Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866911372690849792 |
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| author | Ganguly, Debdip Gupta, Diksha Sreenadh, K. |
| author_facet | Ganguly, Debdip Gupta, Diksha Sreenadh, K. |
| contents | The paper is concerned with positive solutions to problems of the type
\begin{equation*}
-Δ_{\mathbb{B}^N} u - λu = 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, $1<p<2^*-1:=\frac{N+2}{N-2}$, $\;λ< \frac{(N-1)^2}{4}$, 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 $μ( \{ x : a(x) \neq 1\}) > 0.$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_12574 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space Ganguly, Debdip Gupta, Diksha Sreenadh, K. Analysis of PDEs The paper is concerned with positive solutions to problems of the type \begin{equation*} -Δ_{\mathbb{B}^N} u - λu = 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, $1<p<2^*-1:=\frac{N+2}{N-2}$, $\;λ< \frac{(N-1)^2}{4}$, 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 $μ( \{ x : a(x) \neq 1\}) > 0.$ 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. |
| title | Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2209.12574 |