A note on Hopf's lemma and strong minimum principle for nonlocal equations with non-standard growth

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be any open set and $u$ be a weak supersolution of $\mathcal{L}u=c(x)g(\mathrm{\mid }u\mathrm{\mid })\frac{u}{\mathrm{\mid }u\mathrm{\mid }}$ where $$\mathcal{L}u(x)=\text{p.v.}{\int }_{{\mathbb{R}}^n}g\left(\frac{\mathrm{\mid }u(x)-u(y)\mathrm{\mid }}{\mathrm{\mid }x-y{\mathrm{\mid }}^s}\right)\frac{u(x)-u(y)}{\mathrm{\mid }u(x)-u(y)\mathrm{\mid }}K(x,y)\frac{dy}{\mathrm{\mid }x-y{\mathrm{\mid }}^s}$$ and $g=G^{\mathrm{\prime }}$ for some Young function $G\mathrm{.}$ This note imparts a Hopf’s type lemma and strong minimum principle for $u$ when $c(x)$ is continuous in $\bar{\mathrm{\Omega }}$ that extend the results of Del Pezzo and Quaas (JDE-2017) in fractional Orlicz-Sobolev setting.