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 KdV equation which lie in $L^p(\mathbb{T})$, $p<2$. In addition, we demonstrate this result is sharp in the sense that if $u\in L^2(\mathbb{T})$ is a weak solution then $u\in C^{\mathrm{\infty }}(\mathbb{T})$.