Let's $A_n=\{ x\in\Omega : u_n(x) \geq 0\}$ and $A=\{ x\in\Omega : u(x) \geq 0\}$. We have $u_n^+(x)=u_n(x)\cdot 1_{A_n}$ and $u^+(x)=u(x)\cdot 1_A$.
Then $u_n\rightharpoonup u$ and $1_{A_n}\rightharpoonup 1_A$ implies $u_n^+\rightharpoonup u^+$. By cause in all metric space if $g_n\to g$ and $h_n\to h$ we have $g_n\cdot h_n \to g\cdot h$.
Now if we not have $1_{A_n}\rightharpoonup 1_A$ then $weak\,lim 1_{A_n}\neq 1_A$ end $|weak\,lim 1_{A_n}-1_A|=1$. And for all linear fuctional $F:L^p\to \mathbb{R}$ we have $ |F(u_n^+)-F(u^+)|=|F(u_n^{+})-F(u_n\cdot 1_{A})+F(u_n\cdot 1_{A})-F(u^+)| $ and implies for $N$ big $ |F(u_n^+)-F(u^+)|\geq |F(u_n\cdot 1_{A_n})-F(u_n\cdot 1_A)|= |F(u_n)|\cdot| 1_A-1_{A_n}|, \quad \forall n> N $ If $weak\,lim \inf\{u_n(x): x\in\Omega\}\neq 0$ not have convergence.