# American Institute of Mathematical Sciences

March  2019, 39(3): 1585-1594. doi: 10.3934/dcds.2018122

## Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus

 1 School of Mathematics and Statistics, Central South University, Changsha, China 2 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, China 3 Department of Mathematics, California State University Northridge, Northridge, CA 91330, United States

* Corresponding author: yaorf5812@stu.xjtu.edu.cn

Received  July 2017 Revised  November 2017 Published  April 2018

Fund Project: The first author is supported by Tian Yuan Special Funds of the National Science Foundation of China (No.11626182).

In this paper, we show the following equation
 $\begin{cases} Δ u+u^{p}+λ u = 0&\text{ in }Ω,\\ u = 0&\text{ on }\partialΩ, \end{cases}$
has at most one positive radial solution for a certain range of
 $λ>0$
. Here
 $p>1$
and
 $Ω$
is the annulus
 $\{x∈{{\mathbb{R}}^{n}}:a<|x| , $0
. We also show this solution is radially non-degenerate via the bifurcation methods.
Citation: Ruofei Yao, Yi Li, Hongbin Chen. Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1585-1594. doi: 10.3934/dcds.2018122
##### References:
 [1] C. V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differential Equations, 54 (1984), 429-437.  doi: 10.1016/0022-0396(84)90153-0. [2] C. V. Coffman, Uniqueness of the positive radial solution on an annulus of the Dirichlet problem for $Δ u-u+u^{3} = 0$, J. Differential Equations, 128 (1996), 379-386.  doi: 10.1006/jdeq.1996.0100. [3] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2. [4] P. Felmer, S. Martínez and K. Tanaka, Uniqueness of radially symmetric positive solutions for $-Δ u+u = u^{p}$ in an annulus, J. Differential Equations, 245 (2008), 1198-1209.  doi: 10.1016/j.jde.2008.06.006. [5] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125. [6] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{n}$, Adv. in Math. Suppl. Stud., 7a (1981), 369-402. [7] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196. [8] F. Gladiali, M. Grossi, F. Pacella and P. N. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, Calc. Var. Partial Differential Equations, 40 (2011), 295-317.  doi: 10.1007/s00526-010-0341-3. [9] M. Grossi, F. Pacella and S. L. Yadava, Symmetry results for perturbed problems and related questions, Topol. Methods Nonlinear Anal., 21 (2003), 211-226.  doi: 10.12775/TMNA.2003.013. [10] J. Jang, Uniqueness of positive radial solutions of $Δ u+f(u) = 0$ in $\mathbb{R}^N, N≥2$, Nonlinear Anal., 73 (2010), 2189-2198.  doi: 10.1016/j.na.2010.05.045. [11] K. Kabeya and K. Tanaka, Uniqueness of positive radial solutions of semilinear elliptic equations in $R^{N}$ and Séré's non-degeneracy condition, Comm. Partial Differential Equations, 24 (1999), 563-598.  doi: 10.1080/03605309908821434. [12] P. Korman, On the multiplicity of solutions of semilinear equations, Math. Nachr., 229 (2001), 119-127.  doi: 10.1002/1522-2616(200109)229:1<119::AID-MANA119>3.0.CO;2-P. [13] M. K. Kwong, Uniqueness of positive solutions of $Δ u-u+u^p = 0$ in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502. [14] M. K. Kwong and Y. Li, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 333 (1992), 339-363.  doi: 10.1090/S0002-9947-1992-1088021-X. [15] Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differential Equations, 83 (1990), 348-367.  doi: 10.1016/0022-0396(90)90062-T. [16] W. M. Ni and R. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $Δ u+f(u,r) = 0$, Comm. Pure Appl. Math., 38 (1985), 67-108.  doi: 10.1002/cpa.3160380105. [17] T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems, J. Differential Equations, 146 (1998), 121-156.  doi: 10.1006/jdeq.1998.3414. [18] P. N. Srikanth, Uniqueness of solutions of nonlinear Dirichlet problems, Differential Integral Equations, 6 (1993), 663-670. [19] M. Struwe, Variational Methods, Applications to nonlinear partial differential equations and Hamiltonian systems, 4th edition. Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-74013-1. [20] M. X. Tang, Uniqueness of positive radial solutions for $Δ u-u+u^p=0$ on an annulus, J. Differential Equations, 189 (2003), 148-160.  doi: 10.1016/S0022-0396(02)00142-0. [21] S. L. Yadava, Uniqueness of positive radial solutions of the Dirichlet problems $-Δ u=u^{p}± u^{q}$ in an annulus, J. Differential Equations, 139 (1997), 194-217.  doi: 10.1006/jdeq.1997.3283. [22] L. Q. Zhang, Uniqueness of positive solutions of $Δ u+ u+u^{p}=0$ in a ball, Comm. Partial Differential Equations, 17 (1992), 1141-1164.  doi: 10.1080/03605309208820880.

