American Institute of Mathematical Sciences

Advanced
October  2016, 36(10): 5789-5800. doi: 10.3934/dcds.2016054

Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations

Chang-Lin Xiang 1,

1. 

University of Jyvaskyla, Department of Mathematics and Statistics, P.O. Box 35, FI-40014 University of Jyvaskyla, Finland

Received  September 2015 Revised  March 2016 Published  July 2016

In this paper, we answer affirmatively the problem proposed by A. Selvitella in his work "Nondegeneracy of the ground state for quasilinear Schrödinger equations" (see Calc. Var. Partial Differential Equations, 53 (2015), 349-364): every ground state of the quasilinear Schrödinger equation \begin{eqnarray*}-\Delta u-u\Delta |u|^2+\omega u-|u|^{p-1}u=0&&\text{in }\mathbb{R}^N\end{eqnarray*} is nondegenerate for $1< p <3$, where $\omega > 0$ is a given constant and $N \ge1$.
Keywords: Quasilinear Schrödinger equations, linearized operator, nondegeneracy, Perron-Frobenius property., ground states.
Mathematics Subject Classification: Primary: 35J62, 35Q55; Secondary: 35J6.
Citation: Chang-Lin Xiang. Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5789-5800. doi: 10.3934/dcds.2016054
References:
[1]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbfR^n$, Progress in Mathematics, 240. Birkhäuser Verlag, Basel, 2006.  Google Scholar

[2]

S.-M. Chang, S. Gustafson, K. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves,, SIAM J. Math. Anal., 39 (): 1070.  doi: 10.1137/050648389.  Google Scholar

[3]

M. Colin, On the local well-posedness of quasilinear Schrödinger equations in arbitrary space dimension, Comm. Partial Differ. Equ., 27 (2002), 325-354. doi: 10.1081/PDE-120002789.  Google Scholar

[4]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.  Google Scholar

[5]

M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385. doi: 10.1088/0951-7715/23/6/006.  Google Scholar

[6]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbbR$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9.  Google Scholar

[7]

R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., (2015). doi: 10.1002/cpa.21591.  Google Scholar

[8]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbfR^n$, in Mathematical analysis and applications, Part A, pp. 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.  Google Scholar

[9]

Q. Han and F.-H. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, 1. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.  Google Scholar

[10]

P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory. With Applications to Schrödinger Operators, Applied Mathematical Sciences, 113. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0741-2.  Google Scholar

[11]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasi-linear Schrödinger equations, Invent. Math., 158 (2004), 343-388. doi: 10.1007/s00222-004-0373-4.  Google Scholar

[12]

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

[13]

H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasi-linear Schrödinger equations, Comm. Partial Differ. Equ., 24 (1999), 1399-1418. doi: 10.1080/03605309908821469.  Google Scholar

[14]

R. S. Laugesen, Spectral Theory of Partial Differential Equations - Lecture Notes, preprint,, , ().   Google Scholar

[15]

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27. doi: 10.2140/apde.2009.2.1.  Google Scholar

[16]

E. H. Lieb and M. Loss, Analysis,, $2^{nd}$ edition, ().  doi: 10.1090/gsm/014.  Google Scholar

[17]

J.-Q. Liu and Z.-Q. Wang, Solitons solutions for quasi-linear Schrödinger equations, I, Proc. Am. Math. Soc., 131 (2003), 441-448. doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[18]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasi-linear Schrödinger equations II, J. Differ. