Next Article in Journal
Distributed Traffic Signal Optimization at V2X Intersections
Next Article in Special Issue
A Nonlinear ODE Model for a Consumeristic Society
Previous Article in Journal
Examining Differences of Invariance Alignment in the Mplus Software and the R Package Sirt
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Multiplicity of Normalized Solutions for the Fractional Schrödinger Equation with Potentials

1
College of Mathematica and Statistics, Chongqing Jiaotong University, Chongqing 400074, China
2
Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121 Brescia, Italy
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(5), 772; https://doi.org/10.3390/math12050772
Submission received: 18 January 2024 / Revised: 22 February 2024 / Accepted: 1 March 2024 / Published: 5 March 2024
(This article belongs to the Special Issue Problems and Methods in Nonlinear Analysis)

Abstract

:
We are concerned with the existence and multiplicity of normalized solutions to the fractional Schrödinger equation ( Δ ) s u + V ( ε x ) u = λ u + h ( ε x ) f ( u ) i n R N , R N | u | 2 d x = a , , where ( Δ ) s is the fractional Laplacian, s ( 0 , 1 ) , a , ε > 0 , λ R is an unknown parameter that appears as a Lagrange multiplier, h : R N [ 0 , + ) are bounded and continuous, and f is L 2 -subcritical. Under some assumptions on the potential V, we show the existence of normalized solutions depends on the global maximum points of h when ε is small enough.
MSC:
35A15; 35B33; 35Q55

1. Introduction

1.1. Background and Motivation

In this paper, we investigate the multiplicity of normalized solutions for the fractional Schrödinger equation as follows:
i ψ t = ( Δ ) s ψ + V ( x ) ψ g ( | ψ | 2 ) ψ in R N ,
where 0 < s < 1 , i denotes the imaginary unit and ψ ( x , t ) is a complex wave. A solution of (1) is called a standing wave solution if it has the form ψ ( x , t ) = e i λ t u ( x ) for some λ R . ( Δ ) s stands for the fractional Laplacian, and if u is small enough, it can be computed by the following singular integral:
( Δ ) s u = C ( N , s ) P . V . R N u ( x ) u ( y ) | x y | N + 2 s d y .
Here, the symbol P . V . is the Cauchy principal value and C ( N , s ) is a suitable positive normalizing constant.
The operator ( Δ ) s can be seen as the infinitesimal generators of Lévy stable diffusion processes [1], it originates from describing various phenomena in the field of applied science, such as fractional quantum mechanics, the barrier problem, Markov processes, and the phase transition phenomenon, see [2,3,4,5]. In recent decades, the study of the fractional Schrödinger equation has attracted wide attention, see, e.g., [6,7,8,9] and the references therein.
In [10], Alves considered the following class of elliptic problems with a L 2 -subcritical nonlinear term:
Δ u = λ u + h ( ε x ) f ( u ) in R N , R N | u | 2 d x = a .
By using the variational approaches, the author shows that problem (2) admits multiple normalized solutions if ε is small enough. Particularly, the numbers of the normalized solutions are at least the numbers of the global maximum points of h. Moreover, for the following class of problem:
Δ u + V ( ε x ) u = λ u + f ( u ) in R N , R N | u | 2 d x = a ,
a similar result is also obtained for some negative and continuous potential V.
Motivated by [10], our interest is mainly focused on the fractional case with both potentials and weights. Actually, our purpose of this paper is devoted to the multiplicity of normalized solutions for the fractional Schrödinger equation
( Δ ) s u + V ( ε x ) u = λ u + h ( ε x ) f ( u ) in R N , R N | u | 2 d x = a ,
where s ( 0 , 1 ) , a , ε > 0 , λ R is an unknown parameter that appears as a Lagrange multiplier.
In the local case, when s = 1 , the fractional Laplace ( Δ ) s reduces to the local differential operator Δ . If V ( x ) 0 , Jeanjean’s [11] exploited the mountain pass geometry to deal with the existence of normalized solutions in purely L 2 -supercritical, we refer [12,13,14,15] for more results in this type of problems. In [16], they considered the related problem for q = 2 + 4 N . The multiplicity of normalized solutions for the Schrödinger equation or systems has also been extensively investigated, see [17,18,19].
For the non-potential case, a large body of literature is devoted to the following problem:
Δ u = λ u + g ( u ) in R N , R N | u | 2 d x = a 2 .
In particular, for the case g ( u ) = | u | p 1 u , by assuming the H 1 -precompactness of any minimizing sequences, Cazenave and Lions [20] showed the attainability of the L 2 -constraint minimization problem and the orbital stability of global minimizers, it is assumed that E α < 0 for all α > 0 , and then the strict subadditivity condition as follows holds.
E α + β < E α + E β .
However, when dealing with the general function g, it is difficult to show if (5) holds. Shibata [19] proved the subadditivity condition (5) using a scaling argument.
In addition, if V ( x ) 0 , Ikoma and Miyamoto [21] studied the existence and nonexistence of a minimizer of the L 2 -constraint minimization problem as follows:
e ( a ) = inf { E ( u ) | u H 1 ( R N ) , | u | 2 2 = a } ,
where
E ( u ) = 1 2 R N ( | u | 2 d x + V ( x ) | u | 2 ) d x   R N F ( u ) d x ,
V and f satisfy some suitable assumptions. They performed a careful analysis to exclude dichotomy and proved the precompactness of the modified minimizing sequence. When dealing with general nonlinear terms in mass subcritical cases, one can apply the subadditive inequality to prove the compactness of the minimizing sequence.
Zhong and Zou in [22] studied the existence of a ground state normalized solution to Schrödinger equations with potential under different assumptions, and presented a new approach to establish the strict subadditive inequality. Alves and Thin [23] studied the existence of multiple normalized solutions to the following class of elliptic problems:
Δ u + V ( ε x ) u = λ u + f ( u ) in R N , R N | u | 2 d x = a ,
where ε > 0 , V : R N [ 0 , ) is a continuous function, and f is a differentiable function with L 2 -subcritical growth. For the normalized solutions of the nonlinear Schrödinger equation with potential, we also see [24,25,26] and the references therein.
In the case 0 < s < 1 , few results are available. In the paper [27], the author proved some existence and asymptotic results for the fractional nonlinear Schrödinger equation. For the particular case of a combined nonlinearity of power type, namely, f ( t ) = μ | u | q 2 u + | u | p 2 u , h ( x ) = 1 and V ( x ) 0 , i.e., 2 < q < p < 2 s * , Luo and Zhang [28] proved some existence and nonexistence results about the normalized solutions for L 2 -subcritical, L 2 -critical, and L 2 -supercritical. Dinh [29] studied the existence and nonexistence of normalized solutions for the fractional Schrödinger equations as follows:
( Δ ) s u + V ( x ) u = | u | p 2 u , in R N .
By using the concentration–compactness principle, he showed a complete classification for the existence and nonexistence of normalized solutions for the problem (7). For more results about the fractional Schrödinger equations, we can refer to [30,31] and the references therein.

