Multiplicity of Solutions for Fractional-Order Differential Equations via the κ ( x ) -Laplacian Operator and the Genus Theory

: In this paper, we investigate the existence and multiplicity of solutions for a class of quasi-linear problems involving fractional differential equations in the χ -fractional space H γ , β ; χ κ ( x ) ( ∆ ) . Using the Genus Theory, the Concentration-Compactness Principle, and the Mountain Pass Theorem, we show that under certain suitable assumptions the considered problem has at least k pairs of non-trivial solutions.

In the present paper, we consider the following class of quasi-linear fractional-order problems with variable exponents: where κ(x) < γ < 1 and type β (0 β 1), λ > 0; κ, n : ∆ → R are Lipschitz functions such that: for all x ∈ ∆; • (p 2 ) The set A = x ∈ ∆ : n(x) = κ * γ (x) is not empty. We now make several assumptions which are detailed below. Let f : ∆ × R → R be a function provided by with ζ ∈ L ∞ (∆) and the function ψ : ∆ × R → R satisfying the following conditions: • (g 1 ) The function g is odd with respect to t, that is, ψ(x, −t) = −ψ(x, t) for all (x, t) ∈ ∆ × R; ψ(x, t) = o(|t| κ(x)−1 ) when |t| → 0 uniformly in x; ψ(x, t) = o(|t| n(x)−1 ) when |t| → ∞ uniformly in x. In addition, consider the following: • (H 1 ) There exists γ > 0 such that with κ(x) = κ > 1 being a real constant. In recent years, there has been growing interest in the study of equations with growth conditions involving variable exponents. The study of such problems has been stimulated by their applications in elasticity [13], electro-rheological fluids [14,15] and image restoration [16]. Lebesgue spaces with variable exponents appeared for the first time as early as 1931 in the work of Orlicz [17]. Applications of this work include clutches, damper and rehabilitation equipment, and more [18][19][20][21].
In 2005, Chabrowkski and Fu [32] considered the following κ(x)-Laplacian problem: . In fact, Chabrowkski and Fu [32] investigated the existence of solutions in W 1,κ(x) 0 (∆) in the superlinear and sublinear cases using the Mountain Pass Theorem (MPT). Subsequently, in 2016, Alves and Ferreira [25] discussed the existence of  solutions for a class of quasi-linear problems involving variable exponents by applying the  Ekeland variational principle and the Mountain Pass Theorem (MPT). In 2022, Taarabti [33] investigated the existence of positive solutions of the following equation: For further details about the parameters and functions of problem (3), see [33]. Over the years, interest in fractional differential equations involving variational techniques has been gaining remarkable popularity and attention from researchers. However, works in this direction remain very limited, especially those involving the χ-Hilfer fractional derivative operators (see [34][35][36]). The pioneering work involving the χ-Hilfer fractional derivative operator, the m-Laplacian, and the Nehari manifold was conducted by da Costa Sousa et al. [37] in 2018. We mention here that classical variational techniques have been applied in partial differential equations involving fractional derivatives; see, for example, [38][39][40][41].
Recently, Zhang and Zhang [42] investigated the properties of the following problem: is a continuous function with sκ(x, y) < N for any (x, y) ∈ ∆ × ∆ and 0 f ∈ L 1 (∆).
Research on fractional Laplacian operators has been fairly productive in recent years. For example, in 2019 Xiang et al. [41] carried out interesting work on a multiplicity of solutions for the variable-order fractional Laplacian equation with variable growth using the MPT and Ekeland's variational principle. For other interesting results on multiplicity of solutions, see the works by Ayazoglu et al. [43], Xiang et al. [44], Colasuono et al. [45], and Mihailescu and Radulescu [46], as well as the references cited in each of these publications.
In the year 2020, da Costa Sousa et al. [48] considered a mean curvature type problem involving a χ-H operator and variable exponents provided by For further details about the parameters and functions of problem (5), see [48]. Motivated by the above-mentioned works, our main object in this paper is to investigate the multiplicity of solutions to problem (2) by applying the Concentration-Compactness Principle (CCP), the Mountain Pass Theorem (MPT) for paired functionals, and the genus theory. More precisely, we present the following theorem. Theorem 1. Suppose that a function g satisfies the conditions (g 1 ) and (g 2 ) and that the conditions (p 1 ), (p 2 ), (H 1 ), and (H 2 ) are satisfied. Then there exists a sequence {λ k } ⊂ (0, +∞) with λ k > λ k+1 for all k ∈ N such that, for λ ∈ (λ k+1 , λ k ), the problem (2) has at least k pairs of non-trivial solutions.
The rest of this paper is organized as follows. In Section 2, we present important definitions results needed for the further development of the paper. In Section 3, we present technical lemmas and discuss the main result of the paper, that is, the multiplicity of solutions for a class of quasi-linear fractional-order problems in (2) via the Genus Theory, the Concentration-Compactness Principle, and the Mountain Pass Theorem (MPT). Finally, in Section 4, we conclude the paper by presenting several closing remarks and observations.

