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

For s₁, \, s₂ (0, \, 1) and p, \, q (1, \, ), we study the following nonlinear Dirichlet eigenvalue problem with parameters, \, R driven by the sum of two nonlocal operators: \ (-) ^s₁ₚ u+ (-) ^s₂q u=|u|^p-2u+|u|^q-2u\ in, u=0\ in Rᵈ, (P) \ where Rᵈ is a bounded open set. Depending on the values of, \, , we completely describe the existence and non-existence of positive solutions to (P). We construct a continuous threshold curve in the two-dimensional (, \, ) -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.