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Article

# Existence and Multiplicity of Solutions for a Bi-Non-Local Problem

by
Jiabin Zuo
1,*,
Tianqing An
2,
Alessio Fiscella
3 and
Chungen Liu
1,*
1
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2
College of Science, Hohai University, Nanjing 210098, China
3
Dipartimento di Matematica e Applicazioni, Universita degli Studi di Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy
*
Authors to whom correspondence should be addressed.
Mathematics 2022, 10(12), 1973; https://doi.org/10.3390/math10121973
Submission received: 9 May 2022 / Revised: 3 June 2022 / Accepted: 6 June 2022 / Published: 8 June 2022

## Abstract

:
The aim of this paper is to investigate the existence and multiplicity of solutions for a bi-non-local problem. Precisely, we show that the above problem admits at least a non-trivial positive energy solution by using the mountain pass theorem. Furthermore, with the help of the fountain theorem, we obtain the existence of infinitely many positive energy solutions, assuming a symmetric condition for g. The main feature and difficulty of this paper is the presence of a double non-local term involving two variable parameters.
MSC:
35R11; 47G20; 35S15; 35J60

## 1. Introduction

Recently, Lorenzo and Hartley in [1] came up with the fractional variable-order derivatives that are used to describe different processes of nonlinear diffusion. Indeed, the variable order problem of non-local integro-differential operators can better reflect the temperature change of an object. Therefore, a large number of researchers have begun to pay attention to fractional variable-order spaces. See [2,3,4,5] and the references therein.
In this paper, we study the following variable $s ( · )$-order fractional Kirchhoff type problem
where the domain $Ω ⊂ R N$ is bounded and smooth with $N > p ( x , y ) s ( x , y )$ for any $( x , y ) ∈ Ω ¯ × Ω ¯$, where $μ$ is a real parameter, $s ( · ) : R N × R N → ( 0 , 1 )$ and $p ( · ) : R N × R N → ( 1 , ∞ )$, exponent $p ¯ ( x ) = p ( x , x )$ for $x ∈ Ω ¯$. Here, $( − Δ ) p ( · ) s ( · )$ is a $p ( · )$-Laplace operator with fractional variable $s ( · )$-order, which is given by
$( − Δ ) p ( · ) s ( · ) φ ( x ) = P . V . ∫ R N | φ ( x ) − φ ( y ) | p ( x , y ) − 2 ( φ ( x ) − φ ( y ) ) | x − y | N + p ( x , y ) s ( x , y ) d y , x ∈ R N ,$
along any $φ ∈ C 0 ∞ ( R N )$, where P.V. is the Cauchy principal value.
For the sake of convenience, we denote
$s − = inf ( x , y ) ∈ R 2 N s ( x , y ) , s + = sup ( x , y ) ∈ R 2 N s ( x , y ) , p − = inf ( x , y ) ∈ R 2 N p ( x , y ) , p + = sup ( x , y ) ∈ R 2 N p ( x , y ) , p s * ( x ) = N p ¯ ( x ) N − s ¯ ( x ) p ¯ ( x ) with p ¯ ( x ) = p ( x , x ) , s ¯ ( x ) = s ( x , x ) , p ¯ − = inf x ∈ R N p ¯ ( x ) , p ¯ + = sup x ∈ R N p ¯ ( x ) .$
The continuous function $M : R + → R +$ fulfills the following conditions.
$( M 1 )$There exist $h 2 ≥ h 1 > 0$ and $β > 1$, with $p + < β p −$, such that
$h 1 t β − 1 ≤ M ( t ) ≤ h 2 t β − 1 for all t ∈ R + .$
Furthermore, we assume the function $g : Ω × R → R$ is continuous and verifies the following two conditions.
$( G 1 )$There exist $c 1 > 0$ and $q ∈ C ( Ω ¯ )$ satisfying
$| g ( x , t ) | ≤ c 1 | t | q ( x ) − 1 , for all ( x , t ) ∈ Ω × R$
and
$β p + < q − = inf x ∈ Ω ¯ q ( x ) < q ( x ) < p s * ( x ) , for all x ∈ Ω ,$
where β is given in $( M 1 )$ above.
$( G 2 )$For $h 1$, $h 2$, and β given in $( M 1 )$, there exist $t 0$ and , such that
$0 < λ G ( x , t ) ≤ t g ( x , t ) , for all t ∈ R with | t | ≥ t 0 , and x ∈ Ω ,$
where $G ( x , t ) = ∫ 0 t g ( x , s ) d s .$
Furthermore, we propose the following condition on the function g.
$( G 3 ) :$$g ( x , − t ) = − g ( x , t )$ for all $( x , t ) ∈ Ω × R$.