Remark 3. If we define
and · 1 are equivalent norms in the space H γ,β;χ We now present several definitions and a version of the Lions' Compactness-Concentration Principle in the setting of the χ-H operator.
Definition 2 (see [50]). A finite measure µ on ∆ is a continuous linear functional over C 0 (∆), and the respective norm is defined by We denote by M(∆) and M + (∆) the spaces of finite measures and positive finite measures over ∆, respectively. There are two important convergence properties in M(∆), as detailed below.
Lemma 3 (see [20], Simon Inequality). Let x, y ∈ R N . Then, there is a constant C = C(κ) such that where ·, · denotes the usual inner product in R N . Lemma 4 (see [51], Strauss Compactness Lemma). Let P : R N × R → R and Q : R N × R → R be continuous functions such that Then, for every bounded Borel set B ⊂ R N , Lemma 5 (see [27]). Let µ and ν be two non-negative and bounded measures on ∆ such that for 1 κ(x) < r(x) < ∞ there exists a constant c > 0 satisfying the following inequality: .
Then, there exist x j j∈J ⊂ ∆ and ν j j∈J ⊂ (0, ∞) such that for all x ∈ ∆, and let the functions κ and n be log-Hölder continuous. Then, there is a continuous embedding H γ,β;χ Lemma 6 (see [27]). Let Ξ n → Ξ a.e. and Ξ n → Ξ in L κ(x) (∆). Then, Lemma 7 (sse [27]). For the sequence {ν j } j∈N , let ν be a non-negative and finite Radon measure in ∆ such that ν j → ν weakly * in the sense of measurement. Then, Considering Ξ ∈ C ∞ (∆), from the Poincaré inequality for variable exponents we obtain If we take the limit as j → ∞ in (9), from Lemma 7 we have Theorem 2. Let q, r ∈ C(∆) be such that where {x j } j∈Λ ⊂ ϕ and S is the best constant provided by Proof. Let Ξ ∈ C ∞ (∆) and let v j = (ϕ j ) − ϕ. Then, by using Lemma 6, we have Moreover, by using the Hölder measure inequality (10) and Lemma 5 and after taking limits, we obtain the following representation: Suppose that , and thus |(ϕ j )| n(x) → |x| n(x) strongly in L 1 (B). This is a contradiction to our assumption that x 1 ∈ B.
Next, by applying (9) to Ξ(ϕ j ) and taking into account the fact that (ϕ We consider Ξ ∈ C ∞ 0 (R N ) such that 0 Ξ 1 and assume that it is supported in the unit ball of R N . For a fixed j ∈ I, we let ε > 0 be arbitrary. We set We use .
it follows that Thus, by means of Proposition 3 and Corollary 1, we have meaning that by using the Hölder inequality (see Proposition 5), it follows that and Next, by applying the relation Thus, clearly, we have Therefore, we obtain As κ and n are continuous functions and n(x i ) = κ * γ (x i ), upon letting ε → ∞, we obtain Finally, we prove that From the weakly lower semi-continuity of the norm, we find that .
is orthogonal to µ, we arrive at the desired result. This completes the proof of Theorem 2.