1.2. Main Results

In what follows, we assume f C 1 ( R N , R ) is odd, continuous, and satisfies the following assumptions on f:
( f 1 )
lim t 0 | f ( t ) | | t | q 1 = c > 0 , where 2 < q < p ¯ = 2 + 4 s N ;
( f 2 )
lim t | f ( t ) | | t | p 1 = 0 , where 2 < p < p ¯ = 2 + 4 s N ;
( f 3 )
There exist α , β R satisfying 2 < α β < p ¯ such that
0 < α F ( t ) t f ( t ) F ( t ) β for   any t > 0 .
Moreover, h and V satisfy the following assumptions.
( A 1 )
h C ( R N , R + ) , 0 < h = lim | x | + h ( x ) < max x R N h ( x ) = h ( a i ) for 1 i k with a 1 = 0 and a j a i if i j ;
( A 2 )
V C ( R N , R ) , V ( a i ) = inf x R N V ( x ) < lim | x | + V ( x ) = 0 for 1 i k .
The problem (3) is variational, and the associated energy functional is given by the following:
I ε ( u ) = 1 2 R N | ( Δ ) s 2 u | 2 d x + 1 2 R N V ( ε x ) u 2 d x R N h ( ε x ) F ( u ) d x , u H s ( R N ) ,
with
R N | ( Δ ) s 2 u | 2 d x = R 2 N | u ( x ) u ( y ) | 2 | x y | N + 2 s d x d y .
It is easy to know that I ε C 1 ( H s ( R N ) , R ) and
I ε ( u ) φ = R N ( Δ ) s 2 u ( Δ ) s 2 φ d x + R N V ( ε x ) u φ d x R N h ( ε x ) f ( u ) φ d x , φ H s ( R N ) .
The solutions to (3) can be characterized as critical points of the function I ε ( u ) constrained on the sphere as follows:
S a = u H s ( R N ) : R N | u | 2 d x = a .
Now, we are ready to state the main result of this paper.
Theorem 1.
Suppose ( A 1 ) , ( A 2 ) , ( f 1 ) ( f 3 ) hold, then there exists ε 1 > 0 such that problem (3) admits at least k couples ( u j , λ j ) H s ( R N ) × R of weak solutions for ε ( 0 , ε 1 ) with R N | u j | 2 d x = a , λ < 0 and I ε ( u j ) < 0 for j = 1 , 2 , , k .
The paper is organized as follows: In Section 2, we study the autonomous problem and give some useful results, which will be used later. Section 3 is devoted to the non-autonomous problem. In Section 4, the proof of Theorem 1, is given.

