Eigenvalues of nonlinear $(p,q)$-fractional Laplace operators under nonlocal Neumann conditions

TL;DR

This paper characterizes the eigenvalues of a sum of nonlocal fractional $(p,q)$-Laplacian operators with Neumann boundary conditions on smooth domains, showing the eigenvalues form a specific interval and eigenfunctions are bounded.

Contribution

It provides a complete description of the eigenvalue spectrum for the sum of nonlocal fractional $(p,q)$-Laplacians with Neumann conditions, including the structure of eigenvalues and boundedness of eigenfunctions.

Findings

Eigenvalues form the interval {0} ∪ (λ₁(s₂,q), ∞)

Eigenfunctions are globally bounded

Eigenvalues are characterized by the first nonzero eigenvalue of the fractional q-Laplacian

Abstract

In this paper, we investigate on a bounded open set of $\mathbb{R}^N$ with smooth boundary, an eigenvalue problem involving the sum of nonlocal operators $(-\Delta)_p^{s_1}+ (-\Delta)_q^{s_2}$ with $s_1,s_2\in (0,1)$, $p,q\in (1,\infty)$ and subject to the corresponding homogeneous nonlocal $(p,q)$-Neumann boundary condition. A careful analysis of the considered problem leads us to a complete description of the set of eigenvalues as being the precise interval $\{0\}\cup(\lambda_{1}(s_2,q),\infty)$, where $\lambda_{1}(s_2,q)$ is the first nonzero eigenvalue of the homogeneous fractional $q$-Laplacian under nonlocal $q$-Neumann boundary condition. Furthermore, we establish that every eigenfunctions is globally bounded.