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

Abstract

Abstract Let Ω ⊂ R n \Omega\subset\mathbb{R}^{n} be any open set and 𝑢 a weak supersolution of L ⁢ u = c ⁢ ( x ) ⁢ g ⁢ ( | u | ) ⁢ u | u | \mathcal{L}u=c(x)g(\lvert u\rvert)\frac{u}{\lvert u\rvert} , where L ⁢ u ⁢ ( x ) = p.v. ⁢ ∫ R n g ⁢ ( | u ⁢ ( x ) − u ⁢ ( y ) | | x − y | s ) ⁢ u ⁢ ( x ) − u ⁢ ( y ) | u ⁢ ( x ) − u ⁢ ( y ) | ⁢ K ⁢ ( x , y ) ⁢ d ⁢ y | x − y | s \mathcal{L}u(x)=\textup{p.v.}\int_{\mathbb{R}^{n}}g\biggl{(}\frac{\lvert u(x)-u(y)\rvert}{\lvert x-y\rvert^{s}}\biggr{)}\frac{u(x)-u(y)}{\lvert u(x)-u(y)\rvert}K(x,y)\frac{dy}{\lvert x-y\rvert^{s}} and g = G ′ g=G^{\prime} for some Young function 𝐺. This note imparts a Hopf type lemma and strong minimum principle for 𝑢 when c ⁢ ( x ) c(x) is continuous in Ω ¯ \overline{\Omega} that extend the results of Del Pezzo and Quaas [A Hopf’s lemma and a strong minimum principle for the fractional 𝑝-Laplacian, J. Differential Equations 263 (2017), 1, 765–778] in fractional Orlicz–Sobolev setting.