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

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.