Robin problem involving the p (x) -Laplacian operator without Ambrosetti-Rabinowizt condition

The paper deals with the following Robin problem- M (_ 1p (x) u ^p (x) dx + a (x) p (x) u ^p (x) d) div (u ^p (x) -2 u) = h (x, u) \ \ in, \\ u ^p (x) -2 u + a (x) u ^p (x) -2 u &=0 \ on. aligned. The goal is to determine the precise positive interval of for which the problem admits at least two nontrivial solutions via variational approach for the above problem without assuming the Ambrosetti-Rabinowitz condition. Next, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the fountain theoreom with Cerami condition.