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The jumping problem for nonlocal singular problems
On the uniqueness of bound state solutions of a semilinear equation with weights
Departamento de Matemática, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile |
$ \mbox{div}\big(\mathsf A\, \nabla v\big)+\mathsf B\, f(v) = 0\, , \quad\lim\limits_{|x|\to+\infty}v(x) = 0, \quad x\in\mathbb R^n, ~~~~{(P)} $ |
$ n>2 $ |
$ \mathsf A $ |
$ \mathsf B $ |
$ \mathbb R^n\setminus\{0\} $ |
$ f\in C(-c, c) $ |
$ 0<c\le\infty $ |
$ b>0 $ |
$ (0, b) $ |
$ (b, c) $ |
$ (0, c) $ |
References:
[1] |
C. C. Chen and C. S. Lin,
Uniqueness of the ground state solutions of $\Delta u+f(u) = 0$ in $\mathbb R^N, $ $N\ge 3, $, Comm. in Partial Differential Equations, 16 (1991), 1549-1572.
doi: 10.1080/03605309108820811. |
[2] |
C. V. Coffman,
Uniqueness of the ground state solution of $\Delta u-u+u^3 = 0$ and a variational characterization of other solutions, Archive Rat. Mech. Anal., 46 (1972), 81-95.
doi: 10.1007/BF00250684. |
[3] |
C. Cortázar, P. Felmer and M. Elgueta,
On a semilinear elliptic problem in $\mathbb R^N$ with a non Lipschitzian nonlinearity, Advances in Differential Equations, 1 (1996), 199-218.
|
[4] |
C. Cortázar, P. Felmer and M. Elgueta,
Uniqueness of positive solutions of $\Delta u+f(u) = 0$ in $\mathbb R^N$, $N\ge 3$, Archive Rat. Mech. Anal., 142 (1998), 127-141.
doi: 10.1007/s002050050086. |
[5] |
C. Cortázar and M. García-Huidobro,
On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian, Comm. Pure. Appl. Anal., 5 (2006), 813-826.
doi: 10.3934/cpaa.2006.5.813. |
[6] |
C. Cortázar, J. Dolbeault, M. García-Huidobro and R. Manásevich,
Existence of sign changing solutions for an equation with a weighted p-Laplace operator, Nonlinear Anal., 110 (2014), 1-22.
doi: 10.1016/j.na.2014.07.016. |
[7] |
C. Cortázar, M. García-Huidobro and C. Yarur,
On the uniqueness of the second bound state solution of a semilinear equation, Annales de l'Institut Henri Poincaré - Analyse non linéaire, 26 (2009), 2091-2110.
doi: 10.1016/j.anihpc.2009.01.004. |
[8] |
C. Cortázar, M. García-Huidobroand and C. Yarur,
On the uniqueness of sign changing bound state solutions of a semilinear equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 599-621.
doi: 10.1016/j.anihpc.2011.04.002. |
[9] |
C. Cortázar, M. García-Huidobro and C. Yarur,
On the existence of sign changing bound state solutions of a quasilinear equation, J. Differential Equations, 254 (2013), 2603-2625.
doi: 10.1016/j.jde.2012.12.015. |
[10] |
L. Erbe and M. Tang,
Uniqueness theorems for positive solutions of quasilinear elliptic equations in a ball, J. Diff. Equations, 138 (1997), 351-379.
doi: 10.1006/jdeq.1997.3279. |
[11] |
B. Franchi, E. Lanconelli and J. Serrin,
Existence and Uniqueness of nonnegative solutions of quasilinear equations in $\mathbb R^n$, Advances in Mathematics, 118 (1996), 177-243.
doi: 10.1006/aima.1996.0021. |
[12] |
M. García-Huidobro and D. Henao,
On the uniqueness of positive solutions of a quasilinear equation containing a weighted $p$-Laplacian, Comm. in Contemp. Math., 10 (2008), 405-432.
doi: 10.1142/S0219199708002831. |
[13] |
R. Kajikiya,
Necessary and sufficient condition for existence and uniqueness of nodal solutions to sublinear elliptic equations., Adv. Differential Equations, 6 (2001), 1317-1346.
|
[14] |
M. K. Kwong,
Uniqueness of positive solutions of $\Delta u-u+u^p = 0$, Archive Rat. Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[15] |
K. McLeod,
Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in $\mathbb R^N $ ,Ⅱ, Trans. Amer. Math. Soc., 339 (1993), 495-505.
|
[16] |
K. McLeod and J. Serrin,
Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in $\mathbb R^N $, Arch. Rational Mech. Anal., 99 (1987), 115-145.
