Advanced Search
Article Contents
Article Contents

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

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

Abstract Full Text(HTML) Related Papers Cited by
  • 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) $.

    Mathematics Subject Classification: 35J61, 35A02.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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. CortázarP. 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. CortázarP. 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. CortázarJ. DolbeaultM. 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. CortázarM. 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. CortázarM. 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. CortázarM. 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. FranchiE. 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. McLeodW. 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.
  • 加载中

Article Metrics

HTML views(327) PDF downloads(248) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint