Notation. $H^k(\mathbb R^n)$ denotes the Sobolev space $W^{k,2}(\mathbb R^n)$. The Sobolev space $W^{k,p}(\mathbb R^n)$ is the Banach space of functions in $L^p$ with all derivatives up to order $k$ in $L^p$.

The PDE

Consider a very simple semilinear elliptic PDE: $$ \left\{\begin{aligned} &-\Delta u=\partial_{x_1}(u^2+f)& &\text{in } \mathbb R^n,\\ & \,\,\,u(x)\to 0\,,& &x\to\infty. \end{aligned}\right. $$ where $x=(x_1,...,x_n)$. The boundary conditions are not rigorous, it is just to describe the kind of solutions we want. The critical Lebesgue spaces with respect to the natural scaling symmetry are: $$ u\in L^{n}(\mathbb R^n), f\in L^{n/2}(\mathbb R^n). $$

Strong solutions

Assume $n=3$ to fix ideas. By using the fixed point theorem, it is possible to show that if $f\in L^{3/2}(\mathbb R^3)$ is small enough, then there exists a unique small solution $u\in L^3(\mathbb R^3)$, which moreover satisfies $$ \|\nabla u\|_{L^{3/2}(\mathbb R^3)}\lesssim \|f\|_{L^{3/2}(\mathbb R^3)} $$ (note that $\dot W^{1,3/2}(\mathbb R^3)\hookrightarrow L^3(\mathbb R^3)$).

Weak solutions

If I instead consider a scaling-subcritical datum, like $f\in L^2(\mathbb R^3)$, I cannot use the fixed point theorem. But notice that multiplying the equation by $u$ and integrating by parts, we obtain the a priori bound

$$ \|\nabla u\|_{L^2(\mathbb R^3)}\lesssim \|f\|_{L^2(\mathbb R^3)}. $$

Using this a-priori bound (please correct me if I am wrong*) it is possible to prove the existence of weak solutions $u\in \dot H^1(\mathbb R^3)$ for any $f\in L^2(\mathbb R^3)$ (we have $\dot H^1(\mathbb R^3)\hookrightarrow L^6(\mathbb R^3)$).

My question

Now, there are two very natural question one could ask:

1. Does a solution exist for large data $f\in L^3(\mathbb R^3)$ (the critical space)? Is it unique, assuming it exists?
2. Is the weak solution unique if $f\in L^2(\mathbb R^3)$? If no, how can one find a counterexample?

The answer to question number (1), I would say, is probably "no" concerning the existence. This is probably related to blowup of solutions of hyperbolic PDE for large data. My question is not about that.

My concern is more about the second question. In a case where we have a priori bounds like in the PDE I wrote, do we have non-uniqueness in case of subcritical data? Note that this is not a regularity prolem: data in $L^2$ are "more regular" than data in $L^{3/2}$. The problem is that data grow more slowly at infinity than data in the critical space. I could ask the same question for data $f\in H^{100}(\mathbb R^3)$, and the answer would still be non trivial.

I feel likely that there are data that lead to two different solutions, but I know no technique that would do that.

_{*I think there should be more than one way of doing that. The way I thought would be to approximate the Cauchy problem with that of the same PDE on a half plane (with Dirichlet boundary conditions), where the "edge" of the plane goes to infinity. On a half plane, you can look at the PDE as an evolution equation via the Poisson kernel, so you can show that a solution exists, and then you can use the uniform bound and weak compactness to show the existence of the solution of the original problem... Probably not the easiest way. I have to say I am not an elliptic person, I am mostly dispersive, sometimes parabolic... In any case, feel free to comment on this point.}