2. The Autonomous Problem

In this section, we focus on the existence of a normalized solution for the autonomous problem.
( Δ ) s u + η u = λ u + μ f ( u ) in R N , R N | u | 2 d x = a ,
where s ( 0 , 1 ) , a , μ > 0 , η 0 , and λ R is an unknown parameter that appears as a Lagrange multiplier. With the assumptions ( f 1 ) ( f 3 ) , it is standard to show that the solutions to (10) can be characterized as critical points of the function as follows:
J ( u ) = 1 2 R N | ( Δ ) s 2 u | 2 d x + η 2 R N u 2 d x μ R N F ( u ) d x ,
restricted to the sphere S a given in (9). Meanwhile, set
J 0 ( u ) = 1 2 R N | ( Δ ) s 2 u | 2 d x μ R N F ( u ) d x ,
and
Υ a = inf S a J 0 ( u ) .
Theorem 2.
Suppose that f satisfies the conditions ( f 1 ) ( f 3 ) . Then, problem (10) has a couple ( u , λ ) solution, where u is positive, radial and λ < η .
The proof of Theorem 2 is standard. For the sake of convenience, we give the details. Before the proof, some lemmas are given below.
Lemma 1.
Assume u is a solution to (10), then u S a P , where
P : = u H s ( R N ) | R N | ( Δ ) s 2 u | 2 d x + N μ s R N F ( u ) d x N μ 2 s R N f ( u ) u d x = 0 .
Proof. 
Let u be a solution (10), then we obtain the following:
R N | ( Δ ) s 2 u | 2 d x + ( η λ ) R N u 2 d x μ R N f ( u ) u d x = 0 .
In addition, one can show that u satisfies the Pohozaev identity as follows:
( N 2 s ) R N | ( Δ ) s 2 u | 2 d x + N ( η λ ) R N u 2 d x 2 N μ R N F ( u ) = 0 ,
which combined with (12) gives the following:
R N | ( Δ ) s 2 u | 2 d x + N μ s R N F ( u ) d x N μ 2 s R N f ( u ) u d x = 0 .
Lemma 2.
Assume ( f 1 ) ( f 2 ) , then we have the following:
( i )
J is bounded from below on S a ;
( i i )
Any minimizing sequence for J is bounded in H s ( R N ) .
Proof. 
(i) According to the assumptions ( f 1 ) ( f 2 ) , there exist C 1 , C 2 > 0 such as the following:
| F ( t ) | C 1 | t | q + C 2 | t | p , t R ,
where q , p ( 2 , 2 + 4 s N ) . By the fractional Gagliardo–Nirenberg–Sobolev inequality [32],
R N | u | α C ( s , N , α ) R N | ( Δ ) s 2 u | 2 N ( α 2 ) 4 s R N | u | 2 α 2 N ( α 2 ) 4 s ,
for some positive constant C ( s , N , α ) > 0 . Then, (13) and (14) give the following:
J ( u ) 1 2 R N ( | ( Δ ) s 2 u | 2 + η u 2 ) d x μ C 1 C ( s , N , q ) q a q 2 N ( q 2 ) 4 s R N | ( Δ ) s 2 u | 2 d x N ( q 2 ) 4 s μ C 2 C ( s , N , p ) p a p 2 N ( p 2 ) 4 s R N | ( Δ ) s 2 u | 2 d x N ( p 2 ) 4 s .
Since q , p ( 2 , 2 + 4 s N ) , we infer that 0 < N ( q 2 ) 4 s , N ( p 2 ) 4 s < 1 . Therefore J ( u ) is bounded from below on S a .
(ii) Since u S a , the conclusion immediately follows from (15). □
The lemma above guarantees the following:
E a = inf u S a J ( u ) ,
is well-defined. Now, we study the properties of the function J defined in (10) restrict to S a and prove Theorem 2.
Lemma 3.
For any a > 0 and η 0 , there holds E a < 0 . In particular, we have E a < η a 2 .
Proof. 
From the condition ( f 1 ) , we know that lim t 0 q F ( t ) t q = c > 0 , then there exists ζ > 0 as follows:
q F ( t ) t q c 2 , t [ 0 , ζ ] .
In fact, taking u S a L ( R N ) as a fixed non-negative function, we define the following:
( τ u ) ( x ) = e N 2 τ u ( e τ x ) , for all x R N and all τ R ,
then τ u S a and
R N F ( ( τ u ) ( x ) ) d x = e N τ R N F ( e N τ 2 u ( x ) ) d x .
Moreover, for τ < 0 and | τ | large enough, we have the following:
0 e N 2 τ u ( x ) ζ , x R N ,
which combines with (16) to give the following:
R N F ( ( τ u ) ( x ) ) d x c 2 q e ( q 2 ) N τ 2 R N | u | q d x .
Hence, as follows:
J ( τ u ) = 1 2 R N | ( Δ ) s 2 ( τ u ) | 2 d x + η a 2 μ R N F ( τ u ) d x 1 2 e 2 s τ R N | ( Δ ) s 2 u | 2 d x + η a 2 μ c 2 q e ( q 2 ) N τ 2 R N | u | q d x .
Since q ( 2 , 2 + 4 s N ) , increasing | τ | if necessary, we deduce the following:
1 2 e 2 s τ R N | ( Δ ) s 2 u | 2 d x μ c 2 q e ( q 2 ) N τ 2 R N | u | q d x = K τ < 0 .
Hence, we obtain the following:
J ( τ u ) K τ + η a 2 < 0 ,
and then E a < 0 . In particular, we have E a < η a 2 . The proof is complete. □
In the following, we adopt some idea introduced in [22] to obtain the subadditive inequality:
Lemma 4.
For μ > 0 , η 0 and let a , b > 0 , then
( i )
a E a is nonincreasing;
( i i )
a E a is continuous;
( i i i )
E a + b E a + E b . If E a or E b can be attained, then E a + b < E a + E b .
Proof. 
( i ) For any ε > 0 small, there exist u S a C 0 ( R N ) and v S b a C 0 ( R N ) such that
J ( u ) E a + ε , J 0 ( v ) Υ b a + ε .
Since u and v have compact support, by using a parallel translation, we can take R large enough, satisfying the following:
v ˜ ( x ) = v ( x R ) , s u p p u s u p p v ˜ = .
Then u + v ˜ S b and
E b J ( u + v ˜ ) = 1 2 R 2 N | ( u + v ˜ ) ( x ) ( u + v ˜ ) ( y ) | 2 | x y | N + 2 s d x d y + η 2 | u + v ˜ | 2 2 μ R N F ( u + v ˜ ) d x = J ( u ) + J ( v ˜ ) + R 2 N ( u ( x ) u ( y ) ) ( v ˜ ( x ) v ˜ ( y ) ) | x y | N + 2 s d x d y .
Suppose that
s u p p u B R ( 0 ) a n d s u p p v ˜ B 3 R ( 0 ) B 2 R ( 0 ) ,
we obtain the following:
R 2 N ( u ( x ) u ( y ) ) ( v ˜ ( x ) v ˜ ( y ) ) | x y | N + 2 s d x d y = R 2 N u ( x ) v ˜ ( x ) 2 u ( x ) v ˜ ( y ) + u ( y ) v ˜ ( y ) | x y | N + 2 s d x d y = R 2 N 2 u ( x ) v ˜ ( y ) | x y | N + 2 s d x d y .
Noting that | x y | R is large enough, we have the following:
E b J ( u + v ˜ ) J ( u ) + J ( v ˜ ) + ε J ( u ) + J 0 ( v ) + ε E a + Υ b a + 3 ε E a + 3 ε .
Here, we used the fact Υ b a < 0 . Then by (18) and the arbitrariness of ε , we obtain that E b E a for any b > a > 0 .
( i i ) We prove the following two claims:
Claim 1: lim h 0 + E a h E a .
For ε > 0 , by the definition of E a , there exists u S a such that
E a J ( u ) E a + ε .
Setting
t = t ( h ) = ( a h a ) 1 N ,
and u t ( x ) = u ( x t ) , we obtain the following:
lim h 0 + t = 1 and | u t | 2 2 = t N a = a h .
Then, by using ( i ) , we have J ( u t ) E a h . In addition,
J ( u t ) = t N 2 s 2 R N | ( Δ ) s 2 u | 2 d x + η t N 2 R N u 2 d x μ t N R N F ( u ) d x = t N 2 s 2 R N | ( Δ ) s 2 u | 2 d x + t N J ( u ) 1 2 R N | ( Δ ) s 2 u | 2 d x = t N J ( u ) + t N 2 s ( 1 t 2 s ) 2 R N | ( Δ ) s 2 u | 2 d x ,
by (19) and (20), we obtain the following:
lim h 0 + E a h E a + ε .
Since ε is arbitrary, the claim holds.
Claim 2: lim h 0 + E a + h E a .
Actually, we consider the case h = 1 n , n N . Take u n S a + 1 n such that J ( u n ) E a + 1 n + 1 n . Set
v n ( x ) : = n a n a + 1 u n ( x ) .
By Lemma 2, we know { u n } is bounded in H s ( R N ) . Moreover, we have the following:
| v n | 2 2 = n a n a + 1 | u n | 2 2 = n a n a + 1 a + 1 n = a .
Hence, we obtain u n S a . On the other hand,
| | v n u n | | H s ( R N ) = 1 n a n a + 1 | | u n | | H s ( R N ) 0 as n + .
Then
E a lim inf n + J ( v n ) = lim inf n + [ J ( u n ) + o n ( 1 ) ] = lim h 0 + E a + h .
Thus, we obtain the following:
lim h 0 + E a + h E a .
Moreover, E a h E a E a + h holds due to ( i ) . Hence, we obtain the following:
lim h 0 + E a h E a lim h 0 + E a + h .
We complete the proof of ( i i ) .
( i i i ) Firstly, we prove that
E θ a θ E a for θ > 1 closing   to 1 .
For any ε > 0 , we take u S a P such that
J ( u ) E a + ε .
Setting u ˜ ( x ) = u ( ν 1 N x ) for ν 1 , by the assumption, we have | u ˜ | 2 2 = ν a and the following:
J ( u ˜ ) = 1 2 R N | ( Δ ) s 2 u ˜ | 2 d x + η 2 R N u ˜ 2 d x μ R N F ( u ˜ ) d x = 1 2 ν N 2 s N R N | ( Δ ) s 2 u | 2 d x + η ν 2 R N u 2 d x μ ν R N F ( u ) d x .
