How to express this expression with numerator and denominator?

I have an expression as following


and the constraint is as follows
\gamma_u \ge \gamma_{th},\forall u

Here, \alpha_{u,c} is a binary variable and 0\le P_c\le Pmax is a continuous variable.

We have multiplication of binary variable and continuous variables both in numerator and denominator in the form of \alpha_{u,c}P_c.

We can introduce auxiliary variable z_{u,c}=\alpha_{u,c}P_c and introduce these following constraints

\alpha_{u,c}p_l \le z_{u,c} \le \alpha_{u,c}p_u

P_c - p_l(1-\alpha_{u,c}) \le z_{u,c} \le P_c + p_u(1-\alpha_{u,c})

where p_u and p_l are upper and lower bound for P_c. Is this linearisation correct?

Also, how can I express the numerator, denominator expressions and the constraint?

Leaving aside the indices, I think your 2nd constraint needs to be
pl*(1-alpha) <= P-z <= pu*(1-alpha)

Presuming the denominator is nonnegative, move it to the RHS.