The paper is concerned with positive solutions to problems of the type

where $\mathbb {B}^N$ denotes the hyperbolic space, $1< p<2^*-1:=\frac {N+2}{N-2}$, $\;\lambda < \frac {(N-1)^2}{4}$, and $f \in H^{-1}(\mathbb {B}^{N})$ ($f \not \equiv 0$) is a non-negative functional. The potential $a\in L^\infty (\mathbb {B}^N)$ is assumed to be strictly positive, such that $\lim _{d(x, 0) \rightarrow \infty } a(x) \rightarrow 1,$ where $d(x,\, 0)$ denotes the geodesic distance. First, the existence of three positive solutions is proved under the assumption that $a(x) \leq 1$. Then the case $a(x) \geq 1$ is considered, and the existence of two positive solutions is proved. In both cases, it is assumed that $\mu ( \{ x : a(x) \neq 1\}) > 0.$ Subsequently, we establish the existence of two positive solutions for $a(x) \equiv 1$ and prove asymptotic estimates for positive solutions using barrier-type arguments. The proofs for existence combine variational arguments, key energy estimates involving hyperbolic bubbles.

In this article, we investigate numerical schemes for solvinga three component Cahn-Hilliard model. The space discretization isperformed by usinga Galerkin formulation and the finite element method.Concerning the time discretization,the main difficulty is to write a scheme ensuring,at the discrete level, the decrease of the free energyand thus the stability of the method.We study three different schemes and proveexistence and convergence theorems. Theoretical results areillustrated by various numerical examples showing that the new semi-implicitdiscretization that we propose seems to be a good compromise between robustnessand accuracy.

We consider the coupling between three-dimensional(3D) and one-dimensional (1D) fluid-structure interaction (FSI) models describing blood flow inside compliant vessels. The 1D model is a hyperbolicsystem of partial differential equations.The 3D model consists of the Navier-Stokes equationsfor incompressible Newtonian fluids coupled with a model for the vessel wall dynamics. A non standard formulationfor the Navier-Stokes equations is adopted tohave suitable boundary conditions for the couplingof the models. With this we derive an energy estimatefor the fully 3D-1D FSI coupling. We consider several possiblemodels for the mechanics of the vessel wall in the 3D problemand show how the 3D-1D coupling depends on them.Several comparative numerical tests illustrating the coupling are presented.