Nonlinear elliptic problems with right hand side measure appear in many engineering applications such as petroleum engineering, coupling electric conduction and thermical conduction and electromagnetic induction heating. In this thesis, our goal is to study the existence, the regularity and uniqueness of solutions for this type of problems. In Chapter 1, we present a brief overview of basic concepts and results of the Lp, Sobolev and measure spaces. In Chapter 2, we study the existence and regularity of the weak solution of this problems. Also, we give a counterexample for the non uniqueness of the weak solution. In Chapter 3, we present and discuss the notion of entropy solution when the measure is absolutely continuous with respect to the pcapacity. In Chapter 4, we generalize the notion of entropy solution to new concept called renormalized solution when the measure is a general Radon measure. In Chapter 5, we present another approach to the existence result of renormalized solution by using just the Dirac measure which ensures the existence and stability of the solution..