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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 $|x| \to +\infty$.

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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.

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