Study of generalized double phase logarithmic problem with $ \tau $-Laplacian operator

This study investigates a class of $ \phi $-Hilfer generalized fractional nonlinear double-phase problems with Dirichlet boundary conditions, focusing on the existence of nonnegative solutions with logarithmic nonlinearity. By using Nehari manifold minimization techniques and variational approaches, we demonstrate the existence of positive solutions dependent on the parameter $ \xi $ in appropriates $ \phi $-Hilfer fractional spaces. Our findings are novel and contribute to expanding the knowledge base on double-phase problems and $ \phi $-Hilfer generalized fractional differential equations with Dirichlet boundary conditions.