On the singular problem involving $g$-Laplacian

TL;DR

This paper proves the existence of positive weak solutions to a fractional $g$-Laplacian equation with singular nonlinearity, extending to mixed fractional $(p,q)$-Laplacian cases, and shows solutions are locally H"older continuous.

Contribution

It establishes existence and regularity results for solutions to a broad class of fractional $g$-Laplacian equations with singular nonlinearities, including mixed fractional $(p,q)$-Laplacian.

Findings

Existence of positive weak solutions in bounded domains.

Solutions are locally H"older continuous.

Includes mixed fractional $(p,q)$-Laplacian as a special case.

Abstract

In this paper, we show that the existence of a positive weak solution to the equation $(-\Delta_g)^s u=f u^{-q(x)}\;\mbox{in}\; \Omega,$ 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 H\"older continuous.