This paper is concerned with the study of the nonlinear elliptic equations in a bounded subset Ω ⊂ RN Au = f, where A is an operator of Leray-Lions type acted from the space W1,p(·)0(Ω) into its dual. when the second term f belongs to Lm(·), with m(·) > 1 being small. we prove existence and regularity of weak solutions for this class of problems p(x)-growth conditions. The functional framework involves Sobolev spaces with variable exponents as well as Lebesgue spaces with variable exponents.