In the operator $( − Δ ) p ( · ) s ( · )$, we suppose that continuous functions $s ( · ) : R N × R N → ( 0 , 1 )$ and $p ( · ) : R N × R N → ( 1 , ∞ )$ fulfill
(H1):
$0 < s − ≤ s + < 1 < p − ≤ p +$;
(H2):
$s ( · )$ and $p ( · )$ are symmetric, i.e., $s ( x , y ) = s ( y , x )$ and $p ( x , y ) = p ( y , x )$ for any $( x , y ) ∈ R N × R N$.
Clearly, the operator in (2) reduces to the fractional p-Laplacian $( − Δ ) p s$ as $p ( x , y ) ≡ p$ and $s ( x , y ) ≡ s$; see [6,7,8], and the references therein. In particular, we point out that An et al. in [9] studied the existence of infinitely many solutions for a class of fractional p-Laplacian equations by using the fountain theorem. They also investigated a fractional p-Laplacian system with the help of the Nehari manifold method in [10]. This type of operator has a widespread application in various sciences, such as mechanics [11], finance [12], and so on. For a Kirchhoff situation, we recall [13] where the authors construct a stationary fractional Kirchhoff problem, which is excellent pioneering work. It is worth noting that a typical non-local equation that has attracted attention is the Kirchhoff type equation, which is a physical model given by Kirchhoff [14] in 1883, i.e.,
where u denotes the displacement of the string, $ρ$ denotes the mass density, $P 0$ denotes the initial tension, h denotes the area of the cross section, E denotes the Young’s modulus of the material, and L denotes the length of the string. For more physical phenomena described by classical Kirchhoff theory, see [15].
In addition, in the scope of ordinary differential equation research, non-local problems have also received extensive attention, and we specifically point out two excellent studies [16,17].
Very recently, great interest has been devoted to the investigation of fractional problems involving possibly variable order and variable exponent. The classic example is from Chen, Levine, and Rao [18], and it concerns applications to image restoration. We also refer to [19,20] for a multiplicity result for a problem driven by $( − Δ ) s ( · )$, that is, operator (2) with $p ( x , y ) ≡ 2$. In [21,22,23,24], different approaches are described to handle a fractional operator $( − Δ ) p ( · ) s$, with $s ( x , y ) ≡ s$. Papers [25,26,27,28] introduce variational techniques and properties for the local version of operator $( − Δ ) p ( · ) s$, that is, with the integral in (2) set on $Ω$ instead of $R N$. Finally, the authors in [29,30] try to consider problems involving a variable-order fractional operator with variable exponent $p ( · )$.
Motivated by the above papers, we study a new double variable order fractional Kirchhoff type problem (1). As far as we know, very few papers have studied the infinite number of solutions to such a bi-non-local equation. Indeed, in [22], the authors considered a class of fractional $p ( · )$-Kirchhoff type problems, such as (1) but with $s ( x , y ) ≡ s$, $μ = 0$ and g of a model form. Thus, our main results stated below generalize ([22], Theorems 3.1 and 3.3) in several directions, and somehow the existence results in [21,24].
Theorem 1.
If the conditions $( H 1 ) − ( H 2 )$, $( M 1 )$, and $( G 1 ) − ( G 2 )$ hold, then, there exists $μ * > 0$ such that for any $μ ∈ ( − ∞ , μ * ]$ problem (1) admits a non-trivial weak solution.
By further assuming the symmetric condition $( G 3 )$, we obtain the following multiplicity result.
Theorem 2.
If the conditions $( H 1 ) − ( H 2 )$, $( M 1 )$, and $( G 1 ) − ( G 3 )$ hold, then, for any $μ ∈ R$ problem (1) has infinitely many weak solutions with unbounded positive energy.
The remaining sections are organized as follows. Section 2 introduces some lemmas and knowledge of space theory. Section 3 verifies the Palais–Smale condition. Section 4 gives the proof of Theorem 1. Section 5 proves Theorem 2.

