On a problem of Pillai involving S-units and Lucas numbers

Let $\{L_n{\}}_{n\ge 0}$ be the sequence of Lucas numbers. In this paper, we look at the exponential Diophantine equation $L_n-2^x{3}^y=c$, for $n,x,y\in {\mathbb{Z}}_{\ge 0}$. We treat the cases $c\in -\mathbb{N}$, $c=0$ and $c\in \mathbb{N}$ independently. In the cases that $c\in \mathbb{N}$ and $c\in -\mathbb{N}$, we find all integers $c$ such that the Diophantine equation has at least three solutions. These cases are treated independently since we employ quite different techniques in proving the two cases.