Existence of three weak solutions for a fractional problem with logarithmic nonlinearity involving the ψ-Hilfer operator

In this paper, we study a nonlinear fractional boundary value problem involving a p -Laplacian-type ψ-Hilfer operator with a logarithmic source term under homogeneous Dirichlet boundary conditions in a one-dimensional domain Ω ⊂ ℝ {Ω} . The logarithmic nonlinearity introduces slow-growth effects and additional analytical difficulties compared to standard polynomial-type nonlinearities. Using a variational framework, we associate the problem with an energy functional defined on a suitable Banach space (which is reflexive) and establish its main properties, including coercivity and the Palais–Smale condition. By applying the Bonanno–Marano-type three critical points theorem, we prove the existence of at least three distinct weak solutions. The results extend previous works in the ψ-Hilfer setting by explicitly treating logarithmic nonlinearities and highlighting the differences with existing polynomial-growth models.