Study of Double Phase-Choquard Problem in Generalized ψ- Hilfer Fractional Derivative Spaces with p-Laplacian Operator

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Authors: E. ARHRRABI AND H. EL-HOUARI

DOI: 10.46793/KgJMat2607.1087A

Abstract:

In this paper, our focus is on a specific class of non-linear ψ-Hilfer fractional generalized double phase-Choquard differential equations involving the p-Laplacian operator with Dirichlet boundary conditions. The equation is given by:

( ( ∫ ) { γ,β;ψ G-(u(x)) ℒ u = Ω|x− y|λdx g(u(y)), in Ω, (u = 0, on ∂Ω,

with ℒγ,β;ψ is defined as:

ℒγ,β;ψu := ????γ,β;ψ(|???? γ,+β;ψu |p−2 ???? γ,β+;ψu+ a(x)|???? γ+,β;ψ u|q−2 ????γ,β+;ψu), T 0 0 0 0

where ????Tγ,β;ψ and ????0+γ,β;ψ are ψ-Hilfer fractional derivatives of order 1p < γ < 1 and type 0 ≤ β ≤ 1 and a(⋅) is non-negative weight function, and G(⋅) represents Choquard nonlinearities satisfying a certain growth conditions. By employing the mountain pass theorem without the Palais-Smale condition, along with the Hardy-Littlewood-Sobolev inequality, we establish the existence of a weak solution to the aforementioned problem. Our main results are novel and contribute to the literature on problems involving ψ-Hilfer derivatives with the p-Laplacian operator. This investigation enhances the scope of understanding in this specific class of problems.



Keywords:

Generalized ψ-Hilfer derivative, double phase-Choquard equation, mountain pass theorem, Hardy-Littlewood-Sobolev inequality.



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