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
$\{x∈{{\mathbb{R}}^{n}}:a<|x|<b\}$
,
$0<a<b$
. 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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

P. FelmerS. 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.  Google Scholar

[5]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[6]

B. GidasW. 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.   Google Scholar

[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.  Google Scholar

[8]

F. GladialiM. GrossiF. 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.  Google Scholar

[9]

M. GrossiF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

P. N. Srikanth, Uniqueness of solutions of nonlinear Dirichlet problems, Differential Integral Equations, 6 (1993), 663-670.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

P. FelmerS. 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.  Google Scholar

[5]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[6]

B. GidasW. 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.   Google Scholar

[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.  Google Scholar

[8]

F. GladialiM. GrossiF. 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.  Google Scholar

[9]

M. GrossiF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

P. N. Srikanth, Uniqueness of solutions of nonlinear Dirichlet problems, Differential Integral Equations, 6 (1993), 663-670.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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