For $s_1,s_2\in(0,1)$ and $p,q \in (1, \infty)$, we study the following nonlinear Dirichlet eigenvalue problem with parameters $\alpha, \beta \in \mathbb{ R}$ driven by the sum of two nonlocal operators: \begin{ equation*} (-\Delta)^{ s_1}_p u+(-\Delta)^{ s_2}_q u=\alpha|u|^{ p-2}u+\beta|u|^{ q-2}u\;\;\text{ in }\Omega, \quad u=0\;\;\text{ in } \mathbb{ R}^d \setminus \Omega, \ \ \ \qquad \quad \mathrm{ (P)} \end{ equation*} where $\Omega \subset \mathbb{ R}^d$ is a bounded open set. Depending on the values of $\alpha,\beta$, we completely describe the existence and non-existence of positive solutions to (P). We construct a continuous threshold curve in the two-dimensional $(\alpha, \beta)$-plane, which separates the regions of the existence and non-existence of positive solutions. In addition, we prove that the first Dirichlet eigenfunctions of the fractional $p$-Laplace and fractional $q$-Laplace operators are linearly independent, which plays an essential role in the formation of the curve. Furthermore, we establish that every nonnegative solution of (P) is globally bounded.
Categories: math.AP