Smooth subsolutions of the discounted Hamilton-Jacobi equations

For the discounted Hamilton-Jacobi equation \[ \lambda u+H(x,d_{x}u)=0,\quad \lambda >0 \] on compact, boundless manifold $M$, we construct $C^{1,1}$ subsolutions which are exactly solutions on the projected Aubry set. The smoothness of such subsolutions can be improved if additional hyperbolicity of the Aubry set is assumed. As applications, such subsolutions can be used to identify the maximal global attractor of the associated conformally symplectic flow, and to control the convergent speed of the Lax-Oleinik semigroup.