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_λ, λu_λ(x))+λV(x,λ)=c \quad $\text{in }M$, \end{equation*} where $M$ is a closed manifold and $\lambda >0$ is a perturbation parameter. The Hamiltonian $H(x,p,u):T^{\ast }M\times \mathbb{R}\to \mathbb{R}$ 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.