{"title":"Bifurcation Method for Solving Positive Solutions to a Class of Semilinear Elliptic Equations and Stability Analysis of Solutions","authors":"Hailong Zhu, Zhaoxiang Li","volume":50,"journal":"International Journal of Nuclear and Quantum Engineering","pagesStart":129,"pagesEnd":136,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/9287","abstract":"
Semilinear elliptic equations are ubiquitous in natural sciences. They give rise to a variety of important phenomena in quantum mechanics, nonlinear optics, astrophysics, etc because they have rich multiple solutions. But the nontrivial solutions of semilinear equations are hard to be solved for the lack of stabilities, such as Lane-Emden equation, Henon equation and Chandrasekhar equation. In this paper, bifurcation method is applied to solving semilinear elliptic equations which are with homogeneous Dirichlet boundary conditions in 2D. Using this method, nontrivial numerical solutions will be computed and visualized in many different domains (such as square, disk, annulus, dumbbell, etc).<\/p>\r\n","references":"[1] Amann H. Supersolution, monotone iteration and stability. J Differential\r\nEquations, 21: 367-377 (1976).\r\n[2] Amann H, Crandall M G. on some existence theorems for semilinear\r\nelliptic equations. Indian Univ Math J, 27: 779-790 (1978).\r\n[3] Chang K C. Infinite Dimensional Morse Theory and Multiple Solution\r\nProblems. Boston: Birkhauser, (1933).\r\n[4] Struwe M. Variational Methods, A Series of Modern Surveys in Math.\r\nBerlin: Springer-Verlag, (1966).\r\n[5] Pao C V. Block monotone iterative methods for numerical solutions of\r\nnonlinear elliptic equations. Numer Math, 72: 239-262 (1995).\r\n[6] Deng Y, Chen G, Ni W M, et al. Boundary element monotone iteration\r\nscheme for semilinear elliptic partial differential equations. Math Comput,\r\n65: 943-982 (1996).\r\n[7] Choi Y S, McKenna P J. A mountain pass method for the numerical\r\nsolutions of semilinear elliptic problems. Nonlinear Anal, 20: 417-437\r\n(1993).\r\n[8] Ding Z H, Costa D, Chen G. A high-linking algorithm for sign-changing\r\nsolutions of semilinear elliptic equations. Nonlinear Anal, 38: 151-172\r\n(1999).\r\n[9] Li Y, Zhou J X. A minimax method for finding multiple critical points\r\nand its applications to semilinear PDEs. SIAM J Sci Comput, 23: 840-865\r\n(2002).\r\n[10] Yao X D, Zhou J X. A minimax method for finding multiple critical\r\npoints in Banach spaces and its application to quasi-linear elliptic PDE.\r\nSIAM J Sci Comput, 26: 1796-1809 (2005).\r\n[11] Chen C M, Xie Z Q. Search-extension method for multiple solutions of\r\nnonlinear problem. Comp Math Appl, 47: 327-343 (2004).\r\n[12] Yang Z H, Li Z X, Zhu H L. Bifurcation method for solving multiple\r\npositive solutions to Henon equation Science in China Series A: Mathematics,\r\nDec, Vol. 51, No. 12, 2330-2342 (2008).\r\n[13] Yang Z H, Li Z X, Zhu H L, Shen J. Bifurcation method to compute\r\nmultiple solution of Henon equation(in Chinese). Journal of Shanghai\r\nNormal University (Natural Sciences), 36(1): 1-6. (2007).\r\n[14] Chandrasekhar S. An Introduction to the Study of Stellar Structure.\r\nUniversity of Chicago Press, (1939).\r\n[15] Fowler R H. Further Studies on Emdens and similar differertial equations.\r\nQuart.J.Math, 2: 259-288(1931).\r\n[16] Henon M. Numerical experiments on the stability of spherical stellar\r\nsystems. Astro. Astrophys, 24:229-238 (1973).\r\n[17] Lieb E,Yao,H -T. The Chandrasekhar theory of stellar collapse as the\r\nlimit of quantum mechanics. Commun.Math.Phys, 112: 147-174(1987).\r\n[18] Yang Z H. Non-linear Bifurcation: Theory and Computation (in Chinese).\r\nBeijing: Science Press, (2007).\r\n[19] M Golubitsky, D G Schaeffer. Singularities and Groups in Bifurcation\r\nTheory. Vol.1, Springer-Verlag, (1985).\r\n[20] Kantorovich L V, Akilov G P. Functional Analysis in Normal Spaces.\r\nPergamon Press, (1964).\r\n[21] Decker D W, Keller H B. Path following near bifurcation. Comm\r\nPure.Appl.Math, 34:149-175(1981).\r\n[22] Hansjorg Kielhofer. Bifurcation Theory: An Introduction with Applications\r\nto PDEs.Springer-Verlag, (2004).\r\n[23] Tang Y. Theoretical priciple of symmetry bifurcation(in Chinese). Beijing:\r\nScience Press, (1995).\r\n[24] Golubitsky M, Schaeffer D G. Singularities and Groups in Bifurcation\r\nTheory, Vol.1. New York: Springer, (1986).","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 50, 2011"}