⁢ ( 0 ) ) , for every 𝜗 ∈ [ 𝑁 / 𝑠 , ∞ ) and 𝑅

0 , and 𝔳 𝑛 → 𝔳 a.e. in ℝ 𝑁 . Clearly, (3.34) implies that

∫ 𝐵 𝑅 ⁢ ( 0 ) | 𝔳 | ℘ ⁢ 𝑑 𝑥 ≥ 𝛿 / 2

0 , ℘ ∈ { 𝑝 , 𝑞 } ,

hence, 𝔳 ≢ 0 .

Arguing similarly to the proof of Thin et al. [36, Lemma 13], we get that ℰ 𝜗 ′ ⁢ ( 𝔳 )

0 and 𝔳 is indeed a ground state solution of problem ( 𝒬 𝜗 ). This completes the proof of Proposition 3.1. ∎

4.The auxiliary problem ( 𝒬 𝜀 )

Using the transformation 𝑥 ↦ 𝜀 ⁢ 𝑥 , problem ( 𝒬 ) can be rewritten as follows

( − Δ ) 𝑝 𝑠 ⁢ 𝑢 + ( − Δ ) 𝑞 𝑠 ⁢ 𝑢 + 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ ( | 𝑢 | 𝑝 − 2 ⁢ 𝑢 + | 𝑢 | 𝑞 − 2 ⁢ 𝑢 )

[ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑢 ) ] ⁢ 𝑓 ⁢ ( 𝑢 ) .

( 𝒬 𝜀 )

Inspired by the work of del Pino and Felmer [20], we introduce a penalized function which will play an essential role to obtain our main results. In general, we assume that 0 ∈ Ω and 𝑍 ⁢ ( 0 )

𝑍 0 . Fix ℏ 0

0 . We define

𝑓 ^ ⁢ ( 𝑡 ) := {

𝑓 ⁢ ( 𝑡 ) if ⁢ 𝑡 ≤ 𝑎 ,

𝑍 0 ℏ 0 ⁢ 𝑡 𝑞 − 1 if ⁢ 𝑡

𝑎 ,

and

𝑔 ^ ⁢ ( 𝑡 ) := {
𝑔 ⁢ ( 𝑥 , 𝑡 )

𝜒 Ω ⁢ ( 𝑥 ) ⁢ 𝑓 ⁢ ( 𝑡 ) + ( 1 − 𝜒 Ω ⁢ ( 𝑥 ) ) ⁢ 𝑓 ^ ⁢ ( 𝑡 ) if ⁢ 𝑡

0 ,

0 if ⁢ 𝑡 ≤ 0 .

To study problem ( 𝒬 𝜀 ), we introduce the Euler-Lagrange functional ℐ 𝜀 : 𝒲 𝜀 → ℝ by

ℐ 𝜀 ⁢ ( 𝑢 )

∑ ℘ ∈ { 𝑝 , 𝑞 } 1 ℘ ⁢ ‖ 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , ℘ ⁢ ( ℝ 𝑁 ) ℘ − 1 2 ⁢ ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐺 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝐺 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 .

By conditions ( 𝒵 1 ) , ( 𝒵 2 ) and ( 𝑓 1 ) , the functional ℐ 𝜀 is well defined on 𝒲 𝜀 and of class 𝐶 2 ⁢ ( 𝒲 𝜀 ) . Moreover, the critical points of ℐ 𝜀 are exactly the (weak) solutions of problem ( 𝒬 𝜀 ). Associated to the energy functional ℐ 𝜀 , we denote the Nehari manifold 𝒩 𝜀 by

𝒩 𝜀

{ 𝑣 ∈ 𝒲 𝜀 ∖ { 0 } : ⟨ ℐ 𝜀 ′ ⁢ ( 𝑣 ) , 𝑣 ⟩

0 } ,

where

⟨ ℐ 𝜀 ′ ⁢ ( 𝑣 ) , 𝜑 ⟩

∑ ℘ ∈ { 𝑝 , 𝑞 } ∫ ℝ 2 ⁢ 𝑁 | 𝑣 ⁢ ( 𝑥 ) − 𝑣 ⁢ ( 𝑦 ) | ℘ − 2 ⁢ ( 𝑣 ⁢ ( 𝑥 ) − 𝑣 ⁢ ( 𝑦 ) ) ⁢ ( 𝜑 ⁢ ( 𝑥 ) − 𝜑 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦

• ∫ ℝ 𝑁 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ ( | 𝑣 | 𝑝 − 2 ⁢ 𝑣
• | 𝑣 | 𝑞 − 2 ⁢ 𝑣 ) ⁢ 𝜑 ⁢ 𝑑 𝑥 − ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐺 ⁢ ( 𝑣 ⁢ ( 𝑦 ) ) ⁢ 𝑔 ⁢ ( 𝑣 ⁢ ( 𝑥 ) ) ⁢ 𝜑 ⁢ ( 𝑥 ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

for every 𝑣 , 𝜑 ∈ 𝒲 𝜀 .

Lemma 4.1.

Suppose that conditions ( 𝑓 1 ) − ( 𝑓 5 ) hold. The the following statements hold:

( 𝑔 1 )

lim 𝑡 → 0 + 𝑔 ⁢ ( 𝑥 , 𝑡 ) 𝑡 𝑞 − 1

0 uniformly with respect to 𝑥 ∈ ℝ 𝑁 ;

( 𝑔 2 ) ( 𝑖 )

0 < 𝑔 ⁢ ( 𝑥 , 𝑡 ) ≤ 𝑓 ⁢ ( 𝑡 ) for every 𝑡

0 , for every 𝑥 ∈ ℝ 𝑁 ;

( 𝑖 ⁢ 𝑖 )

𝑔 ⁢ ( 𝑥 , 𝑡 )

0 for every 𝑡 ≤ 0 , for every 𝑥 ∈ ℝ 𝑁 ;

( 𝑔 3 ) ( 𝑖 )

0 < 𝜃 ⁢ 𝐺 ⁢ ( 𝑥 , 𝑡 ) ≤ 𝑔 ⁢ ( 𝑥 , 𝑡 ) ⁢ 𝑡 , for every 𝑥 ∈ Ω and 𝑡

0 ;

( 𝑖 ⁢ 𝑖 )

0 ≤ 𝑞 ⁢ 𝐺 ⁢ ( 𝑥 , 𝑡 ) ≤ 𝑔 ⁢ ( 𝑥 , 𝑡 ) ⁢ 𝑡 ≤ 𝑍 0 ℏ 0 ⁢ ( 𝑡 𝑝 + 𝑡 𝑞 ) for every 𝑥 ∈ ℝ 𝑁 ∖ Ω and 𝑡

0 ;

( 𝑔 4 )

for every 𝑥 ∈ Ω , the function 𝑡 ↦ 𝑔 ⁢ ( 𝑥 , 𝑡 ) 𝑡 𝑞 2 − 1 is strictly increasing on ( 0 , + ∞ ) ;

( 𝑔 5 )

for every 𝑥 ∈ ℝ 𝑁 ∖ Ω , function 𝑡 ↦ 𝑔 ⁢ ( 𝑥 , 𝑡 ) 𝑡 𝑞 2 − 1 is strictly increasing on ( 0 , 𝑎 ) .

Proof.

We shall only give the proof of ( 𝑔 3 ) − ( 𝑖 ⁢ 𝑖 ) . The rest of the properties can be verified by the definition of 𝑔 . Using ( 𝑓 5 ) , we obtain that

𝑓 ⁢ ( 𝑡 ) 𝑡 𝑞 ≤ 𝑓 ⁢ ( 𝑎 ) 𝑎 𝑞

𝑍 0 ℏ 0 for every 𝑡 ∈ [ 0 , 𝑎 ] .

Consequently,

𝑔 ⁢ ( 𝑥 , 𝑡 ) ⁢ 𝑡

𝑓 ⁢ ( 𝑡 ) ⁢ 𝑡 ≤ 𝑍 0 ℏ 0 ⁢ 𝑡 𝑞 ≤ 𝑍 0 ℏ 0 ⁢ ( 𝑡 𝑝 + 𝑡 𝑞 ) ⁢  for every  ⁢ 𝑡 ∈ [ 0 , 𝑎 ] .

If 𝑡 ∈ ( 𝑎 , + ∞ ) , we have

𝑔 ⁢ ( 𝑥 , 𝑡 )

𝑍 0 ℏ 0 ⁢ 𝑡 𝑞 < 𝑍 0 ℏ 0 ⁢ ( 𝑡 𝑝 + 𝑡 𝑞 ) .

From ( 𝑓 2 ) and ( 𝑔 2 ) , we obtain that

𝑔 ⁢ ( 𝑥 , 𝑡 ) ⁢ 𝑡

𝑓 ⁢ ( 𝑡 ) ⁢ 𝑡 ≥ 𝜃 ⁢ 𝐹 ⁢ ( 𝑡 )

𝑞 ⁢ 𝐹 ⁢ ( 𝑡 ) ≥ 𝑞 ⁢ 𝐺 ⁢ ( 𝑥 , 𝑡 )

0 ⁢  for every  ⁢ 𝑡 ∈ [ 0 , 𝑎 ] .

In addition, if 𝑡 ∈ ( 𝑎 , + ∞ ) , we have

𝑔 ⁢ ( 𝑥 , 𝑡 )

𝑓 ^ ⁢ ( 𝑡 )

𝑍 0 ℏ 0 ⁢ 𝑡 𝑞 − 1 .

Hence

𝐺 ⁢ ( 𝑥 , 𝑡 )

𝑍 0 𝑞 ⁢ ℏ 0 ⁢ 𝑡 𝑞

and

𝑔 ⁢ ( 𝑥 , 𝑡 ) ⁢ 𝑡

𝑍 0 ℏ 0 ⁢ 𝑡 𝑞

𝑞 ⁢ 𝐺 ⁢ ( 𝑥 , 𝑡 ) .

This completes the proof of Lemma 4.1. ∎

Lemma 4.2.

(see [24, Proposition 3.1]) Suppose that conditions ( 𝒵 1 ) − 𝒵 2 and ( 𝑓 1 ) − ( 𝑓 3 ) hold. Then there is a real number 𝔯 ∗

0 such that

‖ 𝑢 ‖ 𝒲 𝜀 ≥ 𝔯 ∗

0 ⁢ for every  ⁢ 𝑢 ∈ 𝒩 𝜀 .

Lemma 4.3.

Suppose that conditions ( 𝒵 1 ) − ( 𝒵 2 ) and ( 𝑓 1 ) − ( 𝑓 5 ) hold. Then ℐ 𝜀 satisfies the following the geometric conditions:

( 𝑖 )

There exist real numbers 𝛼 ∗ > 0 , 𝜌 ∗ > 0 such that for every 𝑢 ∈ 𝒲 𝜀 : ‖ 𝑢 ‖ 𝒲 𝜀

𝜌 ∗ , we have ℐ 𝜀 ⁢ ( 𝑢 ) ≥ 𝛼 ∗

0 ;

( 𝑖 ⁢ 𝑖 )

There exists 𝑢 ∈ 𝒲 𝜀 such that ‖ 𝑢 ‖ 𝒲 𝜀

𝜌 ∗ and ℐ 𝜀 ⁢ ( 𝑢 ) < 0 .

