We consider the following boundary value problem <p align="center"> $ -\Delta u= g(x,u) + f(x,u)\quad x\in \Omega $ <p align="left" class="times"> <p align="center"> $u=0\quad x\in \partial \Omega$ <p align="left" class="times"> where $g(x,-\xi )=-g(x,\xi)$ and $g$ has subcritical exponential growth in $\mathbb R^2$. Using the method developed by Bolle, we prove that this problem has infinitely many solutions under suitable conditions on the growth of $g(u)$ and $f(u)$.