Regularizing effects concerning elliptic equations with a superlinear gradient term

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 with 1\le m <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<N/2 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.