Nontrivial Solutions of the Asymmetric Beam System with Jumping Nonlinear Terms

We investigate the existence of multiple nontrivial solutions ξ, η for perturbations b1 u 2 − 2 and b2 u 3 −3 of the beam systemwith Dirichlet boundary condition Lξ b1 ξ 3η 2 −2 in −π/2, π/2 ×R, Lη b2 ξ 3η 3 − 3 in −π/2, π/2 ×R, where u max {u, 0}, and μ, ν are nonzero constants. Here L is the beam operator in R2 , and the nonlinearity b1 u 2 − 2 b2 u 3 − 3 crosses the eigenvalues of the beam operator.