Weakly coupled Hamilton-Jacobi systems without monotonicity condition: A first step

In this paper, we mainly focus on the existence of the viscosity solutions of \begin{equation*} \left\{ \begin{aligned} &H_1(x,Du_1(x),u_1(x),u_2(x))=0,\\ &H_2(x,Du_2(x),u_2(x),u_1(x))=0. \end{aligned} \right. \end{equation*} The standard assumption for the above system is called the monotonicity condition, which requires that $H_i$ is increasing in $u_i$ and decreasing in $u_j$ for each $i,j\in\{1,2\}$ and $i\neq j$. In this paper, it is assumed that $H_i$ is either increasing or decreasing in $u_i$, and may be non-monotone in $u_j$. The existence of viscosity solutions is proved when $$\chi:=\sup_{u,v,w\in\mathbb R}\bigg|\frac{\partial_{u_2} H_1(x,0,0,u)}{\partial_{u_1} H_1(x,0,v,w)}\bigg|\cdot \sup_{u,v,w\in\mathbb R}\bigg|\frac{\partial_{u_1} H_2(x,0,0,u)}{\partial_{u_2} H_2(x,0,v,w)}\bigg|<1.$$ Then we consider \begin{equation*} \left\{ \begin{aligned} &h_1(x,Du_1(x))+\Lambda_1(x)(u_1(x)-u_2(x))=c,\\ &h_2(x,Du_2(x))+\Lambda_2(x)(u_2(x)-u_1(x))=\alpha(c). \end{aligned} \right. \end{equation*} It turns out that for each $c\in\mathbb R$, there is a unique constant $\alpha(c)\in\mathbb R$ such that the above system has viscosity solutions. The function $c\mapsto \alpha(c)$ is non-increasing and Lipschitz continuous. In the appendix, the large time convergence of the viscosity solution of evolutionary weakly coupled systems is proved when $\chi<1$.