“Fubini instanton.” Consider the functional \(S [ f ] = 2 \pi ^ { 2 } \int _ { 0 } ^ { \infty } d x x ^ { 3 } \left( { \frac { 1 } { 2 } } f ^ { \prime 2 } + V ( f ) \right)\) where \(V ( f ) = - { \frac { \lambda } { 4 } } f ^ { 4 }\) , and \(f \in C ^ { 2 } [ 0 , \infty ]\)
Write the differential equation on f, whose solution is an extremum of \(S [ f ]\)
Find all solutions to this equation satisfying the boundary conditions \(f ( \infty ) =\) \(f ^ { \prime } ( \infty ) = 0\)
Compute the value of S[f] on these solutions.