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

Abstract

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 &= \lambda f(u), \quad u>0 &&\text{in }\Omega,\\ u&=0 &&\text{in }U^c,\\ \mathcal{N}_s(u)&=0 &&\text{in }\mathcal{N},\\ \frac{\partial u}{\partial\nu}&=0 &&\text{on }\partial\Omega\cap\overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_\lambda$} \end{equation} where \( U=\Omega\cup\mathcal{N}\cup \bigl(\partial\Omega\cap\overline{\mathcal{N}}\bigr), \) \(\Omega\subseteq\mathbb{R}^n\) is a bounded open set with smooth boundary, \(\lambda>0\) is a real parameter, \(f\) is continuous function with \(f(0)=0\), and \[ \mathcal{L}=-\Delta+(-\Delta)^s, \qquad s\in(0,1). \] We establish a characterization theorem for the existence of positive weak solutions to problem (P_{\lambda}). Motivated by the classical elliptic framework developed by Molica Bisci and R\u{a}dulescu \cite{MolicaBisciRadulescu2017}, we establish a corresponding characterization result for mixed local-nonlocal operators under mixed boundary conditions.