Generalized double-phase logarithmic problem with the p-Laplacian operator involving the $$\phi $$-Hilfer fractional derivative via the min–max method

This study investigates a class of nonlinear $$\phi $$ ϕ -Hilfer fractional generalized double-phase problems governed by the p-Laplace operator under Dirichlet boundary conditions. More precisely, we establish the existence of nontrivial solutions in the presence of both logarithmic nonlinearities and singular perturbation terms. To do this, we combine the min–max method with variational techniques. We rigorously demonstrate the existence of nontrivial weak solutions to the proposed class of problems. The theoretical results obtained are novel and provide a significant generalization of several existing contributions in the literature. Beyond their theoretical interest, such fractional double-phase mo dels have several applications in various fields of applied mathematics and physics. They can be used to describe heterogeneous materials with nonstandard growth properties, anomalous diffusion processes in viscoelastic media, phase transition phenomena, and nonlinear heat conduction in materials with memory effects. Moreover, the presence of the $$\phi $$ ϕ -Hilfer fractional operator allows a more accurate modeling of systems exhibiting both local and nonlocal interactions, bridging both the classical and fractional dynamics and the analytical framework.