Existence theorems for quasilinear elliptic eigenvalue problems in unbounded domains

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue problem depending on a real parameter λ under Robin boundary conditions in unbounded domains, with (possibly noncompact) smooth boundary. The problem involves a weighted p-Laplacian operator and subcritical nonlinearities and even in the case p=2 the main existence results are new. Denoted by λ1 the first eigenvalue of the underlying Robin eigenvalue problem, we prove the existence of (weak) solutions, with different methods, according to the case λ≥λ1 or λ<λ1. In the first part of the paper we show the existence of a nontrivial solution for all λ in R for the problem under Ambrosetti-Rabinowitz type conditions on the nonlinearities involved in the model. In details, we apply the Mountain Pass theorem of Ambrosetti and Rabinowitz if λ<λ1, while we use mini-max methods and linking structures over cones, as in Degiovanni [On topological and metric critical point theory, J. Fixed Point Theory Appl. 7 (2010), 85-102] and in Degiovanni and Lancelotti [Linking over cones and nontrivial solutions for p-Laplacian equations with p-superlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), 907-919], if λ≥λ1. In the latter part of the paper we do not require any longer the Ambrosetti-Rabinowitz condition at infinity, but the so called Szulkin-Weth conditions and we obtain the same result for all λ in R. More precisely, using the Nehari manifold method for C1 functionals developed by Szulkin and Weth [The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, D.Y. Gao and D. Motreanu eds., International Press, Boston, 2010, 597-632], we prove existence of ground states, multiple solutions and least energy sign-changing solutions, whenever λ<λ1. On the other hand, in the case λ≥λ1, we establish the existence of solutions again by linking methods.