Existence results for a Kirchhoff-type equations involving the fractional $$p_{1}(x)$$ & $$p_{2}(x)$$-Laplace operator

Abstract

In this paper, we use variational approaches to establish the existence of weak solutions for a class of Kirchhoff-type equations with fractional $$p_{1}(x)$$ & $$p_{2}(x)$$ -Laplacian operator, for $$1\le p_{1}(x,y)<p_{2}(x,y)$$ , $$sp_{2}(x,y)<N$$ for all $$(x,y)\in {\overline{\varOmega }}\times {\overline{\varOmega }}$$ , and a Caratheodory reaction term which does not satisfy the Ambrosetti–Rabinowitz type growth condition. By mountain pass theorem with Cerami condition and the theory of the fractional variable exponent Sobolev space, we prove the existence of nontrivial solution for the problems in an appropriate space of functions. Furthermore, a multiplicity result of the problem is proved for odd nonlinearity.