I know the following result is true in the case of strong convergence. But I don't know whether it is true in the case of weak convergence also. Let $p>1$. Suppose that each $x_n$ is a non negative sequence such that $\x_n\_p=1$ and $\stackrel{w}{x_{n}\rightarrow x}$ in $\ell^p$. Is it true then that $\stackrel{w}{x_n^p\rightarrow x^p}$ in $\ell^1$.

No. Consider the unit vector basis of $\ell_2$. 

