Nontrivial solutions to singular fractional differential equations with φ-Hilfer derivatives in variable exponent Sobolev spaces

Abstract

In this work, we investigate the existence of weak solutions for a new class of fractional differential equations containing a singular term and the generalized φ-Hilfer fractional derivative with variable exponent nonlinearities. Such problems arise naturally in several applied fields, including electrorheological fluids, image processing, elasticity, and models exhibiting Lavrentiev-type phenomena, where variable exponent structures play a crucial role. The studied problem is further complicated by the presence of singular terms, which pose analytical challenges due to their behavior near the origin. To address this difficulty, we employ advanced variational techniques, particularly the Min–Max method analyzed in the framework of variable exponent Sobolev spaces, which allow us to treat nonstandard growth and spatial heterogeneity. By carefully constructing an appropriate energy functional and verifying the necessary compactness and coercivity conditions, we establish the existence of a nontrivial weak solution. The use of the Min–Max method is justified by the geometry of the associated functional, and critical point theory is employed to overcome difficulties caused by the lack of smoothness and the singular behavior of the nonlinearities. Moreover, we provide explicit examples of admissible nonlinearities illustrating the applicability of our main theorem. Our results extend and generalize several previous works by incorporating both the φ-Hilfer derivative an operator unifying many classical and modern fractional derivatives and the flexibility of variable exponent analysis. This paper contributes to the theory of fractional variational problems and offers potential applications in modeling physical systems with spatially variable features and memory effects.