On the singular problem involving $g$-Laplacian

In this paper, we show that the existence of a positive weak solution to the equation $(-{\mathrm{\Delta }}_g{)}^{s}u=f{u}^{-q(x)}in\mathrm{\Omega },$ where $\mathrm{\Omega }$ is a smooth bounded domain in $R^N$, $q\in C^1(\bar{\mathrm{\Omega }})$, and $(-{\mathrm{\Delta }}_g{)}^s$ is the fractional $g$-Laplacian with $g$ is the antiderivative of a Young function and $f$ in suitable Orlicz space subjected to zero Dirichlet condition. This includes the mixed fractional $(p,q)-$Laplacian as a special case. The solution so obtained is also shown to be locally Hölder continuous.