Proof.

One can apply a similar discussion as in Lemmas 3.3 and 3.4, combined with the Trudinger-Moser inequality, so we shall omit the details here. ∎

By virtue of Lemma 4.2 and the Mountain Pass Theorem, there exists a ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜀 sequence { 𝔲 𝑛 } 𝑛 ⊂ 𝒲 𝜀 , that is,

ℐ 𝜀 ⁢ ( 𝔲 𝑛 ) → 𝑐 𝜀 and ℐ 𝜀 ′ ⁢ ( 𝔲 𝑛 ) → 0 ,

where

𝑐 𝜀 := inf 𝛾 ∈ Γ max 𝑡 ∈ [ 0 , 1 ] ⁡ ℐ 𝜀 ⁢ ( 𝛾 ⁢ ( 𝑡 ) )

and Γ

{ 𝛾 ∈ ( 𝐶 0 ⁢ [ 0 , 1 ] , 𝒲 𝜀 ⁢ ( ℝ 𝑁 ) ) : 𝛾 ⁢ ( 0 )

0 , ℐ 𝜀 ⁢ ( 𝛾 ⁢ ( 1 ) ) } < 0 .

We give the definition of 𝒯 𝜀 + as follows:

𝒯 𝜀 + := { 𝑢 ∈ 𝒲 𝜀 ⁢ ( ℝ 𝑁 ) : | supp ⁢ ( 𝑢 + ) ∩ Ω 𝜀 |

0 } ⊂ 𝒲 𝜀 ⁢ ( ℝ 𝑁 ) ,

Ω 𝜀 := { 𝑥 ∈ ℝ 𝑁 : 𝜀 ⁢ 𝑥 ∈ Ω } . Let 𝕊 𝜀 be a the unit sphere in 𝒲 𝜀 ⁢ ( ℝ 𝑁 ) and denote by 𝕊 𝜀 +

𝕊 𝜀 ∩ 𝒯 𝜀 + . Note that 𝕊 𝜀 + is an incomplete 𝐶 1 , 1 -manifold of codimension 1, modelled on 𝒲 𝜀 ⁢ ( ℝ 𝑁 ) and contained in the open 𝒯 𝜀 + . Thus, 𝒲 𝜀

𝑇 𝑢 ⁢ 𝕊 𝜀 + ⊕ ℝ ⁢ 𝑢 , for every 𝑢 ∈ 𝕊 𝜀 + , where

𝑇 𝑢 ⁢ 𝕊 𝜀 +

{ 𝑣 ∈ 𝒲 : ∑ ℘ ∈ { 𝑝 , 𝑞 } ∬ ℝ 2 ⁢ 𝑁 | 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) | ℘ − 2 ⁢ ( 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) ) ⁢ ( 𝜑 ⁢ ( 𝑥 ) − 𝜑 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 𝑑 𝑥 𝑑 𝑦

+ ∫ ℝ 𝑁 𝑍 ( 𝜀 𝑥 ) ( | 𝑢 | 𝑝 − 2 𝑢 + | 𝑢 | 𝑞 − 2 𝑢 ) 𝜑 𝑑 𝑥

0 } .

As in Lemmas 3.5 and 3.6, the following results can be obtained.

Lemma 4.4.

Suppose that conditions ( 𝒵 1 ) − ( 𝒵 2 ) and ( 𝑓 1 ) − ( 𝑓 5 ) hold. Then the following statements are true:

( 𝑖 )

There exists a unique 𝑡 𝑢 ⁢ 𝑢 ∈ 𝒩 𝜀 and ℐ 𝜀 ⁢ ( 𝑡 𝑢 ⁢ 𝑢 )

max 𝑡

0 ⁡ ℐ 𝜀 ⁢ ( 𝑡 ⁢ 𝑢 ) for every 𝑢 ∈ 𝒯 𝜀 + . Moreover, we have 𝑐 𝜀 ≥ 𝜁

0 and

𝑐 𝜀

inf 𝑢 ∈ 𝒩 𝜀 ℐ 𝜀 ⁢ ( 𝑢 )

inf 𝑢 ∈ 𝒯 𝜀 + max 𝑡 > 0 ⁡ ℐ 𝜀 ⁢ ( 𝑡 ⁢ 𝑢 )

inf 𝑢 ∈ 𝕊 𝜀 + max 𝑡

0 ⁡ ℐ 𝜀 ⁢ ( 𝑡 ⁢ 𝑢 ) .

( 𝑖 ⁢ 𝑖 )

𝒩 𝜀 is bounded away from 0, and there exists 𝛼

0 such that 𝑡 𝑢 ≥ 𝛼 , for every 𝑢 ∈ 𝕊 𝜀 . Moreover, for every compact subset 𝒦 ⊂ 𝕊 𝜀 + , there exists 𝐶 𝒦

0 such that 𝑡 𝑢 ≤ 𝐶 𝒦 , for every 𝑢 ∈ 𝒦 .

( 𝑖 ⁢ 𝑖 ⁢ 𝑖 )

The continuous map 𝑚 ^ 𝜀 : 𝒲 𝜀 → 𝒩 𝜀 is given by 𝑚 ^ 𝜀 ⁢ ( 𝑢 )

𝑡 𝑢 ⁢ 𝑢 and 𝑚 𝜀 := 𝑚 ^ 𝜀 | 𝕊 𝜀 + is a homeomorphism between 𝕊 𝜀 + and 𝒩 𝜀 , and 𝑚 𝜀 − 1 ⁢ ( 𝑢 )

𝑢 ‖ 𝑢 ‖ 𝒲 𝜀 .

Exploring the functional Θ 𝜀 ⁢ ( 𝑢 ) := ℐ 𝜀 ⁢ ( 𝑚 𝜀 ⁢ ( 𝑢 ) ) , together with argument similar to the proof of Szulkin and Weth [34, Corollary 3.3], we obtain the following results.

Lemma 4.5.

Suppose that conditions ( 𝒵 1 ) − ( 𝒵 2 ) and ( 𝑓 1 ) − ( 𝑓 5 ) hold. Then the following statements are true:

( 𝑖 )

If { 𝔲 𝑛 } 𝑛 is a ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜀 sequence for Θ 𝜀 , then { 𝑚 𝜀 ⁢ ( 𝔲 𝑛 ) } 𝑛 is a ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜀 sequence for ℐ 𝜀 . Moreover, if { 𝔲 𝑛 } 𝑛 ⊂ 𝒩 𝜀 is a bounded ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜀 sequence for ℐ 𝜀 , then { 𝑚 𝜀 − 1 ⁢ ( 𝔲 𝑛 ) } 𝑛 is a ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜀 sequence for Θ 𝜀 .

( 𝑖 ⁢ 𝑖 )

𝑢 is a critical point of Θ 𝜀 if and only if 𝑚 𝜀 ⁢ ( 𝑢 ) is a nontrivial critical point of ℐ 𝜀 . Furthermore, inf 𝒩 𝜀 ℐ 𝜀

inf 𝕊 𝜀 + Θ 𝜀 .

Lemma 4.6.

𝑐 𝜀 and 𝑐 𝜗 satisfy the following inequalities

lim sup 𝜀 → 0 + 𝑐 𝜀 ≤ 𝑐 𝜗 ≤ 𝐶 𝛾 1 .

(4.1) Proof.

Let 𝜑 ∈ 𝐶 0 ∞ ⁢ ( ℝ 𝑁 ) such that 𝜑 ≡ 1 on 𝐵 𝛿 / 2 ⁢ ( 0 ) , supp ( 𝜑 ) ⊂ 𝐵 𝛿 ⁢ ( 0 ) ⊂ Ω for some 𝛿

0 and 𝜑 ≡ 0 on 𝐵 𝛿 ⁢ ( 0 ) 𝑐 . We define

𝑢 𝜀 ⁢ ( 𝑥 ) := 𝜑 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ 𝑢 ⁢ ( 𝑥 ) for every 𝜀

0 ,

where 𝑢 is the ground state solution of problem ( 𝒬 𝜗 ) obtained in Proposition 3.1. We know that supp ( 𝑢 𝜀 ) ⊂ Ω 𝜀 and 𝑢 𝜀 → 𝑢 in 𝒲 (see Ambrosio and Isernia [6, Lemma 2.4]). If we assume that 𝑡 𝜀

0 with 𝑡 𝜀 ⁢ 𝑢 𝜀 ∈ 𝒩 𝜀 , then

𝑐 𝜀 ≤ ℐ 𝜀 ⁢ ( 𝑡 𝜀 ⁢ 𝑣 𝜀 )

𝑡 𝜀 𝑝 𝑝 ⁢ ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 𝜀 𝑞 𝑞 ⁢ ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − 1 2 ⁢ ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐺 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑦 ) ) ⁢ 𝐺 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑥 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

𝑡 𝜀 𝑝 𝑝 ⁢ ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 𝜀 𝑞 𝑞 ⁢ ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − 1 2 ⁢ ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑦 ) ) ⁢ 𝐹 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑥 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 .

Since 𝑡 𝜀 ⁢ 𝑢 𝜀 ⊂ 𝒩 𝜀 , we have

𝑡 𝜀 𝑝 ⁢ ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 𝜀 𝑞 ⁢ ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑥 ) ) ⁢ 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑥 ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 .

(4.2)

Moreover, it follows from ( 𝑓 3 ) that

ℐ 𝜀 ⁢ ( 𝑡 𝜀 ⁢ 𝑣 𝜀 )

𝑡 𝜀 𝑝 𝑝 ⁢ ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 𝜀 𝑞 𝑞 ⁢ ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − 1 2 ⁢ ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑦 ) ) ⁢ 𝐹 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑥 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

≥ ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ( 𝑡 𝜀 𝑢 𝜀 ( 𝑦 ) | 𝑥 − 𝑦 | 𝜇 ⁢ [ 1 𝑞 ⁢ 𝑓 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ) ⁢ 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ 𝑑 ⁢ 𝑥 − 1 2 ⁢ 𝐹 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ) ] ⁢ 𝑑 𝑥 ≥ 0 .

(4.3)

This fact implies that { 𝑡 𝜀 } is bounded, as 𝜀 → 0 + . Indeed, suppose to the contrary, that { 𝑡 𝜀 } is unbounded when 𝜀 → 0 + . Consequently, together with ( 𝑓 4 ) , we would have

ℐ 𝜀 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ) ≥ 𝑡 𝜀 𝑝 𝑝 ⁢ ‖ 𝑢 𝜀 ‖ 𝜗 , 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 𝜀 𝑞 𝑞 ⁢ ‖ 𝑢 𝜀 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − 𝛾 1 2 ⁢ 𝑡 𝜀 2 ⁢ 𝜃 ⁢ ‖ | 𝑢 𝜀 | 𝜃 ‖ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 2 → − ∞

which is a contradiction with (4). Thus, we may suppose that 𝑡 𝜀 → 𝑡 0 , as 𝜀 → 0 + . Using the Vitali’s Theorem, we obtain

lim sup 𝜀 → 0 + 𝑐 𝜀
≤ 𝑡 0 𝑝 𝑝 ⁢ ‖ 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 0 𝑞 𝑞 ⁢ ‖ 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − 1 2 ⁢ ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝑡 0 ⁢ 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝐹 ⁢ ( 𝑡 0 ⁢ 𝑢 ⁢ ( 𝑥 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

ℰ 𝜗 ⁢ ( 𝑡 0 ⁢ 𝑢 ) .

