On the singular problem involving $g$-Laplacian

Abstract

In this paper, we show that the existence of a positive weak solution to the equation $(-Δ_g)^s u=f u^{-q(x)}\;\mbox{in}\; Ω,$ where $\Omega$ is a smooth bounded domain in $R^N$, $q\in C^1(\overline{\Omega })$, and $(-{\Delta }_g{)}^s$ 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.

15pp