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

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