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doi: 10.3934/cpaa.2021029

## On the uniqueness of solutions of a semilinear equation in an annulus

 1 Departamento de Matemática, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile 2 Mathematical Institute, Tohoku University, Aoba 6–3, Aramaki, Aoba-ku, Sendai 980–8578, Japan

* Corresponding author

Received  September 2020 Revised  January 2021 Published  March 2021

Fund Project: This research was supported by FONDECYT-1190102 for the first and second author, and FONDECYT- 1170665 for the third author and by JSPS KAKENHI Grant Number 19K03595 and 17H01095 for the fourth author

We establish the uniqueness of positive radial solutions of
 \begin{align} \begin{cases} \Delta u +f(u) = 0, \quad x\in A \\ u(x) = 0 \qquad \qquad x\in \partial A \end{cases} \;\;\;\; (P)\end{align}
where
 $A: = A_{a, b} = \{ x\in {\mathbb R}^n : a<|x| , $ 0
. We assume that the nonlinearity
 $f\in C[0, \infty)\cap C^1(0, \infty)$
is such that
 $f(0) = 0$
and satisfies some convexity and growth conditions, and either
 $f(s)>0$
for all
 $s>0$
, or has one zero at
 $B>0$
, is non positive and not identically 0 in
 $(0, B)$
and it is positive in
 $(B, \infty)$
.
Citation: Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021029
##### References:
 [1] J. Byeon, Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli, J. Differ. Equ., 136 (1997), 136-165.  doi: 10.1006/jdeq.1996.3241.  Google Scholar [2] J. Cheng and L. Guang, Uniqueness of positive radial solutions for Dirichlet problems on annular domains, J. Math. Anal. Appl., 338 (2008), 416-426.  doi: 10.1016/j.jmaa.2007.05.027.  Google Scholar [3] C. V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differ. Equ., 54 (1984), 429-437.  doi: 10.1016/0022-0396(84)90153-0.  Google Scholar [4] C. V. Coffman, Uniqueness of the positive radial solution on an annulus of the Dirichlet problem for $\Delta u-u+u^p = 0$, J. Differ. Equ., 128 (1996), 379-386.  doi: 10.1006/jdeq.1996.0100.  Google Scholar [5] C. Cortázar, M. García-Huidobro and C. Yarur, On the uniqueness of the second bound state solution of a semilinear equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2091-2110.  doi: 10.1016/j.anihpc.2009.01.004.  Google Scholar [6] L. Erbe and M. Tang, Uniqueness theorems for positive solutions of quasilinear elliptic equations in a ball, J. Differ. Equ., 138 (1997), 351-379.  doi: 10.1006/jdeq.1997.3279.  Google Scholar [7] L. Erbe and M. Tang, Uniqueness of positive radial solutions of $\Delta u + f(|x|, u) = 0$, Differ. Integral Equ., 11 (1998), 725-743.   Google Scholar [8] P. Felmer, S. Martinez and K. Tanaka, Uniqueness of radially symmetric positive solutions for $-\Delta u+u = u^p$ in an annulus, J. Differ. Equ., 245 (2008), 1198-1209.  doi: 10.1016/j.jde.2008.06.006.  