In the present paper, we study an anisotropic $\overrightarrow{p}(\cdot )$-Laplace equation with combined effects of variable singular and sublinear nonlinearities. Using the Ekeland's variational principle and a constrained minimization, we show the existence of a positive solution for the case where the variable singularity $\beta (x)$ assumes its values in the interval $(1,\mathrm{\infty })$.

This study proposes a new stochastic model where the diffusion coefficient involves a state-dependent variable exponent function $p(\cdot )$. This new theoretically flexible framework generalizes the classical Cox–Ingersoll–Ross model. The existence, uniqueness, and higher moment properties of solutions are analyzed. The validity and efficiency of the model is illustrated with numerical experiments. Finally, a detailed analysis for Itô vs. Stratonovich interpretations of the proposed model is provided.