We shall now verify that 𝑡 0

1 . For every 𝜀

0 , there exists 𝑡 𝜀

0 such that

ℐ 𝜀 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 )

max 𝑡 ≥ 0 ⁡ ℐ 𝜀 ⁢ ( 𝑡 ⁢ 𝑢 𝜀 ) .

Therefore, ⟨ ℐ 𝜀 ′ ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ) , 𝑢 𝜀 ⟩

0 and we have

𝑡 𝜀 𝑝 ⁢ ‖ 𝑢 𝜀 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 𝜀 𝑞 ⁢ ‖ 𝑢 𝜀 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑥 ) ) ⁢ 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑥 ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

which means that

‖ 𝑢 𝜀 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 𝜀 𝑞 − 𝑝 ⁢ ‖ 𝑢 𝜀 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝑡 𝜀 ⁢ 𝑢 𝜀 ⁢ ( 𝑥 ) ) ⁢ 𝑢 𝜀 ⁢ ( 𝑥 ) 𝑡 𝜀 𝑝 − 1 ⁢ | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 .

(4.4)

Passing to the limit as 𝜀 → 0 in (4.4) and using the fact that 𝑢 𝜀 → 𝑢 in 𝑊 𝑠 , 𝑁 𝑠 ⁢ ( ℝ 𝑁 ) (see Ambrosio and Isernia [6, Lemma 2.4]), we obtain

‖ 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 𝑡 0 𝑞 − 𝑝 ⁢ ‖ 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝑡 0 ⁢ 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝑓 ⁢ ( 𝑡 0 ⁢ 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝑢 ⁢ ( 𝑥 ) 𝑡 0 𝑝 − 1 ⁢ | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 .

Together with this fact, 𝑢 ∈ ℳ 𝜗 and ( 𝑓 5 ) , we deduce that 𝑡 0

1 . Therefore,

lim sup 𝜀 → 0 + 𝑐 𝜀 ≤ ℰ 𝜗 ⁢ ( 𝑣 )

𝑐 𝜗 .

Combining with Lemma 3.10, we can obtain that the last inequality hold in (4.1). This completes the proof of Lemma 4.6. ∎

Lemma 4.7.

The functional ℐ 𝜀 satisfies the Palais-Smale condition at level 𝑐 𝜀 , for every 𝜀 ∈ ( 0 , 𝜀 0 ) .

Proof.

Let { 𝔲 𝑛 } 𝑛 ⊂ 𝒲 𝜀 ⁢ ( ℝ 𝑁 ) be a ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜀 sequence for ℐ 𝜀 , i.e.,

ℐ 𝜀 ⁢ ( 𝔲 𝑛 ) → 𝑐 𝜀 and ℐ 𝜀 ′ ⁢ ( 𝔲 𝑛 ) → 0 as 𝑛 → ∞ .

We shall complete the proof of this lemma by the following two claims.

Claim 1. { 𝔲 𝑛 } 𝑛 is bounded in 𝒲 𝜀 ⁢ ( ℝ 𝑁 ) .

Indeed, by ( 𝑓 3 ) , we have

𝑐 𝜀 + 𝑜 𝑛 ⁢ ( 1 )

ℐ 𝜀 ⁢ ( 𝔲 𝑛 ) − 1 2 ⁢ 𝑞 ⁢ ⟨ ℐ 𝜀 ′ ⁢ ( 𝔲 𝑛 ) , 𝔲 𝑛 ⟩

( 1 𝑝 − 1 2 ⁢ 𝑞 ) ⁢ ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 1 2 ⁢ 𝑞 ⁢ ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

+ 1 2 ⁢ ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐹 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ [ 1 𝑞 ⁢ 𝑓 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝐹 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) ) ] ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

≥ 1 2 ⁢ 𝑞 ⁢ ( ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 )

1 2 ⁢ 𝑞 ⁢ ‖ 𝔲 𝑛 ‖ 𝒲 𝜀

which implies that { 𝔲 𝑛 } 𝑛 is bounded in 𝒲 𝜀 ⁢ ( ℝ 𝑁 ) . Moreover, we obtain

lim sup 𝑛 → ∞ ‖ 𝔲 𝑛 ‖ 𝒲 𝜀 𝑝 ≤ 2 ⁢ 𝑞 ⁢ 𝑐 𝜀 .

(4.5)

Together with this fact and Lemma 3.10, we obtain that

lim sup 𝑛 → ∞ ‖ 𝔲 𝑛 ‖ 𝒲 𝜀 ≤ 2 ⁢ 𝑞 ⁢ 𝑐 𝜗 ≤ 2 ⁢ 𝑞 ⁢ 𝐶 𝛾 1

2 ⁢ 𝑞 ⁢ 𝑎 ⁢ ( 1 − 𝑁 2 ⁢ 𝜃 ⁢ 𝑠 ) ⁢ ( 𝑎 ⁢ 𝑁 2 ⁢ 𝜃 ⁢ 𝑠 ⁢ 𝑏 ) 𝑁 / ( 2 ⁢ 𝜃 ⁢ 𝑠 − 𝑁 ) := 𝒢

(4.6)

for 𝛾 1 ≥ max ⁡ { 𝛾 ∗ , 𝑎 } , where 𝛾 ∗ , 𝑎 are given in (3.31) and Lemma 3.10, respectively. Therefore, going to a subsequence if necessary, we may assume that 𝔲 𝑛 ⇀ 𝑢 in 𝒲 𝜀 ,

𝔲 𝑛 → 𝑢 in 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 𝑞 ⁢ ( ℝ 𝑁 ) for every 𝑞 ∈ [ 𝑁 𝑠 , + ∞ ) and 𝔲 𝑛 → 𝑢 a.e. in ℝ 𝑁 .

Claim 2. ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜀 condition holds in 𝒲 𝜀 .

We shall divide the proof into three steps. We first verify that 𝑢 is a critical point of ℐ 𝜀 . For every 𝜑 ∈ 𝐶 𝑐 ∞ ⁢ ( ℝ 𝑁 ) , we have

∑ ℘ ∈ { 𝑝 , 𝑞 } ∬ ℝ 2 ⁢ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) | ℘ − 2 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ ( 𝜑 ⁢ ( 𝑥 ) − 𝜑 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦

→ ∑ ℘ ∈ { 𝑝 , 𝑞 } ∬ ℝ 2 ⁢ 𝑁 | 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) | ℘ − 2 ⁢ ( 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) ) ⁢ ( 𝜑 ⁢ ( 𝑥 ) − 𝜑 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦

and

∫ ℝ 𝑁 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ | 𝔲 𝑛 | ℘ − 2 ⁢ 𝔲 𝑛 ⁢ 𝜑 ⁢ 𝑑 𝑥 → ∫ ℝ 𝑁 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ | 𝑢 | ℘ − 2 ⁢ 𝑢 ⁢ 𝜑 ⁢ 𝑑 𝑥 for every ℘ ∈ { 𝑝 , 𝑞 } .

Step 1. We prove that

∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐺 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ 𝑔 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝜑 ⁢ ( 𝑥 ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 → ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐺 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝑔 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) ⁢ 𝜑 ⁢ ( 𝑥 ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 .

(4.7)

Since the boundedness of { 𝐺 ⁢ ( 𝜀 ⁢ 𝑥 , 𝔲 𝑛 ) } 𝑛 in 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ⁢ ( ℝ 𝑁 ) ,

𝔲 𝑛 → 𝑢 a.e. in ℝ 𝑁 , and 𝑡 ↦ 𝐺 ⁢ ( ⋅ , 𝑡 ) is continuous, hence

𝐺 ⁢ ( 𝜀 ⁢ 𝑥 , 𝔲 𝑛 ) ⇀ 𝐺 ⁢ ( 𝜀 ⁢ 𝑥 , 𝑢 ) ⁢ in ⁢ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ⁢ ( ℝ 𝑁 ) .

From Lemma 3.2, we obtain the linear bounded operator

1 | 𝑥 | 𝜇 ∗ 𝐹 ∈ 𝐿 2 ⁢ 𝑁 𝜇 ⁢ ( ℝ 𝑁 ) for every ⁢ 𝐹 ∈ 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ⁢ ( ℝ 𝑁 ) ,

from 𝐿 2 ⁢ 𝑁 2 ⁢ 𝑁 − 𝜇 ⁢ ( ℝ 𝑁 ) to 𝐿 2 ⁢ 𝑁 𝜇 ⁢ ( ℝ 𝑁 ) . Therefore, 1 | 𝑥 | 𝜇 ∗ 𝐺 ⁢ ( 𝜀 ⁢ 𝑦 , 𝔲 𝑛 ) ⇀ 1 | 𝑥 | 𝜇 ∗ 𝐺 ⁢ ( 𝜀 ⁢ 𝑦 , 𝑢 ) in 𝐿 2 ⁢ 𝑁 𝜇 ⁢ ( ℝ 𝑁 ) . Since 𝑔 ⁢ ( 𝔲 𝑛 ) → 𝑔 ⁢ ( 𝑢 ) in 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 𝜈 ⁢ ( ℝ 𝑁 ) , for every 𝜈 ∈ [ 𝑝 , + ∞ ) , we deduce that (4.7) is true. Consequently, in view of ⟨ ℐ 𝜀 ′ ⁢ ( 𝔲 𝑛 ) , 𝜙 ⟩

𝑜 𝑛 ⁢ ( 1 ) , for every 𝜙 ∈ 𝐶 𝑐 ∞ ⁢ ( ℝ 𝑁 ) , we obtain ⟨ ℐ 𝜀 ′ ⁢ ( 𝑢 ) , 𝜙 ⟩

0 , for every 𝜙 ∈ 𝐶 𝑐 ∞ ⁢ ( ℝ 𝑁 ) . Since 𝐶 𝑐 ∞ ⁢ ( ℝ 𝑁 ) is dense in 𝒲 𝜀 , we obtain that 𝑢 is a critical point of ℐ 𝜀 .

Step 2. We shall prove that for every 𝜉 , there exists 𝑅

𝑅 ⁢ ( 𝜉 )

0 such that

lim sup 𝑛 → ∞ ∫ 𝐵 𝑅 𝑐 ( ∑ ℘ ∈ { 𝑝 , 𝑞 } ∫ ℝ 𝑁 | 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) | ℘ | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ⁢ 𝑑 𝑦 + 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ | 𝔲 𝑛 | 𝑝 + 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ | 𝔲 𝑛 | 𝑞 ) ⁢ 𝑑 𝑥 < 𝜉 .

