A quasilinear Dirichlet problem with quadratic growth respect to the gradient and L1L1 data

For an open, bounded set Ω⊂RNΩ⊂RN, measurable bounded functions a(x),b(x)a(x),b(x) which are strictly positive and p,q>0p,q>0, we prove the existence of a weak solution of the quasilinear b.v.p {−div[(a(x)+|u|q)∇u(x)]+b(x)u|u|p−1|∇u|2=f(x),x∈Ω;u(x)=0,x∈∂Ω. The datum ff is assumed to be in L1(Ω)L1(Ω) and does not satisfy any sign assumption.