Eigenvalues of the (p, q, r)-Laplacian with a parametric boundary condition

Consider in a bounded domain Omega subset of R-N, N >= 2, with smooth boundary partial derivative Omega the following nonlinear eigenvalue problem {-Sigma(alpha is an element of{p, q, r}) rho(alpha)Delta(alpha) u = lambda a(x) vertical bar u vertical bar(r-2) u in Omega, (Sigma(alpha is an element of{p, q, r}) rho(alpha) vertical bar del u vertical bar(alpha-2)) partial derivative u/partial derivative v = lambda b(x) vertical bar u vertical bar(r - 2) u on partial derivative Omega where p, q, r is an element of (1, +infinity), q < p, r is not an element of {p, q}; rho(p), rho(q), rho(r) are positive constants; Delta(alpha) is the usual alpha-Laplacian, i.e., Delta(alpha)u = div (vertical bar del u vertical bar(alpha-2)del u); v is the unit outward normal to partial derivative Omega; 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. Such a triple-phase problem is motivated by some models arising in mathematical physics. If r is not an element of (q, p), we determine a positive number lambda(r) such that the set of eigenvalues of the above problem is precisely {0} boolean OR (lambda(r), +infinity). On the other hand, in the complementary case r is an element of (q, p) with r < q(N - 1)/(N - q) if q < N, 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 the above problem, while the set (-infinity, 0) boolean OR (0, lambda(*)) contains no eigenvalue lambda of the problem.