## 2. Functional Analytic Setup and Preliminaries

Let
For any $r ∈ C + ( Ω ¯ )$ we recall the variable exponent Lebesgue space
with the norm
Then $( L r ( · ) ( Ω ) , ∥ · ∥ r ( · ) )$ is a separable reflexive Banach space (see [31], Theorem 2.5 and Corollaries 2.7 and 2.12).
Let $r ˜ ∈ C + ( Ω ¯ )$ be the conjugate exponent of r, that is
Then we have the following Hölder inequality (see [31], Theorem 2.1).
Lemma 1.
Suppose that $u ∈ L r ( · ) ( Ω )$ and $v ∈ L r ˜ ( · ) ( Ω )$. Then
Defining the modular functional $ρ r ( · ) : L r ( · ) ( Ω ) → R$, by
$ρ r ( · ) ( u ) = ∫ Ω | u ( x ) | r ( x ) d x ,$
we have the next crucial result given in [32].
Proposition 1.
Suppose that $u ∈ L r ( · ) ( Ω )$ and ${ u j } ⊂ L r ( · ) ( Ω )$. Then
(1)
$∥ u ∥ r ( · ) < 1 ( r e s p . = 1 , > 1 ) ⇔ ρ r ( · ) ( u ) < 1 ( r e s p . = 1 , > 1 ) ,$
(2)
$∥ u ∥ r ( · ) < 1 ⇒ ∥ u ∥ r ( · ) r + ≤ ρ r ( · ) ( u ) ≤ ∥ u ∥ r ( · ) r − ,$
(3)
$∥ u ∥ r ( · ) > 1 ⇒ ∥ u ∥ r ( · ) r − ≤ ρ r ( · ) ( u ) ≤ ∥ u ∥ r ( · ) r + ,$
(4)
$lim j → ∞ ∥ u j ∥ r ( · ) = 0 ( ∞ ) ⇔ lim j → ∞ ρ r ( · ) ( u j ) = 0 ( ∞ ) ,$
(5)
$lim j → ∞ ∥ u j − u ∥ r ( · ) = 0 ⇔ lim j → ∞ ρ r ( · ) ( u j − u ) = 0 .$
The variable order fractional Sobolev spaces with variable exponent is defined by
with the norm $∥ u ∥ s ( · ) , p ( · ) = ∥ u ∥ p ¯ ( · ) + [ u ] s ( · ) , p ( · )$, where
We define the new variable order fractional Sobolev spaces with variable exponent (see more details in reference [29]):
where $Q : = R 2 N ∖ ( Ω c × Ω c )$. The space X is endowed with the norm
$∥ u ∥ X = ∥ u ∥ p ¯ ( · ) + [ u ] X ,$
where
We notice that the norms $∥ · ∥ s ( · ) , p ( · )$ and $∥ · ∥ X$ are not the same because $Ω × Ω ⊂ Q$ and $Ω × Ω ≠ Q$. This makes $W s ( · ) , p ( · ) ( Ω )$ not sufficient for studying the kind of problem like (1).
For this, we set our Banach space workspace as
which is separable and reflexive (see [30], Proposition 3.7), with respect to the norm
where the last equality is a consequence of the fact that $u = 0$ a.e. in $R N ∖ Ω$.
We are ready to introduce an embedding theorem for $X 0$, given in ([29], Theorem 2.5).
Lemma 2.
Let $Ω ⊂ R N$ be a bounded smooth domain. Let $p ( · )$ and $s ( · )$ satisfy $( H 1 ) − ( H 2 )$, such that $N > p ( x , y ) s ( x , y )$ for all $( x , y ) ∈ Ω ¯ × Ω ¯$. Then for any $r ∈ C + ( Ω ¯ )$ with $1 < r ( x ) < p s * ( x )$ for $x ∈ Ω ¯$, there exists a positive constant $C r = C r ( N , s , p , r , Ω )$ such that
$∥ u ∥ r ( x ) ≤ C r ∥ u ∥ X 0$
for any $v ∈ X 0$. Furthermore, the embedding $X 0 ↪ L r ( · ) ( Ω )$ is compact.
We note that $∥ · ∥ X 0$ and $∥ · ∥ X$ are equivalent norms on $X 0$. We define the fractional modular functional $ϱ p ( · ) s ( · ) : X 0 → R$, by
$ϱ p ( · ) s ( · ) ( u ) = ∫ ∫ R 2 N | u ( x ) − u ( y ) | p ( x , y ) | x − y | N + p ( x , y ) s ( x , y ) d x d y .$
Then, similar to Proposition 1, we get
Proposition 2.
([30], Lemmas 3.4 and 3.5). Suppose that $u ∈ X 0$ and ${ u j } ⊂ X 0$. Then
(1)
$∥ u ∥ X 0 < 1 ( r e s p . = 1 , > 1 ) ⇔ ϱ p ( · ) s ( · ) ( u ) < 1 ( r e s p . = 1 , > 1 ) ,$
(2)
$∥ u ∥ X 0 < 1 ⇒ ∥ u ∥ X 0 p + ≤ ϱ p ( · ) s ( · ) ( u ) ≤ ∥ u ∥ X 0 p − ,$
(3)
$∥ u ∥ X 0 > 1 ⇒ ∥ u ∥ X 0 p − ≤ ϱ p ( · ) s ( · ) ( u ) ≤ ∥ u ∥ X 0 p + ,$
(4)
$lim j → ∞ ∥ u j ∥ X 0 = 0 ( ∞ ) ⇔ lim j → ∞ ϱ p ( · ) s ( · ) ( u j ) = 0 ( ∞ ) ,$
(5)
$lim j → ∞ ∥ u j − u ∥ X 0 = 0 ⇔ lim j → ∞ ϱ p ( · ) s ( · ) ( u j − u ) = 0 .$
A function $u ∈ X 0$ is a weak solution of problem (1), if
for all $ϕ ∈ X 0$, where
$δ p ( · ) ( u ) = ∫ ∫ R 2 N 1 p ( x , y ) | u ( x ) − u ( y ) | p ( x , y ) | x − y | N + p ( x , y ) s ( x , y ) d x d y .$
Considering the following functional associated with problem (1), defined by $I μ : X 0 → R$
where $M ˜ ( t ) = ∫ 0 t M ( τ ) d τ .$ Clearly, it follows from the continuity of M that $I μ$ is well defined and of class $C 1$ on $X 0$. Furthermore, we have
for all u, $ϕ ∈ X 0$. Hence, the weak solutions of problem (1) are the critical points of $I μ$. If such a weak solution exists and is non-trivial, then $μ$ is an eigenvalue of problem (1).
We conclude this section by presenting a technical result that is useful in studying the compactness of $I μ$. The proof of this proposition is similar to ([26], Lemma 4.2) and working on $X 0$.
Proposition 3.
We consider the following functional $A : X 0 → X 0 *$, with $X 0 *$ the dual space of $X 0$, such that
$〈 A ( u ) , ϕ 〉 = ∫ ∫ R 2 N | u ( x ) − u ( y ) | p ( x , y ) − 2 ( u ( x ) − u ( y ) ) ( ϕ ( x ) − ϕ ( y ) ) | x − y | N + p ( x , y ) s ( x , y ) d x d y ,$
for any u, $ϕ ∈ X 0$. Then:
(i)
The operator $A$ is bounded and strictly monotone;
(ii)
$A$ is a mapping of type $( S + )$, that is, if $u j ⇀ u ∈ X 0$ and $lim sup j → ∞ A ( u j ) ( u j − u ) ≤ 0 ,$ then $u j → u ∈ X 0$;
(iii)
$A : X 0 → X 0 *$ is a homeomorphism.
Throughout this paper, for simplicity, we use ${ c i , i ∈ N }$ to denote different non-negative or positive constants. In addition, we denote with $c +$ and $c −$, respectively, the positive part and negative part of a number $c ∈ R$.

