
Minimax Theory and its Applications 08 (2023), No. 1, 061084 Copyright Heldermann Verlag 2023 Nonlinear Semigroup Approach to the HamiltonJacobi Equation  a Toy Model Liang Jin School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, China jl@njust.edu.cn Jun Yan School of Mathematical Sciences, Fudan University, Shanghai, China yanjun@fudan.edu.cn Kai Zhao School of Mathematical Sciences, Fudan University, Shanghai, China zhao_kai@fudan.edu.cn [Abstractpdf] We discuss the existence and multiplicity problem of viscosity solution to the HamiltonJacobi equation $$h(x,d_x u)+\lambda(x)u=c,\quad x\in M,$$ where $M$ is a closed manifold and $\lambda:M\rightarrow\mathbb{R}$ changes signs on $M$, via nonlinear semigroup method. It turns out that a bifurcation phenomenon occurs when the parameter $c$ strides over some critical value. As an application of the main result, we analyse the structure of the set of viscosity solutions of an onedimensional example in detail. Keywords: HamiltonJacobi equations, viscosity solutions, bifurcation phenomenon. MSC: 35F21,35D40,35A02,35B32. [ Fulltextpdf (655 KB)] for subscribers only. 