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

Ruofei Yao; Yi Li; Hongbin Chen

doi:10.3934/dcds.2018122

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

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

Ruofei Yao ^1,2,, , Yi Li ^3, and Hongbin Chen ^2,

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.

Keywords: Uniqueness, bifurcation methods, positive radial solution.

Mathematics Subject Classification: 35J61, 35J25, 35B32.

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, 4^th 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.

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.
[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, 4^th 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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Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus

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