Abstract
We prove the existence and uniqueness of positive radially symmetric solutions of the Dirichlet problem for a nonlinear elliptic system of second order. A numerical method for constructing such solutions is also given.
Similar content being viewed by others
References
E. Mitidieri and S. I. Pokhozhaev, “A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities,” in Trudy Mat. Inst. Steklov (Nauka, Moscow, 2001), Vol. 234, pp. 3–383 [Proc. Steklov Inst. Math. 234, 2001].
S. I. Pokhozhaev, “On a problem of L. V. Ovsyannikov,” Zh. Prikl. i Mekh. Tekhn. Fiz. No. 2, 5–10 (1989) [J. Appl. Mech. Tech. Phys. 30 (2), 169–174 (1989)].
E. I. Galakhov, “Positive solutions of quasilinear elliptic equations,” Mat. Zametki 78(2), 202–211 (2005) [Math. Notes 78 (2), 185–193 (2005)].
K.-S. Cheng and J.-T. Lin, “On the elliptic equations δu = K(x)u α and δu = K(x)e 2u,” Trans. Amer. Math. Soc. 304(2), 639–668 (1987).
B. Gidas and J. Spruck, “Global and local behavior of positive solutions of nonlinear elliptic equations,” Comm. Pure Appl. Math. 34(4), 525–598 (1981).
H. Zou, “A priori estimates, existence and non-existence for quasilinear cooperative elliptic systems,” Mat. Sb. 199(4), 83–106 (2008) [Sb. Math. 199 (4), 557–578 (2008)].
E. N. Dancer and J. Shi, “Uniqueness and nonexistence of positive solutions to semipositone problems,” Bull. London Math. Soc. 38(6), 1033–1044 (2006).
N. Kawano, J. Satsuma, and S. Yotsutani, “On the positive solutions of an Emden-type elliptic equation,” Proc. Japan Acad. Ser. A Math. Sci. 61(6), 186–189 (1985).
J. Jiang, “On radially symmetric solutions to singular nonlinear Dirichlet problems,” Nonlinear Anal.: Theory, Methods & Appl. 24(No. 2), 159–163 (1995).
É. I. Abduragimov, “Uniqueness of a positive radially symmetric solution of the Dirichlet problem for certain nonlinear differential equation of the second order in a ball,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 12, 3–6 (2008) [Russian Math. (Iz. VUZ) 52 (12), 1–3(2008)].
É. I. Abduragimov, “On positive radial symmetric solution of the Dirichlet problem for a certain nonlinear equation and a numerical method for its evaluation,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 3–6 (1997) [RussianMath. (Iz. VUZ) No. 5, 1–3 (1997)].
T. Na, Computational Methods in Engineering Boundary-Value Problems (Moscow, 1982) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © É. I. Abduragimov, 2013, published in Matematicheskie Zametki, 2013, Vol. 93, No. 1, pp. 3–12.
Rights and permissions
About this article
Cite this article
Abduragimov, É.I. Uniqueness of positive radially symmetric solutions of the Dirichlet problem for a nonlinear elliptic system of second order. Math Notes 93, 3–11 (2013). https://doi.org/10.1134/S000143461301001X
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000143461301001X