## 3. Palais–Smale Condition

We now recall the definition of the Palais–Smale condition. We say that $I μ$ fulfills the Palais–Smale condition at the level $c ∈ R$ if any sequence $u j ⊂ X 0$ fulfilling
$I μ ( u j ) → c and I μ ′ ( u j ) → 0 in X 0 * as j → ∞ ,$
possesses a convergent subsequence in $X 0$.
Lemma 3.
Suppose that $( M 1 )$, $( G 1 ) − ( G 2 )$, and $( H 1 ) − ( H 2 )$ hold. Then for any $μ ∈ R$ the functional $I μ$ fulfills the Palais–Smale condition for any $c ∈ R$.
Proof.
Let $μ ∈ R$. Suppose a sequence ${ u j } ⊂ X 0$ verifying (6). We argue in two steps.
Step 1. We first show that the sequence ${ u j }$$X 0$ is bounded. For this end, by $( M 1 )$, $( G 2 )$, Propositions 1 and 2, and Lemma 2, we get
and recall that $λ > p + ≥ p ¯ ( x ) ≥ p −$ for $x ∈ Ω ¯$, by $( G 2 )$. Thus from (6), there exists $σ μ > 0$ such that as $j → ∞$, there holds
which implies that ${ u j }$ is bounded in $X 0$, as $1 < p − ≤ p ¯ − ≤ p ¯ + ≤ p + < β p − ≤ β p +$ by $( G 1 )$.
Step 2. We will show that ${ u j }$ converges strongly in $X 0$. In view of Lemma 2 and the reflexivity of $X 0$, that there exists a subsequence, still denoted by ${ u j }$, and $u ∈ X 0$ such that
$u j ⇀ u in X 0 , u j → u in L r ( · ) ( Ω ) , u j ( x ) → u ( x ) a . e . in Ω ,$
for any $r ∈ C + ( Ω ¯ )$, with $1 < r ( x ) < p s * ( x )$ for $x ∈ Ω ¯$. Using the Hölder inequality (Lemma 1) and (7) with $r ≡ p ¯$, from $p + < β p + < p s * ( x )$ for $x ∈ Ω ¯$, by $( G 1 )$, we get
$| ∫ Ω | u j | p ¯ ( x ) − 2 u j ( u j − u ) d x | ≤ ∫ Ω | u j | p ¯ ( x ) − 1 | u j − u | d x ≤ 2 ∥ | u j | p ¯ ( x ) − 1 ∥ p ¯ ( x ) p ¯ ( x ) − 1 ∥ u j − u ∥ p ¯ ( x ) → 0 as j → ∞ .$
Therefore,
$lim j → ∞ ∫ Ω | u j | p ¯ ( x ) − 2 u j ( u j − u ) d x = 0 .$
According to $( G 1 )$, (7) with $r ≡ q$ and the Hölder inequality (Lemma 1), we have
$| ∫ Ω g ( x , u j ) ( u j − u ) d x | ≤ c 1 ∫ Ω | u j | q ( x ) − 1 | u j − u | d x ≤ 2 c 1 ∥ | u j | q ( x ) − 1 ∥ q ( x ) q ( x ) − 1 ∥ u j − u ∥ q ( x ) → 0 as j → ∞ ,$
which implies that
$lim j → ∞ ∫ Ω g ( x , u j ) ( u j − u ) d x = 0 .$
By virtue of (6), we get
$〈 I μ ′ ( u j ) , u j − u 〉 → 0 .$
As ${ u j }$ is bounded in $X 0$, and in view of Proposition 2, passing to subsequence, if necessary, we suppose that
$δ p ( · ) ( u j ) → κ ≥ 0 , as j → ∞ .$
If $κ = 0$, then ${ u j }$ strongly converges to $u = 0$ in $X 0$ and the proof is complete.
If $κ > 0$, in view of the function M is continuous, we know
Thus, it follows from $( M 1 )$ that
From (8)–(11), we obtain
Now together with (7), we have
$u j ⇀ u ∈ X 0 , lim sup j → ∞ A ( u j ) ( u j − u ) ≤ 0 .$
Therefore, $A is a mapping of type ( S + )$, which implies that $u j → u$ in $X 0$ from Proposition 3. This concludes the proof of the Palais–Smale compactness condition.

