Ricerche di Matematica https://doi.org/10.1007/s11587-019-00452-0 Entropy solutions of anisotropic elliptic nonlinear obstacle problem with measure data Abdelhafid Salmani 1 · Youssef Akdim 2 · Hicham Redwane 3 Received: 16 March 2019 / Revised: 2 June 2019 © Università degli Studi di Napoli "Federico II" 2019 Abstract We prove the existence of an entropy solution for a class of nonlinear anisotropic elliptic unilateral problem associated to the following equation − N  i =1 ∂ i a i (x , u , ∇u ) − N  i =1 ∂ i φ i (u ) = μ, where the right hand side μ belongs to L 1 () + W −1,  p ′ (). The operator − ∑ N i =1 ∂ i a i (x , u , ∇u ) is a Leray–Lions anisotropic operator and φ i ∈ C 0 (R, R). Keywords Entropy solutions · Anisotropic elliptic equations · Anisotropic Sobolev space Mathematics Subject Classification 35J60 · 35J87 · 35J66 B Abdelhafid Salmani slmnhfd@gmail.com Youssef Akdim akdimyoussef@yahoo.fr Hicham Redwane redwanehicham@yahoo.fr 1 Laboratory LSI, Polydisciplinary Faculty of Taza, Sidi Mohamed ben Abdellah University, Road of Oujda, B.P. 1223 Taza, Morocco 2 Laboratory LAMA Faculty of Sciences, Dhar-Mahrez, Sidi Mohamed ben Abdellah University, B.P. 1796 Fez Atlas, Morocco 3 Faculty of Juridical, Economic and Social Sciences, Hassan 1st University, Km 3, Road of Casablanca, B.P. 784 Settat, Morocco 123