Then, we obtain the following:
d d ν J ( u ˜ ) = N 2 s 2 N ν 2 s N R N | ( Δ ) s 2 u | 2 d x + η 2 R N u 2 d x μ R N F ( u ) d x .
Since u P , we know the following:
R N | ( Δ ) s 2 u | 2 d x + N μ s R N F ( u ) N μ 2 s R N f ( u ) u d x = 0 .
Thus
d d ν J ( u ˜ ) J ( u ) = ( N 2 s 2 N ν 2 s N 1 2 ) R N | ( Δ ) s 2 u | 2 d x = ( N 2 s 2 N ν 2 s N 1 2 ) N μ s R N [ 1 2 f ( u ) u F ( u ) ] d x = ( N 2 s 2 s μ ν 2 s N N μ 2 s ) R N [ 1 2 f ( u ) u F ( u ) ] d x .
Obviously, if ξ > 0 is small, it follows that
N 2 s 2 s μ ν 2 s N N μ 2 s = N μ ( ν 2 s N 1 ) 2 s μ ν 2 s N < 0 , for ν [ 1 , 1 + ξ ] .
Then by (21) and the condition ( f 3 ) , we obtain the following:
d d ν J ( u ˜ ) J ( u ) ( N 2 s 2 s μ ν 2 s N N μ 2 s ) ( α 2 2 ) R N F ( u ) d x < 0 .
Namely, the following:
d d ν J ( u ˜ ) J ( u ) < 0 , for ν [ 1 , 1 + ξ ] .
Therefore, for any θ ( 1 , 1 + ξ ) , we have the following:
J ( u ˜ ) J ( u ) = 1 θ d d ν J ( u ˜ ) d ν < 1 θ J ( u ) d ν = J ( u ) ( θ 1 ) .
Then, it is easy to see that
E θ a J ( u ˜ ) θ J ( u ) θ ( E a + ε ) .
Since the arbitrariness of ε , we obtain the following:
E θ a θ E a , θ ( 1 , 1 + ξ ) .
If E a is attained, we can take u as a minimizer in the above step, and obtain the strictly inequality as follows:
E θ a J ( u ˜ ) < θ J ( u ) = θ E a , θ ( 1 , 1 + ξ ) .
Furthermore, following the proof of ( i ) , since E a is nonincreasing, if E a < 0 , for any b ( a , + ) , we can obtain some uniform ξ > 0 satisfying
E θ c θ E c , θ [ 1 , 1 + ξ ) , c [ a , b ] .
Now, for any a > 0 with E a < 0 and θ > 1 , we take ξ > 0 such that
E ( 1 + k ) c ( 1 + k ) E c , k [ 0 , ξ ) , c [ a , θ b ] .
Then, we may choose k 0 ( 0 , ξ ) and n N such that
( 1 + k 0 ) n < θ < ( 1 + k 0 ) n + 1 ,
and so
E θ a = E ( 1 + k 0 ) θ 1 + k 0 a ( 1 + k 0 ) E θ 1 + k 0 a ( 1 + k 0 ) 2 E θ ( 1 + k 0 ) 2 a ( 1 + k 0 ) n E θ ( 1 + k 0 ) n a ( 1 + k 0 ) n θ ( 1 + k 0 ) n E a = θ E a .
Then, if E a is attained, we obtain that E θ a < θ E a for any θ > 1 . For 0 < b a , we obtain the following:
E a + b = E a + b a a a + b a E a = E a + b a E a = E a + b a E a b b E a + E b .
If E a or E b is attained, we obtain the following:
E a = E a b b < a b E b ,
and thus E a + b < E a + E b . The proof is complete. □
The next compactness lemma on S a is useful in the study of the autonomous problem as well as the non-autonomous problem.
Lemma 5.
Let { u n } S a be a minimizing sequence with respect to E a . Then, for some subsequence, one of the following alternatives holds:
( i )
{ u n } is strongly convergent;
( i i )
There exists { y n } S a with | y n | such that the sequence v n ( x ) = u n ( x + y n ) is strongly convergent to a function v S a with J ( v ) = E a .
Proof. 
By Lemma 2, we know J is coercive on S a , the sequence { u n } is bounded, so u n u in H s ( R N ) for some subsequence. Now we consider the following three possibilities:
( 1 ) If u 0 and | u | 2 2 = b a , we must have b ( 0 , a ) . Set v n = u n u , by the Brézis-Lieb Lemma [33], we obtain the following:
R 2 N | u n ( x ) u n ( y ) | 2 | x y | N + 2 s d x d y = R 2 N | v n ( x ) v n ( y ) | 2 | x y | N + 2 s d x d y + R 2 N | u ( x ) u ( y ) | 2 | x y | N + 2 s d x d y + o n ( 1 ) .
Since F is a C 1 function and has a subcritical growth in the Sobolev sense, we can see the following:
R N F ( u n ) d x = R N F ( u n u ) d x + R N F ( u ) d x + o n ( 1 ) .
Furthermore, setting d n = | v n | 2 2 , and by using
| u n | 2 2 = | v n | 2 2 + | u n | 2 2 + o n ( 1 ) ,
we obtain that d n ( 0 , a ) for n large enough and | v n | 2 2 d with a = b + d . Hence, the following:
E a + o n ( 1 ) = J ( u n ) = 1 2 R 2 N | v n ( x ) v n ( y ) | 2 | x y | N + 2 s d x d y + η 2 | v n | 2 2 μ R N F ( v n ) d x + 1 2 R 2 N | u ( x ) u ( y ) | 2 | x y | N + 2 s d x d y + η 2 | u | 2 2 μ R N F ( u ) d x + o n ( 1 ) = J ( v n ) + J ( u ) + o n ( 1 ) E d n + E b + o n ( 1 ) .
Letting n + , by Lemma 4, we find the following:
E a E d + E b > E a ,
which is absurd. This possibility can not exist.
( 2 ) If | u n | 2 2 = | u | 2 2 = a , it is well-known that u n u in L 2 ( R N ) . Then, by (13) and (14), we have the following:
R N F ( u n u ) d x C 1 R N | u n u | q d x + C 2 R N | u n u | p d x C ( R N | u n u | 2 d x ) q 2 N ( q 2 ) 4 s + C ( R N | u n u | 2 d x ) p 2 N ( p 2 ) 4 s .
Hence, we obtain R N F ( u n u ) d x 0 . From (24), we obtain the following:
R N F ( u n ) d x R N F ( u ) d x .
These limits together with E a = lim n + J ( u n ) provide the following:
E a = lim n + 1 2 R N | ( Δ ) s 2 u n | 2 + η u n 2 ) d x μ R N F ( u ) d x 1 2 R N | ( Δ ) s 2 u | 2 + η u 2 ) d x μ R N F ( u ) d x = J ( u ) E a .
Since u S a , we infer that E a = J ( u ) , then | | u n | | 2 | | u | | 2 , where | | | | denotes the usual norm in H s ( R N ) . Thus, u n u in H s ( R N ) , which implies that ( i ) occurs.
( 3 ) If u 0 ; that is, u n 0 in H s ( R N ) . We claim that there exists β > 0 such that
lim inf n + sup y R N B R ( y ) | u n | 2 d x β , for   some R > 0 .
Indeed, otherwise by Lemma 2.2 of [34], we have u n 0 in L l ( R N ) for all l ( 2 , 2 N N 2 s ) . Thus
E a + o n ( 1 ) = J ( u n ) = 1 2 R N | ( Δ ) s 2 u n | 2 d x + η 2 R N u n 2 d x μ R N F ( u n ) d x = 1 2 R N | ( Δ ) s 2 u n | 2 d x + η 2 R N u n 2 d x + o n ( 1 ) ,
which contradicts the Lemma 3.
Hence, from this case, (25) holds and | y n | + , then we consider u ˜ n ( x ) = u ( x + y n ) , obviously { u ˜ n } S a and it is also a minimizing sequence with respect to J a . Moreover, there exists u ˜ H s ( R N ) { 0 } such that u ˜ n ( x ) u ˜ in H s ( R N ) . Following as in the first two possibilities of the proof, we infer that u ˜ n ( x ) u ˜ in H s ( R N ) , which implies that ( i i ) occurs. This proves the lemma. □
In what follows, we begin to prove Theorem 2.
Proof of Theorem 2.
By Lemmas 2 and 3, there exists a bounded minimizing sequence { u n } S a satisfying J ( u n ) E a . Then applying Lemma 5, there exists u S a such that J ( u ) = E a . By the Lagrange multiplier, there exists λ R such that
J ( u ) = λ Φ ( u ) in ( H s ( R N ) ) ,
where Φ ( u ) : H s ( R N ) R is given by the following:
Φ ( u ) = 1 2 R N | u | 2 d x , u H s ( R N ) .
Therefore, from (26), we have the following:
( Δ ) s u + η u = λ u + μ f ( u ) in R N .
By Lemma 1, we can obtain the following:
( λ η ) R N u 2 d x = R N | ( Δ ) s 2 u | 2 d x μ R N f ( u ) u d x = N μ s R N F ( u ) d x + N μ 2 s R N f ( u ) u d x μ R N f ( u ) u d x = μ s R N ( N F ( u ) N 2 s 2 f ( u ) u ) d x .
Furthermore, according to the condition ( f 3 ) and the claim 3, we must have λ < η .
Next, we will prove that u can be chosen to be positive. Obviously, we have J ( u ) = J ( | u | ) . Moreover, since u S a shows that | u | S a , we infer that
E a = J ( u ) = J ( | u | ) E a ,
which implies that J ( | u | ) = E a , and so, we can replace u by | u | . Furthermore, if u * denotes the symmetrization radial decreasing rearrangement of u (see Section 1 [35]), we observe the following:
R 2 N | u * ( x ) u * ( y ) | 2 | x y | N + 2 s d x d y R 2 N | u ( x ) u ( y ) | 2 | x y | N + 2 s d x d y , R N | u | 2 d x = R N | u * | 2 d x and R N F ( u ) d x = R N F ( u * ) d x ,
then u * S a and J ( u * ) = E a , it follows that we can replace u by u * . Similarly, as in [36], one can show that u ( x ) > 0 for any x R . This completes the proof. □

