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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 $$h(x,d_x u)+λ(x)u=c,\quad x\in M,$$ where $M$ is a closed manifold and $λ:M\rightarrow\mathbb{R}$ 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.