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February  2020, 19(2): 1037-1055. doi: 10.3934/cpaa.2020048

Local uniqueness problem for a nonlinear elliptic equation

Miao Chen 1, , Youyan Wan 2, and  Chang-Lin Xiang 1,,

1. 

School of Information and Mathematics, Yangtze University, Jingzhou 434023, China
2. 

Department of Mathematics, Jianghan University, Wuhan, Hubei, 430056, China

* Corresponding author

Received  March 2019 Revised  March 2019 Published  October 2019

Fund Project: Wan is supported by Scientific Research Fund of Hubei Provincial Education Department (B2013155). The corresponding author Xiang is financially supported by NSFC (No. 11701045) and the Yangtze Youth Fund (No. 2016cqn56).

In this paper, we consider the following nonlinear Schrödinger equation
$ \begin{eqnarray*} - \varepsilon^{2}\Delta u_{ \varepsilon}+u_{ \varepsilon} = K(x)u_{ \varepsilon}^{p-1} & & {\rm{in\;}}\mathbb{R}^{N}, \end{eqnarray*} $
where
$ N\ge3 $
 and
$ 2<p<2N/(N-2) $
. Under mild assumptions on the function
$ K $
 and using the local Pohozaev identity method developed by Deng, Lin and Yan [10], we show that multi-peak solutions to the above equation are unique for
$ \varepsilon>0 $
 sufficiently small.
Keywords: Nonlinear Schrödinger equation, concentrating solutions, local uniqueness, local Pohozaev identity.
Mathematics Subject Classification: Primary: 35A02; Secondary: 35J20, 35J61.
Citation: Miao Chen, Youyan Wan, Chang-Lin Xiang. Local uniqueness problem for a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1037-1055. doi: 10.3934/cpaa.2020048
References:
[1]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.  Google Scholar

[2]

A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math., 18 (2005), 317-348.  doi: 10.1007/BF02790279.  Google Scholar

[3]

A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Diff. Int. Equats., 18 (2005), 1321-1332.   Google Scholar

[4]

A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.  doi: 10.1002/cpa.3160410302.  Google Scholar

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T. Bartsch and S. Peng, Semiclassical symmetric Schrödinger equations: existence of solutions concentrating simultaneously on several spheres, Z. Angew. Math. Phys., 58 (2007), 778-804.  doi: 10.1007/s00033-006-5111-x.  Google Scholar

[6]

D. Cao and H. P. Heinz, Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations, Math. Z., 243 (2003), 599-642.  doi: 10.1007/s00209-002-0485-8.  Google Scholar

[7]

D. Cao, S. Li and P. Luo, Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 4037-4063.  doi: 10.1007/s00526-015-0930-2.  Google Scholar

[8]

D. Cao, E. S. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. Royal Soc. Edinburgh, 129 (1999), 235-264.  doi: 10.1017/S030821050002134X.  Google Scholar

[9]

D. Cao and S. Peng, Semi-classical bound states for Schrödinger equations with potentials vanishing or unbounded at infinity, Comm. Partial Differential Equations, 34 (2009), 1566-1591.  doi: 10.1080/03605300903346721.  Google Scholar

[10]

Y. Deng, C.-S. Lin and S. Yan, On the prescribed scalar curvature problem in $\mathbb R^{N}$, local uniqueness and periodicity, J. Math. Pures Appl., 104 (2015), 1013-1044.  doi: 10.1016/j.matpur.2015.07.003.  Google Scholar

[11]

M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Cal. Var. PDE, 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[12]

M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 127-149.  doi: 10.1016/S0294-1449(97)89296-7.  Google Scholar

[13]

A. Floer and A. Weinstein, Nonspeading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[14]

L. Glangetas, Uniqueness of positive solutions of a nonlinear equation involving the critical exponent, Nonlinear Anal. TMA, 20 (1993), 115-178.  doi: 10.1016/0362-546X(93)90039-U.  Google Scholar

[15]

M. Grossi, On the number of single-peak solutions of the nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Lineairé, 19 (2002), 261-280.  doi: 10.1016/S0294-1449(01)00089-0.  Google Scholar

[16]

C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Commun. Part. Differ. Equ., 21 (1996), 787-820.  doi: 10.1080/03605309608821208.  Google Scholar

[17]

