Eigenvalues of −Δp − Δq Under Neumann Boundary Condition

Abstract

AbstractThe eigenvalue problem −Δpu − Δqu = λ|u|q−2u with p ∊ (1,∞), q ∊ (2,∞), p ≠ q subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from ℝN with N ≥ 2. A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval (λ1, ∞) plus an isolated point λ = 0. This comprehensive result is strongly related to our framework, which is complementary to the well-known case p = q ≠ 2 for which a full description of the set of eigenvalues is still unavailable.