Bifurcation results and multiple solutions for the fractional $(p,q)$-Laplace operators

We investigate a nonlinear nonlocal eigenvalue problem involving the sum of fractional $(p,q)$-Laplace operators $(-\mathrm{\Delta }{)}_p^{s_1}+(-\mathrm{\Delta }{)}_q^{s_2}$ with $s_1,s_2\in (0,1)$; $p,q\in (1,\mathrm{\infty })$ and subject to Dirichlet boundary conditions in an open bounded set of ${\mathbb{R}}^N$. We prove bifurcation results from trivial solutions and from infinity for the considered nonlinear nonlocal eigenvalue problem. We also show the existence of multiple solutions of the nonlinear nonlocal problem using variational methods.