Integration by Parts for not so regular Sobolev functions

Question

I am concerned with the following question: Let us assume we have some nice bounded domain $\Omega$ and $u\in W^{2,p}(\Omega)$ for some $1<p<2$. Let us further assume that we know that $-\Delta u\cdot u$ is in $L^1(\Omega)$ such that the integral $$ \int_\Omega -\Delta u\cdot u <\infty $$ is well-defined. Observe that this is not trivial since it might be that $p$ is very close to one such that the Sobolev embeddings do not yield $-\Delta u\cdot u\in L^1$. Now my question is the following: Can we, with this further information about integrability, deduce something like $|\nabla u \nabla u|^2\in L^1(\Omega)$, $\nabla u\cdot \vec{n}\cdot u\in L^1(\partial \Omega)$ and the usual integration by parts: $$ \int_\Omega -\Delta u\cdot u = \int_\Omega |\nabla u|^2+\int_{\partial \Omega} \nabla u\cdot \vec{n} \cdot u <\infty $$ Calculating the Sobolev embeddings this is clear for $p\geq \frac{6}{5}$ but can we get it even for $1<p<\frac{6}{5}$? Maybe there is some literature for this stuff, but I am not sure what the appropriate key words are. Thank you for any help!