An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains

We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p–Laplacian operator and subcritical nonlinearities satisfying Ambrosetti–Rabinowitz type conditions. Using Morse theory and a cohomological local splitting as in Degiovanni et al. (Commun Contemp Math 12:475–486, 2010), we prove the existence of a nontrivial weak solution for all (real) values of the eigenvalue parameter. Our result is new even in the semilinear case p = 2 and complements some recent results obtained in Autuori et al. (Adv Anal Equ 18:1–48, 2013).