## 4. Proof of Theorem 1

The next two lemmas verify the mountain pass geometry of $I μ$.
Lemma 4.
Suppose that $( M 1 )$, $( G 1 )$, and $( H 1 ) − ( H 2 )$ hold. Then, there exist numbers $ρ > 0$, $μ * = μ * ( ρ ) > 0$ and $α = α ( ρ ) > 0$ such that $I μ ( u ) ≥ α > 0$ for any $u ∈ X 0$ with $∥ u ∥ X 0 = ρ$, and for any $μ ∈ ( − ∞ , μ * ]$.
Proof.
Let $u ∈ X 0$ be such that , with $C p ¯$ and $C q$ given in Lemma 2. In view of $( G 1 )$, Propositions 1 and 2, and Lemma 2, we have that
Let us consider
Then, for any $u ∈ X 0$ with and all $μ ∈ ( − ∞ , μ * ]$, since $β p + < q −$ by $( G 1 )$ we have
concluding the proof. □
Lemma 5.
Suppose that $( M 1 )$, $( G 2 )$, and $( H 1 ) − ( H 2 )$ hold. Then, for any $μ ∈ R$ there exists $u ∈ X 0$ with $∥ u ∥ X 0 > ρ$, where $ρ > 0$ is given in Lemma 4, such that $I μ ( u ) < 0$.
Proof.
Let $μ ∈ R$. By $( G 2 )$, we have that for all $D > 0$, there exists $C D > 0$ such that
$G ( x , t ) ≥ D | t | λ − C D , for all ( x , t ) ∈ Ω × R .$
Take $φ ∈ C 0 ∞ ( R N )$, with $φ > 0$. Let $t > 1$. From (12) and $( M 1 )$ we get
Since $h 2 ≥ h 1$ and $p + ≥ p −$, we get $λ > β p + ≥ β p − > p −$, we deduce that $I μ ( t φ ) → − ∞$ as $t → ∞$. Then for $t > 1$ sufficiently large, we can let $u = t φ$ such that $∥ u ∥ X 0 > ρ$ and $I μ ( u ) < 0$. □
Proof of Theorem 1.
According to Lemmas 3–5, considering also that $I μ ( 0 ) = 0$, our functional $I μ$ fulfills all conditions of the mountain pass theorem. Thus, problem (1) has a non-trivial weak solution. □

