January/February 2013 Existence theorems for quasilinear elliptic eigenvalue problems in unbounded domains
Giuseppina Autuori, Patrizia Pucci, Csaba Varga
Adv. Differential Equations 18(1/2): 1-48 (January/February 2013). DOI: 10.57262/ade/1355867480


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.


Download Citation

Giuseppina Autuori. Patrizia Pucci. Csaba Varga. "Existence theorems for quasilinear elliptic eigenvalue problems in unbounded domains." Adv. Differential Equations 18 (1/2) 1 - 48, January/February 2013. https://doi.org/10.57262/ade/1355867480


Published: January/February 2013
First available in Project Euclid: 18 December 2012

zbMATH: 1260.35085
MathSciNet: MR3052709
Digital Object Identifier: 10.57262/ade/1355867480

Primary: 35J20 , 35J25 , 35J66 , 35J70

Rights: Copyright © 2013 Khayyam Publishing, Inc.


This article is only available to subscribers.
It is not available for individual sale.

Vol.18 • No. 1/2 • January/February 2013
Back to Top