On a ψ-Hilfer fractional p -Laplacian differential equation: Existence and multiplicity results

Abstract

In this paper, we investigate the existence and multiplicity of weak solutions for a class of nonlinear boundary value problems driven by a [Formula: see text]-Hilfer fractional p-Laplacian operator. By introducing an appropriate fractional derivative Banach space, we establish a variational framework that allows the application of critical point theory to prove the existence of at least one weak solution under sub-linear growth assumptions. Then, under super-linear conditions of Ambrosetti-Rabinowitz type, we obtain the existence of nontrivial solutions via the Mountain Pass Theorem. The main novelty of our results is that they hold for every [Formula: see text] (0, 1), thereby removing the restrictive condition [Formula: see text], that is required in most previous variational approaches to fractional p-Laplacian problems ([14], [22], [23], [26]). This extension is achieved through a refined compact embedding theorem (Theorem 3.7) valid for all [Formula: see text] (0, 1) when [Formula: see text]. Furthermore, assuming the nonlinearity to be even, we employ Krasnoselskii’s genus theory to prove the existence of infinitely many weak solutions. Our framework unifies several classical fractional derivatives including Riemann-Liouville, Caputo and Hadamard as special cases through appropriate choices of the kernel function [Formula: see text], thereby extending and unifying many well-known results in the literature. Finally, we give an example illustrating the validation of our results.