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Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations
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 |
$\begin{cases} Δ u+u^{p}+λ u = 0&\text{ in }Ω,\\ u = 0&\text{ on }\partialΩ, \end{cases}$ |
$λ>0$ |
$p>1$ |
$Ω$ |
$\{x∈{{\mathbb{R}}^{n}}:a<|x|<b\}$ |
$0<a<b$ |
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. |
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, 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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