Existence of two solutions for Kirchhoff-double phase problems with a small perturbation without (AR)-condition

This paper deals with a class of Kirchhoff type problems on a double phase setting with a small perturbation.It is well known that (AR)condition is an important technique to apply the Mountain Pass theorem.A legitimate question arises, can we ensure the existence of weak solutions in the case where (AR)-condition is not satisfied?Our goal is to give a positive answer to this question.We present a new sufficient assumption weaker than (AR)-condition for which the considered problem admits at least two weak solutions.The proof rely on variational arguments based on the Mountain Pass Theorem with Cerami condition.Our results extend the results already proven by other authors.