3. The Non-Autonomous Problem

In this section, we first give some properties of the functional I ε ( u ) given by (8) restricted to the sphere S a , and then prove Theorem 1. Define the following energy functionals:
I ( u ) = 1 2 R N | ( Δ ) s 2 u | 2 d x h R N F ( u ) d x ,
and for i = 1 , 2 , , k ,
I a i ( u ) = 1 2 R N | ( Δ ) s 2 u | 2 d x + V ( a i ) 2 R N u 2 d x h ( a i ) R N F ( u ) d x .
Moreover, denoted by E ε , a , E a i , a , and E , a the following real numbers:
E ε , a = inf u S a I ε ( u ) , E a i , a = inf u S a I a i ( u ) , E , a = inf u S a I ( u ) .
The next two lemmas establish some crucial relations involving the levels E ε , a , E , a , and E a i , a . For any α , β R , set
J α β ( u ) = 1 2 R N | ( Δ ) s 2 u | 2 d x + β 2 R N u 2 d x α R N F ( u ) d x ,
where
E α β , a = inf u S a J α β ( u ) .
Lemma 6.
Fix a > 0 , let 0 < h 1 < h 2 and V 2 < V 1 0 . Then E h 2 V 2 , a < E h 1 V 1 , a < 0 .
Proof. 
The proof is standard and we omit the details. □
Lemma 7.
lim sup ε 0 + E ε , a E a i , a < E , a < 0 , i = 1 , 2 , , k .
Proof. 
By the proof of the Theorem 2, choose u 0 S a such that I a i ( u 0 ) = E a i , a . For 1 i k , we define
u = u 0 x a i ε , x R N .
Then u S a for all ε > 0 , we have the following:
E ε , a I ε ( u ) = 1 2 | ( Δ ) s 2 u 0 | 2 2 + 1 2 R N V ( ε x + a i ) u 0 2 d x R N h ( ε x + a i ) F ( u 0 ) d x .
Letting ε 0 + , by the Lebesgue dominated convergence theorem, we deduce the following:
lim sup ε 0 + E ε , a lim ε 0 + I ε ( u ) = I a i ( u 0 ) = E a i , a .
Noting that E , a can be achieved, due to 0 < h < h ( a i ) and V ( a i ) < 0 , we have the following:
E a i , a < E , a < 0 .
It completes the proof. □
Hence, by Lemma 7, there exists ε 1 > 0 satisfying E ε , a < E , a for all ε ( 0 , ε 1 ) . In the following, we always assume that ε ( 0 , ε 1 ) . The next three lemmas will be used to prove the ( P S ) c condition for I ε restricts to S a at some levels.
Lemma 8.
Assume { u n } S a such that I ε ( u n ) c as n + with c < E , a < 0 , then
δ : = lim inf n sup y R N B ( y , 1 ) | u n ( x ) | 2 d x > 0 .
Proof. 
We argue by contradiction and assume that δ = 0 , then up to a subsequence, we have u n 0 in L l ( R N ) for all l ( 2 , 2 N N 2 s ) , by the Lebesgue dominated convergence theorem and ( f 1 ) ( f 2 ) , we infer the following:
R N h ( ε x ) F ( u n ) d x 0 as n + .
Since V ( x ) 0 as | x | , one can show the following:
R N V ( x ) u n 2 d x = o n ( 1 ) ,
which combined with (30) gives the following:
0 > c = I ε ( u n ) + o ( 1 ) = 1 2 R N | ( Δ ) s 2 u n | 2 d x + o ( 1 ) 0 ,
which is a contradiction. □
Lemma 9.
Under the assumption of Lemma 8, assume u n u in H s ( R N ) , then u 0 .
Proof. 
By Lemma 8, we have that
lim inf n sup y R N B r ( y ) | u n ( x ) | 2 d x > 0 .
So if u 0 , there exists { y n } satisfying | y n | , let u ˜ n = u n ( x + y n ) , obviously { u ˜ n } S a , we have the following:
c + o n ( 1 ) = I ε ( u n ) = 1 2 R N | ( Δ ) s 2 u n | 2 d x + 1 2 R N V ( ε x ) u n 2 d x R N h ( ε x ) F ( u n ) d x = 1 2 R N | ( Δ ) s 2 u ˜ n | 2 d x + 1 2 R N V ( ε x + ε y n ) u ˜ n 2 d x R N h ( ε x + ε y n ) F ( u ˜ n ) d x = I ( u ˜ n ) + 1 2 R N ( V ( ε x + ε y n ) V ) u ˜ n 2 d x + R N ( h h ( ε x + ε y n ) ) F ( u ˜ n ) d x = I ( u ˜ n ) + o n ( 1 ) E , a + o n ( 1 ) ,
which is absurd, because c < E , a < 0 . This proves the lemma. □
Lemma 10.
Let { u n } S a be a ( P S ) c sequence of I ε restricted to S a with c < E , a < 0 and u n u ε in H s ( R N ) . If v n = u n u ε 0 in H s ( R N ) , there exists β > 0 independent of ε ( 0 , ε 1 ) such that
lim inf n + | u n u ε | 2 2 β .
Proof. 
Setting the functional Φ : H s ( R N ) R given by the following:
Φ ( u ) = 1 2 R N | u | 2 d x .
It follows that S a = Φ 1 ( { a / 2 } ) . Then, by Willem (Proposition 5.12 [33]), there exists { λ n } R such that
| | I ε ( u n ) λ n Φ ( u n ) | | ( H s ( R N ) ) 0 as n + .
By the boundedness of { u n } in H s ( R N ) , we know { λ n } is a bounded sequence, thus there exists λ ε such that λ n λ ε as n + . This together with (31) leads to the following:
I ε ( u ε ) λ ε Φ ( u ε ) = 0 in ( H s ( R N ) ) ,
and then
| | I ε ( v n ) λ n Φ ( v n ) | | ( H s ( R N ) ) 0 as n + .
By a straightforward calculation, we have the following:
E , a > lim inf n + I ε ( u n ) = lim inf n + I ε ( u n ) 1 2 I ε ( u n ) u n + 1 2 λ n a + o n ( 1 ) = lim inf n + R N h ( ε x ) 2 f ( u n ) u n d x R N h ( ε x ) F ( u n ) d x + 1 2 λ n a + o n ( 1 ) 1 2 λ ε a ,
implying the following:
λ ε 2 E , a a < 0 , for all ε ( 0 , ε 1 ) .
From (32), we obtain the following:
| ( Δ ) s 2 v n | 2 2 + R N V ( ε x ) | v n | 2 d x λ ε | v n | 2 2 R N h ( ε x ) f ( v n ) v n d x = o n ( 1 ) ,
which combined with (33) gives the following:
| ( Δ ) s 2 v n | 2 2 + R N V ( ε x ) | v n | 2 d x 2 E , a a R N | v n | 2 d x R N h ( ε x ) f ( v n ) v n d x + o n ( 1 ) .
Then
R N | ( Δ ) s 2 v n | 2 d x + C 3 R N | v n | 2 d x C 2 R N | v n | p d x + o n ( 1 ) ,
for some constant C 3 > 0 that does not depend on ε ( 0 , ε 1 ) . If u n u ε in H s ( R N ) ; that is v n 0 in H s ( R N ) , the last inequality ensures that there exists C 0 > 0 independent of ε such that
lim inf n + | v n | p p C 0 .
Then, by the fractional Gagliardo-Nirenberg-sobolev inequality,
R N | v n | α C ( s , N , α ) ( R N | ( Δ ) s 2 v n | 2 ) N ( α 2 ) 4 s ( R N | v n | 2 ) α 2 N ( α 2 ) 4 s ,
for some positive constant C ( s , N , α ) > 0 . We have
lim inf n + R N | v n | p C ( s , N , p ) ( R N | ( Δ ) s 2 v n | 2 ) N ( p 2 ) 4 s ( lim inf n + R N | v n | 2 ) p 2 N ( p 2 ) 4 s C ( s , N , p ) K N ( p 2 ) 4 s ( lim inf n + R N | v n | 2 ) p 2 N ( p 2 ) 4 s ,