(4.8)

For every 𝑅 > 0 , let 𝜙 𝑅 ∈ 𝐶 𝑐 ∞ ⁢ ( ℝ 𝑁 ) such that 0 ≤ 𝜙 𝑅 ≤ 1 , 𝜙 𝑅

0 in 𝐵 𝑅 ⁢ ( 0 ) , 𝜙 𝑅

1 in 𝐵 2 ⁢ 𝑅 𝑐 ⁢ ( 0 ) , and | ∇ 𝜙 𝑅 | ≤ 𝐶 𝑅 for some constant 𝐶 > 0 independent of 𝑅 . Since { 𝜙 𝑅 ⁢ 𝔲 𝑛 } 𝑛 is bounded in 𝒲 𝜀 ⁢ ( ℝ 𝑁 ) , it follows that ⟨ ℐ 𝜀 ′ ⁢ ( 𝔲 𝑛 ) , 𝜙 𝑅 ⁢ 𝔲 𝑛 ⟩

𝑜 𝑛 ⁢ ( 1 ) , i.e.,

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) | 𝑝 | 𝑥 − 𝑦 | 𝑁 + 𝑠 ⁢ 𝑝 ⁢ 𝜙 𝑅 ⁢ ( 𝑥 ) ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 + ∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) | 𝑞 | 𝑥 − 𝑦 | 𝑁 + 𝑠 ⁢ 𝑞 ⁢ 𝜙 𝑅 ⁢ ( 𝑥 ) ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

+ ∫ ℝ 𝑁 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ | 𝔲 𝑛 ⁢ ( 𝑥 ) | 𝑝 ⁢ 𝑑 𝑥 + ∫ ℝ 𝑁 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ | 𝔲 𝑛 ⁢ ( 𝑥 ) | 𝑞 ⁢ 𝑑 𝑥

𝑜 𝑛 ⁢ ( 1 ) + ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐺 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ 𝑔 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝜙 𝑅 ⁢ ( 𝑥 ) ⁢ 𝔲 𝑛 ⁢ ( 𝑥 ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

− ∬ ℝ 2 ⁢ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) | 𝑝 − 2 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ ( 𝜙 𝑅 ⁢ ( 𝑥 ) − 𝜙 𝑅 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + 𝑠 ⁢ 𝑝 ⁢ 𝔲 𝑛 ⁢ ( 𝑦 ) ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦

− ∬ ℝ 2 ⁢ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) | 𝑞 − 2 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ ( 𝜙 𝑅 ⁢ ( 𝑥 ) − 𝜙 𝑅 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + 𝑠 ⁢ 𝑞 ⁢ 𝔲 𝑛 ⁢ ( 𝑦 ) ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦 .

(4.9)

By Lemma 3.7, then there exists ℏ 0

0 such that

sup 𝔲 𝑛 ∈ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) | | 𝑥 | − 𝜇 ∗ 𝐺 ⁢ ( 𝔲 𝑛 ) | ℏ 0 < 1 2 .

(4.10)

Let 𝑅

0 be such that Ω 𝜀 ⊂ 𝐵 𝑅 . By the definition of 𝜙 𝑅 , ( 𝑔 3 ) − ( 𝑖 ⁢ 𝑖 ) and (4.10), we get

∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) | 𝑝 | 𝑥 − 𝑦 | 𝑁 + 𝑠 ⁢ 𝑝 ⁢ 𝜙 𝑅 ⁢ ( 𝑥 ) ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 + ∫ ℝ 𝑁 ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) | 𝑞 | 𝑥 − 𝑦 | 𝑁 + 𝑠 ⁢ 𝑞 ⁢ 𝜙 𝑅 ⁢ ( 𝑥 ) ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥

• 1 2 ⁢ ∫ ℝ 𝑁 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ ( | 𝔲 𝑛 ⁢ ( 𝑥 ) | 𝑝
• | 𝔲 𝑛 ⁢ ( 𝑥 ) | 𝑞 ) ⁢ 𝑑 𝑥

≤ 𝑜 𝑛 − ∬ ℝ 2 ⁢ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) | 𝑝 − 2 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ ( 𝜙 𝑅 ⁢ ( 𝑥 ) − 𝜙 𝑅 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + 𝑠 ⁢ 𝑝 ⁢ 𝔲 𝑛 ⁢ ( 𝑦 ) ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦

− ∬ ℝ 2 ⁢ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) | 𝑞 − 2 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ ( 𝜙 𝑅 ⁢ ( 𝑥 ) − 𝜙 𝑅 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + 𝑠 ⁢ 𝑞 ⁢ 𝔲 𝑛 ⁢ ( 𝑦 ) ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦 .

(4.11)

For ℘ ∈ { 𝑝 , 𝑞 } , by virtue of the Hölder inequality and the boundedness of { 𝔲 𝑛 } 𝑛 in 𝒲 𝜀 , we have

∬ ℝ 2 ⁢ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) | ℘ − 2 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ ( 𝜙 𝑅 ⁢ ( 𝑥 ) − 𝜙 𝑅 ⁢ ( 𝑦 ) ) | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ⁢ 𝔲 𝑛 ⁢ ( 𝑦 ) ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦

≤ 𝐶 ⁢ ( ∬ ℝ 2 ⁢ 𝑁 | 𝜙 𝑅 ⁢ ( 𝑥 ) − 𝜓 𝑅 ⁢ ( 𝑦 ) | ℘ | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ⁢ | 𝔲 𝑛 ⁢ ( 𝑦 ) | ℘ ⁢ 𝑑 𝑦 ) 1 ℘ .

(4.12)

Next, by the definition of 𝜙 𝑅 , polar coordinates and the boundedness of { 𝔲 𝑛 } 𝑛 in 𝒲 𝜀 , we obtain

∬ ℝ 2 ⁢ 𝑁 | 𝜙 𝑅 ⁢ ( 𝑥 ) − 𝜙 𝑅 ⁢ ( 𝑦 ) | ℘ | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ⁢ | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦

∫ ℝ 𝑁 ∫ | 𝑦 − 𝑥 |

𝑅 | 𝜙 𝑅 ⁢ ( 𝑥 ) − 𝜙 𝑅 ⁢ ( 𝑦 ) | ℘ | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ⁢ | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦 + ∫ ℝ 𝑁 ∫ | 𝑦 − 𝑥 | ≤ 𝑅 | 𝜙 𝑅 ⁢ ( 𝑥 ) − 𝜙 𝑅 ⁢ ( 𝑦 ) | ℘ | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ⁢ | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ 𝑑 𝑥 ⁢ 𝑑 𝑦

≤ 𝐶 ⁢ ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ ( ∫ | 𝑦 − 𝑥 |

𝑅 𝑑 ⁢ 𝑦 | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 ) ⁢ 𝑑 𝑥 + 𝐶 𝑅 ℘ ⁢ ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ ( ∫ | 𝑦 − 𝑥 | ≤ 𝑅 𝑑 ⁢ 𝑦 | 𝑥 − 𝑦 | 𝑁 + ℘ ⁢ 𝑠 − ℘ ) ⁢ 𝑑 𝑥

≤ 𝐶 ⁢ ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ ( ∫ | 𝑧 |

𝑅 𝑑 ⁢ 𝑧 | 𝑧 | 𝑁 + ℘ ⁢ 𝑠 ) ⁢ 𝑑 𝑥 + 𝐶 𝑅 𝑡 ⁢ ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ ( ∫ | 𝑧 | ≤ 𝑅 𝑑 ⁢ 𝑧 | 𝑧 | 𝑁 + ℘ ⁢ 𝑠 − ℘ ) ⁢ 𝑑 𝑥

≤ 𝐶 ⁢ ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ 𝑑 𝑥 ⁢ ( ∫ 𝑅 ∞ 𝑑 ⁢ 𝜌 𝜌 𝑠 ⁢ ℘ + 1 ) + 𝐶 𝑅 ℘ ⁢ ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ 𝑑 𝑥 ⁢ ( ∫ 0 𝑅 𝑑 ⁢ 𝜌 𝜌 𝑠 ⁢ ℘ − ℘ + 1 )

≤ 𝐶 𝑅 𝑠 ⁢ ℘ ⁢ ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ 𝑑 𝑥 + 𝐶 𝑅 ℘ ⁢ 𝑅 − 𝑠 ⁢ ℘ + ℘ ⁢ ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ 𝑑 𝑥 ≤ 𝐶 𝑅 𝑠 ⁢ ℘ ⁢ ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) | ℘ ⁢ 𝑑 𝑥

≤ 𝐶 𝑅 𝑠 ⁢ ℘ → 0 as 𝑅 → ∞ ,  where  ⁢ ℘ ∈ { 𝑝 , 𝑞 } .

(4.13)

Gathering (4)-(4), we infer that (4.8) is satisfied.

Step 3. We verify that 𝔲 𝑛 → 𝑢 in 𝒲 𝜀 as 𝑛 → ∞ .

In view of (4.8), we obtain 𝔲 𝑛 → 𝑢 in 𝐿 𝜈 ⁢ ( ℝ 𝑁 ) , for every 𝜈 ∈ [ 𝑝 , + ∞ ) . Indeed, fixed 𝜉 > 0 , there exists 𝑅

𝑅 ⁢ ( 𝜉 )

0 such that (4.8) holds. Using the compactness embedding 𝒲 𝜀 ↪ ↪ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 𝜈 ( ℝ 𝑁 ) and ( 𝒵 1 ) , we see

lim sup 𝑛 → ∞ | 𝔲 𝑛 − 𝑢 | 𝐿 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝

lim sup 𝑛 → ∞ [ | 𝔲 𝑛 − 𝑢 | 𝐿 𝑝 ⁢ ( 𝐵 𝑅 ⁢ ( 0 ) ) 𝑝 + | 𝔲 𝑛 − 𝑢 | 𝐿 𝑝 ⁢ ( 𝐵 𝑅 𝑐 ⁢ ( 0 ) ) 𝑝 ]

≤ 2 𝑝 − 1 ⁢ lim sup 𝑛 → ∞ ( | 𝔲 𝑛 | 𝐿 𝑝 ⁢ ( 𝐵 𝑅 𝑐 ⁢ ( 0 ) ) 𝑝 + | 𝑢 | 𝐿 𝑝 ⁢ ( 𝐵 𝑅 𝑐 ⁢ ( 0 ) ) 𝑝 )

≤ 2 𝑝 − 1 𝑍 0 ⁢ lim sup 𝑛 → ∞ ∫ 𝐵 𝑅 𝑐 ⁢ ( 0 ) ( ∫ ℝ 𝑁 | 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝔲 𝑛 ⁢ ( 𝑦 ) | 𝑝 | 𝑥 − 𝑦 | 2 ⁢ 𝑁 ⁢ 𝑑 𝑦 + 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ | 𝔲 𝑛 | 𝑝 ) ⁢ 𝑑 𝑥

• 2 𝑝 − 1 𝑍 0 ⁢ ∫ 𝐵 𝑅 𝑐 ⁢ ( 0 ) ( ∫ ℝ 𝑁 | 𝑢 ⁢ ( 𝑥 ) − 𝑢 ⁢ ( 𝑦 ) | 𝑝 | 𝑥 − 𝑦 | 2 ⁢ 𝑁 ⁢ 𝑑 𝑦
• 𝑍 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ | 𝑢 | 𝑝 ) ⁢ 𝑑 𝑥 < 2 𝑝 𝑍 0 ⁢ 𝜉 .

