Existence of solutions for quasilinear elliptic equations with jumping nonlinearities under the Neumann boundary condition

By variational methods, we prove the existence of a non-trivial solution for the quasilinear elliptic equations with jumping nonlinearities under the Neumann boundary condition. We also provide existence results for positive, negative and non-trivial multiple solutions. The studied equations contain the p-Laplacian problems as a special case.