Lane-Emden equation

A Lane-Emden equation is a semilinear elliptic equation of the formFor a radial function it becomes $u^{\prime \prime }+(d-1)u^{\prime }/r+u^p=0$.

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.