How to convert a PDE on a sphere to a PDE in $\mathbb{R}^n?$

Question

Consider the following equation, $$\Delta_{\mathbb{S}^n} u + u = u^3 $$ on the unit Sphere $\mathbb{S}^n.$ How do we convert this equation to an equation in $\mathbb{R}^{n}?$ I read about Stereographic projections that is a map of the following form $F:\mathbb{R}^n \to \mathbb{S}^n\setminus\{(0,\cdots, 1)\}$, $$F(x) = \left(\frac{2x}{1+|x|^2},\frac{1- |x|^2}{1+|x|^2}\right).$$ But I am not sure how to come up with a PDE such that if $u$ solves the equation written above on the sphere, then $\hat{u}=G[u(F(x))])$ solves a PDE in $\mathbb{R}^n.$ Any ideas/comments will be much appreciated.