Year: 2015
Author: Haiyang He
Journal of Partial Differential Equations, Vol. 28 (2015), Iss. 2 : pp. 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 $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N)$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v28.n2.2
Journal of Partial Differential Equations, Vol. 28 (2015), Iss. 2 : pp. 120–127
Published online: 2015-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 8
Keywords: Supercritical