Parametric nonlinear resonant Robin problems

We consider a nonlinear Robin problem driven by the $p$-Laplacian. In the reaction we have the competing effects of two nonlinearities. One term is parametric, strictly $(p-1)$-sublinear and the other one is $(p-1)$-linear and resonant at any nonprincipal variational eigenvalue. Using variational tools from the critical theory (critical groups), we show that for all large enough values of parameter $\lambda$ the problem has at least five nontrivial smooth solutions.