A new class of multiple nonlocal problems with two parameters and variable-order fractional $p(\cdot )$-Laplacian

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:

$$\begin{array}{r}\{\begin{array}{l}M({\sigma }_{p(x,y)}(u))(-\mathrm{\Delta }{)}_{p(\cdot )}^{s(\cdot )}u(x)=\lambda |u{|}^{q(x)-2}u{\left({\int }_{\mathrm{\Omega }}\frac{1}{q(x)}|u{|}^{q(x)}dx\right)}^{k_1}+\beta |u{|}^{r(x)-2}u{\left({\int }_{\mathrm{\Omega }}\frac{1}{r(x)}|u{|}^{r(x)}dx\right)}^{k_2}\text{in}\mathrm{\Omega },\\ u=0\text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$$

where the nonlocal term is defined as

$${\sigma }_{p(x,y)}(u)={\int }_{\mathrm{\Omega }\times \mathrm{\Omega }}\frac{1}{p(x,y)}\frac{|u(x)-u(y){|}^{p(x,y)}}{|x-y{|}^{N+s(x,y)p(x,y)}}dxdy.$$

Here, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ represents a bounded smooth domain with at least $N\ge 2$. The function $M(s)$ is given by $M(s)=a-b{s}^{\gamma }$, where $a\ge 0$, $b>0$, 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.