Generalized comparison principle for contact Hamilton-Jacobi equations

In this paper, we discuss all the possible pairs $(u,c)\in C(M,\mathbb R)\times\mathbb R$ solving (in the sense of viscosity) the contact Hamilton-Jacobi equation $$H (x, d_xu, u) = c,\quad x\in M$$ of which $M$ is a closed manifold and the continuous Hamiltonian $H: (x,p,u)\in T^*M\times\mathbb R\rightarrow\mathbb R$ is convex, coercive in $p$ but merely non-decreasing in $u$. Firstly, we propose a comparison principle for solutions by using the dynamical information of Mather measures. We then describe the structure of $\mathfrak C$ containing all the $c\in\mathbb R$ makes previous equation solvable. We also propose examples to verify the optimality of our approach.