If $\phi \in H^{2,p'}$ and $\phi^+ \in H^1_0$, is it true that $\phi \in L^p$? And that $\phi$ in $H^{1,2}$?

Question

Let $ \Omega \in R^N$ be a limited open set, $N \ge 3$, and $p \in (1,\alpha]$ where $\alpha=\frac{2N}{N-2}$, $\phi:\Omega \rightarrow R$. Let $\phi \in H^{2,p'}(\Omega)$ where $\frac{1}{p} + \frac{1}{p'} =1$, and suppose $\phi^+ \in H^1_0(\Omega)$, where $\phi^+$ is the positive part of $\phi$.

Is it true that $\phi \in L^p$ and that $\phi \in H^{1,2}$?