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

Abstract: In this paper we utilize a convex integration scheme to construct non-trivial solutions to the stationary KdV equation which lie in $L<sup>p(\mathbb{T})$, $p &lt; 2$. In addition, we demonstrate this result is sharp in the sense that if $u \in L<sup>2(\mathbb{T})$ is a weak solution then $u \in C<sup>\infty(\mathbb{T})$.