# Article

MSC: 35B09, 35J57, 35J91
Full entry | PDF   (0.3 MB)
Keywords:
Elliptic system, uniqueness, solutions structure
Summary:
In this note, we consider some elliptic systems on a smooth domain of \$R^n\$. By using the maximum principle, we can get a more general and complete results of the identical property of positive solution pair, and thus classify the structure of all positive solutions depending on the nonlinarities easily.
References:
[1] Caffarelli, L. A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42(1989), no. 3, 271–297. DOI 10.1002/cpa.3160420304 | MR 0982351
[2] Chern, Jann-Long, Kawano, Nichiro, Yotsutani, Shoji: Approximations and Analysis of Positive Solutions for Some Elliptic Systems. Preprint, 2017.
[3] Chen, W., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63 (1991), 615–622. MR 1121147
[4] Chen, C.-C., Lin, C.-S.: Uniqueness of the ground state solutions of \$\Delta u + f(u)=0\$ in \$\Bbb R^n\$, \$n\ge3\$. Comm. Partial Diff. Eqns 16 (1991), 1549-1572. MR 1132797
[5] Figueiredo, D.G. de, Lopes, O.: Solitary waves for some nonlinear Schr\"odinger systems. Ann. I. H. Poincaré Anal. Nonlinéaire 25 (2008), 149-161. DOI 10.1016/j.anihpc.2006.11.006 | MR 2383083
[6] Gidas, B., Ni, W. M., HASH(0x1e4a5c8), Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 209-243. DOI 10.1007/BF01221125 | MR 0544879
[7] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften. vol. 224, Springer-Verlag, Berlin, 1983. MR 0737190
[8] Kanna, T., Lakshmanan, M.: Effect of phase shift in shape changing collision of solitonsin coupled nonlinear Schrodinger equations. Topical issue on geometry, integrability and nonlinearity in condensed matter physics. Eur. Phys. J. B Condens. Matter Phys. 29 (2002), no. 2, 249–254. MR 1949959
[9] Li, C.: Local asymptotic symmetry of singular solutions to nonlinear elliptic equations. Invent. Math. 123 (1996), no. 2, 221–231. DOI 10.1007/s002220050023 | MR 1374197
[10] Li, C., Ma, L.: Uniqueness of positive bound states to Schrodinger systems with critical exponents. SIAM J. Math. Anal. 40 (2008), no. 3, 1049–1057. DOI 10.1137/080712301 | MR 2452879
[11] Lin, T., Wei, J.: Ground state of N coupled nonlinear Schrodinger equations in \$\Bbb R^n\$, \$n\le3\$. Comm. Math. Phys. 255 (2005), 629–653. DOI 10.1007/s00220-005-1313-x | MR 2135447
[12] Lin, T., Wei, J.: Spikes in two coupled nonlinear Schrodinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 403–439. DOI 10.1016/j.anihpc.2004.03.004 | MR 2145720
[13] Li, Y.: On the positive solutions of the Matukuma equation. Duke Math, J. 70 (1993), no. 3, 575-589. DOI 10.1215/S0012-7094-93-07012-3 | MR 1224099

Partner of