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

Abstract

Let Omega subset of R-n be any open set and u a weak supersolution of Lu = c(x)g(vertical bar u vertical bar) u/vertical bar u vertical bar, where |u|, where Lu(x) = p.v. (Rn)integral g vertical bar u(x)-u(y)vertical bar/vertical bar x-y vertical bar(s)) u(x) -u (y)/vertical bar u(x) - u(y)vertical bar K(x;y)dy/vertical bar x -y vertical bar(s) and g = G' for some Young function G. This note imparts a Hopf type lemma and strong minimum principle for u when c(x) is continuous in (Omega) over bar 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), no. 1, 765-778] in fractional Orlicz-Sobolev setting.