A scalar field equation on hyperbolic space with indefinite sign nonlinearity
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866909017321766912 |
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| author | Karmakar, Debabrata Manna, Atanu Manna, Bhakti Bhusan |
| author_facet | Karmakar, Debabrata Manna, Atanu Manna, Bhakti Bhusan |
| contents | In this article, we study threshold phenomena for the semilinear double-power elliptic equation $$-Δ_{\mathbb{B}^N} u - λu = |u|^{p-1}u - |u|^{q-1}u, \quad u \in H^1(\mathbb{B}^N),$$
on the hyperbolic space $\mathbb{B}^N$ for $N \ge 3$. For parameters $1 < p \le 2^*-1$ (though we occasionally allow for supercritical exponents) and $q > 0$, we seek to identify the optimal spectral regimes for $λ\in \mathbb{R}$ that delineate the existence and non-existence of positive-energy solutions. We achieve a complete resolution of these thresholds across all exponent configurations: $p < q$, $0 < q < 1 < p$, and $1 < q < p$. Our results demonstrate that the boundary separating these regimes is governed by an explicit critical spectral parameter, which depends on $p$, $q$, and $N$ in the regime where $p < q$, but depends solely on $N$ in the remaining cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04687 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A scalar field equation on hyperbolic space with indefinite sign nonlinearity Karmakar, Debabrata Manna, Atanu Manna, Bhakti Bhusan Analysis of PDEs Primary: 35A01, Secondary: 35A15, 35B08, 35B33, 58J05 In this article, we study threshold phenomena for the semilinear double-power elliptic equation $$-Δ_{\mathbb{B}^N} u - λu = |u|^{p-1}u - |u|^{q-1}u, \quad u \in H^1(\mathbb{B}^N),$$ on the hyperbolic space $\mathbb{B}^N$ for $N \ge 3$. For parameters $1 < p \le 2^*-1$ (though we occasionally allow for supercritical exponents) and $q > 0$, we seek to identify the optimal spectral regimes for $λ\in \mathbb{R}$ that delineate the existence and non-existence of positive-energy solutions. We achieve a complete resolution of these thresholds across all exponent configurations: $p < q$, $0 < q < 1 < p$, and $1 < q < p$. Our results demonstrate that the boundary separating these regimes is governed by an explicit critical spectral parameter, which depends on $p$, $q$, and $N$ in the regime where $p < q$, but depends solely on $N$ in the remaining cases. |
| title | A scalar field equation on hyperbolic space with indefinite sign nonlinearity |
| topic | Analysis of PDEs Primary: 35A01, Secondary: 35A15, 35B08, 35B33, 58J05 |
| url | https://arxiv.org/abs/2605.04687 |