On subspaces of the spaces $L^p_{loc}$ and of their strong duals

For $\Omega$ an open subset of $\R^N$ and $1<p<\infty$ we prove that any complemented subspace $E$ of $L^p_{loc}(\Omega)$ contains either a complemented copy of $(\ell^2)^\N$ or a complemented copy of $(\ell^p)^\N$, provided $E\not\simeq F$, $\not\simeq \omega$ and $\not\simeq F\oplus \omega$ with $F$ Banach space. We also prove that any complemented subspace $E$ of $(L^p_{loc}(\Omega))'_\beta$ contains either a complemented copy of $(\ell^2)^{(\N))$ or a complemented copy of $(\ell^p)^{(\N)}$, provided $E\not\simeq F$, $\not\simeq \varphi$ and $\not\simeq F\oplus \varphi$ with $F$ Banach space.