## 5. Proof of Theorem 2

The proof of Theorem 2 is based on the application of the fountain theorem, which can be found in [33]. For this, as the real Banach space $X 0$ is reflexive and separable, there exist ${ w i } ⊂ X 0$ and ${ w i * } ⊂ X 0 *$ such that
$X 0 = span { w i : i ∈ N + } ¯ , X 0 * = span { w i * : i ∈ N + } ¯$
and
$X 0 i = span { w i } , Y j = ⨁ i = 1 j X 0 i , Z j = ⨁ i = j ∞ X 0 i ¯ , j = 1 , 2 , …$
Now we are ready to introduce the fountain theorem.
Theorem 3.
([33]) Consider an even functional $I ∈ C 1 ( X 0 , R )$. Assume that for every $j ∈ N$, there exist $ρ j > γ j > 0$ such that
(I1)
$a j : = max u ∈ Y j , ∥ u ∥ X 0 = ρ j I ( u ) ≤ 0 ,$
(I2)
$b j : = inf u ∈ Z j , ∥ u ∥ X 0 = γ j I ( u ) → + ∞ , j → ∞ ,$
(I3)
I fulfills the Palais–Smale condition for every $c > 0$.
Then I has an unbounded sequence of critical values.
Lemma 6.
Suppose that $( H 1 )$$( H 2 )$ hold. Let $r ∈ C + ( Ω ¯ )$, with $1 < r ( x ) < p s * ( x )$ for any $x ∈ Ω ¯$, and denote
Then, $ξ j → 0$ as $j → ∞$
Proof.
By definition of $Z j$ we have $Z j + 1 ⊂ Z j$ and so $0 < ξ j + 1 ≤ ξ j$ for any $j ∈ N$. Thus $ξ j → ξ ≥ 0$ as $j → ∞$. Moreover, by (13) there exists $v j ∈ Z j$ such that
$∥ u j ∥ X 0 = 1 , 0 ≤ ξ j − ∥ u j ∥ r ( · ) < 1 j .$
As ${ u j }$ is bounded in $X 0$, there exists a subsequence of ${ u j }$, still denoted by $u j$, such that $u j ⇀ u$ in $X 0$ and $〈 w i * , u 〉 = lim j → ∞ 〈 w i * , u j 〉 = 0$ for $i ∈ N + .$ Hence we have $u = 0$. Furthermore, by Lemma 2 we obtain that $u j → 0$ in $L r ( · ) ( Ω )$. Therefore, by (14) we have $ξ j → 0$ as $j → ∞$. □
Proof of Theorem 2.
We know that the functional $I μ$ fulfills the Palais–Smale condition by Lemma 3. In what follows, we show that $I μ$ satisfies all conditions of Theorem 3, step by step.
In view of $( G 1 )$ and $( G 2 )$, there exist two positive numbers $c 4$ and $c 5$ such that
$| G ( x , t ) | ≥ c 4 | t | λ − c 5 | t | , for all ( x , t ) ∈ Ω × R .$
For $u ∈ Y j$, with $∥ u ∥ X 0 > 1$, by (15) and $( M 1 )$, we get
On a finite dimensional space $Y j$ all the norms are equivalent, so there are three positive constants $c 6$, $c 7$, and $c 8$ such that
$∥ u ∥ p ¯ ( · ) p ¯ − ≥ c 6 ∥ u ∥ X 0 p ¯ − , ∥ u ∥ λ λ ≥ c 7 ∥ u ∥ X 0 λ , ∥ u ∥ 1 ≥ c 8 ∥ u ∥ X 0 .$
Consequently, from the above inequalities and Propositions 1 and 2, for any $u ∈ Y j$ with , we have
$I μ ( u ) ≤ h 2 β ( p − ) β ∥ u ∥ X 0 β p + − μ − p + c 6 ∥ u ∥ X 0 p − − c 4 c 7 ∥ u ∥ X 0 λ + c 5 c 8 ∥ u ∥ X 0 .$
Since $λ > β p + > p − > 1$ by $( G 2 )$, by choosing large enough, we get
$a j : = max u ∈ Y j , ∥ u ∥ X 0 = ρ j I μ ( u ) ≤ 0 .$
Therefore, the condition $( I 1 )$ of Theorem 3 holds.
According to $( G 1 )$, (13), and Propositions 1 and 2, we get for any $u ∈ Z j$ with $∥ u ∥ X 0 > 1$
We can suppose $ξ j < 1$ for j sufficiently large, in view of Lemma 6. Let us define
then since $γ j → + ∞$ as $j → + ∞$ by Lemma 6 and the fact that $q + ≥ q − > β p + ≥ β p −$ by $( G 1 )$, we can assume that $γ j > 1$ for j larger. Hence, by (16) applied for any $u ∈ Z j$ with $∥ u ∥ X 0 = γ j$, we obtain
as $j → ∞$, by Lemma 6, as also $p + < β p −$ by $( M 1 )$ and $q − > β p + > p +$ by $( G 1 )$. Thus, the condition $( I 2 )$ of Theorem 3 holds. So for j large enough, $b j > 0$. Theorem 3.5 of [33] implies then the existence of a sequence ${ u n } ⊂ X 0$ fulfilling
$I μ ( u n ) → c j and I μ ′ ( u n ) → 0 in X 0 * as n → ∞ .$
It follows from the condition $( I 3 )$ that $c j$ is a critical value of $I μ$. According to $c j ≥ b j$ and $b j → + ∞ , j → ∞$, the proof of Theorem 2 is complete, considering that $I μ$ is even by $( G 3 )$. □

## 6. Conclusions

In this work, the existence of a solution is obtained by the mountain pass lemma, and the existence of infinitely many solutions with positive energy to Equation (1) is established by using the fountain theorem. We consider a class of complex bi-nonlocal problems, which improves the previous results. In order to overcome the difficulties arising from such problems, we use more sophisticated analytical techniques. This kind of equation has a wide range of applications in many fields, and interested readers may refer to the thin obstacle problem [34,35], ultra-relativistic limits of quantum mechanics [11], finance [12] and so on.