doi: 10.1007/BF00275874. |
[17] |
K. McLeod, W. C. Troy and F. B. Weissler,
Radial solutions of $\Delta u+f(u) = 0$ with prescribed numbers of zeros, J. Differential Equations, 83 (1990), 368-378.
doi: 10.1016/0022-0396(90)90063-U. |
[18] |
L. Peletier and J. Serrin,
Uniqueness of positive solutions of quasilinear equations, Archive Rat. Mech. Anal., 81 (1983), 181-197.
doi: 10.1007/BF00250651. |
[19] |
L. Peletier and J. Serrin,
Uniqueness of nonnegative solutions of quasilinear equations, J. Diff. Equat., 61 (1986), 380-397.
doi: 10.1016/0022-0396(86)90112-9. |
[20] |
P. Pucci, M. Garca-Huidobro, R. Mansevich and J. Serrin, Qualitative properties of ground states for singular elliptic equations with weights, Ann. Mat. Pura Appl., (4) 185 (2006), S205–S243.
doi: 10.1007/s10231-004-0143-3. |
[21] |
P. R. Pucci and J. Serrin,
Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J., 47 (1998), 501-528.
|
[22] |
J. Serrin and M. Tang,
Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.
|
[23] |
S. Tanaka,
On the uniqueness of solutions with prescribed numbers of zeros for a two-point boundary value problem, Differential Integral Equations, 20 (2007), 93-104.
|
[24] |
S. Tanaka,
Uniqueness of nodal radial solutions of superlinear elliptic equations in a ball, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1331-1343.
doi: 10.1017/S0308210507000431. |
[25] |
S. Tanaka,
Uniqueness and nonuniqueness of nodal radial solutions of sublinear elliptic equations in a ball, Nonlinear Anal., 71 (2009), 5256-5267.
doi: 10.1016/j.na.2009.04.009. |
[26] |
S. Tanaka,
Uniqueness of sign-changing radial solutions for $\Delta u-u+|u|^{p-1}u=0$ in some ball and annulus, J. Math. Anal. Appl., 439 (2016), 154-170.
doi: 10.1016/j.jmaa.2016.02.036. |
[27] |
M. Tang,
Uniqueness of positive radial solutions for $\Delta u-u+u^p=0$ on an annulus, J. Differential Equations, 189 (2003), 148-160.
doi: 10.1016/S0022-0396(02)00142-0. |
[28] |
W. Troy,
The existence and uniqueness of bound state solutions of a semilinear equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2941-2963.
doi: 10.1098/rspa.2005.1482. |
[29] |
W. Troy,
Uniqueness of positive ground state solutions of the logarithmic Schrdinger equation, Arch. Ration. Mech. Anal., 222 (2016), 1581-1600.
doi: 10.1007/s00205-016-1028-5. |
show all references
References:
[1] |
C. C. Chen and C. S. Lin,
Uniqueness of the ground state solutions of $\Delta u+f(u) = 0$ in $\mathbb R^N, $ $N\ge 3, $, Comm. in Partial Differential Equations, 16 (1991), 1549-1572.
doi: 10.1080/03605309108820811. |
[2] |
C. V. Coffman,
Uniqueness of the ground state solution of $\Delta u-u+u^3 = 0$ and a variational characterization of other solutions, Archive Rat. Mech. Anal., 46 (1972), 81-95.
doi: 10.1007/BF00250684. |
[3] |
C. Cortázar, P. Felmer and M. Elgueta,
On a semilinear elliptic problem in $\mathbb R^N$ with a non Lipschitzian nonlinearity, Advances in Differential Equations, 1 (1996), 199-218.
|
[4] |
C. Cortázar, P. Felmer and M. Elgueta,
Uniqueness of positive solutions of $\Delta u+f(u) = 0$ in $\mathbb R^N$, $N\ge 3$, Archive Rat. Mech. Anal., 142 (1998), 127-141.
doi: 10.1007/s002050050086. |
[5] |
C. Cortázar and M. García-Huidobro,
On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian, Comm. Pure. Appl. Anal., 5 (2006), 813-826.
doi: 10.3934/cpaa.2006.5.813. |
[6] |
C. Cortázar, J. Dolbeault, M. García-Huidobro and R. Manásevich,
Existence of sign changing solutions for an equation with a weighted p-Laplace operator, Nonlinear Anal., 110 (2014), 1-22.
doi: 10.1016/j.na.2014.07.016. |
[7] |
C. Cortázar, M. García-Huidobro and C. Yarur,
On the uniqueness of the second bound state solution of a semilinear equation, Annales de l'Institut Henri Poincaré - Analyse non linéaire, 26 (2009), 2091-2110.
