Let Ω ⊂ ℝN, N ≥ 2, be a bounded domain with smooth boundary ∂Ω. Consider the following generalized Robin-Steklov eigenvalue problem associated with the operator 𝒜u = − Δpu − Δqu
\left\{ {\matrix{ {\mathcal{A}u + {\rho _1}\left( x \right){{\left| u \right|}^{p - 2}}u + {\rho _2}\left( x \right){{\left| u \right|}^{q - 2}}u = \lambda \alpha \left( x \right){{\left| u \right|}^{r - 2}}u,\,\,\,x \in \Omega ,} \cr {{{\partial u} \over {\partial {v_A}}} + {\gamma _1}\left( x \right){{\left| u \right|}^{p - 2}}u + {\gamma _2}\left( x \right){{\left| u \right|}^{q - 2}}u = \lambda \beta \left( x \right){{\left| u \right|}^{r - 2}}u,\,\,x \in \partial \Omega ,} \cr } } \right.
where p, q, r ∈ (1, ∞), p < q; α, ρi ∈ L∞(Ω) and β, γi ∈ L∞(∂Ω) are nonnegative functions satisfying ∫Ω α dx + ∫∂Ω β dσ > 0 and ∫Ω ρi dx + ∫∂Ω γi dσ > 0, i = 1, 2.
We show that, if either r < p or r > q with r < q(N − 1)/(N − q) in case q < N, then the eigenvalue set (spectrum) of the above problem is precisely (0, ∞). If r ∈ {p, q} then the corresponding spectrum is a smaller interval (d, ∞), d > 0. On the other hand, if r ∈ (p, q) with r < p(N − 1)/(N − p) in case p < N, then we are able to identify an interval of eigenvalues [λ*, ∞), where λ* is a positive number depending on r.
Obviously, the spectrum of the above problem coincides with the spectra of the Neumann-like, Robin-like, and Steklov-like eigenvalue problems corresponding to the cases when some of the functions α, β, γi vanish.