Explicit Radial Bratu Solutions in Dimension n = 1, 2

We find the general solution \(u(x)\), \(x\in \mathbb{R}\) of the 1-D Bratu equation \(u_{xx} + \lambda e^u = 0\) and the general solution \(U(r)\), \(r>0\), of 2-D radial Bratu equation \(rU_{rr} + U_r + \lambda r e^U = 0\) . We use these results to obtain explicit solutions and bifurcation patterns for boundary value problems involving these equations, and in particular, solutions of the Bratu boundary value problem used in combustion theory. We review the history of Liouville's idea which made these results possible, and correct historical errors found in recent literature.