Kirchhoff-type problems on a geodesic ball of the hyperbolic space. (September 2019)

• Record Type: Journal Article
• Authors: Molica Bisci, Giovanni
• Is Part Of: Nonlinear analysis. Volume 186(2019)
• Page Start: 55
• Page End: 73
• Publication Date: 2019-09
• Subjects: primary 58J32 35J60 49K10 -- secondary 35A01 35R01; Elliptic problems on manifolds -- Hyperbolic space -- Poincaré model -- Variational methods -- Multiple solutions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.11.003
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we study the existence of (weak) solutions for some Kirchhoff-type problems whose simple prototype is given by − a + b ∫ B | ∇ H u ( σ ) | 2 d μ Δ H u = λ f ( u ) in B R u = 0 on ∂ B R, where Δ H denotes the Laplace–Beltrami operator on the ball model of the Hyperbolic space B N (with N ≥ 3 ), a, b and λ are real parameters, B R ⊂ B N is a geodesic ball centered in zero of radius R and f is a subcritical continuous function. The Kirchhoff term is allowed to vanish at the origin covering the degenerate case. The main technical approach is based on variational and topological methods.