where K > 0 is a suitable constant independent of ε ( 0 , ε 1 ) satisfying the condition R N | ( Δ ) s 2 v n | 2 K for all n N . Now, the lemma follows from (36) and (37). □
Next we will give the compactness lemma.
Lemma 11.
Let
0 < ρ 0 < min E , a E a i , a , β a ( E , a E a i , a ) .
Then, for each ε ( 0 , ε 1 ) , the functional I ε satisfies the ( P S ) c condition restricts to S a if c < E a i , a + ρ 0 .
Proof. 
Let { u n } be a ( P S ) c sequence for I ε restricts to S a and c < E a i , a + ρ 0 . It follows that c < E , a < 0 , since { u n } is bounded in H s ( R N ) , we let u n u ε in H s ( R N ) . By Lemma 9, u ε 0 . Denote v n = u n u ε , if u n u ε in H s ( R N ) , the proof is complete. If u n u ε in H s ( R N ) , by Lemma 10,
lim inf n + | v n | 2 2 β .
Set b =   | u ε | 2 2 , d n = | v n | 2 2 and suppose that | v n | 2 2 d > 0 , then we obtain d β > 0 and a = b + d . From d n ( 0 , a ) for n large enough, we obtain the following:
c + o n ( 1 ) = I ε ( u n ) = I ε ( v n ) + I ε ( u ε ) + o n ( 1 ) .
Since v n 0 in H s ( R N ) , we can follow the lines in the proof of Lemma 9. Then
I ε ( v n ) E , d n + o n ( 1 ) ,
by (38) and (39), we obtain the following:
c + o n ( 1 ) = I ε ( u n ) E , d n + I ε ( u ε ) + o n ( 1 ) E , d n + E a i , b + o n ( 1 ) .
Letting n , by the in Equation (22), we have the following:
c E , d + E a i , b d a E , a + b a E a i , a = E a i , a + d a ( E , a E a i , a ) E a i , a + β a ( E , a E a i , a ) ,
which is a contradiction, because c < E a i , a + β a ( E , a E a i , a ) . Therefore, we can obtain u n u ε in H s ( R N ) .
In what follows, let us fix ρ ¯ , r ¯ > 0 satisfying the following:
(1)
B ρ ¯ ( a i ) ¯ B ρ ¯ ( a j ) ¯ for i j and i , j { 1 , k } ;
(2)
i = 1 k B ρ ¯ ( a i ) B r ¯ ( 0 ) ;
(3)
Q ρ ¯ 2 = i = 1 l B ρ ¯ 2 ( a i ) ¯ .
We set the function G ε : H s ( R N ) { 0 } R N by
G ε ( u ) = R N χ ( ε x ) | u | 2 d x R N | u | 2 d x ,
where χ : R N R N denotes the characteristic function; that is, the following:
χ ( x ) = x , if | x | r ¯ , r ¯ x | x | , if | x | > r ¯ .
The next two lemmas will be useful to obtain important ( P S ) sequences for I ε restricted to S a .
Lemma 12.
For ε ( 0 , ε 1 ) , there exists δ 1 > 0 such that if u S a and I ε ( u ) E a i , a + δ 1 , then
G ε ( u ) Q ρ ¯ 2 , ε ( 0 , ε 1 ) .
Proof. 
If the lemma does not occur, there must be δ n 0 , ε n 0 and { u n } S a such that
I ε n ( u n ) E a i , a + δ n and G ε n ( u n ) Q ρ ¯ 2 , ε ( 0 , ε 1 ) .
So we have
E a i , a I a i ( u n ) I ε n ( u n ) E a i , a + δ n ,
then
{ u n } S a and I a i ( u n ) E a i , a .
According to Lemma 5, we have one of the following two cases:
( i )
u n u in H s ( R N ) for some u S a ;
( i i )
There exists { y n } S a with | y n | such that the sequence v n ( x ) = u n ( x + y n ) in H s ( R N ) to some v S a .
For ( i ) : By the Lebesgue dominated convergence theorem,
G ε n ( u n ) = R N χ ( ε x ) | u n | 2 d x R N | u n | 2 d x R N χ ( 0 ) | u | 2 d x R N | u | 2 d x = 0 Q ρ ¯ 2 .
Then G ε n ( u n ) Q ρ ¯ 2 for n large enough, which contradicts (40).
For ( i i ) : We will study the following two cases: ( I ) | ε n y n | + ; ( II ) ε n y n y for some y R N .
If ( I ) holds, the limit v n v in H s ( R N ) provides the following:
I ε n ( u n ) = 1 2 | ( Δ ) s 2 v n | 2 2 + 1 2 R N V ( ε n x + ε n y n ) | v n | 2 d x R N h ( ε n x + ε n y n ) F ( v n ) d x I ( v ) as n + .
Since I ε ( u n ) E a i , a + δ n , we deduce the following:
E , a I ( v ) E a i , a ,
which contradicts E a i , a < E , a in Lemma 7.
If ( II ) holds, by (41), we obtain the following:
I ε n ( u n ) I h ( y ) V ( y ) ( v ) as n + ,
and then E h ( y ) V ( y ) , a I h ( y ) V ( y ) ( v ) E a i , a . By Lemma 6, we must have h ( y ) = h ( a i ) and V ( y ) = V ( a i ) . Namely, y = a i for some i = 1 , 2 , , k . Hence,
G ε n ( u n ) = R N χ ( ε n x ) | u n | 2 d x R N | u n | 2 d x = R N χ ( ε n x + ε n y n ) | v n | 2 d x R N | v n | 2 d x R N χ ( y ) | v | 2 d x R N | v | 2 d x = 0 Q ρ ¯ 2 ,
which implies that G ε n ( u n ) Q ρ ¯ 2 for n large enough, which contradicts (40). The proof is complete. □
From now on, we will use the following notations:
θ ε i := { u S a : | G ε ( u ) a i | ρ ¯ } ;
θ ε i := { u S a : | G ε ( u ) a i | = ρ ¯ } ;
β ε i = inf u θ ε i I ε ( u ) ;
β ¯ ε i = inf u θ ε i I ε ( u ) .
Lemma 13.
Let ρ 0 be defined in lemma 11. Then,
β ε i < E a i , a + ρ 0 a n d β ε i < β ¯ ε i , f o r ε ( 0 , ε 1 ) .
Proof. 
Let u S a satisfy
I a i ( u ) = E a i , a .
For 1 i k , we define the following:
u ^ ε i ( x ) : = u x a i ε , x R N .
Then u ^ ε i ( x ) S a for all ε > 0 and 1 i k . Direct calculations give the following:
I ε ( u ^ ε i ) = 1 2 | ( Δ ) s 2 u | 2 2 + 1 2 R N V ( ε x + a i ) | u | 2 d x R N h ( ε x + a i ) F ( u ) d x ,
and then
lim ε 0 I ε ( u ^ ε i ) = I a i ( u ) = E a i , a ,
we know
G ε ( u ^ ε i ) = R N χ ( ε x + a i ) | u | 2 d x R N | u | 2 d x a i as ε 0 + .
So u ^ ε i ( x ) θ ε i for ε small enough, which combined with (42) implies the following:
I ε ( u ^ ε i ) < E a i , a + δ 1 2 , ε ( 0 , ε 1 ) .
Decreasing δ 1 if necessary, we know the following:
β ε i < E a i , a + ρ 0 , ε ( 0 , ε 1 ) .
For any u θ ε i , which is u S a and | G ε ( u ) a i | = ρ ¯ , we obtain the following: | G ε ( u ) | Q ρ ¯ 2 .
Then by Lemma 12,
I ε ( u ) > E a i , a + δ 1 , for all u θ ε i and ε ( 0 , ε 1 ) ,
which implies the following:
β ¯ ε i = inf u θ ε i I ε ( u ) E a i , a + δ 1 .
Then, we have
β ε i < β ¯ ε i , for all ε ( 0 . ε 1 ) .

