The selection problem for a new class of perturbations of Hamilton-Jacobi equations and its applications

This paper studies a perturbation problem given by the equation: \begin{equation*} H(x, d_xu_\lambda, \lambda u_\lambda(x))+\lambda V(x,\lambda)=c \quad \text{in $M$}, \end{equation*} where $M$ is a closed manifold and $\lambda>0$ is a perturbation parameter. The Hamiltonian $H(x,p,u):T^*M\times \mathbb{R}\to \mathbb{R}$ satisfies certain convexity, superlinearity, and monotonicity conditions. $\lambda V(\cdot,\lambda):M\to\mathbb{R}$ converges to zero as $\lambda\to 0$. First, we study the asymptotic behavior of the viscosity solution $u_\lambda:M\to\mathbb{R}$ as $\lambda$ approaches zero. This perturbation problem explores the combined effects of both the vanishing discount process and potential perturbations, leading to a new selection principle that extends beyond the classical vanishing discount approach. Additionally, we apply this principle to Hamilton-Jacobi equations with $u$-independent Hamiltonians, resulting in the introduction of a new solution operator. This operator provides new insights into the variational characterization of viscosity solutions and Mather measures.