Multiplicity Results for some Perturbed and Unperturbed “Zero Mass” Elliptic Problems in Unbounded Cylinders

Abstract

AbstractWe study the following nonlinear elliptic problem \( \left\{ \begin{array}{clclclcllc}{{-\Delta u}=g(x,u)+f(x) \;\;\; \rm {in} \; \Omega} \\ {\quad u=0 \qquad \qquad \qquad \; \rm {on} \; \partial \Omega}\end{array}\right. \) on unbounded cylinders \( \Omega = \tilde{\Omega}\times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}, N-m \geq 2, m \geq 1, \) under suitable conditions on g and f. In the unperturbed case \( f(x) \equiv 0, \) by means of the Principle of Symmetric Criticality by Palais and some compact imbeddings in spherically symmetric spaces, existence and multiplicity results are proved by applying Mountain Pass Theorem and its Symmetric version. Multiplicity results are also proved in the perturbed case \( f(x) \equiv 0, \) f(x)≢0 by using Bolle’s Perturbation Methods and suitable growth estimates on min-max critical levels. To this aim, we improve a classical estimate of the number N_(-∆ + V) of the negative eigenvalues of the operator -∆+V(x) when the potential V is partially spherically symmetric.KeywordsNonlinear elliptic equationszero mass caseunbounded cylindersvariational and perturbative methodscompact imbeddings