Elliptic problems with superlinear convection terms

TL;DR

This paper investigates elliptic equations with superlinear convection terms, establishing conditions for the existence and non-existence of solutions, along with a comparison principle, in the context of divergence form equations.

Contribution

It provides new existence and non-existence results for elliptic equations with superlinear first order divergence terms, expanding understanding of solution behavior under these conditions.

Findings

Existence results under certain growth conditions.

Non-existence results for specific parameter ranges.

A comparison principle for solutions.

Abstract

In this manuscript we deal with elliptic equations with superlinear first order terms in divergence form of the following type \[ -\mbox{div}(M(x)\nabla u)= -\mbox{div}(h(u)E(x))+f(x), \] where $M$ is a bounded elliptic matrix, the vector field $E$ and the function $f$ belong to suitable Lebesgue spaces, and the function $s\to h(s)$ 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.