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

Abstract

The paper deals with the following Robin problem$$\left\lbrace\begin{aligned}- \mathcal{M} \left( \int _{\Omega} \frac{1}{p(x)} \vert \nabla u \vert ^{p(x)} dx + \int _{\partial \Omega } \frac{a(x)}{p(x)} \vert \nabla u \vert ^{p(x)} d \sigma \right) \mathop{\rm div} (\vert \nabla u \vert ^{p(x)-2} \nabla u) = \lambda h(x,u) \ \ \text{ in } \Omega,\\\vert \nabla u \vert ^{p(x)-2} \frac{\partial u}{\partial \nu} + a(x) \vert u \vert ^{p(x)-2} u &=0 \quad \quad \quad \ \text{ on } \partial \Omega .\end{aligned}\right.$$The goal is to determine the precise positive interval of $\lambda $ 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.