Nontrivial weak solutions of the stationary KdV equation in sharp L p spaces

In this paper we utilize a convex integration scheme to construct non-trivial solutions to the stationary Korteweg–de Vries equation which lie in L p ( T ) , p < 2 . In addition, we demonstrate this result is sharp in the sense that if u ∈ L 2 ( T ) is a weak solution then u ∈ C ∞ ( T ) .