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

TL;DR

This paper investigates an indefinite weighted eigenvalue problem involving mixed local and nonlocal operators, establishing existence, uniqueness, and properties of eigenvalues and eigenfunctions using variational and topological methods.

Contribution

It introduces a novel analysis of a mixed-type eigenvalue problem combining p-Laplacian and fractional p-Laplacian operators, proving existence, uniqueness, and spectral properties.

Findings

Existence and uniqueness of principal eigenvalue

A nondecreasing sequence of positive eigenvalues tending to infinity

The set of positive eigenvalues is closed and eigenfunctions are bounded

Abstract

We study an {\it indefinite weighted eigenvalue problem} for an operator of {\it mixed-type} (that includes both the classical {\it $p$-Laplacian} and the {\it fractional $p$-Laplacian}) in a bounded open subset $\Omega\subset \mathbb{R}^N \,(N\geq2)$ with {\it Lipschitz boundary} $\partial \Omega$, which is given by \begin{align*} -\Delta_p u + (-\Delta_p)^su+V(x)|u|^{p-2}u&=\lambda g(x)|u|^{p-2}u~\text{in}~\Omega, u&=0~\text{in}~\mathbb{R}^N\setminus\Omega, \end{align*} where $\lambda >0$ is a parameter, exponents $0<s<1<p<N$, and $V, g\in L^q(\Omega)$ for $q\in \left(\frac{N}{sp}, \infty\right)$ with $V\geq 0, g > 0$ a.e. in $\Omega$. Using the variational tools together with a {\it weak comparison} and {\it strong maximum principles}, we investigate the existence and uniqueness of {\it principal eigenvalue} and discuss its qualitative properties. Moreover, with the help of {\it Ljusternik-Schnirelman category theory}, it is proved that there exists a {\it nondecreasing sequence of positive eigenvalues} which goes to infinity. Further, we show that {\it the set of all positive eigenvalues is closed}, and {\it eigenfunctions} associated with every {\it positive eigenvalue} are bounded.