The existence of solutions to certain quasilinear elliptic equations

Let Lu=−∑i,j=1Naij(x,u)Diju+c(x,u)u. Consider the quasilinear elliptic equation Lu=f(x,u,∇u)Lu=f(x,u,∇u) on a bounded smooth domain ΩΩ in RNRN, where c(x,r)≥α>0c(x,r)≥α>0, f(x,r,ξ)=o[|r|+h(|r|)|ξ|2]f(x,r,ξ)=o[|r|+h(|r|)|ξ|2]. It is shown that if the oscillation of aij(x,r)aij(x,r) with respect to rr is sufficiently small, then there exists a solution u∈W2,p(Ω)∩W01,p(Ω) to the equation Lu=f(x,u,∇u)Lu=f(x,u,∇u).