Let's say I have an equation of the form $\Delta A = J$ where $J=u\nabla u + A|u|^2$ (Clarification: We are on $\mathbb{R}^3$ and $u$ is assumed to be in $H^1(\mathbb{R}^3)$). Then I could simply infer from Hardy-Littlewood-Sobolev and Hölder that
$$\|A\|\_6 \leq \|J\|\_{6/5} \leq \|u\|\_3\|\nabla u\|\_2 +\|A\|\_6 \|u\|\_3^2$$
and then from Sobolev
$$\cdots \leq \|u\|\_{H^1}^2 + \|A\|\_6 \|u\|\_{H^1}^2$$
Can I somehow infer from that that $\|A\|\_6$ is controlled by some norm of $u$?


EDIT: A thought that came to mind: Using $2ab \leq \varepsilon a^2 +\frac{1}{\varepsilon} b^2$ we could write
$$\cdots \leq \varepsilon \|A\|^2\_6 + (1+ C\_{\epsilon}\|u\|\_{H^1}^2) \|u\|\_{H^1}^2$$
for $\epsilon >0$. Is this sensible?


EDIT 2: The problem that arises is that from the inequality above we obtain
$$\|A\|\_6(1-\|u\|\_{H^1}^2) \leq \|u\|\_{H^1}^2$$
so if $||u||^2\_{H^1} = 1 $ we get no information on $\|A\|\_6$. Also if $||u||^2\_{H^1} > 1 $ Then the LHS becomes negative.



Estimates for an elliptic PDE
This is a way to get an a-priori estimate, if I understood correctly the question. Multiply by $A$ and integrate by parts the left-hand-side. Then
$$\int\_{R^3}(A^2u^2+|\nabla A|^2)=-\int\_{R^3}Au\nabla u\le \|Au\|\_2\|\nabla u\|\_2
$$ and then both $\|Au\|\_2, \|\nabla A\|\_2 \le \|\nabla u\|\_2$. Since $2^\*=6$, $\|A\|\_6 \le C\|\nabla A\|\_2 \le C\|\nabla u\|\_2$.


