-
Previous Article
Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions
- CPAA Home
- This Issue
-
Next Article
Elliptic approximation of forward-backward parabolic equations
Local uniqueness problem for a nonlinear elliptic equation
1. | School of Information and Mathematics, Yangtze University, Jingzhou 434023, China |
2. | Department of Mathematics, Jianghan University, Wuhan, Hubei, 430056, China |
$ \begin{eqnarray*} - \varepsilon^{2}\Delta u_{ \varepsilon}+u_{ \varepsilon} = K(x)u_{ \varepsilon}^{p-1} & & {\rm{in\;}}\mathbb{R}^{N}, \end{eqnarray*} $ |
$ N\ge3 $ |
$ 2<p<2N/(N-2) $ |
$ K $ |
$ \varepsilon>0 $ |
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. |
[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. |
[3] |
A. Ambrosetti and Z. Q. Wang,
Nonlinear Schrödinger equations with vanishing and decaying potentials, Diff. Int. Equats., 18 (2005), 1321-1332.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[23] |
P.H. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
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. |
[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. |
[3] |
A. Ambrosetti and Z. Q. Wang,
Nonlinear Schrödinger equations with vanishing and decaying potentials, Diff. Int. Equats., 18 (2005), 1321-1332.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[23] |
P.H. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[1] |
Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104 |
[2] |
Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082 |
[3] |
Gyu Eun Lee. Local wellposedness for the critical nonlinear Schrödinger equation on $ \mathbb{T}^3 $. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2763-2783. doi: 10.3934/dcds.2019116 |
[4] |
D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563 |
[5] |
Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825 |
[6] |
Shaoming Guo, Xianfeng Ren, Baoxiang Wang. Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4207-4253. doi: 10.3934/dcds.2021034 |
[7] |
Razvan Mosincat, Haewon Yoon. Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 47-80. doi: 10.3934/dcds.2020003 |
[8] |
Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803 |
[9] |
Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003 |
[10] |
Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214 |
[11] |
Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257 |
[12] |
Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448 |
[13] |
Takeshi Wada. A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934/cpaa.2019066 |
[14] |
Xavier Ros-Oton, Joaquim Serra. Local integration by parts and Pohozaev identities for higher order fractional Laplacians. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2131-2150. doi: 10.3934/dcds.2015.35.2131 |
[15] |
Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072 |
[16] |
Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123 |
[17] |
Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations and Control Theory, 2022, 11 (1) : 301-324. doi: 10.3934/eect.2021014 |
[18] |
Xuan Liu, Ting Zhang. Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2721-2757. doi: 10.3934/dcdsb.2021156 |
[19] |
Mégane Bournissou. Local controllability of the bilinear 1D Schrödinger equation with simultaneous estimates. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022027 |
[20] |
Yi He, Gongbao Li. Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 731-762. doi: 10.3934/dcds.2016.36.731 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]