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

This paper is devoted to dealing with a kind of new Kirchhoff-type problem in ${\mathbb{R}}^N$ that involves a general double-phase variable exponent elliptic operator $\mathit{\varphi }$. Specifically, the operator $\mathit{\varphi }$ 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. Rădulescu, J. Math. Pures Appl., 118 (2018), 159–203.] and other related results in the literature.