  We introduce tow assumptions weaker than the classical Ambrosetti-Rabinowitz
and the subcritical polynomial growth conditions to obtain the Palais-Smale
Condition. Therefore, we improve the Ambrosetti- Rabinowitz existence theorems.
Also, we prove some existence results under a weaken subcritical polynomial
growth conditions where the nonlinearity does not satisfy the super-quadratic
condition and even it does not have any limits at infinity.
