The authors consider the Hamilton-Jacobi equation \(H(x, Du) + \lambda(x)u = c\), where \(H: T^*M \times \mathbb{R} \rightarrow \mathbb{R}\) is a contact Hamiltonian, \(M\) is a connected, closed and smooth manifold, \(x \in M\), \(D\) denotes the spatial derivative with respect to \(x\), and where \(H(x, p)\) and \(\lambda(x)\) are continuous. They also assume that \(H(x,p)\) is convex and coercive with respect to \(p\) and that \(\lambda(x)\) changes signs. This means that there exist \(x_1\) and \(x_2\) such that \(\lambda(x_1) >0 \) and \(\lambda(x_2) < 0\), while \(H(x,p)\) coercive in \(p\) is means that limit \(H(x,p)_{\|p\|_x} \rightarrow \infty, \) where \(\|\cdot \|_x\) represents norms induced by the Riemannian metric on \(TM\) and \(T^*M\).
As the authors claim, the results in this paper are motivated by [L. Jin et al., Minimax Theory Appl. 8, No. 1, 61–84 (2023; Zbl 1514.35110)]. The main result is that there is a value \(c_0\) such that for \(c \geq c_0\), there exist maximal and minimal elements in the set of solutions of the original equation. The authors use the nonlinear Lax-Oleinik semigroup to extend the results of [loc. cit.], and to explore the detailed structure of the viscosity solutions and their long term behavior.