Multiplicity Results for Mixed Local Nonlocal Equations With Indefinite Concave-Convex Type Nonlinearity

Abstract: In this article we examine the multiplicity of non-negative solutions to mixed local-nonlocal equations involving ((-\Delta_p) + (-\Delta^{s}_{q})) in a bounded smooth domain. The nonlinearity incorporates a parameter (\lambda > 0), a sublinear term, and a superlinear term, with sign-changing weight functions (a(x)) and (b(x)). Under suitable conditions, we establish the existence of at least two distinct nontrivial non-negative solutions in both the subcritical and critical regimes via fibering map analysis and constrained minimization on the Nehari manifold. Additionally, for (p \not = q), we obtain a nonexistence result for large (\lambda) by analyzing the associated generalized eigenvalue problem.