Y. Guo, S. Peng and S. Yan, Local uniqueness and periodicity induced by concentration, Proc. Lond. Math. Soc., 114 (2017), 1005-1043.  doi: 10.1112/plms.12029.  Google Scholar

[18]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^{p} = 0$ in $\mathbf{R}^{n}$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[19]

G. Li, P. Luo, S. Peng, C. Wang and C.-L. Xiang, Uniqueness and nondegeneracy of positive solutions to Kirchhoff equations and its applications in singular perturbation problems, arXiv: 1703.05459. doi: 10.1017/prm.2018.108.  Google Scholar

[20]

E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2000), 213-227.  doi: 10.1112/S002461070000898X.  Google Scholar

[21]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class $(V)_{a}$, Commun. Part. Differ. Equ., 13 (1988), 1499-1519.  doi: 10.1080/03605308808820585.  Google Scholar

[22]

Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Commun. Math. Phys., 131 (1990), 223-253.   Google Scholar

[23]

P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

show all references

References:
[1]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.  Google Scholar

[2]

A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math., 18 (2005), 317-348.  doi: 10.1007/BF02790279.  Google Scholar

[3]

A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Diff. Int. Equats., 18 (2005), 1321-1332.   Google Scholar

[4]

A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.  doi: 10.1002/cpa.3160410302.  Google Scholar

[5]

T. Bartsch and S. Peng, Semiclassical symmetric Schrödinger equations: existence of solutions concentrating simultaneously on several spheres, Z. Angew. Math. Phys., 58 (2007), 778-804.  doi: 10.1007/s00033-006-5111-x.  Google Scholar

[6]

D. Cao and H. P. Heinz, Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations, Math. Z., 243 (2003), 599-642.  doi: 10.1007/s00209-002-0485-8.  Google Scholar

[7]

D. Cao, S. Li and P. Luo, Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 4037-4063.  doi: 10.1007/s00526-015-0930-2.  Google Scholar

[8]

D. Cao, E. S. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. Royal Soc. Edinburgh, 129 (1999), 235-264.  doi: 10.1017/S030821050002134X.  Google Scholar

[9]

D. Cao and S. Peng, Semi-classical bound states for Schrödinger equations with potentials vanishing or unbounded at infinity, Comm. Partial Differential Equations, 34 (2009), 1566-1591.  doi: 10.1080/03605300903346721.  Google Scholar

[10]

Y. Deng, C.-S. Lin and S. Yan, On the prescribed scalar curvature problem in $\mathbb R^{N}$, local uniqueness and periodicity, J. Math. Pures Appl., 104 (2015), 1013-1044.  doi: 10.1016/j.matpur.2015.07.003.  Google Scholar

[11]

M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Cal. Var. PDE, 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[12]

M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 127-149.  doi: 10.1016/S0294-1449(97)89296-7.  Google Scholar

[13]

A. Floer and A. Weinstein, Nonspeading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[14]

L. Glangetas, Uniqueness of positive solutions of a nonlinear equation involving the critical exponent, Nonlinear Anal. TMA, 20 (1993), 115-178.  doi: 10.1016/0362-546X(93)90039-U.  Google Scholar

[15]

M. Grossi, On the number of single-peak solutions of the nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Lineairé, 19 (2002), 261-280.  doi: 10.1016/S0294-1449(01)00089-0.  Google Scholar

[16]

C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Commun. Part. Differ. Equ., 21 (1996), 787-820.  doi: 10.1080/03605309608821208.  Google Scholar

[17]

Y. Guo, S. Peng and S. Yan, Local uniqueness and periodicity induced by concentration, Proc. Lond. Math. Soc., 114 (2017), 1005-1043.  doi: 10.1112/plms.12029.  Google Scholar

[18]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^{p} = 0$ in $\mathbf{R}^{n}$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[19]

G. Li, P. Luo, S. Peng, C. Wang and C.-L. Xiang, Uniqueness and nondegeneracy of positive solutions to Kirchhoff equations and its applications in singular perturbation problems, arXiv: 1703.05459. doi: 10.1017/prm.2018.108.  Google Scholar

[20]

E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2000), 213-227.  doi: 10.1112/S002461070000898X.  Google Scholar

[21]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class $(V)_{a}$, Commun. Part. Differ. Equ., 13 (1988), 1499-1519.  doi: 10.1080/03605308808820585.  Google Scholar

[22]

Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Commun. Math. Phys., 131 (1990), 223-253.   Google Scholar

[23]

P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

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