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

In this article we examine the multiplicity of non-negative solutions to mixed local-nonlocal equations involving $(-{\mathrm{\Delta }}_p)+(-{\mathrm{\Delta }}_q^s)$ 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 $p≠q$, we obtain a nonexistence result for large $\lambda$ by analyzing the associated generalized eigenvalue problem.