Variational approach to the Bratu's problem

Khuri (A new approach to Bratu's problem, Applied Mathematics and Computation, 147(2004), 131-136) applied Adomian decomposition method to study the problem, the procedure is tedious though the expression is a lovely closed-form, furthermore the procedure can only locate one branch of the solutions. The problem, however, can be solved with easy to find the two branch solutions and identify the bifurcation point by the variational method.