Consider a nonlinear PDE for $p(x,y)$ defined on $\mathbb{R}^2$:
\begin{equation} N p(x,y)=0, \:\:\:\:\:\:\:Eq. (1)\end{equation}
where $N$ is a 2nd order nonlinear operator. The variable $p$ is like "density". The only BCs I have is that $p$ is zero at infinity.
Now a particular solution, $\hat{p}$, turns out to be independent of $y$, i.e, $\hat{p}=\hat{p}(x)$. That is,
$N\hat{p}(x)=0$.
However, due to the physical motivation behind this problem, I want to find a solution with compact support in $y$. The reason is that I want the integral of $\hat{p}$ over all of $\mathbb{R}^2$ to be bounded.
My Question:
Let $q(x,y)=\hat{p}(x)\mathbb{I}_{\Omega}(y)$, where $\mathbb{I}_{\Omega}$ is the characteristic function of a well-chosen set $\Omega\in \mathbb{R}$. Can I claim that the $q(x,y)$ is also a weak solution of Eq. (1)?
I am not sure how to rigorously justify this construction. Is the theory of distributions applicable here ? Clearly $q(x,y)$ satisfies the equation except at set of measure 0, which is the boundary of $\Omega$.