Google Scholar [9] B. Franchi, E. Lanconelli and J. Serrin, Existence and Uniqueness of nonnegative solutions of quasilinear equations in ${\mathbb R}^n$, Adv. math., 118 (1996), 177-243.  doi: 10.1006/aima.1996.0021.  Google Scholar [10] C. C. Fu and S. S. Lin, Uniqueness of positive radial solutions for semilinear elliptic equations on annular domains, Nonlinear Anal., 44 (2001), 749-758.  doi: 10.1016/S0362-546X(99)00303-X.  Google Scholar [11] X. Garaizar, Existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differ. Equ, 70 (1987), 69-92.  doi: 10.1016/0022-0396(87)90169-0.  Google Scholar [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>3.3.CO;2-G.  Google Scholar [13] M. K. Kwong and L. Q. Zhang, Uniqueness of the positive solution of $\Delta u+f(u) = 0$ in an annulus, Differ. Integral Equ., 4 (1991), 583-599.   Google Scholar [14] C. Li and Y. Zhou, Uniqueness of positive solutions to a class of semilinear elliptic equations, Bound. Value Probl., 2011 (2011), 9 pp. doi: 10.1186/1687-2770-2011-38.  Google Scholar [15] Y. Y. Li, Existence of many positive solutions of semilinear equations on annulus, J. Differ. Equ., 83 (1990), 348-367.  doi: 10.1016/0022-0396(90)90062-T.  Google Scholar [16] W. M. Ni, Uniqueness of solutions of nonlinear Dirichlet problems, J. Differ. Equ., 50 (1983), 289-304.  doi: 10.1016/0022-0396(83)90079-7.  Google Scholar [17] W. M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u, r) = 0$, Commun. Pure Appl. Math., 38 (1985), 67-108.  doi: 10.1002/cpa.3160380105.  Google Scholar [18] J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.  doi: 10.1512/iumj.2000.49.1893.  Google Scholar [19] N. Shioji, S. Tanaka and K. Watanabe, Uniqueness of positive radial solutions of superlinear elliptic equations in annuli, preprint  Google Scholar [20] N. Shioji and K. Watanabe, A generalized Pohožaev identity and uniqueness of positive radial solutions of $\Delta u + g(r)u + h(r)u^p = 0$, J. Differ. Equ., 255 (2013), 4448-4475.  doi: 10.1016/j.jde.2013.08.017.  Google Scholar [21] N. Shioji and K. Watanabe, Uniqueness and nondegeneracy of positive radial solutions of ${\rm{div}} (\rho\nabla u)+\rho(-gu+hu^p) = 0$, Calc. Var. Partial Differ. Equ., 55 (2016), 42 pp. doi: 10.1007/s00526-016-0970-2.  Google Scholar [22] M. Tang, Uniqueness of positive radial solutions for $\Delta u-u+u^p=0$ on an annulus, J. Differ. Equ., 189 (2003), 148-160.  doi: 10.1016/S0022-0396(02)00142-0.  Google Scholar [23] S. L. Yadava, Uniqueness of positive radial solutions of the Dirichlet problems $-\Delta u=u^p\pm u^q$ in an annulus, J. Differ. Equ., 139 (1997), 194-217.  doi: 10.1006/jdeq.1997.3283.  Google Scholar [24] S. L. Yadava, Uniqueness of positive radial solutions of a semilinear Dirichlet problem in an annulus, Proc. Roy. Soc. Edinb. Sect. A, 130 (2000), 1417-1428.  doi: 10.1017/S0308210500000755.  Google Scholar

