Multiple solutions with precise sign for nonlinear parametric Robin problems

We consider a parametric nonlinear Robin problem driven by the p  -Laplacian. We show that if the parameter λ>λˆ2= the second eigenvalue of the Robin p  -Laplacian, then the problem has at least three nontrivial solutions, two of constant sign and the third nodal. In the semilinear case (p=2)(p=2), we show that we can generate a second nodal solution. Our approach uses variational methods, truncation and perturbation techniques, and Morse theory. In the process we produce two useful remarks about the first two eigenvalues of the Robin p-Laplacian.