Assuming $p \neq q \neq r$, (otherwise there are infinitely many solutions, see p=q=2, r=r), the only solutions is $(p,q,r)=(2,3,5)$ and its permutations, when one of the $p,q,r$ equals to 7, the maximum value $LHS$ can get is $\frac{1}{2}+\frac{1}{3}+\frac{1}{7}= \frac{41}{42} <1$ therefore $p,q,r \le 5$