That doesn't work because $H_0^1$ functions are small near the boundary, so testing against them won't detect bad behavior of $u$ near $\partial\Omega$.

For a concrete example, take $\Omega$ as the unit ball and $u(x)=1/(1-|x|)\notin L^1$. Then $$ \int |uv|\, dx \le \left( \int \frac{v^2\, dx}{(1-|x|)^{3/2}} \int \frac{dx}{(1-|x|)^{1/2}} \right)^{1/2} . $$ If $v\in H_0^1$ is also smooth, then we can estimate the first integral in the same way as in this related question (by just integrating the gradient, starting from the boundary, to bound $v$). This gives $\int v^2/(1-|x|)^{3/2}\lesssim \|v\|^2_{H^1}$, so $\int |uv| \lesssim \|v\|_{H^1}$ for all such $v$, and by density of the smooth functions, this also holds for arbitrary $v\in H_0^1$.

That doesn't work because $H_0^1$ functions are small near the boundary, so testing against them won't detect bad behavior of $u$ near $\partial\Omega$.

For a concrete example, take $\Omega$ as the unit ball and $u(x)=1/(1-|x|)\notin L^1$. Then $$ \int |uv|\, dx \le \left( \int \frac{v^2\, dx}{(1-|x|)^{3/2}} \int \frac{dx}{(1-|x|)^{1/2}} \right)^{1/2} . $$ If $v\in H_0^1$ is also smooth, then we can estimate the first integral in the same way as in this related question (by just integrating the gradient, starting from the boundary, to bound $v$). This gives $\int v^2/(1-|x|)^{3/2}\lesssim \|v\|^2_{H^1}$, so $\int |uv| \lesssim \|v\|_{H^1}$ for all such $v$, and by density of the smooth functions, this also holds for arbitrary $v\in H_0^1$.

That doesn't work because $H_0^1$ functions are small near the boundary, so testing against them won't detect bad behavior of $u$ near $\partial\Omega$.

For a concrete example, take $\Omega$ as the unit ball and $u(x)=1/(1-|x|)\notin L^1$. Then, if $v\in H_0^1$ is also smooth, then (by just integrating the gradient, starting from the boundary; see for example this related question) $$ |v(x)|\le (1-|x|)^{1/2}\|v\|_{H_0^1} . $$ So $\int |uv| \lesssim \|v\|_{H_0^1}$ for all such $v$, and by density of the smooth functions, this also holds for arbitrary $v\in H_0^1$.

That doesn't work because $H_0^1$ functions are small near the boundary, so testing against them won't detect bad behavior of $u$ near $\partial\Omega$.

For a concrete example, take $\Omega$ as the unit ball and $u(x)=1/(1-|x|)\notin L^1$. Then $$ \int |uv|\, dx \le \left( \int \frac{v^2\, dx}{(1-|x|)^{3/2}} \int \frac{dx}{(1-|x|)^{1/2}} \right)^{1/2} . $$ If $v\in H_0^1$ is also smooth, then we can estimate the first integral in the same way as in this related question (by just integrating the gradient, starting from the boundary, to bound $v$). This gives $\int v^2/(1-|x|)^{3/2}\lesssim \|v\|^2_{H^1}$, so $\int |uv| \lesssim \|v\|_{H^1}$ for all such $v$, and by density of the smooth functions, this also holds for arbitrary $v\in H_0^1$.