Pre-print, Working paper: We use Ekeland’s variational principle together with Pontryagin’s maximum principle to solve an optimal spatiotemporal economic growth model with a state constraint (no-negative capital stock) where capital law of motion follows a diffusion equation. We obtain the set of necessary optimal conditions for the solution to meet the state constraints for all time and locations. The maximum principle allows to reduce the infinite-horizon optimal control problem into a finite-horizon one ultimately leading to prove the uniqueness of the optimal solution with positive capital, and non-existence of the optimal solution with eventually strictly positive capital when the time discount rate is too large or too small.