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

In this paper, we discuss the existence and multiplicity problem of viscosity solution to the Hamilton-Jacobi equation $$h(x,d_x u)+\lambda(x)u=c,\quad x\in M,$$ 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 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.