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

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 $(-\mathrm{\Delta }{)}_p^{s_1}+(-\mathrm{\Delta }{)}_q^{s_2}$ with $s_1,s_2\in (0,1)$, $p,q\in (1,\mathrm{\infty })$ 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 $\{0\}\cup ({\lambda }_1(s_2,q),\mathrm{\infty })$, 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.