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

Abstract

In this paper, we investigate on a bounded open set of $R^N$ with smooth boundary, an eigenvalue problem involving a 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 $\left\{0\right\}\cup ({\lambda }_1(s_2,q),\infty )$, where ${\lambda }_1(s_2,q)$ is the first nonzero eigenvalue of the homogeneous fractional $q$-Laplacian under nonlocal $q$-Neumann boundary condition. Due to the nonlocal feature of the operators appearing in the equations, some purely nonlocal situations occur and bring in a difference in the study of nonlocal problems compared to local ones. Furthermore, we establish that every eigenfunctions is globally bounded.