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

For $s_1,s_2\in (0,1)$ and $p,q\in (1,\mathrm{\infty })$, 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*} (-Δ)^{s_1}_p u+(-Δ)^{s_2}_q u=α|u|^{p-2}u+β|u|^{q-2}u;;$\text{in }$Ω, \quad u=0;;$\text{in }$ $\mathbb{R}$^d \setminus Ω, \ \ \ \qquad \quad $(\mathrm{P})$ \end{equation*} where $\mathrm{\Omega }\subset {\mathbb{R}}^d$ 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.