  For the discounted Hamilton-Jacobi equation,$$\lambda u+H(x,d_x u)=0, \ x \in
M, $$we construct $C^{1,1}$ subsolutions which are indeed solutions on the
projected Aubry set. The smoothness of such subsolutions can be improved under
additional hyperbolicity assumptions. As applications, we can use such
subsolutions to identify the maximal global attractor of the associated
conformally symplectic flow and to control the convergent speed of the
Lax-Oleinik semigroups
