Eigenvalues of -Δp - Δq Under Neumann Boundary Condition

The eigenvalue problem -Delta(p)u - Delta(q)u = lambda vertical bar u vertical bar(q-2)u with p is an element of (1, infinity), q is an element of (2, infinity), p not equal q subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from R-N 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 (lambda(1), +infinity) plus an isolated point lambda = 0. This comprehensive result is strongly related to our framework, which is complementary to the well-known case p = q not equal 2 for which a full description of the set of eigenvalues is still unavailable.