ON THE EIGENVALUE SET OF THE (p, q)-LAPLACIAN WITH A NEUMANN-STEKLOV BOUNDARY CONDITION

Consider in a bounded domain Omega subset of R-N, N >= 2, with smooth boundary.O, the following eigenvalue problem -Delta(p)u -Delta(q)u =lambda a(x)|u|(r-2) u in Omega, |del u|(p-2) + |.del u|(q-2) )partial derivative u/partial derivative v=lambda b(x)|u|(r-2) u on partial derivative Omega, where 1 < q < r < p < infinity, with r < q(N - 1)/(N - q) if q < N; a is an element of L-infinity(Omega), b is an element of L-infinity(partial derivative Omega) are given nonnegative functions satisfying integral(Omega) a dx + integral(partial derivative Omega) b d sigma > 0. Under these assumptions, we prove that there exist two positive constants lambda(*) <lambda(*) such that any lambda is an element of {0}boolean OR [lambda*,infinity) is an eigenvalue of this problem, while the set (-infinity, 0)boolean OR(0,lambda(*)) contains no eigenvalue of the problem. This result is complementary to previous results related to the above eigenvalue problem.