A weighted eigenvalue problem for mixed local and nonlocal operators with potential

We study an indefinite weighted eigenvalue problem for an operator of mixed‐type (that includes both the classical p$p$‐Laplacian and the fractional p$p$‐Laplacian) in a bounded open subset Ω⊂RN(N≥2)$\Omega \subset \mathbb {R}^N \,(N\ge 2)$ with Lipschitz boundary ∂Ω$\partial \Omega$, which is given by −Δpu+(−Δp)su+V(x)|u|p−2u=λg(x)|u|p−2uinΩ,u=0inRN∖Ω,$$\begin{align*} -\Delta _p u + (-\Delta _p)^su+V(x)|u|^{p-2}u&=\lambda g(x)|u|^{p-2}u\nobreakspace \text{in}\nobreakspace \Omega,\cr u&=0\nobreakspace \text{in}\nobreakspace \mathbb {R}^N\setminus \Omega, \end{align*}$$where λ>0$\lambda >0$ is a parameter, exponents 0 0$V\ge 0, g > 0$ a.e. in Ω$\Omega$. Using the variational tools together with a weak comparison and strong maximum principles, we investigate the existence and uniqueness of principal eigenvalue and discuss its qualitative properties. Moreover, with the help of Ljusternik–Schnirelman category theory, it is proved that there exists a nondecreasing sequence of positive eigenvalues, which goes to infinity. Further, we show that the set of all positive eigenvalues is closed, and eigenfunctions associated with every positive eigenvalue are bounded.