Liouville–Bratu–Gelfand equation

In two dimension with Cartesian Coordinates ${\displaystyle (x,y)}$, Joseph Liouville proposed a solution in 1853 as

${\displaystyle \lambda e^{\psi }(u^{2}+v^{2}+1)^{2}=2\left[\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial u}{\partial y}}\right)^{2}\right]}$

where ${\displaystyle f(z)=u+iv}$ is an arbitrary analytic function with ${\displaystyle z=x+iy}$. In 1915, G.W. Walker[6] found a solution by assuming a form for ${\displaystyle f(z)}$. If ${\displaystyle r^{2}=x^{2}+y^{2}}$, then Walker's solution is

${\displaystyle 8e^{-\psi }=\lambda \left[\left({\frac {r}{a}}\right)^{n}+\left({\frac {a}{r}}\right)^{n}\right]^{2}}$

where ${\displaystyle a}$ is some finite radius. This solution decays at infinity for any ${\displaystyle n}$, but becomes infinite at the origin for ${\displaystyle n<1}$ , becomes finite at the origin for ${\displaystyle n=1}$ and becomes zero at the origin for ${\displaystyle n>1}$. Walker also proposed two more solutions in his 1915 paper.

If the system to be studied is radially symmetric, then the equation in ${\displaystyle n}$ dimension becomes

${\displaystyle \psi ''+{\frac {n-1}{r}}\psi '+\lambda e^{\psi }=0}$

where ${\displaystyle r}$ is the distance from the origin. With the boundary conditions

${\displaystyle \psi '(0)=0,\quad \psi (1)=0}$

and for ${\displaystyle \lambda \geq 0}$, a real solution exists only for ${\displaystyle \lambda \in [0,\lambda _{c}]}$, where ${\displaystyle \lambda _{c}}$ is the critical parameter called as Frank-Kamenetskii parameter. The critical parameter is ${\displaystyle \lambda _{c}=0.8785}$ for ${\displaystyle n=1}$, ${\displaystyle \lambda _{c}=2}$ for ${\displaystyle n=2}$ and ${\displaystyle \lambda _{c}=3.32}$ for ${\displaystyle n=3}$. For ${\displaystyle n=1,\ 2}$, two solution exists and for ${\displaystyle 3\leq n\leq 9}$ infinitely many solution exists with solutions oscillating about the point ${\displaystyle \lambda =2(n-2)}$. For ${\displaystyle n\geq 10}$, the solution is unique and in these cases the critical parameter is given by ${\displaystyle \lambda _{c}=2(n-2)}$. Multiplicity of solution for ${\displaystyle n=3}$ was discovered by Israel Gelfand in 1963 and in later 1973 generalized for all ${\displaystyle n}$ by Daniel D. Joseph and Thomas S. Lundgren.[7]

The solution for ${\displaystyle n=1}$ that is valid in the range ${\displaystyle \lambda \in [0,0.8785]}$ is given by

${\displaystyle \psi =-2\ln \left[e^{-\psi _{m}/2}\cosh \left({\frac {\sqrt {\lambda }}{\sqrt {2}}}e^{-\psi _{m}/2}r\right)\right]}$

where ${\displaystyle \psi _{m}=\psi (0)}$ is related to ${\displaystyle \lambda }$ as

${\displaystyle e^{\psi _{m}/2}=\cosh \left({\frac {\sqrt {\lambda }}{\sqrt {2}}}e^{-\psi _{m}/2}\right).}$

The solution for ${\displaystyle n=2}$ that is valid in the range ${\displaystyle \lambda \in [0,2]}$ is given by

${\displaystyle \psi =\ln \left[{\frac {64e^{\psi _{m}}}{(\lambda e^{\psi _{m}}r^{2}+8)^{2}}}\right]}$

where ${\displaystyle \psi _{m}=\psi (0)}$ is related to ${\displaystyle \lambda }$ as

${\displaystyle (\lambda e^{\psi _{m}}+8)^{2}-64e^{\psi _{m}}=0.}$