4. Proof of Theorem 1

Proof. 
By Lemma 13, for each i { 1 , 2 , , k } , we can use the Ekeland’s variational principle to find a sequence { u n i } S a satisfying
I ε ( u n i ) β ε i and I ε ( w ) I ε ( u n i ) 1 n | | w u n i | | , w θ ε i .
Recalling Lemma 13, β ε i < β ¯ ε i , and so u n i θ ε i θ ε i for n large enough.
Let w T u n i S a , there exists δ > 0 such that the path γ : ( δ , δ ) S a defined by the following:
γ ( t ) = a ( u n i + t w ) | u n i + t w | 2 ,
belongs to C 1 ( ( δ , δ ) , S a ) and satisfies
γ ( t ) θ ε i θ ε i t ( δ , δ ) , γ ( 0 ) = u n i and γ ( 0 ) = w .
Then for any t ( 0 , δ ) ,
I ε ( γ ( t ) ) I ε ( γ ( 0 ) ) t = I ε ( γ ( t ) ) I ε ( u n i ) t 1 n γ ( t ) u n i t = 1 n γ ( t ) γ ( 0 ) t .
Taking the limit of t 0 + , we obtain the following: I ε ( u n i ) ( w ) 1 n | | w | | . Replacing w by w , we obtain | I ε ( u n i ) ( w ) | 1 n | | w | | .
Then, we have the following:
s u p { | I ε ( u n i ) ( w ) | : | | w | | δ n } 1 n .
Consequently,
I ε ( u n i ) β ε i as n + and | | I ε | S a ( u n i ) | | 0 as n + ,
that is, { u n i } is a ( P S ) β ε i for I ε restricts to S a . Since β ε i < E a i , a + ρ 0 , it follows from Lemma 11 that there exists u i such that u n i u i in H s ( R N ) . Then, we obtain the following:
u i θ ε i , I ε ( u n i ) = β ε i and I ε | S a ( u n i ) = 0 .
Moreover,
G ε ( u i ) B ρ ¯ ( a i ) ¯ , G ε ( u j ) B ρ ¯ ( a j ) ¯ ,
and
B ρ ¯ ( a i ) ¯ B ρ ¯ ( a j ) ¯ = f o r i j ,
which implies that u i u j for i j while 1 i , j k , we can understand I ε has at least k nontrivial critical points for any ε ( 0 , ε 1 ) . Therefore, we obtain the theorem. □

Author Contributions

Methodology, M.S.; writing—original draft preparation, X.Z.; writing—review and editing, J.Z.; supervision, M.S. All authors have read and agreed to the published version of the manuscript.

Funding

Marco Squassina is supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni, while Xue Zhang and Jianjun Zhang are supported by Joint Training Base Construction Project for Graduate Students in Chongqing (JDLHPYJD2021016).

