Generalized solutions for a class of Hamilton-Jacobi equations

We study the Dirichlet problem for Hamilton-Jacobi equations of the form \begin{displaymath} \begin{cases} H(Du(x)) - a(x) = 0 & \ \textrm{in} \ \O\cr u(x)=\varphi(x) & \ \textrm{on} \ \partial \O, \end{cases} \end{displaymath} under suitable hypotheses on the function $H$ and without continuity assumptions on the map $a$. We find a class of maps $a$ contained in the space $L^\infty(\O)$ for which the problem admits a (maximal) generalized solution, providing a generalization of the notion of viscosity solution.