On a nonlinear transmission eigenvalue problem with a Neumann-Robin boundary condition

Let Omega be a bounded domain in R-N, N >= 2, with smooth boundary Sigma and let Omega(1) be a subdomain of Omega with smooth boundary Gamma,such that (Omega) over bar (1) subset of Omega. Denote Omega(2)=Omega\(Omega) over bar (1).Consider the transmission eigenvalue problem {-Delta(p)u(1)+ gamma(1)(x)vertical bar u(1)vertical bar(r-2)u(1)=lambda vertical bar u(1)vertical bar(p-2)u(1) in Omega(1), -Delta(q)u(2)+ gamma(2)(x)vertical bar u(2)vertical bar(r-2)u(1)=lambda vertical bar u(2)vertical bar(p-2)u(2)in Omega(2), u(1)=u(2), partial derivative u(1)/partial derivative v(p)= partial derivative u(2)/partial derivative v(q) on Gamma, partial derivative u(2)/partial derivative v(q)+beta(x)vertical bar u(2)vertical bar(zeta-2)u(2)=0 on Sigma, where lambda is a real parameter, p, q, r, s, zeta is an element of(1,infinity), and gamma(i) is an element of L-infinity(Omega(i)), i = 1,2, beta is an element of L-infinity(Sigma), beta >= 0 a. e. on Sigma.Under additional suitable assumptions on p, q, r, s, zeta, we prove the existence of a sequence of eigenvalues(lambda(n))(n), lambda(n) -> infinity.The proof is based on Lusternik-Schnirelmann theory on C-1-manifolds.