For s1, s2 ∈ (0, 1) and p, q ∈ (1,∞), we study the following nonlinear Dirichlet eigenvalue
problem with parameters α, β ∈ R driven by the sum of two nonlocal operators:
(−Δ)s1p u + (−Δ)s2q u = α|u|p−2u + β|u|q−2u in Ω, u = 0 in Rd \ Ω,
where Ω ⊂ Rd is a bounded open set. Depending on the values of α, β, we investigate the existence and non-existence of positive solutions to the above problem. A continuous curve in the two-dimensional (α, β)-plane is constructed, 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.