On generalized eigenvalue problems of fractional $(p,q)$-Laplace operator with two parameters

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.