Due to the arbitrariness of 𝜉 ,

𝔲 𝑛 → 𝑢 in 𝐿 𝑝 ⁢ ( ℝ 𝑁 ) . By interpolation, 𝔲 𝑛 → 𝑢 in 𝐿 𝜈 ⁢ ( ℝ 𝑁 ) for every 𝜈 ∈ [ 𝑝 , + ∞ ) , as desired. Arguing similarly as in the proof of [43, Lemma 22], we can obtain that

∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐺 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑦 ) ) ⁢ 𝑔 ⁢ ( 𝔲 𝑛 ⁢ ( 𝑥 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 → ∫ ℝ 𝑁 ∫ ℝ 𝑁 𝐺 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) ⁢ 𝑔 ⁢ ( 𝑢 ⁢ ( 𝑥 ) ) | 𝑥 − 𝑦 | 𝜇 ⁢ 𝑑 𝑦 ⁢ 𝑑 𝑥 .

(4.14)

Therefore, it follows from ⟨ ℐ 𝜀 ′ ⁢ ( 𝔲 𝑛 ) , 𝔲 𝑛 ⟩

𝑜 𝑛 ⁢ ( 1 ) and ⟨ ℐ 𝜀 ′ ⁢ ( 𝑢 ) , 𝑢 ⟩

𝑜 𝑛 ⁢ ( 1 ) that

‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 𝑝 + ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 𝑞

‖ 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 𝑝 + ‖ 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 𝑞 + 𝑜 𝑛 ⁢ ( 1 ) .

By the Brézis-Lieb lemma, we get that

‖ 𝔲 𝑛 − 𝑢 ‖ 𝒲 𝜀 ℘

‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑠 , ℘ ℘ − ‖ 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , ℘ ℘ + 𝑜 𝑛 ⁢ ( 1 ) for every ℘ ∈ { 𝑝 , 𝑞 } .

Therefore

‖ 𝔲 𝑛 − 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑝 𝑝 + ‖ 𝔲 𝑛 − 𝑢 ‖ 𝑊 𝑍 , 𝜀 𝑠 , 𝑞 𝑞

𝑜 𝑛 ⁢ ( 1 ) .

Moreover, we obtain 𝔲 𝑛 → 𝑢 in 𝒲 𝜀 . This completes the proof of Lemma 4.7. ∎

Lemma 4.8.

(see Ambrosio [5, Corollary 3.1]) The functional Φ 𝜀 satisfies the ( 𝑃 ⁢ 𝑆 ) 𝑐 𝜀 condition at level 𝑐 𝜀 , for every 𝜀 ∈ ( 0 , 𝜀 0 ) on 𝕊 𝜀 .

5.Multiplicity and concentration of positive solutions of problem ( ⁢ Q ⁢ )

In this section, we shall prove the main results. To this end, we shall give some notations and useful results which will be used later. Fix 𝛿 > 0 . Let 𝔴 be a ground state solution of equation ( 𝒬 𝑍 0 ) , so that ℰ 𝑍 0 ⁢ ( 𝔴 )

𝑐 𝑍 0 and ℰ 𝑍 0 ′ ⁢ ( 𝔴 )

0 . Let 𝜂 be a smooth nonincreasing cut-off function in ℝ 0 + such that 𝜂 ⁢ ( 𝑡 )

1 if 0 ≤ 𝑡 ≤ 𝛿 / 2 and 𝜂 ⁢ ( 𝑡 )

0 if 𝑡 ≥ 𝛿 . For 𝜀

0 and any 𝑦 ∈ ℳ , we define

𝜓 𝜀 , 𝑦 ⁢ ( 𝑥 )

𝜂 ⁢ ( | 𝜀 ⁢ 𝑥 − 𝑦 | ) ⁢ 𝔴 ⁢ ( 𝜀 ⁢ 𝑥 − 𝑦 𝜀 ) , 𝑥 ∈ ℝ 𝑁

and Φ 𝜀 : ℳ → 𝒩 𝜀 is given by Φ 𝜀 ⁢ ( 𝑦 )

𝑡 𝜀 ⁢ 𝜓 𝜀 , 𝑦 , when 𝑡 𝜀

0 satisfies

max 𝑡 ≥ 0 ⁡ ℐ 𝜀 ⁢ ( 𝑡 ⁢ 𝜓 𝜀 , 𝑦 )

ℐ 𝜀 ⁢ ( 𝑡 𝜀 ⁢ 𝜓 𝜀 , 𝑦 ) .

We obtain that Φ 𝜀 ⁢ ( 𝑦 ) has compact support in ℝ 𝑁 for every 𝑦 ∈ ℳ .

Lemma 5.1.

(see Liang et al. [24, Lemma 5.1]) The function Φ 𝜀 has the following property

lim 𝜀 → 0 + ℐ 𝜀 ⁢ ( Φ 𝜀 ⁢ ( 𝑦 ) )

𝑐 𝑍 0 ,  uniformly in  ⁢ 𝑦 ∈ ℳ .

For any 𝛿 > 0 , let 𝜚

𝜚 ⁢ ( 𝛿 )

0 be such that ℳ 𝛿 ⊂ 𝐵 𝜚 ⁢ ( 0 ) . We define the function 𝒳 : ℝ 𝑁 → ℝ 𝑁 by

𝒳 ⁢ ( 𝑥 )

{ 𝑥 ,

if ⁢ | 𝑥 | < 𝜚 ,

𝜚 ⁢ 𝑥 | 𝑥 | ,

if ⁢ | 𝑥 | ≥ 𝜚 .

In what follows, let the barycenter map 𝛽 𝜀 : 𝒩 𝜀 → ℝ 𝑁 be defined by

𝛽 𝜀 ⁢ ( 𝑢 )

∫ ℝ 𝑁 𝒳 ⁢ ( 𝜀 ⁢ 𝑥 ) ⁢ ( | 𝑢 ⁢ ( 𝑥 ) | 𝑝 + | 𝑢 ⁢ ( 𝑥 ) | 𝑞 ) ⁢ 𝑑 𝑥 ∫ ℝ 𝑁 ( | 𝑢 | 𝑝 + | 𝑢 | 𝑞 ) ⁢ 𝑑 𝑥 , 𝑢 ∈ 𝒩 𝜀 .

Arguing as in the similar discussion of  [35, Lemma 14], we obtain the following result.

Lemma 5.2.

The map 𝛽 𝜀 ∘ Φ 𝜀 satisfies the following limit

lim 𝜀 → 0 + 𝛽 𝜀 ⁢ ( Φ 𝜀 ⁢ ( 𝑦 ) )

𝑦 , uniformly in ⁢ 𝑦 ∈ ℳ .

(5.1) Lemma 5.3.

Let 𝜀 𝑛 → 0 + and { 𝔲 𝑛 } 𝑛 ⊂ 𝒩 𝜀 𝑛 satisfy ℐ 𝜀 𝑛 ⁢ ( 𝔲 𝑛 ) → 𝑐 𝑉 0 , as 𝑛 → ∞ . Then there exists a sequence { 𝑦 ~ 𝑛 } 𝑛 ⊂ ℝ 𝑁 such that the sequence 𝔳 𝑛

𝔲 𝑛 ( ⋅ + 𝑦 ~ 𝑛 ) has a subsequence which strongly converges in 𝒲 . Furthermore, up to a subsequence, 𝑦 𝑛

𝜀 ⁢ 𝑦 ~ 𝑛 → 𝑦 ∈ ℳ .

Proof.

Since ⟨ ℐ 𝜀 𝑛 ′ ⁢ ( 𝔲 𝑛 ) , 𝔲 𝑛 ⟩

0 and ℐ 𝜀 𝑛 ⁢ ( 𝔲 𝑛 ) → 𝑐 𝑍 0 , we can see that { 𝔲 𝑛 } 𝑛 is a bounded sequence in 𝒲 . Indeed, by ( 𝑔 3 ) − ( 𝑖 ⁢ 𝑖 ) , we have

ℐ 𝜀 𝑛 ⁢ ( 𝔲 𝑛 )

ℐ 𝜀 𝑛 ⁢ ( 𝔲 𝑛 ) − 1 2 ⁢ 𝑞 ⁢ ⟨ ℐ 𝜀 𝑛 ′ ⁢ ( 𝔲 𝑛 ) , 𝔲 𝑛 ⟩

( 1 𝑝 − 1 2 ⁢ 𝑞 ) ⁢ ‖ 𝔲 𝑛 ‖ 𝑍 , 𝑊 𝜀 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 1 2 ⁢ 𝑞 ⁢ ‖ 𝔲 𝑛 ‖ 𝑍 , 𝑊 𝜀 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

• 1 2 ⁢ ∫ ℝ 𝑁 [ 1 | 𝑥 | 𝜇 ∗ 𝐺 ⁢ ( 𝜀 ⁢ 𝑦 , 𝔲 𝑛 ⁢ ( 𝑦 ) ) ] ⁢ ( 1 𝑞 ⁢ 𝑔 ⁢ ( 𝜀 ⁢ 𝑥 , 𝑢 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝔲 𝑛 ⁢ ( 𝑥 ) − 𝐺 ⁢ ( 𝜀 ⁢ 𝑥 , 𝔲 𝑛 ⁢ ( 𝑥 ) ) ) ⁢ 𝑑 𝑥 ≥ 1 2 ⁢ 𝑞 ⁢ ‖ 𝔲 𝑛 ‖ 𝒲 𝜀 𝑛 .

Therefore

lim sup 𝑛 → ∞ ‖ 𝔲 𝑛 ‖ 𝒲 𝜀 ≤ 2 ⁢ 𝑞 ⁢ 𝑐 𝑍 0 .

(5.2)

By conditions ( 𝒵 1 ) and ( 𝒵 2 ) , we obtain that

‖ 𝑢 ‖ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) ≤ min { 1 , 𝑉 0 } 1 𝑝 ⁢ ‖ 𝑢 ‖ 𝒲 𝜀 .

(5.3)

Together with the continuous embedding 𝒲 𝜀 𝑛 ↪ 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) , we get that { 𝔲 𝑛 } 𝑛 is bounded in 𝒲 .

Next, we claim that there exist a sequence { 𝑦 ~ 𝑛 } 𝑛 ⊂ ℝ 𝑁 and constants 𝑅 , 𝛿

0 such that

lim inf 𝑛 → + ∞ ∫ 𝐵 𝑅 ⁢ ( 𝑦 ~ 𝑛 ) | 𝑢 𝑛 | 𝑞 ⁢ 𝑑 𝑥 ≥ 𝛿

0 .

(5.4)

Suppose to the contrary, that for every 𝑅

0 , we deduce that

lim 𝑛 → + ∞ sup 𝑦 ∈ ℝ 𝑁 ∫ 𝐵 𝑅 ⁢ ( 𝑦 ) | 𝔲 𝑛 | 𝑞 ⁢ 𝑑 𝑥

0 .

Together with Lemma 3.8, we obtain that 𝔲 𝑛 → 0 in 𝐿 𝜈 ⁢ ( ℝ 𝑁 ) , for every 𝜈 ∈ ( 𝑝 , + ∞ ) . Using Lemma 3.1 and (5.2), we get

lim 𝑛 → ∞ ∫ ℝ 𝑁 [ 1 | 𝑥 | 𝜇 ∗ 𝐺 ⁢ ( 𝜀 ⁢ 𝑦 , 𝔲 𝑛 ⁢ ( 𝑦 ) ) ] ⁢ 𝐺 ⁢ ( 𝜀 ⁢ 𝑥 , 𝔲 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝑑 𝑥

0 .

