The generalized Pillai equation $\pm r{a}^x\pm s{b}^y=c$

In this paper we consider $N$, the number of solutions $(x,y,u,v)$ to the equation $(-1{)}^{u}r{a}^x+(-1{)}^{v}s{b}^y=c$ in nonnegative integers $x,y$ and integers $u,v\in \{0,1\}$, for given integers $a>1$, $b>1$, $c>0$, $r>0$ and $s>0$. We show that $N\le 2$ when $\gcd (ra,sb)=1$ and $\min (x,y)>0$, except for a finite number of cases that can be found in a finite number of steps. For arbitrary $\gcd (ra,sb)$ and $\min (x,y)\ge 0$, we show that when $(u,v) = (0,1)$wehave$N \le 3$, with an infinite number of cases for which N=3.