Summary: The novelty of this note is to establish existence result for the following anisotropic elliptic equation \[-\text{div}\,B(x,\vartheta,\nabla\vartheta)+H(x,\vartheta)=f-\text{div}\,F\] where the datum \(f\in L^1(\Omega)\) and \(F\in\prod^N_{i=1}L^{p_i'}(\Omega,w^*_i)\) and \(H(x,\vartheta)\in L^1(\Omega)\). Furthermore only the large monotonicity conditions will be assumed on \(B(x,s,\xi)\). To overcome this difficulty we will use the approach of Minty’s lemma in the anisotropic weighted Sobolev spaces.