| Title:
|
A Note on the Uniqueness and Structure of Solutions to the Dirichlet Problem for Some Elliptic Systems (English) |
| Author:
|
Chern, Jang-Long |
| Author:
|
Yotsutani, Shoji |
| Author:
|
Kawano, Nichiro |
| Language:
|
English |
| Journal:
|
Proceedings of Equadiff 14 |
| Volume:
|
Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017 |
| Issue:
|
2017 |
| Year:
|
|
| Pages:
|
283-286 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Elliptic system, uniqueness, solutions structure |
| MSC:
|
35B09 |
| MSC:
|
35J57 |
| MSC:
|
35J91 |
| . |
| Date available:
|
2019-09-27T08:12:19Z |
| Last updated:
|
2019-09-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/703042 |
| . |
| Reference:
|
[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. MR 0982351, 10.1002/cpa.3160420304 |
| Reference:
|
[2] Chern, Jann-Long, Kawano, Nichiro, Yotsutani, Shoji: Approximations and Analysis of Positive Solutions for Some Elliptic Systems., Preprint, 2017. |
| Reference:
|
[3] Chen, W., Li, C.: Classification of solutions of some nonlinear elliptic equations., Duke Math. J. 63 (1991), 615–622. MR 1121147 |
| Reference:
|
[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 |
| Reference:
|
[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. MR 2383083, 10.1016/j.anihpc.2006.11.006 |
| Reference:
|
[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. MR 0544879, 10.1007/BF01221125 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[9] Li, C.: Local asymptotic symmetry of singular solutions to nonlinear elliptic equations., Invent. Math. 123 (1996), no. 2, 221–231. MR 1374197, 10.1007/s002220050023 |
| Reference:
|
[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. MR 2452879, 10.1137/080712301 |
| Reference:
|
[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. MR 2135447, 10.1007/s00220-005-1313-x |
| Reference:
|
[12] Lin, T., Wei, J.: Spikes in two coupled nonlinear Schrodinger equations., Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 403–439. MR 2145720, 10.1016/j.anihpc.2004.03.004 |
| Reference:
|
[13] Li, Y.: On the positive solutions of the Matukuma equation., Duke Math, J. 70 (1993), no. 3, 575-589. MR 1224099, 10.1215/S0012-7094-93-07012-3 |
| . |
