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

TL;DR

This work characterizes the spectrum of the nonlinear eigenvalue problem for the sum of fractional $(p,s_1)$- and $(q,s_2)$-Laplacians under nonlocal $(p,q)$-Neumann boundary conditions on a bounded domain. Using variational methods and Nehari-manifold techniques across several parameter regimes, the authors prove the spectrum consists of $0$ and the open half-line $(\lambda_{1}(s_{2},q),\infty)$, with $\lambda_{1}(s_{2},q)$ the first nonzero eigenvalue of the fractional $q$-Laplacian under nonlocal Neumann conditions. Existence of eigenvalues above this threshold is established by direct minimization on a cone or constrained minimization on a Nehari manifold, depending on the relative size of $s_1,s_2,p,q$, including handling crossing cases. A global $L^{\infty}$ bound for eigenfunctions is obtained via De Giorgi iteration, ensuring uniform boundedness in $\mathbb{R}^N$. Together, these results extend the Neumann spectrum from local to nonlocal mixed operators and provide a complete spectral description for this nonlinear nonlocal setting.

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 $(-Δ)_p^{s_1}+ (-Δ)_q^{s_2}$ 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 $\{0\}\cup(λ_{1}(s_2,q),\infty)$, where $λ_{1}(s_2,q)$ 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.

Theorems & Definitions (37)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Lemma 2.4
• Lemma 2.5
• proof