Suppose we work on the two dimensional torus $\mathbb T^2$. Let $L_a^2$ be the space of square integrable functions with zero space average and $H_a^m$ be the corresponding Sobolev space. Suppose we have an elliptic equation of the form $$ -\Delta \psi - g'(\psi^0) \psi = \phi $$ where $\psi \in H_a^2$ solves this equation, where $g'(\psi^0) \in C^0(\mathbb T^2)$ and $\phi \in H_a^1$, would it be possible to conclude that $\psi \in H_a^3$? Is there a reference for this?
Edit: Let us also assume the following. Let $\psi^0$ be a function that generates the flow $u^0= -\nabla^\perp\psi^0$ on the torus. Also assume that the level sets of $\psi^0$ consist of finitely many closed curves each of which is diffeomorphic to the unit cirle $\mathbb S^1$. Let $L^0$ be the operator of differentiation defined as $L^0g = u \cdot \nabla g$ where $L^0:H_a^1 \to L_a^2$ and $\phi= - (\lambda - L^0)^{-1}\lambda g'(\psi^0)\psi$ where $\lambda$ is in the resolvent set of the operator $L^0$. We note that $g'(\psi^0)$ is constant on the level sets of $\psi^0$.