Definition 5.
When E is an abstract Banach space and I ∈ C 1 (E, R), we say that a sequence {v n } in E is a Palais-Smale (PS) sequence for I at level c. We denote this by (PS) c when I(v n ) → c and I (v n ) → c in E * as n → ∞. We say that I satisfies the PS condition at level c when every sequence (PS) c has a subsequence convergent in E.
Theorem 3 (see [52]). Let U and V be an infinite-dimensional space and a finite-dimensional space (U being a Banach space), respectively, with being an even functional with I(0) = 0 satisfying the following conditions: (I 1 ) There are constants δ, σ > 0 such that I(ϕ) δ > 0 for each ϕ ∈ ∂B σ ∩ W; (I 2 ) There exists ϕ > 0 such that I satisfies the condition (PS) c for 0 < c < ϕ; Suppose that {e 1 , · · · , e k } is a basis for the vector space V. For m k, choose inductively e m+1 / ∈ U m := span{e 1 , · · · , e m }. Let R m = R(U m ) and D m = B Rm (0) ∩ U m . Define the following sets: where Σ is the family of the sets Ξ ⊂ U \ {0} such that Ξ is closed in U and symmetric with respect to 0; that is, If 0 < β j j+1 for j > k and, if j > k and j < ϕ, then j is a critical value for I. Furthermore, if j = j+1 = · · · = j+l = < ϕ (j > k), then q(K ) l + 1, where K := ϕ ∈ U : I(ϕ) = and I (ϕ) = 0 .

Main Results
Consider the following energy functional of (2) provided by Thus, using condition (g 1 ), it is shown that . Therefore, the critical points of the energy functional E λ (·) are solutions to problem (2).
In our first result in this section (Lemma 8 below), we prove that the functional E λ (·) satisfies the first geometry of the MPT for even functionals.
By continuity, there is a constant M > 0 such that F(x, t) −M for all x ∈ ∆ and |t| R. Therefore, we obtain thereby proving the claim.

By applying Proposition 3, it follows that
Because dim E < ∞, any two norms in E are equivalent, and thus ∃c > 0 (constant) such that Moreover, as κ + < n − , we have Consequently, for a sufficiently large R > 0, the last inequality implies that We now establish a compactness condition for the functional E λ (·). We prove that the (PS) condition holds true below a certain level, provided that the parameter λ is less than 1.
We note that In the above equality, using the hypotheses (g 2 ) and (16), we obtain for sufficiently large n. Provided ε > 0, note that ∃C ε > 0 such that Upon combining the last inequality with (17), we obtain which implies that where c 5 is a positive constant. If we set . Now, using the definition of E λ (·) together with (11), we obtain Of the growth conditions over g, given ε > 0, there exists C ε > 0 such that for all x ∈ ∆ and t ∈ R, meaning that Therefore, for sufficiently large n, we have , where c 7 is a positive constant. If then it follows from Proposition 4 that .
From the reflexivity of H γ,β;χ On the other hand, the immersion H γ,β;χ Consequently, (ϕ n ) → ϕ in L κ(x) (∆). From the CCP, for Lebesgue spaces with variable exponents (see Theorem 2) there are two non-negative measures µ, ν ∈ M(∆), a countable set Λ, points x j j∈J in A, and sequences µ j j∈J and ν j j∈J ⊂ [0, ∞), and thus we have and Our objective is now to establish a lower estimate for {v i }. For this purpose, we need to prove the following lemma.

Proof. We note that
where c p is the constant provided by the Hölder inequality (see Proposition 5). Thus, upon changing the variable, we have Now, the result follows from Proposition 3 and Lemma 1. We have thus completed the proof of Lemma 11.

