We are interested in the existence of solutions for the following fractional $p(x,\cdot )$p ( x , ⋅ ) -Kirchhoff-type problem: $$ \textstyle\begin{cases} M ( \int _{\Omega \times \Omega } {\frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{p(x,y) \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy )(-\Delta )^{s}_{p(x,\cdot )}u = f(x,u), \quad x\in \Omega , \\ u= 0, \quad x\in \partial \Omega , \end{cases} $${ M ( ∫ Ω × Ω | u ( x ) − u ( y ) | p ( x , y ) p ( x , y ) | x − y | N + p ( x , y ) s d x d y ) ( − Δ ) p ( x , ⋅ ) s u = f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $\Omega \subset \mathbb{R}^{N}$Ω ⊂ R N , $N\geq 2$N ≥ 2 is a bounded smooth domain, $s\in (0,1)$s ∈ ( 0 , 1 ) , $p: \overline{\Omega }\times \overline{\Omega } \rightarrow (1, \infty )$p : Ω ‾ × Ω ‾ → ( 1 , ∞ ) , $(-\Delta )^{s}_{p(x,\cdot )}$( − Δ ) p ( x , ⋅ ) s denotes the $p(x,\cdot )$p ( x , ⋅ ) -fractional Laplace operator, $M: [0,\infty ) \to [0, \infty )$M : [ 0 , ∞ ) → [ 0 , ∞ ) , and $f: \Omega \times \mathbb{R} \to \mathbb{R}$f : Ω × R → R are continuous functions. Using variational methods, especially the symmetric mountain pass theorem due to Bartolo–Benci–Fortunato (Nonlinear Anal. 7(9):981–1012, 1983), we establish the existence of infinitely many solutions for this problem without assuming the Ambrosetti–Rabinowitz condition. Our main result in several directions extends previous ones which have recently appeared in the literature.

We study a class of $ p(x) $-Kirchhoff problems which is seldom studied because the nonlinearity has nonstandard growth and contains a bi-nonlocal term. Based on variational methods, especially the Mountain pass theorem and Ekeland's variational principle, we obtain the existence of two nontrivial solutions for the problem under certain assumptions. We also apply the Symmetric mountain pass theorem and Clarke's theorem to establish the existence of infinitely many solutions. Our results generalize and extend several existing results.

In the present manuscript, we focus on a novel tri-nonlocal Kirchhoff problem, which involves the $ p(x) $-fractional Laplacian equations of variable order. The problem is stated as follows: \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{ll} M\Big(\sigma_{p(x, y)}(u)\Big)(-\Delta)^{s(\cdot)}_{p(\cdot)}u(x) = \lambda |u|^{q(x)-2}u\left(\int_{\Omega}\frac{1}{q(x)} |u|^{q(x)}dx \right)^{k_1}+\beta|u|^{r(x)-2}u\left(\int_{\Omega}\frac{1}{r(x)} |u|^{r(x)}dx \right)^{k_2} \quad \mbox{in }\Omega, \\ \ u = 0 \quad \mbox{on }\partial\Omega, \end{array} \right. \end{eqnarray*} $\end{document} where the nonlocal term is defined as \begin{document}$ \sigma_{p(x, y)}(u) = \int_{\Omega\times \Omega}\frac{1}{p(x, y)}\frac{|u(x)-u(y)|^{p(x, y)}}{|x-y|^{N+s(x, y)p(x, y)}} \, dx\, dy. $\end{document} Here, $ \Omega\subset\mathbb{R}^{N} $ represents a bounded smooth domain with at least $ N\geq2 $. The function $ M(s) $ is given by $ M(s) = a - bs^\gamma $, where $ a\geq 0 $, $ b > 0 $, and $ \gamma > 0 $. The parameters $ k_1 $, $ k_2 $, $ \lambda $ and $ \beta $ are real parameters, while the variables $ p(x) $, $ s(\cdot) $, $ q(x) $, and $ r(x) $ are continuous and can change with respect to $ x $. To tackle this problem, we employ some new methods and variational approaches along with two specific methods, namely the Fountain theorem and the symmetric Mountain Pass theorem. By utilizing these techniques, we establish the existence and multiplicity of solutions for this problem separately in two distinct cases: when $ a > 0 $ and when $ a = 0 $. To the best of our knowledge, these results are the first contributions to research on the variable-order $ p(x) $-fractional Laplacian operator.