Nonresonance for problems involving $(p,q)$-Laplacian equations with nonlinear perturbations

We are concerned with the existence of solutions of $(p,q)$-Laplacian problems that include nonlinear perturbation terms in both the differential equations and the boundary. Using variational methods and critical point theory, we prove the existence of weak solutions for the nonlinear problem when the nonlinearities involved remain asymptotically below the infimum of the set of eigenvalues of the $(p,q)$-Laplacian problem with weights and a spectral parameter present in both the differential equation and the boundary. Additionally, we establish an existence result for the nonlinear problem when the nonlinearities involved remain asymptotically below the first Steklov-Neumann eigenvalue-line, which is a line connecting the first Steklov and first Neumann eigenvalues for q-Laplacian problems with weights and a spectral parameter present either in the differential equation or on the boundary.