Geodesic
SYMMETRY AND EXISTENCE OF SOLUTIONS OF SEMI-LINEAR ELLIPTIC SYSTEMS IN HYPERBOLIC SPACE
Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 519-541

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DOI

(0.1)\begin{equation}\label{eq:0.1}\left\{\begin{array}{ll}\displaystyle-\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v x, \\\displaystyle-\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, \\\end{array}\right.\end{equation}in the whole Hyperbolic space HN. We establish decay estimates and symmetry properties of positive solutions. Unlike the corresponding problem in Euclidean space RN, we prove that there is a positive solution pair (u, v) ∈ H1(HN) × H1(HN) of problem (0.1), moreover a ground state solution is obtained. Furthermore, we also prove that the above problem has a radial positive solution. 
HE, HAIYANG. SYMMETRY AND EXISTENCE OF SOLUTIONS OF SEMI-LINEAR ELLIPTIC SYSTEMS IN HYPERBOLIC SPACE. Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 519-541. doi: 10.1017/S0017089514000457
@article{10_1017_S0017089514000457,
     author = {HE, HAIYANG},
     title = {SYMMETRY {AND} {EXISTENCE} {OF} {SOLUTIONS} {OF} {SEMI-LINEAR} {ELLIPTIC} {SYSTEMS} {IN} {HYPERBOLIC} {SPACE}},
     journal = {Glasgow mathematical journal},
     pages = {519--541},
     year = {2015},
     volume = {57},
     number = {3},
     doi = {10.1017/S0017089514000457},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089514000457/}
}
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