Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space

TL;DR

This paper investigates the existence and multiplicity of positive solutions to a class of nonlinear elliptic equations in hyperbolic space, employing variational methods and energy estimates to establish multiple solutions under various conditions on the potential.

Contribution

It proves the existence of multiple positive solutions for the problem in hyperbolic space, including cases where the potential varies and approaches a constant at infinity, using novel variational and barrier techniques.

Findings

Existence of three positive solutions when a(x) ≤ 1.

Existence of two positive solutions when a(x) ≥ 1.

Asymptotic estimates for positive solutions using barrier methods.

Abstract

The paper is concerned with positive solutions to problems of the type \begin{equation*} -\Delta_{\mathbb{B}^N} u - \lambda u = a(x) |u|^{p-1}\;u \, + \, f \, \;\;\text{in}\;\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, \end{equation*} 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.