Liouville–Bratu–Gelfand equation

For Liouville's equation in differential geometry, see Liouville's equation.

In mathematics, Liouville–Bratu–Gelfand equation or Liouville's equation is a non-linear Poisson equation, named after the mathematicians Joseph Liouville, Gheorghe Bratu and Israel Gelfand. The equation reads

${\displaystyle \nabla ^{2}\psi +\lambda e^{\psi }=0}$

The equation appears in thermal runaway as Frank-Kamenetskii theory, astrophysics for example, Emden–Chandrasekhar equation. This equation also describes space charge of electricity around a glowing wire and describes planetary nebula.