In the present article, we study a class of Kirchhoff-type equations driven by the $\left(p\right(x),q(x\left)\right)$-Laplacian. Due to the lack of a variational structure, ellipticity, and monotonicity, the well-known variational methods are not applicable. With the help of the Galerkin method and Brezis theorem, we obtain the existence of finite-dimensional approximate solutions and weak solutions. One of the main difficulties and innovations of the present article is that we consider competing $\left(p\right(x),q(x\left)\right)$-Laplacian, convective terms, and logarithmic nonlinearity with variable exponents, another one is the weaker assumptions on nonlocal term $M_{\upsilon \left(x\right)}$ and nonlinear term g.

In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schrödinger–Kirchhoff type equations ${\left(a+b{\int }_{\mathsf{\Omega }\times \mathsf{\Omega }}\frac{{|\xi \left(x\right)-\xi \left(y\right)|}^{{}^p}}{{|x-y|}^{N+ps(x,y)}}dxdy\right)}^{p-1}{(-\Delta )}_p^{s(·)}\xi +\lambda V\left(x\right){\left|\xi \right|}^{p-2}\xi =f(x,\xi ),x\in \mathsf{\Omega },\xi =0,x\in \partial \mathsf{\Omega },$ where $\mathsf{\Omega }$ is a bounded Lipschitz domain in ${\mathbb{R}}^N$, $1<p<+\infty$, $a,b>0$ are constants, $s(·):{\mathbb{R}}^N\times {\mathbb{R}}^N\to (0,1)$ is a continuous and symmetric function with $N>s(x,y)p$ for all $(x,y)\in \mathsf{\Omega }\times \mathsf{\Omega }$, $\lambda >0$ is a parameter, ${(-\mathsf{\Delta })}_p^{s(·)}$ is a fractional p-Laplace operator with variable-order, $V\left(x\right):\mathsf{\Omega }\to {\mathbb{R}}^{+}$ is a potential function, and $f(x,\xi ):\mathsf{\Omega }\times {\mathbb{R}}^N\to \mathbb{R}$ is a continuous nonlinearity function. Assuming that V and f satisfy some reasonable hypotheses, we obtain the existence of infinitely many solutions for the above problem by using the fountain theorem and symmetric mountain pass theorem without the Ambrosetti–Rabinowitz ((AR) for short) condition.

In this article, we consider the following nonlocal fractional Kirchhoff-type elliptic systems $ \begin{equation*} \left\{\begin{array}{l} -M_{1}\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\eta(x)-\eta(y)|^{^{p(x, y)}}}{p(x, y)|x-y|^{N+p(x, y)s(x, y)}} \ \ \ \ \ dxdy +\int_{\Omega}\frac{|\eta|^{\overline{p}(x)}}{\overline{p}(x)}dx\right) \left(\Delta_{p(\cdot)}^{s(\cdot)}\eta-|\eta|^{\overline{p}(x)}\eta\right)\\ \; \; \; = \lambda F_{\eta}(x, \eta, \xi)+\mu G_{\eta}(x, \eta, \xi), \; \; x \in \Omega, \\ -M_{2}\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\xi(x)-\xi(y)|^{^{p(x, y)}}}{p(x, y)|x-y|^{N+p(x, y)s(x, y)}} \ \ \ \ \ dxdy +\int_{\Omega}\frac{|\xi|^{\overline{p}(x)}}{\overline{p}(x)}dx\right) \left(\Delta_{p(\cdot)}^{s(\cdot)}\xi-|\xi|^{\overline{p}(x)}\xi\right)\\ \; \; \; = \lambda F_{\xi}(x, \eta, \xi)+\mu G_{\xi}(x, \eta, \xi), \; \; x \in \Omega, \\ \; \eta = \xi = 0, \; \; x \in \mathbb{R}^{N}\backslash \Omega, \end{array} \right. \end{equation*} $ where $ M_{1}(t), M_{2}(t) $ are the models of Kirchhoff coefficient, $ \Omega $ is a bounded smooth domain in $ \mathbb R^{N} $, $ (-\Delta)_{p(\cdot)}^{s(\cdot)} $ is a fractional Laplace operator, $ \lambda, \mu $ are two real parameters, $ F, G $ are continuous differentiable functions, whose partial derivatives are $ F_{\eta}, F_{\xi}, G_{\eta}, G_{\xi} $. With the help of direct variational methods, we study the existence of solutions for nonlocal fractional $ p(\cdot) $-Kirchhoff systems with variable-order, and obtain at least two and three weak solutions based on Bonanno's and Ricceri's critical points theorem. The outstanding feature is the case that the Palais-Smale condition is not requested. The major difficulties and innovations are nonlocal Kirchhoff functions with the presence of the Laplace operator involving two variable parameters.

In this paper, we investigate the following Kirchhoff type problem involving the fractional p x -Laplacian operator. a − b ∫ Ω × Ω u x − u y p x , y / p x , y x − y N + s p x , y d x d y L u = λ u q x − 2 u + f x , u x ∈ Ω u = 0 x ∈ ∂ Ω , , where Ω is a bounded domain in ℝ N with Lipschitz boundary, a ≥ b > 0 are constants, p x , y is a function defined on Ω ¯ × Ω ¯ , s ∈ 0 , 1 , and q x > 1 , L u is the fractional p x -Laplacian operator, N > s p x , y , for any x , y ∈ Ω ¯ × Ω ¯ , p x ∗ = p x , x N / N − s p x , x , λ is a given positive parameter, and f is a continuous function. By using Ekeland’s variational principle and dual fountain theorem, we obtain some new existence and multiplicity of negative energy solutions for the above problem without the Ambrosetti-Rabinowitz ((AR) for short) condition.

In this article, we study a class of Kirchhoff-type equation driven by the variable s(x, ⋅)-order fractional p₁(x, ⋅) & p₂(x, ⋅)-Laplacian. With the help of three different critical point theories, we obtain the existence and multiplicity of solutions in an appropriate space of functions. The main difficulties and innovations are the Kirchhoff functions with double Laplace operators in the whole space ℝ^N. Moreover, the approach is variational, but we do not impose any Ambrosetti-Rabinowitz condition for the nonlinear term.