Data Availability Statement

Data is contained within the article.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Applebaum, D. Levy processes-from probability to finance and quantum groups. Not. Am. Math. Soc. 2004, 51, 1336–1347. [Google Scholar]
  2. Guan, Q.Y.; Ma, Z.M. Boundary problems for fractional Laplacians. Stoch. Dyn. 2005, 5, 385–424. [Google Scholar] [CrossRef]
  3. Laskin, N. Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 2000, 268, 298–305. [Google Scholar] [CrossRef]
  4. Silvestre, L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 2007, 60, 67–112. [Google Scholar] [CrossRef]
  5. Sire, Y.; Valdinoci, E. Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result. J. Funct. Anal. 2009, 256, 1842–1864. [Google Scholar] [CrossRef]
  6. Liu, Z.S.; Luo, H.J.; Zhang, Z.T. Dancer-Fučik spectrum for fractional Schrödinger operators with a steep potential well on R N . Nonlinear Anal. 2019, 189, 111565. [Google Scholar] [CrossRef]
  7. Servadei, R.; Valdinoci, E. The Brezis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 2015, 367, 67–102. [Google Scholar] [CrossRef]
  8. Shang, X.; Zhang, J.; Yang, Y. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Commun. Pure Appl. Anal. 2014, 13, 567–584. [Google Scholar]
  9. Yan, S.; Yang, J.; Yu, X. Equations involving fractional Laplacian operator: Compactness and application. J. Funct. Anal. 2015, 269, 47–79. [Google Scholar] [CrossRef]
  10. Alves, C.O. On existence of multiple normalized solutions to a class of elliptic problems in whole R N . Z. Angew. Math. Phys. 2022, 73, 97. [Google Scholar] [CrossRef]
  11. Jeanjean, L. Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. 1997, 28, 1633–1659. [Google Scholar] [CrossRef]
  12. Bonheure, D.; Casteras, J.B.; Gou, T.; Jeanjean, L. Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime. Trans. Am. Math. Soc. 2019, 372, 2167–2212. [Google Scholar] [CrossRef]
  13. Guo, Y.; Wang, Z.Q.; Zeng, X.; Zhou, H. Properties of ground states of attractive Gross?Pitaevskii equations with multi-well potentials. Nonlinearity 2018, 31, 957–979. [Google Scholar] [CrossRef]
  14. Hirata, J.; Tanaka, K. Nonlinear scalar field equations with L2-constraint: Mountain pass and symmetric mountain pass approaches. Adv. Nonlinear Stud. 2019, 19, 263–290. [Google Scholar] [CrossRef]
  15. Li, G.; Luo, X. Normalized solutions for the Chern-Simons-Schrödinger equation in R 2 . Ann. Acad. Sci. Fenn. Math. 2017, 42, 405–428. [Google Scholar] [CrossRef]
  16. Miao, C.X.; Xu, G.X.; Zhao, L.F. The dynamics of the 3D radial NLS with the combined terms. Comm. Math. Phys. 2013, 318, 767–808. [Google Scholar] [CrossRef]
  17. Gou, T.X.; Jeanjean, L. Multiple positive normalized solutions for nonlinear Schrödinger systems. Nonlinearity 2018, 31, 2319–2345. [Google Scholar] [CrossRef]
  18. Jeanjean, L.; Lu, S.S. Nonradial normalized solutions for nonlinear scalar field equations. Nonlinearity 2019, 32, 4942–4966. [Google Scholar] [CrossRef]
  19. Shibata, M. Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term. Manuscripta Math. 2014, 143, 221–237. [Google Scholar] [CrossRef]
  20. Cazenave, T.; Lions, P.-L. Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 1982, 85, 549–561. [Google Scholar] [CrossRef]
  21. Ikoma, N.; Miyamoto, Y. Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities. Calc. Var. Partial. Differ. Equ. 2020, 59, 48. [Google Scholar] [CrossRef]
  22. Zhong, X.X.; Zou, W. A new deduction of the strict sub-additive inequality and its application: Ground state normalized solution to Schrödinger equations with potential. Differ. Integral Equ. 2023, 36, 133–160. [Google Scholar] [CrossRef]
  23. Alves, C.O.; Thin, N.V. On existence of multiple normalized solutions to a class of elliptic problems in whole R N via Lusternik–Schnirelmann category. SIAM J. Math. Anal. 2023, 55, 1264–1283. [Google Scholar]
  24. Bartsch, T.; Molle, R.; Rizzi, M.; Verzini, G. Normalized solutions of mass supercritical Schrödinger equations with potential. Commun. Partial Differ. Equ. 2021, 46, 1729–1756. [Google Scholar] [CrossRef]
  25. Ikoma, N.; Miyamoto, Y. The compactness of minimizing sequences for a nonlinear Schrödinger system with potentials. Commun. Contemp. Math. 2023, 25, 2150103. [Google Scholar] [CrossRef]
  26. Molle, R.; Riey, G.; Verzini, G. Normalized solutions to mass supercritical Schrödinger equations with negative potential. J. Differ. Equ. 2020, 333, 302–331. [Google Scholar] [CrossRef]
  27. Yang, T. Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal L2-critical or L2-supercritical perturbation. J. Math. Phys. 2020, 61, 051505. [Google Scholar] [CrossRef]
  28. Luo, H.; Zhang, Z. Normalized solutions to the fractional Schrödinger equations with combined nonlinearities. Calc. Var. Partial. Differ. Equ. 2020, 59, 143. [Google Scholar] [CrossRef]
  29. Dinh, V.D. Existence, non-existence and blow-up behaviour of minimizers for the mass-critical fractional non-linear Schrödinger equations with periodic potentials. Proc. Roy. Soc. Edinburgh Sect. A 2019, 150, 3252–3292. [Google Scholar] [CrossRef]
  30. Appolloni, L.; Secchi, S. Normalized solutions for the fractional NLS with mass supercritical nonlinearity. J. Differ. Equ. 2021, 286, 248–283. [Google Scholar] [CrossRef]
  31. Feng, B.; Ren, J.; Wang, Q. Existence and instability of normalized standing waves for the fractional Schrödinger equations in the L2-supercritical case. J. Math. Phys. 2020, 61, 071511. [Google Scholar] [CrossRef]
  32. Frank, R.L.; Lenzmann, E.; Silvestre, L. Uniqueness of radial solutions for the fractional Laplacian. Commun. Pure Appl. Math. 2016, 69, 1671–1726. [Google Scholar] [CrossRef]
  33. Willem, M. Minimax Theorems; Birkhauser: Boston, MA, USA, 1996. [Google Scholar]
  34. Felmer, P.; Quaas, A.; Tan, J. Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A Math. 2012, 142, 1237–1262. [Google Scholar] [CrossRef]
  35. Almgren, F.J., Jr.; Lieb, E.H. Symmetric decreasing rearrangement is sometimes continuous. J. Am. Math. Soc. 1989, 2, 683–773. [Google Scholar] [CrossRef]
  36. Liu, Z.S.; Zhang, J.J. Multiplicity and concentration of positive solutions for the fractional Schrödinger-Poisson systems with critical growth. ESAIM COCV 2017, 23, 1515–1542. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Zhang, X.; Squassina, M.; Zhang, J. Multiplicity of Normalized Solutions for the Fractional Schrödinger Equation with Potentials. Mathematics 2024, 12, 772. https://doi.org/10.3390/math12050772

AMA Style

Zhang X, Squassina M, Zhang J. Multiplicity of Normalized Solutions for the Fractional Schrödinger Equation with Potentials. Mathematics. 2024; 12(5):772. https://doi.org/10.3390/math12050772

Chicago/Turabian Style

Zhang, Xue, Marco Squassina, and Jianjun Zhang. 2024. "Multiplicity of Normalized Solutions for the Fractional Schrödinger Equation with Potentials" Mathematics 12, no. 5: 772. https://doi.org/10.3390/math12050772

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop