Kirchhoff-type problems on a geodesic ball of the hyperbolic space

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.