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

Abstract

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 {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 0 and ∫Ω ρi dx + ∫∂Ω γi dσ > 0, i = 1, 2.

Keywords

• eigenvalues
• (p, q)−laplacian
• parametric boundary condition
• nehari manifold
• variational methods
• primary 35j60, 35d30
• secondary 35p30