Lower order terms in divergence form versus lower order terms with natural growth in some Dirichlet problems

Today: Dirichlet problems with singular convection terms without

Ildefonso (see [5,6]).

Abstract

We study the existence of solutions for some nonlinear elliptic boundary value problems, whose general form is:

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\mathrm{\;div}(M(x)\nabla u) + g(u)|\nabla u|^2 = -\mathrm{\;div}(E(x,u)) + f(x) &{} \text{ in }\, \Omega , \\ u = 0 &{} \text{ on }\, \partial \Omega , \end{array}\right. } \end{aligned}$$

where $\Omega$ is an open bounded subset of $\mathbb {R}^N$ and $f\in \,L^1(\Omega )$. Despite this poor summability, we prove the existence of finite energy solutions due to the effect of the presence of the term $g(u)|\nabla u|^2$.