Abstract: By using  the weak convergence methods for nonlinear partial differential equations (PDEs), the author proved the existence of solutions to a class of quasi-linear elliptic equations with gradient term and zero-order term. The main characteristic of the equation was  that the coefficient function of the gradient term b∈L^N(Ω), but its  norm ‖b‖_N,Ω was not required to be sufficiently 
 small. Firstly, by segmenting the bounded domain Ω, the solution sequence {u^t}_0<t<1 was  split into a sum of some subfunctions, and the energy estimate of the subfunction was limited to small  subdomain. Secondly, the author obtained the energy estimate of  {u^t}_0<t<1 on W^1,p₀(Ω) by using  iterative techniques. Finally, with the help of Boccardo-Murat’s technique, the author proved the almost everywhere convergence of the gradient solution sequence {u^t}_0<t<1, and determined the convergence element  of the nonlinear term of the equation based on this convergence.

Key words: elliptic equation, quasilinear, lower order term