Kirchhoff-type equations involving the Fractional $(p,q)-$Laplacian

In this paper, we study the existence and nonexistence of solutions for the following Kirchhoff-type fractional $(p\text{-}q)$-Laplacian problem: \begin{equation*} \begin{cases} M\left([u]^p_{p,s_1}\right)(-\Delta)^{s_1}_p u + M\left([u]^q_{q,s_2}\right)(-\Delta)^{s_2}_q u = \lambda\big[a(x)|u|^{p-2}u + b(x)|u|^{q-2}u\big] + h(x), & \text{in } \Omega, \\ u = 0, & \text{on } \mathbb{R}^N \setminus \Omega, \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^N$ ($N \geq 1$) is a bounded domain with smooth boundary, $0 < s_1 < s_2 < 1$, and $s_1 p < N$. We assume $1 < q \leq p < \theta p < p^{*}_{s_1} := \dfrac{Np}{N - s_1 p}$, and $\lambda \in \mathbb{R}$. The functions $a(x), b(x)$, and $h(x)$ are non-negative, with $a, b \in L^\infty(\Omega)$ and $h \in L^q(\Omega)$. Using variational methods, we establish the existence of at least two weak solutions. The first solution is obtained via the direct minimization of the associated energy functional, and the second is obtained by applying the Mountain Pass Theorem. We also prove a nonexistence result for small values of the parameter $\lambda > 0$.