BALKAN 2019 Problem 1

Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f: \mathbb{P} \rightarrow \mathbb{P}$ such that

$$f(p)^{f(q)}+q^{p}=f(q)^{f(p)}+p^{q}$$

holds for all $p, q \in \mathbb{P}$. Proposed by Albania