Existence of solutions for Kirchhoff ff-double phase anisotropic variational problems with variable exponents

This paper is devoted to dealing with a kind of new Kirchhoff-type problem in R(N )that involves a general double-phase variable exponent elliptic operator Phi. Specifically, the operator Phi has behaviors like |tau|q((x)-2)tau if |tau| is small and like |tau|p((x)-2)tau if |tau| is large, where 1 < p(x) < q(x) < N. By applying some new analytical tricks, we first establish existence results of solutions for this kind of Kirchhoff-double-phase problem based on variational methods and critical point theory. In particular, we also replace the classical Ambrosetti-Rabinowitz type condition with four different superlinear conditions and weaken some of the assumptions in the previous related works. Our results generalize and improve the ones in [Q. H. Zhang, V. D. Radulescu, J. Math. Pures Appl., 118 (2018), 159-203.] and other related results in the literature.