Bifurcation Method for Solving Positive Solutions to a Class of Semilinear Elliptic Equations and Stability Analysis of Solutions

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).

• Hailong Zhu
• Zhaoxiang Li

• Semilinear elliptic equations
• positive solutions
• bifurcation method
• isotropy subgroups.

[1] Amann H. Supersolution, monotone iteration and stability. J Differential
Equations, 21: 367-377 (1976).
[2] Amann H, Crandall M G. on some existence theorems for semilinear
elliptic equations. Indian Univ Math J, 27: 779-790 (1978).
[3] Chang K C. Infinite Dimensional Morse Theory and Multiple Solution
Problems. Boston: Birkhauser, (1933).
[4] Struwe M. Variational Methods, A Series of Modern Surveys in Math.
Berlin: Springer-Verlag, (1966).
[5] Pao C V. Block monotone iterative methods for numerical solutions of
nonlinear elliptic equations. Numer Math, 72: 239-262 (1995).
[6] Deng Y, Chen G, Ni W M, et al. Boundary element monotone iteration
scheme for semilinear elliptic partial differential equations. Math Comput,
65: 943-982 (1996).
[7] Choi Y S, McKenna P J. A mountain pass method for the numerical
solutions of semilinear elliptic problems. Nonlinear Anal, 20: 417-437
(1993).
[8] Ding Z H, Costa D, Chen G. A high-linking algorithm for sign-changing
solutions of semilinear elliptic equations. Nonlinear Anal, 38: 151-172
(1999).
[9] Li Y, Zhou J X. A minimax method for finding multiple critical points
and its applications to semilinear PDEs. SIAM J Sci Comput, 23: 840-865
(2002).
[10] Yao X D, Zhou J X. A minimax method for finding multiple critical
points in Banach spaces and its application to quasi-linear elliptic PDE.
SIAM J Sci Comput, 26: 1796-1809 (2005).
[11] Chen C M, Xie Z Q. Search-extension method for multiple solutions of
nonlinear problem. Comp Math Appl, 47: 327-343 (2004).
[12] Yang Z H, Li Z X, Zhu H L. Bifurcation method for solving multiple
positive solutions to Henon equation Science in China Series A: Mathematics,
Dec, Vol. 51, No. 12, 2330-2342 (2008).
[13] Yang Z H, Li Z X, Zhu H L, Shen J. Bifurcation method to compute
multiple solution of Henon equation(in Chinese). Journal of Shanghai
Normal University (Natural Sciences), 36(1): 1-6. (2007).
[14] Chandrasekhar S. An Introduction to the Study of Stellar Structure.
University of Chicago Press, (1939).
[15] Fowler R H. Further Studies on Emdens and similar differertial equations.
Quart.J.Math, 2: 259-288(1931).
[16] Henon M. Numerical experiments on the stability of spherical stellar
systems. Astro. Astrophys, 24:229-238 (1973).
[17] Lieb E,Yao,H -T. The Chandrasekhar theory of stellar collapse as the
limit of quantum mechanics. Commun.Math.Phys, 112: 147-174(1987).
[18] Yang Z H. Non-linear Bifurcation: Theory and Computation (in Chinese).
Beijing: Science Press, (2007).
[19] M Golubitsky, D G Schaeffer. Singularities and Groups in Bifurcation
Theory. Vol.1, Springer-Verlag, (1985).
[20] Kantorovich L V, Akilov G P. Functional Analysis in Normal Spaces.
Pergamon Press, (1964).
[21] Decker D W, Keller H B. Path following near bifurcation. Comm
Pure.Appl.Math, 34:149-175(1981).
[22] Hansjorg Kielhofer. Bifurcation Theory: An Introduction with Applications
to PDEs.Springer-Verlag, (2004).
[23] Tang Y. Theoretical priciple of symmetry bifurcation(in Chinese). Beijing:
Science Press, (1995).
[24] Golubitsky M, Schaeffer D G. Singularities and Groups in Bifurcation
Theory, Vol.1. New York: Springer, (1986).