What about this example:

f_n:=\frac{1}{n*x}

\forall x \in (0,\frac{1}{2}) \rightarrow |f_n(x)|\ge \frac{1}{n*\frac{1}{2*n}}=2

and \int_{[0,1]} |f_n|^p = \frac{1}{n^p * (1-p)}

I think this one clearly works now, yes?