Since $10^n - 1 = 9 \cdot (10^{n-1} + \dots + 10 + 1)$ we obtain that $11^n - 1$ is divisible by $9$. Considering the residues modulo $9$: $11^n - 1 \equiv 2^n - 1 \pmod{9}$ we get $n = 6k$. But then $10^{6k} - 1$ is divisible by $10^6 - 1$. Hence it is divisible by $10^3 + 1$ and $10 + 1 = 11$ which is not possible.