Elliptic problems with superlinear convection terms

In this manuscript we deal with elliptic equations with superlinear first order terms in divergence form of the following type −div(M(x)∇u)=−div(h(u)E(x))+f(x), where M is a bounded elliptic matrix, the vector field E and the function f belong to suitable Lebesgue spaces, and the function s→h(s) features a superlinear growth at infinity. We provide some existence and non existence results for solutions to the associated Dirichlet problem and a comparison principle.