## Author Contributions

Writing—original draft, J.Z.; Writing—review and editing, T.A., A.F. and C.L. All authors have read and agreed to the published version of the manuscript.

## Funding

This research was funded by National Natural Science Foundation of China, grant number: 12171108 and Guangdong Basic and Applied basic Research Foundation, grant number: 2020A1515011019.

Not applicable.

Not applicable.

Not applicable.

## Acknowledgments

C. Liu is supported by National Natural Science Foundation of China (Grant No. 12171108), Guangdong Basic and Applied Basic Research Foundation (Grant No. 2020A1515011019), Innovation and development project of Guangzhou University. A. Fiscella is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). A. Fiscella realized the manuscript within the auspices of the INdAM-GNAMPA project titled “Equazioni differenziali alle derivate parziali in fenomeni non lineari” (CUP E55F22000270001) and of the FAPESP Thematic Project titled “Systems and partial differential equations” (2019/02512-5).

## Conflicts of Interest

The authors declare no conflict of interest.

## References

1. Lorenzo, C.F.; Hartley, T.T. Initialized fractional calculus. Int. J. Appl. Math. 2000, 3, 249–265. [Google Scholar]
2. Leopold, H.G. Embedding of function spaces of variable order of differentiation. Czechoslov. Math. J. 1999, 49, 633–644. [Google Scholar] [CrossRef] [Green Version]
3. Lorenzo, C.F.; Hartley, T.T. Variable order and distributed order fractional operators. Nonlinear Dynam. 2002, 29, 57–98. [Google Scholar] [CrossRef]
4. Kikuchi, K.; Negoro, A. On Markov processes generated by pseudodifferentail operator of variable order. Osaka J. Math. 1997, 34, 319–335. [Google Scholar]
5. Ruiz-Medina, M.D.; Anh, V.V.; Angulo, J.M. Fractional generalized random felds of variable order. Stoch. Anal. Appl. 2004, 22, 775–799. [Google Scholar] [CrossRef]
6. Korvenpäxax, J.; Kuusi, T.; Lindgren, E. Equivalence of solutions to fractional p-Laplace type equations. J. Math. Pures Appl. 2019, 132, 1–26. [Google Scholar] [CrossRef] [Green Version]
7. Korvenpäxax, J.; Kuusi, T.; Palatucci, G. The obstacle problem for nonlinear integro-differential operators. Calc. Var. Partial Differ. Equ. 2016, 55, 29. [Google Scholar]
8. Wu, L.; Chen, W. The sliding methods for the fractional p-Laplacian. Adv. Math. 2020, 361, 26. [Google Scholar] [CrossRef]
9. Zuo, J.; An, T.; Li, M. Superlinear Kirchhoff-type problems of the fractional p-Laplacian without the (AR) condition. Bound. Value Probl. 2018, 2018, 180. [Google Scholar] [CrossRef]
10. Zuo, J.; An, T.; Yang, L.; Ren, X. The Nehari manifold for a fractional p-Kirchhoff system involving sign-changing weight function and concave-convex nonlinearities. J. Funct. Spaces 2019, 2019, 7624373. [Google Scholar] [CrossRef]
11. Fefferman, C.; de la Llave, R. Relativistic stability of matter-I. I. Rev. Mat. Iberoam. 1986, 2, 119–213. [Google Scholar] [CrossRef] [Green Version]
12. Cont, R.; Tankov, P. Financial Modelling with Jump Processes; Chapman Hall/CRC: Boca Raton, FL, USA, 2004. [Google Scholar]
13. Fiscella, A.; Valdinoci, E. A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 2014, 94, 156–170. [Google Scholar] [CrossRef]
14. Kirchhoff, G. Mechanik; Teubner: Leipzig, Germany, 1883. [Google Scholar]
15. Villaggio, P. Mathematical Models for Elastic Structures; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
16. Sharifov, Y.A.; Zeynally, F.M.; Zeynally, S.M. Existence and uniqueness of solutions for nonlinear fractional differential equations with two-point boundary conditions. Adv. Math. Model. Appl. 2018, 3, 54–62. [Google Scholar] [CrossRef]
17. Gasimov, Y.S.; Jafari, H.; Mardanov, M.J.; Sardarova, R.A.; Sharifov, Y.A. Existence and uniqueness of the solutions of the nonlinear impulse differential equations with nonlocal boundary conditions. Quaest. Math. 2021, 1–14. [Google Scholar] [CrossRef]
