Some results for a kind of Kirchhoff-type problems involving the p(x)-biharmonic operator

Abstract

Our focus in this study revolves around investigating a Kirchhoff problem involving the p(x)-biharmonic operator. The purpose is to study the existence and multiplicity of weak solutions for our problem without assuming the Ambrosetti-Rabinowitz condition. By using the mountain pass theorem with Cerami condition, we show the existence of non-trivial weak solutions for the considered problem. Furthermore, our second purpose is to determine the precise positive interval of $\lambda$ for which the problem admits at least two nontrivial solutions. Finally, the existence of infinitely many solutions is proved by employing the fountain theorem.