Multiplicity of solutions for a class of fractional $p(x,\cdot )$-Kirchhoff-type problems without the Ambrosetti–Rabinowitz condition

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.