An Eigenvalue Problem Involving the (p, q)-Laplacian With a Parametric Boundary Condition

Let Omega R-N, N >= 2, be a bounded domain with smooth boundary partial derivative Omega. Consider the following nonlinear eigenvalue problem {-Delta(p)u - Delta(q)u + rho(x) vertical bar u vertical bar q(-2) u = lambda alpha a(x) vertical bar u vertical bar(|r-2) u in Omega, partial derivative u/partial derivative nu(pq) + gamma(x) vertical bar u vertical bar(|q-2) u = lambda beta(x) vertical bar u vertical bar|(r-2) u on partial derivative Omega , where p, q, r is an element of (1, infinity) with p not equal q; alpha, rho is an element of L-infinity (Omega), beta, gamma is an element of L-infinity (partial derivative Omega), Delta(theta)u := div (parallel to del u parallel to(theta-2) del u), theta is an element of {p, q}, and partial derivative u/partial derivative v(pq) denotes the conormal derivative corresponding to the differential operator -Delta(p) - Delta(p). Under suitable assumptions, we provide the full description of the spectrum of the above problem in eight cases out of ten, and for the other two complementary cases, we obtain subsets of the corresponding spectra. Notice that when some of the potentials alpha, beta, rho, gamma are null functions, the above eigenvalue problem reduces to Neumann-, Robin- or Steklov-type problems, and so we obtain the spectra of these particular eigenvalue problems.