Together with 𝔲 𝑛 ∈ 𝒩 𝜀 𝑛 , we have 𝔲 𝑛 → 0 in 𝒲 𝜀 . Hence ℐ 𝜀 ⁢ ( 𝑢 𝑛 ) → 0 , which is impossible due to 𝑐 𝑍 0 > 0 . We now suppose 𝑣 𝑛 ⁢ ( 𝑥 )

𝔲 𝑛 ⁢ ( 𝑥 + 𝑦 ~ 𝑛 ) , so { 𝑣 𝑛 } 𝑛 is bounded in 𝒲 𝜀 . Therefore, we may assume that 𝑣 𝑛 ⇀ 𝑣 in 𝒲 , as 𝑛 → ∞ . It follows from (5.4) that 𝑣 ≠ 0 .

Let 𝑡 𝑛 > 0 be such that 𝑣 ~ 𝑛 := 𝑡 𝑛 ⁢ 𝑣 𝑛 ∈ 𝒩 𝑍 0 and 𝑦 𝑛 := 𝜀 𝑛 ⁢ 𝑦 ~ 𝑛 . Applying Lemma 4.4, for every 𝑛 there exists a unique 𝑡 𝔲 𝑛 > 0 such that ℐ 𝜀 𝑛 ⁢ ( 𝑡 𝔲 𝑛 ⁢ 𝔲 𝑛 )

sup 𝑡 ≥ 0 ⁢ ℐ 𝜀 𝑛 ⁢ ( 𝑡 ⁢ 𝔲 𝑛 ) . Then 𝑡 𝔲 𝑛 ⁢ 𝔲 𝑛 ∈ 𝒩 𝜀 𝑛 , which yields 𝑡 𝔲 𝑛

1 , due to 𝔲 𝑛 ∈ 𝒩 𝜀 𝑛 . Therefore, sup 𝑡 ≥ 0 ℐ 𝜀 𝑛 ⁢ ( 𝑡 ⁢ 𝔲 𝑛 )

ℐ 𝜀 𝑛 ⁢ ( 𝔲 𝑛 ) . By the change of variable 𝑧

𝑥 + 𝑦 ~ 𝑛 , we deduce that

𝑐 𝑍 0 ≤ ℰ 𝑍 0 ⁢ ( 𝑣 ~ 𝑛 )

1 𝑝 ⁢ ‖ 𝑣 ~ 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 1 𝑞 ⁢ ‖ 𝑣 ~ 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 − ∫ ℝ 𝑁 [ 1 | 𝑥 | 𝜇 ∗ 𝐹 ⁢ ( 𝑣 ~ 𝑛 ⁢ ( 𝑦 ) ) ] ⁢ 𝐹 ⁢ ( 𝑣 ~ 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝑑 𝑥

≤ 1 𝑝 ⁢ ‖ 𝑣 ~ 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 1 𝑞 ⁢ ‖ 𝑣 ~ 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

− ∫ ℝ 𝑁 [ 1 | 𝑥 | 𝜇 ∗ 𝐺 ⁢ ( 𝜀 𝑛 ⁢ 𝑦 + 𝑦 𝑛 , 𝑣 ~ 𝑛 ⁢ ( 𝑦 ) ) ] ⁢ 𝐺 ⁢ ( 𝜀 𝑛 ⁢ 𝑥 + 𝑦 𝑛 , 𝑣 𝑛 ⁢ ( 𝑥 ) ) ⁢ 𝑑 𝑥

ℐ 𝜀 𝑛 ⁢ ( 𝑡 𝑛 ⁢ 𝑢 𝑛 ) ≤ ℐ 𝜀 𝑛 ⁢ ( 𝔲 𝑛 ) ≤ 𝑐 𝑍 0 + 𝑜 𝑛 ⁢ ( 1 )

which yields ℰ 𝑍 0 ⁢ ( 𝑣 ~ 𝑛 ) → 𝑐 𝑍 0 , as 𝑛 → + ∞ . By the fact that { 𝑣 ~ 𝑛 } 𝑛 ⊂ 𝒩 𝑍 0 and ( 𝑓 3 ) , we can pick 𝐶 1

0 such that ‖ 𝑣 ~ 𝑛 ‖ 𝑍 0 ≤ 𝐶 1 , for every 𝑛 ∈ ℕ . In addition, since 𝑣 𝑛 ↛ 0 in 𝒲 , there exists 𝐶 ~ 1

0 such that ‖ 𝑣 𝑛 ‖ 𝑍 0 ≥ 𝐶 ~ 1

0 , for every 𝑛 ∈ ℕ . Therefore,

𝐶 ~ 1 ⁢ 𝑡 𝑛 ≤ ‖ 𝑡 𝑛 ⁢ 𝑣 𝑛 ‖ 𝑍 0 , 𝒲

‖ 𝑣 ~ 𝑛 ‖ 𝑍 0 , 𝒲 ≤ 𝐶 1

which yields 𝑡 𝑛 ≤ 𝐶 1 𝐶 ~ 1 , for every 𝑛 ∈ ℕ . Consequently, going to a subsequence if necessary, we suppose that 𝑡 𝑛 → 𝑡 0 ≥ 0 and 𝑣 ~ 𝑛 ⇀ 𝑣 ~ := 𝑡 0 ⁢ 𝑣 ≢ 0 in 𝒲 and 𝑣 ~ 𝑛 → 𝑣 ~ a.e. in ℝ 𝑁 . If 𝑡 0

0 , then 𝑣 ~ 𝑛 → 0 in 𝒲 . Therefore ℰ 𝑍 0 ⁢ ( 𝑣 ~ ) → 0 , which is impossible since 𝑐 𝑍 0 > 0 , so we get that 𝑡 0 > 0 . Arguing as Proposition 3.1, we obtain ℰ 𝑍 0 ′ ⁢ ( 𝑣 ~ )

0 .

In the sequel, we shall prove that

lim 𝑛 → + ∞ ‖ 𝑣 ~ 𝑛 ‖ 𝑍 0 , 𝒲

‖ 𝑣 ~ ‖ 𝑍 0 , 𝒲 .

(5.5)

Invoking the Fatou lemma, we can deduce

‖ 𝑣 ~ ‖ 𝑍 0 , 𝒲 ≤ lim inf 𝑛 → ∞ ‖ 𝑣 ~ 𝑛 ‖ 𝑍 0 , 𝒲 .

(5.6)

Assume to the contrary, that

‖ 𝑣 ~ ‖ 𝑍 0 , 𝒲 ⁢ < lim sup 𝑛 → ∞ ∥ ⁢ 𝑣 ~ 𝑛 ∥ 𝑍 0 , 𝒲 .

In such a case we would get

𝑐 𝑍 0 + 𝑜 𝑛 ⁢ ( 1 )

ℰ 𝑍 0 ⁢ ( 𝑣 ~ 𝑛 ) − 1 2 ⁢ 𝑞 ⁢ ⟨ ℰ 𝑍 0 ′ ⁢ ( 𝑣 ~ 𝑛 ) , 𝑣 ~ 𝑛 ⟩

( 1 𝑝 − 1 2 ⁢ 𝑞 ) ⁢ ‖ 𝑣 ~ 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 + 1 2 ⁢ 𝑞 ⁢ ‖ 𝑣 ~ 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

• 1 2 ⁢ ∫ ℝ 𝑁 [ 1 | 𝑥 | 𝜇 ∗ 𝐹 ⁢ ( 𝑣 ~ 𝑛 ⁢ ( 𝑦 ) ) ] ⁢ ( 1 𝑞 ⁢ 𝑓 ⁢ ( 𝑣 ~ 𝑛 ) ⁢ 𝑣 ~ 𝑛 − 𝐹 ⁢ ( 𝑣 ~ 𝑛 ) ) ⁢ 𝑑 𝑥

and, by ( 𝑓 3 ) and the Fatou lemma, we would have

𝑐 𝑍 0
≥ lim sup 𝑛 → + ∞ [ ( 1 𝑝 − 1 2 ⁢ 𝑞 ) ⁢ ‖ 𝑣 ~ 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 + 1 2 ⁢ 𝑞 ⁢ ‖ 𝑣 ~ 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 ]

+ 1 2 ⁢ lim inf 𝑛 → + ∞ ∫ ℝ 𝑁 [ 1 | 𝑥 | 𝜇 ∗ 𝐹 ⁢ ( 𝑣 ~ 𝑛 ⁢ ( 𝑦 ) ) ] ⁢ ( 1 𝑞 ⁢ 𝑓 ⁢ ( 𝑣 ~ 𝑛 ) ⁢ 𝑣 ~ 𝑛 − 𝐹 ⁢ ( 𝑣 ~ 𝑛 ) ) ⁢ 𝑑 𝑥

> ( 1 𝑝 − 1 2 ⁢ 𝑞 ) ⁢ ‖ 𝑣 ~ ‖ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞 + 1 2 ⁢ 𝑞 ⁢ ‖ 𝑣 ~ ‖ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

+ 1 2 ⁢ ∫ ℝ 𝑁 [ 1 | 𝑥 | 𝜇 ∗ 𝐹 ⁢ ( 𝑣 ~ ⁢ ( 𝑦 ) ) ] ⁢ ( 1 𝑞 ⁢ 𝑓 ⁢ ( 𝑣 ~ ) ⁢ 𝑣 ~ − 𝐹 ⁢ ( 𝑣 ~ ) ) ⁢ 𝑑 𝑥

ℰ 𝑍 0 ⁢ ( 𝑣 ~ ) − 1 2 ⁢ 𝑞 ⁢ ℰ 𝑍 0 ′ ⁢ ( 𝑣 ~ ) ⁢ ( 𝑣 ~ )

ℰ 𝑍 0 ⁢ ( 𝑣 ~ ) ≥ 𝑐 𝑍 0

which is a contradiction. Therefore, 𝑤 𝑛 ⇀ 𝑤 in 𝒲 and (5.5) implies 𝑣 ~ 𝑛 → 𝑣 ~ in 𝒲 . Moreover, 𝑣 𝑛 → 𝑣 in 𝒲 , as 𝑛 → + ∞ .

In order to complete the proof of this lemma, we explore 𝑦 𝑛

𝜀 𝑛 ⁢ 𝑦 𝑛 . We claim that { 𝑦 𝑛 } 𝑛 allows a subsequence, still denoted the same, such that 𝑦 𝑛 → 𝑦 0 , for some 𝑦 0 ∈ ℳ . In the sequel, we have to verify that the following two claims hold.

Claim 1. { 𝑦 𝑛 } 𝑛 is bounded.

We shall argue by contradiction. Assume that, up to a subsequence, | 𝑦 𝑛 | → ∞ , as 𝑛 → ∞ . Since ⟨ ℐ 𝜀 𝑛 ′ ⁢ ( 𝔲 𝑛 ) , 𝔲 𝑛 ⟩

0 and ℐ 𝜀 𝑛 ⁢ ( 𝔲 𝑛 ) → 𝑐 𝑉 0 , by Lemma 3.7, we can infer that there exists 𝐶 0 ^ ∈ ( 0 , ℏ 0 2 ) such that

| 1 | 𝑥 | 𝜇 ∗ 𝐺 ⁢ ( 𝜀 ⁢ 𝑦 , 𝔲 𝑛 ) | < 𝐶 0 ^ .

