Supercritical Elliptic Equation in Hyperbolic Space

Abstract

" In this paper, we study the following semi-linear elliptic equation $$-\u0394_H^nu=|u|^{p-2}u,\\qquad\\qquad (0.1)$$ in the whole Hyperbolic space $\\mathbb{H}^n$,where n \u2265 3, p \u203a 2n\/(n-2). We obtain some regularity results for the radial singular solutions of problem (0.1). We show that the singular solution $u^\u2217$ with $lim_{t \u2192 0}(sinht)^{\\frac{2}{p-2}}\u22c5u(t)=\u00b1(\\frac{2}{p-2}(n-2-\\frac{2}{p-2})^{\\frac{1}{p-2}}$ belongs to the closure (in the natural topology given by $H\u00b9_{loc}(\\mathbb{H}^N)\u2229L^p_{loc}(H^N))$ 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 $H\u00b9_{loc}(\\mathbb{H}^N)\u2229L^p_{loc}(H^N)$."