5. If $a, b$ satisfy $0<a<b<1$, then which of the following inequalities is correct? ( ).
(A) $a^{a}<b^{b}$
(B) $b^{a}<b^{b}$
(C) $a^{a}<b^{a}$
(D) $b^{b}<a^{a}$
5.C.

According to the monotonicity of power functions and exponential functions, we should have $b^{a}>a^{a}, b^{a}>b^{b}$.
The size relationship between $a^{a}$ and $b^{b}$ is uncertain.