Nonlinear semigroup approach to Hamilton-Jacobi equations -- A toy model

Abstract

In this paper, we discuss the existence and multiplicity problem of viscosity solution to the Hamilton-Jacobi equation $$h(x,d_{x}u)+\lambda (x)u=c,x\in M,$$ where $M$ is a closed manifold and $\lambda :M\to \mathbb{R}$ changes signs on $M$, via nonlinear semigroup method. It turns out that a bifurcation phenomenon occurs when parameter $c$ strides over the critical value. As an application of the main result, we analyse the structure of the set of viscosity solutions of an one-dimensional example in detail.