  We prove existence, non-degeneracy, and exponential decay at infinity of a
non-trivial solution to Emden's equation $-\Delta u = | u |^3$ on an unbounded
$L$-shaped domain, subject to Dirichlet boundary conditions. Besides the direct
value of this result, we also regard this solution as a building block for
solutions on expanding bounded domains with corners, to be established in
future work. Our proof makes heavy use of computer assistance: Starting from a
numerical approximate solution, we use a fixed-point argument to prove
existence of a near-by exact solution. The eigenvalue bounds established in the
course of this proof also imply non-degeneracy of the solution.