Fixed 𝑅

0 such that Λ ⊂ 𝐵 𝑅 ⁢ ( 0 ) , and assume that | 𝑦 𝑛 |

2 ⁢ 𝑅 . Therefore,

| 𝜀 𝑛 ⁢ 𝑥 + 𝑦 𝑛 | ≥ | 𝑦 𝑛 | − | 𝜀 𝑛 ⁢ 𝑥 |

𝑅 ⁢  for every  ⁢ 𝑥 ∈ 𝐵 𝑅 𝜀 𝑛 ⁢ ( 0 ) .

(5.7)

Note that 𝔲 𝑛 ∈ 𝒩 𝜀 𝑛 , so we have

‖ 𝔲 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + ‖ 𝔲 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞
≤ ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑛 𝑠 , 𝑝 𝑝 + ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑛 𝑠 , 𝑞 𝑞

∫ ℝ 𝑁 [ 1 | 𝑥 | 𝜇 ∗ 𝐺 ⁢ ( 𝜀 ⁢ 𝑦 , 𝔲 𝑛 ) ] ⁢ 𝑔 ⁢ ( 𝜀 𝑛 ⁢ 𝑥 , 𝔲 𝑛 ) ⁢ 𝔲 𝑛 ⁢ 𝑑 𝑥 .

Using the change of variables 𝑥 ↦ 𝑥 + 𝑦 ~ 𝑛 and 𝑦 ↦ 𝑦 + 𝑦 ~ 𝑛 , we get

‖ 𝑣 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + ‖ 𝑣 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑛 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + ‖ 𝔲 𝑛 ‖ 𝑊 𝑍 , 𝜀 𝑛 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

≤ ∫ ℝ 𝑁 [ 1 | 𝑥 | 𝜇 ∗ 𝐺 ⁢ ( 𝜀 ⁢ 𝑦 , 𝔲 𝑛 ) ] ⁢ 𝑔 ⁢ ( 𝜀 𝑛 ⁢ 𝑥 , 𝔲 𝑛 ) ⁢ 𝔲 𝑛 ⁢ 𝑑 𝑥

∫ ℝ 𝑁 [ 1 | 𝑥 | 𝜇 ∗ 𝐺 ⁢ ( 𝜀 ⁢ 𝑦 + 𝑦 𝑛 , 𝑣 𝑛 ) ] ⁢ 𝑔 ⁢ ( 𝜀 𝑛 ⁢ 𝑥 + 𝑦 𝑛 , 𝑣 𝑛 ) ⁢ 𝑣 𝑛 ⁢ 𝑑 𝑥 .

By (5.7), the definition of 𝑔 ,

𝑓 ⁢ ( 𝑡 ) ≤ 𝑍 0 ℏ 0 ⁢ 𝑡 𝑝 − 1 ,

𝐶 0 ^ ∈ ( 0 , ℏ 0 2 ) , 𝑣 𝑛 → 𝑣 in 𝒲 𝜀 , the Dominated Convergence Theorem, we have that

‖ 𝑣 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + ‖ 𝑣 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

≤ 𝐶 0 ^ ⁢ ∫ ℝ 𝑁 𝑔 ⁢ ( 𝜀 𝑛 ⁢ 𝑥 + 𝑦 𝑛 , 𝑣 𝑛 ) ⁢ 𝑣 𝑛 ⁢ 𝑑 𝑥

≤ 𝐶 0 ^ ⁢ ∫ 𝐵 𝑅 𝜖 𝑛 ⁢ ( 0 ) 𝑓 ^ ⁢ ( 𝑣 𝑛 ) ⁢ 𝑣 𝑛 ⁢ 𝑑 𝑥 + 𝐶 0 ^ ⁢ ∫ 𝐵 𝑅 𝜖 𝑛 𝑐 ⁢ ( 0 ) 𝑓 ⁢ ( 𝑣 𝑛 ) ⁢ 𝑣 𝑛 ⁢ 𝑑 𝑥

≤ 1 2 ⁢ ∫ 𝐵 𝑅 𝜖 𝑛 ⁢ ( 0 ) 𝑍 0 ⁢ ( | 𝑣 𝑛 | 𝑝 + | 𝑣 𝑛 | 𝑞 ) ⁢ 𝑑 𝑥 + 𝑜 𝑛 ⁢ ( 1 )

which gives

‖ 𝑣 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + ‖ 𝑣 𝑛 ‖ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

𝑜 𝑛 ⁢ ( 1 ) .

Therefore, we have that { 𝑦 𝑛 } 𝑛 is bounded in ℝ 𝑁 .

Claim 2. 𝑦 0 ∈ ℳ .

By Claim 1, up to a subsequence, we can suppose that 𝑦 𝑛 → 𝑦 0 . Once 𝑦 0 ∉ Ω ¯ , which implies the closure of Ω , we can argue as above to get 𝑣 𝑛 → 0 in 𝒲 𝜀 , which is impossible. Therefore, we obtain 𝑦 0 ∈ Ω ¯ . Now, suppose by contradiction that 𝑍 ⁢ ( 𝑦 0 )

𝑍 0 , then by using 𝑣 ~ 𝑛 → 𝑣 in 𝒲 and the Fatou lemma, we can deduce that

𝑐 𝑍 0

ℰ 𝑍 0 ⁢ ( 𝑣 ~ )
< lim inf [ 1 𝑝 ∥ 𝑣 ~ 𝑛 ∥ 𝑍 0 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑝 + 1 𝑞 ∥ 𝑣 ~ 𝑛 ∥ 𝑍 0 , 𝑊 𝑠 , 𝑞 ⁢ ( ℝ 𝑁 ) 𝑞

− 1 2 ∫ ℝ 𝑁 ( 1 | 𝑥 | 𝜇 ∗ 𝐹 ( 𝑣 ~ 𝑛 ) ) 𝐹 ( 𝑣 ~ 𝑛 ) 𝑑 𝑥 ]

≤ lim inf 𝑛 → ∞ ℐ 𝜀 𝑛 ⁢ ( 𝑡 𝑛 ⁢ 𝔲 𝑛 ) ≤ lim inf 𝑛 → ∞ ℐ 𝜀 𝑛 ⁢ ( 𝔲 𝑛 )

𝑐 𝑍 0

which is impossible. Therefore, 𝑍 ⁢ ( 𝑦 0 )

𝑍 0 and 𝑦 0 ∈ Ω ¯ . Thanks to ( 𝒵 2 ) , 𝑦 0 ∉ ∂ Ω , and thus 𝑦 0 ∈ ℳ . This completes the proof of Lemma 5.3. ∎

Let ℎ ⁢ ( 𝜀 ) be any positive function satisfying ℎ ⁢ ( 𝜀 ) → 0 , as 𝜀 → 0 . Define

𝒩 ~ 𝜀

{ 𝑢 ∈ 𝒩 𝜀 : ℐ 𝜀 ⁢ ( 𝑢 ) ≤ 𝑐 𝑍 0 + ℎ ⁢ ( 𝜀 ) } .

For any 𝑦 ∈ ℳ , we deduce from Lemma 5.1 that ℎ ⁢ ( 𝜀 )

| ℐ 𝜀 ⁢ ( Φ 𝜀 ⁢ ( 𝑦 ) ) − 𝑐 𝑍 0 | → 0 , as 𝜀 → 0 . Thus, Φ 𝜀 ⁢ ( 𝑦 ) ∈ 𝒩 𝜀 ~ and 𝒩 𝜀 ~ ≠ 𝜙 for every 𝜀

0 .

Lemma 5.4.

(see Thin [35, Lemma 16]) For every 𝛿

0 ,

lim 𝜀 → 0 + sup 𝑢 ∈ 𝒩 ~ 𝜀 dist ⁢ ( 𝛽 𝜀 ⁢ ( 𝑢 ) , ℳ 𝛿 )

0 .

Lemma 5.5.

Suppose that conditions ( 𝒵 1 ) − ( 𝒵 2 ) and ( 𝑓 1 ) − ( 𝑓 5 ) hold and denote by 𝔳 𝑛 a nontrivial nonnegative solution in  ℝ 𝑁 of

( − Δ ) 𝑁 / 𝑠 𝑠 ⁢ 𝔳 𝑛 + ( − Δ ) 𝑞 𝑠 ⁢ 𝔳 𝑛 + 𝑍 𝑛 ⁢ ( 𝑥 ) ⁢ ( | 𝔳 𝑛 | 𝑁 𝑠 − 2 ⁢ 𝔳 𝑛 + | 𝔳 𝑛 | 𝑞 − 2 ⁢ 𝔳 𝑛 )

[ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝔳 𝑛 ) ] ⁢ 𝑔 ⁢ ( 𝜀 𝑛 ⁢ 𝑥 + 𝜀 𝑛 ⁢ 𝑦 ~ 𝑛 , 𝔳 𝑛 ) ,

(5.8)

where 𝑍 𝑛 ⁢ ( 𝑥 )

𝑍 ⁢ ( 𝜀 𝑛 ⁢ 𝑥 + 𝜀 𝑛 ⁢ 𝑦 ~ 𝑛 ) and 𝜀 𝑛 ⁢ 𝑦 ~ 𝑛 → 𝑦 ∈ ℳ . Then, if ( 𝔳 𝑛 ) 𝑛 is a bounded sequence in 𝒲 satisfying

lim sup 𝑛 → ∞ ‖ 𝑢 𝑛 ‖ 𝜗 , 𝑊 𝑠 , 𝑝 ⁢ ( ℝ 𝑁 ) 𝑁 / ( 𝑁 − 𝑠 ) < 𝛽 ∗ ⁢ 𝔡 ∗ 𝑠 / ( 𝑁 − 𝑠 ) 𝔠 ⁢ 𝛼 0 , with ⁢ 0 < 𝛽 ∗ < 𝛼 ∗

for a suitable constant 𝔠

1 and if 𝔳 𝑛 → 𝔳 in 𝒲 , then each 𝔳 𝑛 ∈ 𝐿 ∞ ⁢ ( ℝ 𝑁 ) and there exists 𝐶

0 such that ‖ 𝔳 𝑛 ‖ 𝐿 ∞ ⁢ ( ℝ 𝑁 ) ≤ 𝐶 for every 𝑛 . Moreover,

lim | 𝑥 | → ∞ 𝔳 𝑛 ⁢ ( 𝑥 )

0 , uniformly in ⁢ 𝑛 .

Proof.

