Ambrosetti-Rabinowitz theorems revisited

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.