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November  2019, 39(11): 6761-6784. doi: 10.3934/dcds.2019294

On the uniqueness of bound state solutions of a semilinear equation with weights

Carmen Cortázar , Marta García-Huidobro and  Pilar Herreros

Departamento de Matemática, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

Received  May 2019 Revised  May 2019 Published  August 2019

Fund Project: This research was supported by FONDECYT-1190102 for the first and second author, FONDECYT-1160540 for the second author and FONDECYT-1170665 for third author

We consider radial solutions of a general elliptic equation involving a weighted Laplace operator. We establish the uniqueness of the radial bound state solutions to
$ \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 $
, where
$ \mathsf A $
 and
$ \mathsf B $
 are two positive, radial, smooth functions defined on
$ \mathbb R^n\setminus\{0\} $
. We assume that the nonlinearity
$ f\in C(-c, c) $
,
$ 0<c\le\infty $
 is an odd function satisfying some convexity and growth conditions, and has a zero at
$ b>0 $
, is non positive and not identically 0 in
$ (0, b) $
, positive in
$ (b, c) $
, and is differentiable in
$ (0, c) $
.
Keywords: Uniqueness, weighted Laplace operator, bound states, semilinear, elliptic.
Mathematics Subject Classification: 35J61, 35A02.
Citation: Carmen Cortázar, Marta García-Huidobro, Pilar Herreros. On the uniqueness of bound state solutions of a semilinear equation with weights. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6761-6784. doi: 10.3934/dcds.2019294
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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

P. R. Pucci and J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J., 47 (1998), 501-528.   Google Scholar

[22]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

P. R. Pucci and J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J., 47 (1998), 501-528.   Google Scholar

[22]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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