On a class of capillarity phenomenon with logarithmic nonlinearity involving $$\theta (\cdot )$$ -Laplacian operator

This research delves into a comprehensive investigation of a class of \(\Im \)-Hilfer generalized fractional nonlinear equation originated from a capillarity phenomenon involving a logarithmic nonlinearity and Dirichlet boundary conditions. The nonlinearity of the problem, in general, do not satisfies the Ambrosetti-Rabinowitz type condition. Using critical point theorem with variational approach and the \((S_{+})\) property of the operator, we establish the existence of positive solutions of our problem with respect to every positive parameter \(\xi \) in appropriate \(\Im \)-fractional spaces. Our main results is novel and its investigation will enhance the scope of the literature on differential equation of \(\Im \)-Hilfer fractional generalized capillary phenomenon with logarithmic nonlinearity.