doi: 10.1016/j.anihpc.2009.01.004. |
[8] |
C. Cortázar, M. García-Huidobroand and C. Yarur,
On the uniqueness of sign changing bound state solutions of a semilinear equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 599-621.
doi: 10.1016/j.anihpc.2011.04.002. |
[9] |
C. Cortázar, M. García-Huidobro and C. Yarur,
On the existence of sign changing bound state solutions of a quasilinear equation, J. Differential Equations, 254 (2013), 2603-2625.
doi: 10.1016/j.jde.2012.12.015. |
[10] |
L. Erbe and M. Tang,
Uniqueness theorems for positive solutions of quasilinear elliptic equations in a ball, J. Diff. Equations, 138 (1997), 351-379.
doi: 10.1006/jdeq.1997.3279. |
[11] |
B. Franchi, E. Lanconelli and J. Serrin,
Existence and Uniqueness of nonnegative solutions of quasilinear equations in $\mathbb R^n$, Advances in Mathematics, 118 (1996), 177-243.
doi: 10.1006/aima.1996.0021. |
[12] |
M. García-Huidobro and D. Henao,
On the uniqueness of positive solutions of a quasilinear equation containing a weighted $p$-Laplacian, Comm. in Contemp. Math., 10 (2008), 405-432.
doi: 10.1142/S0219199708002831. |
[13] |
R. Kajikiya,
Necessary and sufficient condition for existence and uniqueness of nodal solutions to sublinear elliptic equations., Adv. Differential Equations, 6 (2001), 1317-1346.
|
[14] |
M. K. Kwong,
Uniqueness of positive solutions of $\Delta u-u+u^p = 0$, Archive Rat. Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[15] |
K. McLeod,
Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in $\mathbb R^N $ ,Ⅱ, Trans. Amer. Math. Soc., 339 (1993), 495-505.
|
[16] |
K. McLeod and J. Serrin,
Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in $\mathbb R^N $, Arch. Rational Mech. Anal., 99 (1987), 115-145.
doi: 10.1007/BF00275874. |
[17] |
K. McLeod, W. C. Troy and F. B. Weissler,
Radial solutions of $\Delta u+f(u) = 0$ with prescribed numbers of zeros, J. Differential Equations, 83 (1990), 368-378.
doi: 10.1016/0022-0396(90)90063-U. |
[18] |
L. Peletier and J. Serrin,
Uniqueness of positive solutions of quasilinear equations, Archive Rat. Mech. Anal., 81 (1983), 181-197.
doi: 10.1007/BF00250651. |
[19] |
L. Peletier and J. Serrin,
Uniqueness of nonnegative solutions of quasilinear equations, J. Diff. Equat., 61 (1986), 380-397.
doi: 10.1016/0022-0396(86)90112-9. |
[20] |
P. Pucci, M. Garca-Huidobro, R. Mansevich and J. Serrin, Qualitative properties of ground states for singular elliptic equations with weights, Ann. Mat. Pura Appl., (4) 185 (2006), S205–S243.
doi: 10.1007/s10231-004-0143-3. |
[21] |
P. R. Pucci and J. Serrin,
Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J., 47 (1998), 501-528.
|
[22] |
J. Serrin and M. Tang,
Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.
|
[23] |
S. Tanaka,
On the uniqueness of solutions with prescribed numbers of zeros for a two-point boundary value problem, Differential Integral Equations, 20 (2007), 93-104.
|
[24] |
S. Tanaka,
Uniqueness of nodal radial solutions of superlinear elliptic equations in a ball, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1331-1343.
doi: 10.1017/S0308210507000431. |
[25] |
S. Tanaka,
Uniqueness and nonuniqueness of nodal radial solutions of sublinear elliptic equations in a ball, Nonlinear Anal., 71 (2009), 5256-5267.
doi: 10.1016/j.na.2009.04.009. |
[26] |
S. Tanaka,
Uniqueness of sign-changing radial solutions for $\Delta u-u+|u|^{p-1}u=0$ in some ball and annulus, J. Math. Anal. Appl., 439 (2016), 154-170.
doi: 10.1016/j.jmaa.2016.02.036. |
[27] |
M. Tang,
Uniqueness of positive radial solutions for $\Delta u-u+u^p=0$ on an annulus, J. Differential Equations, 189 (2003), 148-160.
doi: 10.1016/S0022-0396(02)00142-0. |
[28] |
W. Troy,
The existence and uniqueness of bound state solutions of a semilinear equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2941-2963.
doi: 10.1098/rspa.2005.1482. |
[29] |
W. Troy,
Uniqueness of positive ground state solutions of the logarithmic Schrdinger equation, Arch. Ration. Mech. Anal., 222 (2016), 1581-1600.
doi: 10.1007/s00205-016-1028-5. |
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