Fractional double-phase nonlocal equation in Musielak-Orlicz Sobolev space

In this paper, we analyze the existence of solutions to a double-phase fractional equation of the Kirchhoﬀ type in Musielak-Orlicz Sobolev space with variable exponents. Our approach is mainly based on the sub-supersolution method and the mountain pass theorem


Introduction and background
In recent years, partial differential equations and variational problems using a doublephase operator have the attracted attention of many researchers; see, for example [14][15][16] and the references therein.It sheds light on various fields of applications, including but not limited to anisotropic materials, Lavrentiev's phenomenon, and elasticity theory.
Equation (1.1) is a generalization of the nonlocal problem ρ ∂ 2 ξ ∂t 2 - which represents a general case of D' Alembert's vibration equation Provided by Kirchhoff [12].Additionally, in [33], a time-related equation was given in the following form: Many authors have worked on problems related to double-phase operators and have obtained several results, including the following: In [15], Liu and Dai proved the existence and multiplicity of solutions to the doublephase problem of the form where is a bounded domain with smooth boundary, N ≥ 2, 1 < p < q, q p < 1 + 1 N , a. : → [0; +∞) is Lipschitz continuous, and f fulfills certain conditions.For more information, one can also see the works by Ragusa and Tachikawa [22] and Wulong et al. [34].
In [19], the existence of positive solutions to a class of double-phase Dirichlet equations that have combined effects of the singular term and the parametric linear term is studied.The reader can be referred to many other papers that discuss double-phase problems, including but not limited to [2,5] and the references therein.
It is worth noting that fractional differential equations have led to the modeling of many phenomena in many fields of science [11,26], and applications of the latter have appeared in engineering, medicine, and mechanics, which increased the researchers' interest in these equations, especially in mathematical aspect; see, for example, [15,36].In [26], the authors were able to construct the ψ-Hilfer fractional operator with several examples.See also [30], where the space H α,β,ψ p ([0, T], R) is created, allowing the study of many of these equations involving the ψ-Hilfer fractional in the appropriate spaces.
In [31], using the Nehari manifolds technique and combining it with fiber maps, the authors presented an analysis of weak solutions by studying a fractional problem of the following form: ), and λ > 0. In [31], the result of bifurcation from infinity to equation (1.4) is also given.
In [23], the authors present the existence and multiplicity of solutions of the Kirchhoff ψ-Hilfer fractional p-Laplacian equation using critical point theory.
Researchers worked on many models of fractional differential equations using variational problems that include fractional operators, for example, Nyamoradi and Tayyebi [18], Ghanmi and Zhang [6], Kamache et al. [10], Sousa et al. [27,30].For example, in [10], Kamache et al. discussed a class of perturbed nonlinear fractional p-Laplacian differential systems and proved the existence of three weak solutions using the variational method and Ricceri's critical points theorems.On the other hand, in [29], the existence and multiplicity of solutions of the following κ(ξ )-Kirchhoff equation are proven using the variational method where L μ,υ,ψ κ(x) φ := R κ(.) p(.),q(.)v, g(x, ξ ) : × R → R is the Caratheodory function, satisfying some conditions, and R(t) is a continuous function.
In [28], the author discusses the multiplicity nontrivial solution for a new class of fractional differential equations of the Kirchhoff type in the ψ-fractional space S α,β,ψ H,0 via critical point result and variational methods.
In [3], Tahar et al. studied the existence and multiplicity of solutions for problem (1.1) with κ(x) = 0,and λ(x) = q(x) proving their results using the mountain pass theorem combined with the sub-supersolution method.
Motivated by these works, we study the existence and multiplicity of solutions for class of fractional fractional Kirchhoff double-phase problem involving a ψ-Hilfer fractional operator with variable exponent using the sub-supersolution method and mountain pass theorem.
We now give our results as follows: Theorem 1.1 Let us consider that (K 0 ) and (g 1 )-(g 2 ) are satisfied.Then, for some α > 0, the problem (1.1) has at least one solution with condition σ ∞ < α .
We arrange our paper in the following manner: In Sect.2, we give some definitions and lemmas for the Lebesgue and Musielak-Orlicz Sobolev spaces.In Sect.3, we present some results that will be needed in our study of the problem (1.1).Sections 4 and 5 deal with the main proofs of the Theorems 1.1 and 1.2, respectively.

Preliminaries
In this section, the basic concepts and ideas on Lebesgue and Musielak-Orlicz Sobolev spaces that we will need in arriving at the results will be presented (see [25]).
Let ν ∈ C( ), with ν > 1, and denote The variable exponent Lebesgue space with the norm is a reflexive and separable Banach space, whose conjugate space is L ν (x) ( ), where ν Assume that (a) is achieved, and let A : the modular associated with A is given by where ( ) is a measurable functions space.Let L A be the Musielak-Orlicz space defined by endowed with the norm

Auxiliary results
We present some important definitions and concepts to create appropriate subsupersolutions to our problem.
Proof Using the Lemmas 3.2, 3.3, and 3.4, there exists a unique solution (0, 0) where l 1 , l 2 , and l are given in Lemma 3.5.Next, consider that K is nondecreasing and there exits α > 0 relying only on l 1 , l 2 , and l such that ξ ∞ ≤ l, provided that σ ∞ < α.Moreover, by Lemma 3.2, ξ ≤ ξ .Let ξ in H 1,A 0 ( ).By (4.1) and (g 1 ), we get From (4.2) and (g 2 ), we have where Hence, we get the expected result.We now highlight the proof of Theorem 1.1: Let ξ , ξ ∈ H 1,A 0 ( ). ∩ L ∞ ( ).According to the previous lemma, we can write We define the problem and the energy functional attached to it I : H 1,A 0 ( ) → R given by where H(x, t) = t 0 h(x, τ ) dτ .Then, I ∈ C 1 , it is clear that the critical points for I are solutions to (4.5).According to (K 0 ), I is coercive and sequentially weakly lower semicontinuous.Hence, I attains its minimum in the weakly closed subset [ξ , ξ ] ∩ H 1,A 0 ( ) at some ξ 0 , which represents a critical point in I.The proof of Theorem 1.1 is complete.

Proof of Theorem 1.2
Let the function f be defined as Also, one can look at the problem To find solutions for (4.5), we follow the approach of identifying critical points of the C 1 -functional J : H 1,A 0 ( ) → R, defined as: where F(x, t) = t 0 f (x, τ ) dτ .
Lemma 5.1 Assuming that the conditions of Theorem 1.2 are satisfied, the functional J satisfies the Palais-Smale condition.
Lemma 5.2 Assuming that the conditions of Theorem 1.2 are satisfied, for σ ∞ sufficiently small, we have ( ) > 2ϑ and J (e) < ς .
We finished the proof of Theorem 1.2.

Conclusion
In this work, we have analyzed the existence of solutions to a double-phase fractional equation of the Kirchhoff type in Musielak-Orlicz Sobolev space with variable exponents.Our approach is mainly based on the sub-supersolution method and the mountain pass theorem.In future work, we will follow the current study with general source terms.