Lane–Emden equation

Remarks

For a radial function $u(x) = \theta(\abs{x})$ on $\R^3$, the first term is the Laplacian $\Delta u$, so the equation is $\Delta u + u^n = 0$. Example: for $n = 0$, $\theta(\xi) = 1 - \xi^2/6$ is the solution. Indeed, $\xi^2 \theta' = -\xi^3/3$, whose derivative divided by $\xi^2$ is $-1$, and $\theta^0 = 1$. Non-example: for $n = 1$, the constant $\theta = 1$ is not a solution, since the left side equals $1$. Closed-form solutions are known for $n = 0$, $n = 1$, where $\theta = \sin \xi / \xi$, and $n = 5$, where $\theta = (1 + \xi^2/3)^{-1/2}$. In PDE the name Lane–Emden equation, or Lane–Emden–Fowler equation, is also used for $-\Delta u = \abs{u}^{p-1} u$ on a domain of $\R^N$ with $p > 1$.