Regularizing effects concerning elliptic equations with a superlinear gradient term

TL;DR

This paper investigates how a superlinear gradient term influences the existence and regularity of solutions to elliptic equations with discontinuous coefficients, especially under measure data and unbounded solutions.

Contribution

It identifies the conditions on the function g and the exponent q that ensure existence and regularization effects for elliptic equations with superlinear gradient terms.

Findings

Presence of g improves solution regularity

Existence results depend on g's behavior at infinity

Results unify with known cases when g is constant or q=2

Abstract

We consider the homogeneous Dirichlet problem for an elliptic equation driven by a linear operator with discontinuous coefficients and having a subquadratic gradient term. This gradient term behaves as $g(u)|\nabla u|^q$, where $1<q<2$ and $g(s)$ is a continuous function. Data belong to $L^m(\Omega)$ with $1\le m <\frac{N}{2}$ as well as measure data instead of $L^1$-data, so that unbounded solutions are expected. Our aim is, given $1\le m<\frac N2$ and $1<q<2$, to find the suitable behaviour of $g$ close to infinity which leads to existence for our problem. We show that the presence of $g$ has a regularizing effect in the existence and summability of the solution. Moreover, our results adjust with continuity with known results when either $g(s)$ is constant or $q=2$.