We consider the existence of unbounded solutions for the asymmetric oscillator $$(\vp_p(x'))'+(p-1)[\al \vp_p(x^+)-\be \vp_p(x^-)]=f(t)\eqno(1)$$where $\vp_p(u)=|u|^{p-2}u, p>1$, $\al$ and $\be$ are positive constants satisfying $$\al^{{-}\frac 1p}+\be^{{-}\frac 1p}=2m/n\eqno(2)$$ with ($m,n)=1,\,\, m, n\in N$ and $x^\pm=\max\{\pm x,0\}$, $f \in L^\infty[0,2\pi_p]$ is $2\pi_p$-periodic, $\pi_p={2\pi}/({p\sin (\pi/p)})$.