Lemma 12.
Under the conditions of Lemma 10, let {(ϕ n )} be a sequence (PS) for the functional E λ (·) and ν j . Then, for each j ∈ Λ, Proof. First, for each ε > 0 let Ξ ε ∈ C ∞ 0 (∆), as in Lemma 11. Therefore, we have for any j ∈ Λ. In addition, by direct calculation we can see that Now, for each δ > 0, by applying the Cauchy-Schwartz and Young inequalities, we obtain Thus, by the Lebesgue Dominated Convergence Theorem and the limit of {(ϕ n )}, we can conclude that Therefore, by applying Lemma 11, it follows that Applying the Strauss Lemma (see Lemma 4) with P (x, t) = ψ(x, t)t and Q(x, t) = |t| κ(x) + |t| n(x) , and using the Lebesgue Dominated Convergence Theorem, we obtain Next, using Equation (19) and Equations (21) to (23), we obtain where C is a constant independent of ε and j. Because Consequently, we have Upon first letting ε → 0 and then δ → 0, we obtain µ j λν j . Hence, This evidently concludes our proof of Lemma 12.
We are now able to demonstrate that the (PS) condition for the functional E λ (·) holds true below a certain level. More precisely, we can prove the following lemma.
Lemma 13. Let the conditions (H 1 ), (H 2 ), (g 1 ), and (g 2 ) be satisfied. If λ < 1, then E λ (·) satisfies the condition (PS) d for Proof. Let the sequence {(ϕ n )} be (PS) d for the energy functional E λ (·) with We observe that Then, it follows from conditions (H 1 ) and (H 2 ) that Now, recalling that . Thus, for λ < 1, we find that which is absurd. Therefore, we must have v j = 0 for all j ∈ Λ, implying that Combining the above limit with Lemma 2, we have Thus, by Proposition 3, (ϕ n ) → ϕ in L n(x) (∆). Now let us denote by {P n } the sequence provided by From the above definition of P n we find that when n → ∞. This implies that On the other hand, because Combining (28) with the Strauss lemma (see Lemma 4), we can conclude that ∆ P n dx → 0 when n → ∞.
Let us now consider the following sets: It follows from Lemma 3 that Consequently, we obtain Now, by applying the Hölder inequality (see Proposition 5), we have , C > 0 (constant). In addition, by direct calculation we can see that is a bounded sequence and From Equations (31), (34), and (35), we deduce that (ϕ n ) → ϕ in H γ,β;χ κ(x) (∆). We thus conclude the proof of Lemma 13.

Lemma 14.
Under conditions (g 1 ) and (H 1 ), there is a sequence {M m } ⊂ (0, ∞) independent of λ with M λ M m+1 such that, for all λ > 0, Proof. First, we observe that Then, by the definition of the set Γ m and by the properties of the infimum of a set, it follows that M m M m+1 . Therefore, as F(x, t) c 1 + 2 |t| n(x) , we can conclude that M m < ∞, proving the result asserted by Lemma 14.
Finally, we prove the main result (Theorem 1) of this paper.
Proof. Proof of Theorem 1: First, λ k for each k ∈ N such that Thus, for λ ∈ (λ k , λ k+1 ] we have Now, by Theorem 3, the levels provided by are the critical values of the functional E λ (·). Thus, if the functional E λ (·) has at least k critical points, meaning that if k j = λ j+1 for some j = 1, 2, · · · , k, it follows from Theorem 3 that K λ j is an infinite set. Consequently, problem (2) has infinite solutions in this case. In either case, therefore, we can see that the problem (2) has at least k pairs of non-trivial solutions. Our proof of Theorem 1 is thus completed.

Remark 4.
We have presented an application of problem (2) in a particular case in the sense of the Liouville-Caputo fractional derivative. However, it is possible for particular choices of β and χ to obtain a new class of particular cases, especially when the limit γ → 1.

Concluding Remarks and Observations
In the investigations presented in this paper, we have successfully addressed a problem involving the multiplicity of solutions for a class of fractional-order differential equations via the κ(x)-Laplacian operator and the Genus Theory. We have first presented several definitions, lemmas and other preliminaries related to the problem. Applying these lemmas and other preliminaries, we have then studied the existence and multiplicity of solutions for a class of quasi-linear problems involving fractional differential equations in the χfractional space H γ,β;χ κ(x) (∆) via the Genus Theory, the Concentration-Compactness Principle (CCP) and the Mountain Pass Theorem (MPT). We have considered a number of corollaries and consequences of the main results in this paper. On the other hand, although we have obtained several results in this paper, many open questions remain about the theory involving the χ-Hilfer fractional derivative. As presented in the introduction, the first work with m-Laplacian via the χ-Hilfer derivative was developed in 2019. It should be noted that there have been few further developments thus far. In this sense, several future questions need to be answered, in particular, those that are itemized below: • Is it possible to discuss the results in Orlicz Spaces and Generalized Orlicz Spaces? • Would it be possible to obtain the existence and multiplicity of solutions of Equation (2) unified with Kirchhoff's problem? • Is it possible to extend these results to more general and global fractional-calculus operators?
Yet another possibility is to further extend this work to the distributed-order Hilfer fractional derivative and the ψ-Hilfer fractional derivative operators with variable exponents. For additional details, see [53,54], and (for recent developments) see [55], which is based upon the Riemann-Liouville, the Liouville-Caputo, and the Hilfer fractional derivatives).
As can be seen, it is a new area and there are many questions yet to be answered. Surely, this calls for the attention of researchers toward the discussing of new and complex problems.