In view of ℐ 𝜀 𝑛 ⁢ ( 𝑢 𝑛 ) ≤ 𝑐 𝑍 0 + ℎ ⁢ ( 𝜀 ) , with ℎ ⁢ ( 𝜀 ) → 0 , as 𝑛 → ∞ . We argue as in the proof of Lemma 5.4 to show that ℐ ⁢ ( 𝜀 𝑛 ) ⁢ ( 𝑢 𝑛 ) → 𝑐 𝑍 0 . Then by Lemma 5.3, there exists { 𝑦 ~ 𝑛 } ⊂ ℝ 𝑁 such that 𝑣 𝑛

𝑢 𝑛 ( ⋅ + 𝑦 ~ 𝑛 ) strongly converges in 𝒲 and 𝜀 𝑛 ⁢ 𝑦 ~ 𝑛 → 𝑦 0 ∈ ℳ . By the boundedness of { 𝑣 𝑛 } 𝑛 in 𝒲 , we can proceed as in the proof of Lemma 3.7 to obtain that there exists 𝐶 ^ 0

0 such that

1 | 𝑥 | 𝜇 ∗ 𝐺 ⁢ ( 𝜀 𝑛 ⁢ 𝑥 + 𝜀 𝑛 ⁢ 𝑦 ~ , 𝑣 𝑛 ) ≤ 𝐶 ^ 0 .

Repeating the same Moser iteration argument developed in the proof of Liang et al. [24, Lemma 5.5], we have that ‖ 𝔳 𝑛 ‖ 𝐿 ∞ ⁢ ( ℝ 𝑁 ) ≤ 𝐶 for every 𝑛 ∈ ℕ . Now, we note that 𝑣 𝑛 satisfies problem (5.8).

Using Ambrosio and R a ˘ dulescu [7, Corollary 2.1 ] and the fact that 𝔳 𝑛 is uniformly bounded in 𝐿 ∞ ⁢ ( ℝ 𝑁 ) ∩ 𝒲 , we can conclude that 𝔳 𝑛 ⁢ ( 𝑥 ) → 0 as | 𝑥 | → ∞ uniformly in 𝑛 ∈ ℕ . This completes the proof of Lemma 5.5. ∎

Proof of Theorem 1.1.

Using the similar arguments to the proof of Ambrosio [3, Theorem 5.2] and [2, Theorem 1.1]. We define 𝛼 𝜀 : ℳ → 𝕊 𝜀 by setting 𝛼 𝜀 ⁢ ( 𝑦 )

𝑚 𝜀 − 1 ⁢ ( Φ 𝜀 ⁢ ( 𝑦 ) ) for every 𝜀

0 . Applying Lemma 5.1 and the definition of Φ 𝜀 , we obtain that

lim 𝜀 → 0 𝜓 𝜀 ⁢ ( 𝛼 𝜀 ⁢ ( 𝑦 ) )

lim 𝜀 → 0 ℐ 𝜀 ⁢ ( Φ 𝜀 ⁢ ( 𝑦 ) )

𝑐 𝑍 0 , uniformly in 𝑦 ∈ ℳ .

Therefore, there exists 𝜀 ~ > 0 such that 𝕊 ¯ 𝜀 := { 𝑤 ∈ 𝕊 𝜀 : 𝜓 𝜀 ⁢ ( 𝑤 ) ≤ 𝑐 𝑍 0 + ℎ ⁢ ( 𝜀 ) } ≠ 𝜙 , for every 𝜀 ∈ ( 0 , 𝜀 ~ ) . With the aid of Lemma 4.4- ( 𝑖 ⁢ 𝑖 ⁢ 𝑖 ) , Lemma 5.1 and Lemma 5.4, for every 𝛿 > 0 , there exists 𝜀 ~

𝜀 ~ 𝛿

0 such that the diagram of continuous mappings

ℳ ⁢ → Φ 𝜀 𝒩 ~ 𝜀 ⁢ → 𝑚 𝜀 − 1 𝕊 𝜀 ⁢ → 𝑚 𝜀 𝒩 ~ 𝜀 ⁢ → 𝛽 𝜀 ℳ 𝛿

𝜀 ~ 𝛿

0

is well-defined, for every 𝜀 ∈ ( 0 , 𝜀 ~ ) . Invoke Lemma 5.2 and take a function 𝜛 ⁢ ( 𝜀 , 𝑦 ) with | 𝜛 ⁢ ( 𝜀 , 𝑦 ) | < 𝛿 2 uniformly in 𝑦 ∈ ℳ , for every 𝜀 ∈ ( 0 , 𝜀 ~ ) such that 𝛽 𝜀 ⁢ ( Φ 𝜀 ⁢ ( 𝑦 ) )

𝑦 + 𝜛 ⁢ ( 𝜀 , 𝑦 ) , for every 𝑦 ∈ ℳ . Therefore, we obtain that ℱ ⁢ ( 𝑡 , 𝑦 )

𝑦 + ( 1 − 𝑡 ) ⁢ 𝜛 ⁢ ( 𝜀 , 𝑦 ) with ( 𝑡 , 𝑦 ) ∈ [ 0 , 1 ] × ℳ is a homotopy between 𝛽 𝜀 ∘ Φ 𝜀

( 𝛽 𝜀 ∘ 𝑚 𝜀 ) ∘ 𝛼 𝜀 and the inclusion map id: ℳ → ℳ 𝛿 . Together with [4, Lemma 6.3.21], we obtain that

𝑐 ⁢ 𝑎 ⁢ 𝑡 𝕊 ~ 𝜀 ⁢ ( 𝕊 𝜀 ~ ) ≥ 𝑐 ⁢ 𝑎 ⁢ 𝑡 ℳ 𝛿 ⁢ ( ℳ ) .

(5.9)

In what follows, we choose a function ℎ ⁢ ( 𝜀 )

0 such that ℎ ⁢ ( 𝜀 ) → 0 , as 𝜀 → 0 and such that 𝑐 𝑍 0 + ℎ ⁢ ( 𝜀 ) is not a critical level for ℐ 𝜀 . From Lemma 4.8, we see that ℐ 𝜀 satisfies the Palais-Smale condition in 𝕊 ~ 𝜀 as 𝜀

0 . Invoking Ambrosio [4, Theorem 6.3.20], we get that 𝜓 𝜀 has at least 𝑐 ⁢ 𝑎 ⁢ 𝑡 𝕊 ~ 𝜀 ⁢ ( 𝕊 ~ 𝜀 ) critical points on 𝕊 ~ 𝜀 . Consequently, by Lemma 4.5 and (5.9), we deduce that ℐ 𝜀 has at least 𝑐 ⁢ 𝑎 ⁢ 𝑡 ℳ 𝛿 ⁢ ( ℳ ) critical points.

Let 𝑢 𝜀 𝑛 be a solution of problem ( 𝒬 𝜀 𝑛 ) , then 𝔳 𝑛 ⁢ ( 𝑥 )

𝑢 𝜀 𝑛 ⁢ ( 𝑥 + 𝑦 ~ 𝑛 ) is also a solution of problem (5.8). Moreover, there exists 𝔳 ∈ 𝒲 , such that, up to a subsequence, 𝔳 𝑛 → 𝔳 in 𝒲 and 𝑦 𝑛

𝜀 𝑛 ⁢ 𝑦 ~ 𝑛 → 𝑦 ∈ ℳ by Lemma 5.3.

We claim that there exists 𝛿 ¯

0 such that ‖ 𝑣 𝑛 ‖ 𝐿 ∞ ⁢ ( ℝ 𝑁 ) ≥ 𝛿 ¯ , for every 𝑛 large enough. In fact, (5.4) in the proof of Lemma 5.3 implies

0 < 𝛽 2 ≤ ∫ 𝐵 𝑟 ⁢ ( 0 ) | 𝔳 𝑛 | 𝑁 / 𝑠 ⁢ 𝑑 𝑥 ≤ | 𝐵 𝑟 ⁢ ( 0 ) | ⁢ ‖ 𝔳 𝑛 ‖ 𝐿 ∞ ⁢ ( ℝ 𝑁 ) 𝑁 / 𝑠

(5.10)

for every 𝑛 large enough. Therefore, we choose 𝛿 ¯

( 𝛽 2 ⁢ | 𝐵 𝑟 ⁢ ( 0 ) | ) 𝑠 / 𝑁 . Applying the fact that 𝔳 𝑛 → 𝔳 in 𝒲 , we have lim | 𝑥 | → ∞ 𝔳 𝑛 ⁢ ( 𝑥 )

0 , uniformly in 𝑛 by Lemma 5.5.

Let 𝔮 𝑛 be a global maximum point of 𝔳 𝑛 . Invoking Lemma 5.5, we see that there exists 𝑅 > 0 such that | 𝔮 𝑛 | ≤ 𝑅 , for every 𝑛 . Consequently, the maximum point of 𝑢 𝜀 𝑛 is denoted by 𝔷 𝜀 𝑛

𝔮 𝑛 + 𝑦 ~ 𝑛 . Furthermore, problem ( 𝒬 ) possesses a nontrivial nonnegative solution 𝑤 𝜀 ⁢ ( 𝑥 )

𝑢 𝜀 ⁢ ( 𝑥 / 𝜀 ) . Therefore, the maximum points 𝜁 𝜀 of 𝑤 𝜀 and 𝔷 𝜀 of 𝑢 𝜀 satisfy 𝜁 𝜀

𝜀 ⁢ 𝔷 𝜀 . We see that

lim 𝜀 → 0 + 𝑍 ⁢ ( 𝜁 𝜀 )

lim 𝑛 → ∞ 𝑍 ⁢ ( 𝜀 𝑛 ⁢ 𝔷 𝜀 𝑛 )

𝑍 0 .

This completes the proof of Theorem 1.1. ∎

Proof of Theorem 1.2.

We know that 𝑤 𝜀 ⁢ ( 𝑥 )

𝑢 𝜀 ⁢ ( 𝑥 / 𝜀 ) is a nontrivial nonnegative solution of problem ( 𝒬 ). Set

𝔳 𝜀 𝑛

𝑤 𝜀 𝑛 ( 𝜀 𝑛 ⋅ + 𝜂 𝜀 𝑛 )

𝑢 𝜀 𝑛 ( ⋅ + 𝔷 𝜀 𝑛 ) .

Therefore, Lemma 5.3 yields that ( 𝔳 𝜀 𝑛 ) 𝑛 → 𝔳 in 𝒲 and 𝔳 is a ground state solution of the following equation

( − Δ ) 𝑝 𝑠 ⁢ 𝑢 + ( − Δ ) 𝑞 𝑠 + 𝑍 0 ⁢ ( | 𝑢 | 𝑝 − 2 ⁢ 𝑣 + | 𝑢 | 𝑞 − 2 ⁢ 𝑢 )

[ | 𝑥 | − 𝜇 ∗ 𝐹 ⁢ ( 𝑢 ⁢ ( 𝑦 ) ) ] ⁢ 𝑓 ⁢ ( 𝑢 ) ⁢ in ⁢ ℝ 𝑁 .

This completes the proof of Theorem 1.2. ∎

Proofs of Corollaries 1.3 and 1.4.

Apply a similar discussion as in Liang et al. [24]. ∎

Acknowledgements. The first two authors were supported by the Science and Technology Development Plan Project of Jilin Province, China (No. 20230101287JC), the Research Foundation of Department of Education of Jilin Province ( No. JJKH20251034KJ), the Young outstanding talents project of Scientific Innovation and entrepreneurship in Jilin (No. 20240601048RC). The third author was supported by the Slovenian Research and Innovation Agency program P1-0292 and grants J1-4031, J1-4001, N1-0278, N1-0114, and N1-0083. We thank the referee for comments and suggestions.

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