18. Chen, Y.; Levine, S.; Rao, M. Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 2006, 66, 1383–1406. [Google Scholar] [CrossRef] [Green Version]
19. Zuo, J.; Choudhuri, D.; Repovs, D. On critical variable-order Kirchhoff type problems with variable singular exponent. J. Math. Anal. Appl. 2022, 514, 126264. [Google Scholar] [CrossRef]
20. Xiang, M.; Zhang, B.; Yang, D. Multiplicity results for variable—Order fractional Laplacian equations with variable growth. Nonlinear Anal. 2019, 178, 190–204. [Google Scholar] [CrossRef]
21. Azroul, E.; Benkirane, A.; Shimi, M. Eigenvalue problems involving the fractional p(x)-Laplacian operator. Adv. Oper. Theory 2019, 4, 539–555. [Google Scholar] [CrossRef] [Green Version]
22. Azroul, E.; Benkirane, A.; Shimi, M.; Srati, M. On a class of fractional p(x)-Kirchhoff type problems. Appl. Anal. 2019, 100, 383–402. [Google Scholar] [CrossRef]
23. Bahrouni, A. Comparison and sub-supersolution principles for the fractional p(x)-Laplacian. J. Math. Anal. Appl. 2018, 458, 1363–1372. [Google Scholar] [CrossRef]
24. Ho, K.; Kim, Y.H. A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional p(·)-Laplacian. Nonlinear Anal. 2019, 188, 179–201. [Google Scholar] [CrossRef]
25. Ali, K.B.; Hsini, M.; Kefi, K.; Chung, N.T. On a Nonlocal Fractional p(·,·)-Laplacian Problem with Competing Nonlinearities. Complex Anal. Oper. Theory 2019, 13, 1377–1399. [Google Scholar] [CrossRef]
26. Bahrouni, A.; Rădulescu, V. On a new fractional Sobolev space and application to non-local variational problems with variable exponent. Discrete Contin. Dyn. Syst. Ser. S 2018, 11, 379–389. [Google Scholar]
27. Kaufmann, U.; Rossi, J.; Vidal, R. Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians. Electron. J. Qual. Theory Differ. Equ. 2017, 76, 1–10. [Google Scholar] [CrossRef]
28. Zhang, C.; Zhang, X. Renormalized solutions for the fractional p(x)-Laplacian equation with L1 data. Nonlinear Anal. 2020, 190, 111610. [Google Scholar] [CrossRef]
29. Biswas, R.; Tiwari, S. On a class of Kirchhoff-Choquard equations involving variable-order fractional p(·)-Laplacian and without Ambrosetti-Rabinowitz type condition. arXiv 2021, arXiv:2005.09221v1. [Google Scholar]
30. Biswas, R.; Tiwari, S. Variable order nonlocal Choquard problem with variable exponents. Complex Var. Elliptic Equ. 2020, 66, 853–875. [Google Scholar] [CrossRef]
31. Kováčik, O.; Rxaxkosník, J. On spaces Lp(x) and W1,p(x). Czechoslov. Math. J. 1991, 41, 592–618. [Google Scholar] [CrossRef]
32. Diening, L.; Harjulehto, P.; Hästö, P.; Ružička, M. Lebesgue and Sobolev Spaces with Variable Exponents. In Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2011; Volume 2017. [Google Scholar]
33. Willem, M. Minimax Theorems. In Progress in Nonlinear Differential Equations and their Applications; Birkhäuser Boston, Inc.: Boston, MA, USA, 1996; Volume 24. [Google Scholar]
34. Silvestre, L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 2007, 60, 67–112. [Google Scholar] [CrossRef]
35. Milakis, E.; Silvestre, L. Regularity for the nonlinear Signorini problem. Adv. Math. 2008, 217, 1301–1312. [Google Scholar] [CrossRef] [Green Version]
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MDPI and ACS Style

Zuo, J.; An, T.; Fiscella, A.; Liu, C. Existence and Multiplicity of Solutions for a Bi-Non-Local Problem. Mathematics 2022, 10, 1973. https://doi.org/10.3390/math10121973

AMA Style

Zuo J, An T, Fiscella A, Liu C. Existence and Multiplicity of Solutions for a Bi-Non-Local Problem. Mathematics. 2022; 10(12):1973. https://doi.org/10.3390/math10121973

Chicago/Turabian Style

Zuo, Jiabin, Tianqing An, Alessio Fiscella, and Chungen Liu. 2022. "Existence and Multiplicity of Solutions for a Bi-Non-Local Problem" Mathematics 10, no. 12: 1973. https://doi.org/10.3390/math10121973

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