# Distributional solutions and test functions with non-compact support

Let's take a simple example. Let $$f\in L_{\rm loc}^1 (\mathbf R^n)$$. A function $$u$$ is called a distributional solution to the Poisson equation $$-\Delta u = f$$ in $$\mathcal D'(\mathbf R^n)$$ if $$-\int_{\mathbf R^n} u \Delta \phi dx = \int_{\mathbf R^n} f \phi dx$$ holds for any test function $$\phi \in C_0^\infty (\mathbf R^n)$$.

My question is: can we prove that the above identity also holds for smooth function $$\phi$$ not necessarily having compact support?

To mimic the idea of compact support, we can assume that $$\phi (x) \to 0$$ uniformly as $$|x| \to +\infty$$.

For any smooth function $$\phi$$ vanishing at infinity there is a sequence of test functions $$\phi_n \in C^\infty_0$$ s.t. $$\phi_n$$ converges to $$\phi$$ in lets say max norm. The only question is that whether $$\int \phi f dx$$ exists or not. Since $$f$$ is only locally integrable it does not need to vanish at infinity, hence it can happen that $$f\phi$$ does not vanish at infinity. So the answer is no in general.