Existence theorems for quasilinear elliptic eigenvalue problems in unbounded domains

Abstract

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue problem depending on a real parameter $\lambda$ 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. Denoting by $\lambda_1$ the first eigenvalue of the underlying Robin eigenvalue problem, we prove the existence of (weak) solutions, with different methods, according to the case $\lambda\ge \lambda_1$ or $\lambda <\lambda_1$. In the first part of the paper we show the existence of a nontrivial solution for all $\lambda\in\mathbb R$ for the problem under Ambrosetti--Rabinowitz-type conditions on the nonlinearities involved in the model. In detail, we apply the mountain-pass theorem of Ambrosetti and Rabinowitzif $\lambda <\lambda_1$, while we use mini-max methods and linking structures over cones, as in Degiovanni [10]and in Degiovanni and Lancelotti [11], if $\lambda\ge \lambda_1$. In the latter part of the paper we do not require any longer the Ambrosetti--Rabinowitz condition at $\infty$, but the so-called Szulkin--Weth conditions, and we obtain the same result for all $\lambda\in\mathbb R$. More precisely, using the Nehari-manifold method for $C^1$ functionals developed by Szulkin and Weth in [38], we prove existence of ground states, multiple solutions, and least-energy sign-changing solutions, whenever $\lambda <\lambda_1$. On the other hand, in the case $\lambda\ge \lambda_1$, we establish the existence of solutions again by linking methods.