Multiplicity results for mixed local-nonlocal variable exponent problem involving singular and superlinear term

In this paper, we study a class of quasilinear elliptic equations involving both local and nonlocal operators with variable exponents. The problem exhibits singular nonlinearities along with a subcritical superlinear growth term and a parameter $\lambda.$ We study the existence of multiple solutions with the help of variational methods by restricting the associated energy functional on appropriate subsets of the Nehari manifold. Using the topological index and the structure of the fibering maps, we analyze a key splitting property of the associated Nehari manifold. This decomposition allows us to establish the existence of two distinct solutions. Additionally, we establish the $L^\infty$-bound for the solutions by adopting the De Giorgi iteration method.