A sharp eigenvalue theorem for mixed elliptic problems under mixed boundary conditions

In this paper, we study a class of eigenvalue problems involving both local and nonlocal operators, namely the classical Laplacian and the fractional Laplacian, under mixed boundary conditions. More precisely, we consider the problem \begin{equation}\label{1} \left{ \begin{aligned} $\mathcal{L}$u &= λf(u), \quad u>0 &&$\text{in }$Ω,\ u&=0 &&$\text{in }$U^c,\ $\mathcal{N}$_s(u)&=0 &&$\text{in }$$\mathcal{N}$,\ \frac{\partial u}{\partialν}&=0 &&$\text{on }$\partialΩ\cap\overline{$\mathcal{N}$}, \end{aligned} \right. \tag{$P_{\lambda }$} \end{equation} where $U=\mathrm{\Omega }\cup \mathcal{N}\cup (\mathrm{\partial }\mathrm{\Omega }\cap \bar{\mathcal{N}}),$ $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ is a bounded open set with smooth boundary, $\lambda >0$ is a real parameter, $f$ is continuous function with $f(0)=0$, and $$\mathcal{L}=-\mathrm{\Delta }+(-\mathrm{\Delta }{)}^s,s\in (0,1)\mathrm{.}$$ We establish a characterization theorem for the existence of positive weak solutions to problem (P_λ). Motivated by the classical elliptic framework developed by Molica Bisci and Rădulescu \cite{MolicaBisciRadulescu2017}, we establish a corresponding characterization result for mixed local-nonlocal operators under mixed boundary conditions.