Lane–Emden equation

Summary

The Lane–Emden equation is the ordinary differential equation $\xi^{-2} (\xi^2 \theta')' + \theta^n = 0$, which describes the density of a self-gravitating ball of gas in equilibrium, called a polytrope. It is the equation $\Delta u + u^n = 0$ for radial functions in $\R^3$. In PDE the name also refers to the semilinear elliptic equation $-\Delta u = u^p$ in any dimension, whose solutions depend on how $p$ compares with critical exponents. It is a standard test case for existence, nonexistence and Liouville-type theorems.