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We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional $ p\& q $ p &q Laplacian operator with indefinite weights \[ \left(-\Delta_p\right)^{\alpha}u + \left(-\Delta_q\right)^{\beta}u = \lambda\left[a \left|u\right|^{p-2}u + b \left|u\right|^{q-2}u \right]\quad{\rm in}\ \Omega, \] (− Δ p) α u + (− Δ q) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where $ \Omega \subseteq \mathbb {R}^N $ Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case $ \Omega =\mathbb {R}^N $ Ω = R N and $ b\equiv 0 $ b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding $ \mathcal {W}^{\alpha, p}\left (\mathbb {R}^N\right) \hookrightarrow \mathcal {W}^{\beta, q}\left (\mathbb {R}^N\right) $ W α , p (R N) ↪ W β , q (R N) in $ \mathbb {R}^N $ R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional $ p\& q $ p &q Laplacian problems in $ \Bbb R^N $ R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for $ (p,q) $ (p , q) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems. [ABSTRACT FROM AUTHOR]