1. We start with the given inequalities:
$$2p^2 - r^2 \geq 49$$
$$2q^2 - r^2 \leq 193$$

2. From these inequalities, we can derive:
$$2q^2 - 193 \leq r^2 \leq 2p^2 - 49$$

3. This implies:
$$2q^2 - 193 \leq 2p^2 - 49$$
Simplifying this, we get:
$$2q^2 - 2p^2 \leq 144$$
$$q^2 - p^2 \leq 72$$

4. We can factorize the left-hand side:
$$(q + p)(q - p) \leq 72$$

5. Since $p$ and $q$ are prime numbers and $5 \leq p < q$, we need to find pairs of primes that satisfy this inequality. We will test possible values for $q$ and $p$.

6. Let's start with the largest possible $q$ less than 37 (since $q$ must be less than $\sqrt{72} + p$ and $p \geq 5$):
- For $q = 31$:
$$(31 + p)(31 - p) \leq 72$$
Testing possible values for $p$:
- $p = 29$:
$$(31 + 29)(31 - 29) = 60 \times 2 = 120 \quad (\text{not valid})$$
- $p = 23$:
$$(31 + 23)(31 - 23) = 54 \times 8 = 432 \quad (\text{not valid})$$
- $p = 19$:
$$(31 + 19)(31 - 19) = 50 \times 12 = 600 \quad (\text{not valid})$$
- $p = 17$:
$$(31 + 17)(31 - 17) = 48 \times 14 = 672 \quad (\text{not valid})$$
- $p = 13$:
$$(31 + 13)(31 - 13) = 44 \times 18 = 792 \quad (\text{not valid})$$
- $p = 11$:
$$(31 + 11)(31 - 11) = 42 \times 20 = 840 \quad (\text{not valid})$$
- $p = 7$:
$$(31 + 7)(31 - 7) = 38 \times 24 = 912 \quad (\text{not valid})$$
- $p = 5$:
$$(31 + 5)(31 - 5) = 36 \times 26 = 936 \quad (\text{not valid})$$

- For $q = 29$:
$$(29 + p)(29 - p) \leq 72$$
Testing possible values for $p$:
- $p = 23$:
$$(29 + 23)(29 - 23) = 52 \times 6 = 312 \quad (\text{not valid})$$
- $p = 19$:
$$(29 + 19)(29 - 19) = 48 \times 10 = 480 \quad (\text{not valid})$$
- $p = 17$:
$$(29 + 17)(29 - 17) = 46 \times 12 = 552 \quad (\text{not valid})$$
- $p = 13$:
$$(29 + 13)(29 - 13) = 42 \times 16 = 672 \quad (\text{not valid})$$
- $p = 11$:
$$(29 + 11)(29 - 11) = 40 \times 18 = 720 \quad (\text{not valid})$$
- $p = 7$:
$$(29 + 7)(29 - 7) = 36 \times 22 = 792 \quad (\text{not valid})$$
- $p = 5$:
$$(29 + 5)(29 - 5) = 34 \times 24 = 816 \quad (\text{not valid})$$

- For $q = 23$:
$$(23 + p)(23 - p) \leq 72$$
Testing possible values for $p$:
- $p = 19$:
$$(23 + 19)(23 - 19) = 42 \times 4 = 168 \quad (\text{not valid})$$
- $p = 17$:
$$(23 + 17)(23 - 17) = 40 \times 6 = 240 \quad (\text{not valid})$$
- $p = 13$:
$$(23 + 13)(23 - 13) = 36 \times 10 = 360 \quad (\text{not valid})$$
- $p = 11$:
$$(23 + 11)(23 - 11) = 34 \times 12 = 408 \quad (\text{not valid})$$
- $p = 7$:
$$(23 + 7)(23 - 7) = 30 \times 16 = 480 \quad (\text{not valid})$$
- $p = 5$:
$$(23 + 5)(23 - 5) = 28 \times 18 = 504 \quad (\text{not valid})$$

- For $q = 19$:
$$(19 + p)(19 - p) \leq 72$$
Testing possible values for $p$:
- $p = 17$:
$$(19 + 17)(19 - 17) = 36 \times 2 = 72 \quad (\text{valid})$$
Now we check $r$:
$$2p^2 - r^2 \geq 49 \quad \text{and} \quad 2q^2 - r^2 \leq 193$$
For $p = 17$ and $q = 19$:
$$2(17)^2 - r^2 \geq 49 \quad \Rightarrow \quad 578 - r^2 \geq 49 \quad \Rightarrow \quad r^2 \leq 529 \quad \Rightarrow \quad r \leq 23$$
$$2(19)^2 - r^2 \leq 193 \quad \Rightarrow \quad 722 - r^2 \leq 193 \quad \Rightarrow \quad r^2 \geq 529 \quad \Rightarrow \quad r \geq 23$$
Therefore, $r = 23$.

The final answer is $\boxed{ p = 17, q = 19, r = 23 }$