Elliptic inequalities with $$L^1$$ L 1 data in Musielak–Orlicz spaces

Abstract

In this paper, we prove a Poincare inequality in Musielak–Orlicz spaces under the assumption that the Musielak function depends only on $$N-1$$ coordinates of the spatial variable x. We will use this inequality to show the existence of solutions for some elliptic inequalities with lower order terms and $$L^1$$ data in Musielak–Orlicz spaces.