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##### References:
 [1] C. V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differential Equations, 54 (1984), 429-437.  doi: 10.1016/0022-0396(84)90153-0. [2] C. V. Coffman, Uniqueness of the positive radial solution on an annulus of the Dirichlet problem for $Δ u-u+u^{3} = 0$, J. Differential Equations, 128 (1996), 379-386.  doi: 10.1006/jdeq.1996.0100. [3] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2. [4] P. Felmer, S. Martínez and K. Tanaka, Uniqueness of radially symmetric positive solutions for $-Δ u+u = u^{p}$ in an annulus, J. Differential Equations, 245 (2008), 1198-1209.  doi: 10.1016/j.jde.2008.06.006. [5] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125. [6] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{n}$, Adv. in Math. Suppl. Stud., 7a (1981), 369-402. [7] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196. [8] F. Gladiali, M. Grossi, F. Pacella and P. N. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, Calc. Var. Partial Differential Equations, 40 (2011), 295-317.  doi: 10.1007/s00526-010-0341-3. [9] M. Grossi, F. Pacella and S. L. Yadava, Symmetry results for perturbed problems and related questions, Topol. Methods Nonlinear Anal., 21 (2003), 211-226.  doi: 10.12775/TMNA.2003.013. [10] J. Jang, Uniqueness of positive radial solutions of $Δ u+f(u) = 0$ in $\mathbb{R}^N, N≥2$, Nonlinear Anal., 73 (2010), 2189-2198.  doi: 10.1016/j.na.2010.05.045. [11] K. Kabeya and K. Tanaka, Uniqueness of positive radial solutions of semilinear elliptic equations in $R^{N}$ and Séré's non-degeneracy condition, Comm. Partial Differential Equations, 24 (1999), 563-598.  doi: 10.1080/03605309908821434. [12] P. Korman, On the multiplicity of solutions of semilinear equations, Math. Nachr., 229 (2001), 119-127.  doi: 10.1002/1522-2616(200109)229:1<119::AID-MANA119>3.0.CO;2-P. [13] M. K. Kwong, Uniqueness of positive solutions of $Δ u-u+u^p = 0$ in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502. [14] M. K. Kwong and Y. Li, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 333 (1992), 339-363.  doi: 10.1090/S0002-9947-1992-1088021-X. [15] Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differential Equations, 83 (1990), 348-367.  doi: 10.1016/0022-0396(90)90062-T. [16] W. M. Ni and R. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $Δ u+f(u,r) = 0$, Comm. Pure Appl. Math., 38 (1985), 67-108.  doi: 10.1002/cpa.3160380105. [17] T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems, J. Differential Equations, 146 (1998), 121-156.  doi: 10.1006/jdeq.1998.3414. [18] P. N. Srikanth, Uniqueness of solutions of nonlinear Dirichlet problems, Differential Integral Equations, 6 (1993), 663-670. [19] M. Struwe, Variational Methods, Applications to nonlinear partial differential equations and Hamiltonian systems, 4th edition. Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-74013-1. [20] M. X. Tang, Uniqueness of positive radial solutions for $Δ u-u+u^p=0$ on an annulus, J. Differential Equations, 189 (2003), 148-160.  doi: 10.1016/S0022-0396(02)00142-0. [21] S. L. Yadava, Uniqueness of positive radial solutions of the Dirichlet problems $-Δ u=u^{p}± u^{q}$ in an annulus, J. Differential Equations, 139 (1997), 194-217.  doi: 10.1006/jdeq.1997.3283. [22] L. Q. Zhang, Uniqueness of positive solutions of $Δ u+ u+u^{p}=0$ in a ball, Comm. Partial Differential Equations, 17 (1992), 1141-1164.  doi: 10.1080/03605309208820880.
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