Acta Mathematica Sinica, English Series Jan., 2015, Vol. 31, No. 1, pp. 151–169 Published online: December 15, 2014 DOI: 10.1007/s10114-015-3555-7 Http://www.ActaMath.com Acta Mathematica Sinica, English Series © Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2015 Entropy and Renormalized Solutions for Nonlinear Elliptic Problem Involving Variable Exponent and Measure Data Mohamed Badr BENBOUBKER National School of Applied Sciences, Abdelmalek Essaadi University, BP 2222 M’hannech T´ etouan, Morocco E-mail : simo.ben@hotmail.com Houssam CHRAYTEH Beirut Arab University, Faculty of Science, Department of Mathematics and Computer Science, Debbieh, Lebanon E-mail : h.chrayteh@yahoo.fr Mostafa EL MOUMNI Hassane HJIAJ Laboratory LAMA, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, B.P 1796 Atlas Fez, Morocco E-mail : mostafaelmoumni@gmail.com hjiajhassane@yahoo.fr Abstract We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type: -div (a(x, u, ∇u)+ φ(u)) + g(x, u, ∇u)= μ, where the right-hand side belongs to L 1 (Ω) + W -1,p ′ (x) (Ω), -div(a(x, u, ∇u)) is a Leray–Lions oper- ator defined from W -1,p ′ (x) (Ω) into its dual and φ ∈C 0 (R, R N ). The function g(x, u, ∇u) is a non linear lower order term with natural growth with respect to |∇u| satisfying the sign condition, that is, g(x, u, ∇u)u ≥ 0. Keywords Nonlinear elliptic problem, Sobolev spaces, variable exponent, entropy solution, renor- malized solution, measure data MR(2010) Subject Classification 35J15, 35J60, 35D05, 46E35 1 Introduction In recent years, there has been an increasing interest in the study of various mathematical prob- lems with variable exponents. These problems are interesting in applications (see [11, 21, 29]) and gave rise to a revival of the interest in Lebesgue and Sobolev spaces with variable expo- nent, the origins of which can be traced back to the work of Orlicz in the 1930’s. In the 1950’s, this study was carried on by Nakano [18] who made the first systematic study of spaces with variable exponent. Later, Polish and Czechoslovak mathematicians investigated the modular function spaces (see for example Musielak [17], Kovacik and Rakosnik [15]). Variable exponent Received October 19, 2013, accepted May 4, 2014