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##### References:
 [1] J. Byeon, Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli, J. Differ. Equ., 136 (1997), 136-165.  doi: 10.1006/jdeq.1996.3241.  Google Scholar [2] J. Cheng and L. Guang, Uniqueness of positive radial solutions for Dirichlet problems on annular domains, J. Math. Anal. Appl., 338 (2008), 416-426.  doi: 10.1016/j.jmaa.2007.05.027.  Google Scholar [3] C. V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differ. Equ., 54 (1984), 429-437.  doi: 10.1016/0022-0396(84)90153-0.  Google Scholar [4] C. V. Coffman, Uniqueness of the positive radial solution on an annulus of the Dirichlet problem for $\Delta u-u+u^p = 0$, J. Differ. Equ., 128 (1996), 379-386.  doi: 10.1006/jdeq.1996.0100.  Google Scholar [5] C. Cortázar, M. García-Huidobro and C. Yarur, On the uniqueness of the second bound state solution of a semilinear equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2091-2110.  doi: 10.1016/j.anihpc.2009.01.004.  Google Scholar [6] L. Erbe and M. Tang, Uniqueness theorems for positive solutions of quasilinear elliptic equations in a ball, J. Differ. Equ., 138 (1997), 351-379.  doi: 10.1006/jdeq.1997.3279.  Google Scholar [7] L. Erbe and M. Tang, Uniqueness of positive radial solutions of $\Delta u + f(|x|, u) = 0$, Differ. Integral Equ., 11 (1998), 725-743.   Google Scholar [8] P. Felmer, S. Martinez and K. Tanaka, Uniqueness of radially symmetric positive solutions for $-\Delta u+u = u^p$ in an annulus, J. Differ. Equ., 245 (2008), 1198-1209.  doi: 10.1016/j.jde.2008.06.006.  Google Scholar [9] B. Franchi, E. Lanconelli and J. Serrin, Existence and Uniqueness of nonnegative solutions of quasilinear equations in ${\mathbb R}^n$, Adv. math., 118 (1996), 177-243.  doi: 10.1006/aima.1996.0021.  Google Scholar [10] C. C. Fu and S. S. Lin, Uniqueness of positive radial solutions for semilinear elliptic equations on annular domains, Nonlinear Anal., 44 (2001), 749-758.  doi: 10.1016/S0362-546X(99)00303-X.  Google Scholar [11] X. Garaizar, Existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differ. Equ, 70 (1987), 69-92.  doi: 10.1016/0022-0396(87)90169-0.  Google Scholar [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>3.3.CO;2-G.  Google Scholar [13] M. K. Kwong and L. Q. Zhang, Uniqueness of the positive solution of $\Delta u+f(u) = 0$ in an annulus, Differ. Integral Equ., 4 (1991), 583-599.   Google Scholar [14] C. Li and Y. Zhou, Uniqueness of positive solutions to a class of semilinear elliptic equations, Bound. Value Probl., 2011 (2011), 9 pp. doi: 10.1186/1687-2770-2011-38.  Google Scholar [15] Y. Y. Li, Existence of many positive solutions of semilinear equations on annulus, J. Differ. Equ., 83 (1990), 348-367.  doi: 10.1016/0022-0396(90)90062-T.  Google Scholar [16] W. M. Ni, Uniqueness of solutions of nonlinear Dirichlet problems, J. Differ. Equ., 50 (1983), 289-304.  doi: 10.1016/0022-0396(83)90079-7.  Google Scholar [17] W. M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u, r) = 0$, Commun. Pure Appl. Math., 38 (1985), 67-108.  doi: 10.1002/cpa.3160380105.  Google Scholar [18] J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.  doi: 10.1512/iumj.2000.49.1893.  Google Scholar [19] N. Shioji, S. Tanaka and K. Watanabe, Uniqueness of positive radial solutions of superlinear elliptic equations in annuli, preprint  Google Scholar [20] N. Shioji and K. Watanabe, A generalized Pohožaev identity and uniqueness of positive radial solutions of $\Delta u + g(r)u + h(r)u^p = 0$, J. Differ. Equ., 255 (2013), 4448-4475.  doi: 10.1016/j.jde.2013.08.017.  Google Scholar [21] N. Shioji and K. Watanabe, Uniqueness and nondegeneracy of positive radial solutions of ${\rm{div}} (\rho\nabla u)+\rho(-gu+hu^p) = 0$, Calc. Var. Partial Differ. Equ., 55 (2016), 42 pp. doi: 10.1007/s00526-016-0970-2.  Google Scholar [22] M. Tang, Uniqueness of positive radial solutions for $\Delta u-u+u^p=0$ on an annulus, J. Differ. Equ., 189 (2003), 148-160.  doi: 10.1016/S0022-0396(02)00142-0.  Google Scholar [23] S. L. Yadava, Uniqueness of positive radial solutions of the Dirichlet problems $-\Delta u=u^p\pm u^q$ in an annulus, J. Differ. Equ., 139 (1997), 194-217.  doi: 10.1006/jdeq.1997.3283.  Google Scholar [24] S. L. Yadava, Uniqueness of positive radial solutions of a semilinear Dirichlet problem in an annulus, Proc. Roy. Soc. Edinb. Sect. A, 130 (2000), 1417-1428.  doi: 10.1017/S0308210500000755.  Google Scholar
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