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SYMMETRY AND EXISTENCE OF SOLUTIONS OF SEMI-LINEAR ELLIPTIC SYSTEMS IN HYPERBOLIC SPACE

Part of: Partial differential equations on manifolds; differential operators Elliptic equations and systems

Published online by Cambridge University Press:  18 December 2014

HAIYANG HE
HAIYANG HE*
Affiliation:
College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing, (Ministry of Education of China), Hunan Normal University, Changsha, Hunan 410081, P.R. China e-mail: [email protected]
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Abstract

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(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.

MSC classification

Primary: 58J05: Elliptic equations on manifolds, general theory
Secondary: 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 3 , September 2015 , pp. 519 - 541
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Alexandrov, A. D., Uniqueness theorems for surfaces in the large V, Amer. Math. Soc. Trans. 21 (1962), 412–416.Google Scholar
2.Almeida, L., Damascelli, L. and Ge, Y., A few symmetry results for nonlinear elliptic PDE on non-compact manifolds, Ann. Inst. Henri Poincaré Anal. Non-Linéaire 19 (2002), 313–342.Google Scholar
3.Almeida, L. and Ge, Y., Symmetry result for positive solutions of some elliptic equations on manifolds, Ann. Global Anal. Geom. 18 (2000), 153–170.CrossRefGoogle Scholar
4.Berestycki, H. and Lions, P. L., Nonlinear scalar field equations (I) Arch. Ration. Mech. Anal. 82 (1983), 313–376.CrossRefGoogle Scholar
5.Bhakta, M. and Sandeep, K., Poincarè Sobolev equations in the hyperbolic space Calc. Var. Partial Differ. Equ. 44 (2012), 247–269.CrossRefGoogle Scholar
6.Bonforte, M., Gazzola, F., Grillo, G. and Vàzquez, J. L., Classification of radial solutions to the Emden-Fowler equation on the Hyperbolic space, Calc. Var. PDE, 46 (1–2) (2013), 375–401. arXiv:1104.3666v2.CrossRefGoogle Scholar
7.Busca, J. and Manasevich, R., A Liouville-type theorem for Lane-Emden systems Indiana Univ. Math. J. 51 (2002), 37–51.Google Scholar
8.Busca, J. and Sirakov, B., Symmetry results for semi-linear elliptic systems in the whole space, J. Differ. Equ. 163 (2000), 41–56.CrossRefGoogle Scholar
9.Caffarelli, L., Gidas, B. and Spruck, J., Asymptotic symmetry and local behavior of semi-linear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math. XLII (1989), 271–297.CrossRefGoogle Scholar
10.Chen, W. and Li, C., Classification of solutions of some nonlinear elliptic equations Duke Math. J. 63 (1991), 615–622.CrossRefGoogle Scholar
11.De Figueiredo, D. G. and Felmer, P., A Liouville-type theorem for elliptic systems, Ann. Sci. Norm. Sup. Pisa 21 (1994), 387–397.Google Scholar
12.De Figueiredo, D. G. and Yang, J., Decay, symmetry and existence of solutions of semi-linear elliptic systems, Nonlinear Anal. 33 (1998), 211–234.CrossRefGoogle Scholar
13.Gidas, B., Ni, W. M. and Nirenberg, L., Symmetry and related properties via the maximum principle, Commun. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
14.Gidas, B. and Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations Commun. Partial Differ. Equ. 6 (1981), 883–901.CrossRefGoogle Scholar
15.Gidas, B. and Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations Commun. Partial Differ. Equ. 17 (1992), 923–940.Google Scholar
16.Hopf, E., Lectures on differential geometry in the large, Seminar lectures New York University and Stanford University (Springer, New York, NY, 1956).Google Scholar
17.Jie, Q., A priori estimates for positive solutions of semi-linear elliptic systems J. Partial Differ. Equ. 1 (1988), 525–598.Google Scholar
18.Lions, H. E. and Magen, E., Nonhomogeneous boundary value problems and applications, vol I. (Springer-Verlag, Berlin, Germany, 1972).Google Scholar
19.Lions, P. L., The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Annales de l'institut Henri Poincaré Analyse non linéaire 1 (1984), 223–283.CrossRefGoogle Scholar
20.Li, S. and Willem, M., Applications of local linking to critical point theory J. Math. Anal. Appl. 189 (1995), 6–32.CrossRefGoogle Scholar
21.Li, G. and Yang, J., Asymptotically linear elliptic systems Commun. Partial Differ. Equ. 29 (2004), 925–954.CrossRefGoogle Scholar
22.Mancini, G. and Sandeep, K., On a semi-linear elliptic equaition in ℍN Ann. Sci. Norm. Super. Pisa Cl. Sci 7 (2008), 635–671.Google Scholar
23.Mitidieri, E., Non-existence of positive solutions of semi-linear elliptic systems in RN Differ. Int. Equ. 9 (1996), 465–479.Google Scholar
24.Serrin, J. and Zou, H., Non-existence of positive solutions of Lane–Emden systems Differ. Int. Equ. 9 (1996), 635–653.Google Scholar
25.Serrin, J., A symmetry problem in potential theory Arch. Ration. Mech. Anal. 43 (1971), 304–318.CrossRefGoogle Scholar
26.Souto, M. A., Sobre a existencia de solucoes positivas para sistemas cooperativos nao lineares, PhD Thesis (Unicamp, Campinas – SP, Brazil, 1992).Google Scholar
27.Triebel, H., Theory of function spaces II, Springer Monographs in Mathematics, vol. 84 (Birkhäuser, Heidelberg, Germany, 1992).CrossRefGoogle Scholar
28.Wolf, J., Spaces of constant curvature (McGraw-Hill, New York, NY, 1967).Google Scholar