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Supercritical Elliptic Equation in Hyperbolic Space

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@Article{JPDE-28-120,
author = {He , Haiyang },
title = {Supercritical Elliptic Equation in Hyperbolic Space},
journal = {Journal of Partial Differential Equations},
year = {2015},
volume = {28},
number = {2},
pages = {120--127},
abstract = { In this paper, we study the following semi-linear elliptic equation $$-Δ_H^nu=|u|^{p-2}u,\qquad\qquad (0.1)$$ in the whole Hyperbolic space $\mathbb{H}^n$,where n ≥ 3, p › 2n/(n-2). We obtain some regularity results for the radial singular solutions of problem (0.1). We show that the singular solution $u^∗$ with $lim_{t → 0}(sinht)^{\frac{2}{p-2}}⋅u(t)=±(\frac{2}{p-2}(n-2-\frac{2}{p-2})^{\frac{1}{p-2}}$ belongs to the closure (in the natural topology given by $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N))$ of the set of smooth classical solutions to the Eq. (0.1). In contrast, we also prove that any oscillating radial solutions of (0.1) on $\mathbb{H}^N$\{0} fails to be in the space $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N)$.},
issn = {2079-732X},
doi = {https://doi.org/10.4208/jpde.v28.n2.2},
url = {http://global-sci.org/intro/article_detail/jpde/5105.html}
}

TY - JOUR
T1 - Supercritical Elliptic Equation in Hyperbolic Space
AU - He , Haiyang
JO - Journal of Partial Differential Equations
VL - 2
SP - 120
EP - 127
PY - 2015
DA - 2015/06
SN - 28
DO - http://doi.org/10.4208/jpde.v28.n2.2
UR - https://global-sci.org/intro/article_detail/jpde/5105.html
KW - Supercritical
KW - singularity
KW - hyperbolic space
AB - In this paper, we study the following semi-linear elliptic equation $$-Δ_H^nu=|u|^{p-2}u,\qquad\qquad (0.1)$$ in the whole Hyperbolic space $\mathbb{H}^n$,where n ≥ 3, p › 2n/(n-2). We obtain some regularity results for the radial singular solutions of problem (0.1). We show that the singular solution $u^∗$ with $lim_{t → 0}(sinht)^{\frac{2}{p-2}}⋅u(t)=±(\frac{2}{p-2}(n-2-\frac{2}{p-2})^{\frac{1}{p-2}}$ belongs to the closure (in the natural topology given by $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N))$ of the set of smooth classical solutions to the Eq. (0.1). In contrast, we also prove that any oscillating radial solutions of (0.1) on $\mathbb{H}^N$\{0} fails to be in the space $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N)$.

Haiyang He. (2019). Supercritical Elliptic Equation in Hyperbolic Space.

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*Journal of Partial Differential Equations*.*28*(2). 120-127. doi:10.4208/jpde.v28.n2.2