On an eigenvalue problem associated with the (p, q) − Laplacian

Let Ω ⊂ ℝ^N, N ≥ 2, be a bounded domain with smooth boundary ∂Ω. Consider the following generalized Robin-Steklov eigenvalue problem associated with the operator 𝒜u = − Δ_pu − Δ_qu { 𝒜u+ρ1(x)| u |p-2u+ρ2(x)| u |q-2u=λα(x)| u |r-2u, x∈Ω,∂u∂vA+γ1(x)| u |p-2u+γ2(x)| u |q-2u=λβ(x)| u |r-2u, x∈∂Ω, \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, ρ_i ∈ L^∞(Ω) and β, γ_i ∈ L^∞(∂Ω) are nonnegative functions satisfying ∫_Ω α dx + ∫_∂Ω β dσ > 0 and ∫_Ω ρ_i dx + ∫_∂Ω γ_i 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, γ_i vanish.