  In this paper, we investigate nonlinear Schr$\ddot{o}$dinger type equations
in $R^N$ under the framework of variable exponent spaces. We propose new
assumptions on the nonlinear term to yield bounded Palais-Smale sequences and
then prove the special sequences we find converge to critical points
respectively. The main arguments are based on the geometry supplied by Fountain
Theorem. Consequently, we show that the equation has a sequence of solutions
with high energies.
