Three Weak Solutions of \((\alpha _1(\cdot ), \ldots , \alpha _N(\cdot ))\)-Laplacian-Schrödinger-Kirchhoff Systems

In this paper, we investigate the existence of multiple weak solutions for a Schrödinger-Kirchhoff type elliptic system involving nonlocal \((\alpha _1(\cdot ), \ldots , \alpha _N(\cdot ))\)-Laplacian operator. The system is modeled as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathfrak {M}_i\left( \int _{\mathbb {R}^N}\frac{1}{\alpha _{i}(y)}|\nabla u_{i}|^{\alpha _{i}(y)} dy+\int _{\mathbb {R}^N}\frac{\mathcal {V}_{i}(y)}{\alpha _{i}(y)}| u_{i}|^{\alpha _{i}(y)} dy\right) \Big (-\Delta _{\alpha _{i}(\cdot )} u_{i} +\mathcal {V}_{i}(y)|u_{i}|^{\alpha _{i}(y)-2}u_{i}\Big ) \\ \quad = \mu \mathcal {F}_{u_i}(y, u_{1}, \ldots , u_{N}) + \nu \mathcal {G}_{u_i}(y, u_{1}, \ldots , u_{N}), \quad \text {in } \mathbb {R}^N, \text { for all } i = 1, \dots , N,\\ (u_{1}, \ldots , u_{N}) \in \mathbb {H}. \end{array}\right. } \end{aligned}$$

We apply the three critical points theorem to establish sufficient conditions for the existence of at least three weak solutions under appropriate assumptions on the system’s parameters and nonlinearity terms. This work extends the analysis of elliptic systems involving variable exponent spaces and nonlocal operators, offering novel insights into their mathematical structure and solution properties.