On a class of critical Schröinger-Kirchhoff-type problems involving anisotropic variable exponent

Abstract

The given problem involves a nonlocal and nonhomogeneous anisotropic elliptic equation of the form :$ \begin{aligned} &-K\left( \int_{\Omega} \sum\limits_{i = 1}^{N} \mathcal{A}_{i}\left(z, \partial_{z_{i}} \vartheta\right)+\frac{\Theta(z)}{p_{K}(z)}|\vartheta|^{p_{K}(z)} \mathrm{\; d} z\right) \\ & \quad\times \left(\sum\limits_{i = 1}^{N} \partial_{z_{i}} a_{i}\left(z, \partial_{z_{i}} \vartheta\right)-\Theta(z)|\vartheta|^{p_{K}(z)-2} \vartheta\right) = \mu f(z, \vartheta), \end{aligned} $where $ \Omega\subset \mathbb{R}^N \; ( N\geq 2) $ is a bounded domain with Lipschitz boundary $ \partial \Omega $, $ K: \mathbb{R}_0^{+}\longrightarrow \mathbb{R}^{+} $ be a nondecreasing and continuous Kirchhoff function, and $ f:\Omega\times \mathbb{R} \longrightarrow \mathbb{R} $ is a Carathéodory function. The existence of weak solutions to this problem is demonstrated through the application of Berkovits and Mustonen's topological degree theory in the framework of anisotropic Sobolev spaces with variable exponent $ W_{0}^{1, \vec{p}(z)}(\Omega) $. The viability of this approach is contingent upon the fulfillment of specific assumptions.