Lane-Emden equation

The classical Lane-Emden equation is the dimensionless equationfor a spherically symmetric polytrope in three dimensions. The Generalized Lane-Emden equation extends it to arbitrary dimensions and to functions without radial symmetry.

The regular dimensionless solution of the Lane-Emden equation with ${\theta }_n(0)=1$ and ${\theta }_n^{\prime }(0)=0$. Its local expansion is ${\theta }_n(\xi )=1-{\xi }^2/6+n{\xi }^4/120+O({\xi }^6)$. If it has a first zero ${\xi }_1$, that zero and $-{\xi }_1^2{\theta }_n^{\prime }({\xi }_1)$ specify radius and mass factors for the corresponding finite-radius stellar polytrope.

A Generalized Lane-Emden equation is a semilinear partial differential equation whose highest-order term is the Laplacian:For a radial function in $d$ dimensions it becomes $u^{\prime \prime }+(d-1)u^{\prime }/r+u^p=0$. The case $d=3$ gives the classical Lane-Emden equation.

When $a=2/(p-1)$ and $a<d-2$, the Lane-Emden equation has the singular scale-invariant solution

At the energy-critical exponent $p=(d+2)/(d-2)$, the normalized Aubin-Talenti bubblesolves $\Delta W+W^p=0$ and optimizes the critical Sobolev embedding theorem.