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 \(\Im \) $\Im$ -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 \((S_{+})\) $(S_{+})$ property of the operator, we establish the existence of positive solutions of our problem with respect to every positive parameter \(\xi \) $\xi$ in appropriate \(\Im \) $\Im$ -fractional spaces. Our main results is novel and its investigation will enhance the scope of the literature on differential equation of \(\Im \) $\Im$ -Hilfer fractional generalized capillary phenomenon with logarithmic nonlinearity.