Ricerche di Matematica https://doi.org/10.1007/s11587-018-0423-7 Existence of solutions for some nonlinear elliptic problems involving Minty’s lemma Mohammed Al-Hawmi 1 · Abdelmoujib Benkirane 1 · Hassane Hjiaj 2 · Abdelfattah Touzani 1 Received: 30 May 2018 / Revised: 25 August 2018 © Università degli Studi di Napoli "Federico II" 2018 Abstract In this paper, we consider the following nonlinear L ϕ ()-elliplic problem: − div a(x , u , ∇u ) = f − div φ(u ) in , in Musielak–Orlicz–Sobolev spaces, when only large monotonicity is satisfied, with f ∈ L 1 () and φ(·) ∈ C 0 (R, R N ). By assuming that ϕ(x , t ) is a generalized Orlicz function, such that the conjugate function ψ(x , t ) satisfies the  2 -condition. We estab- lish the existence of T- L ϕ ()-solutions for our nonlinear problem via Minty’s lemma. Keywords Musielak–Orlicz–Sobolev spaces · Nonlinear elliptic equations · T- L ϕ ()-solution · Truncations · Minty’s lemma Mathematics Subject Classification 35J62 · 35J25 B Hassane Hjiaj hjiajhassane@yahoo.fr Mohammed Al-Hawmi m.alhomi2011@gmail.com Abdelmoujib Benkirane abd.benkirane@gmail.com Abdelfattah Touzani atouzani07@gmail.com 1 Laboratoire d’Analyse Mathématique et Applications (LAMA), Faculté des Sciences Dhar El Mahraz, Université Sidi Mohamed Ben Abdellah, BP 1796, Atlas Fès, Morocco 2 Département de Mathématiques, Faculté des Sciences de Tétouan, Université Abdelmalek Essaadi, BP 2121, Tétouan, Morocco 123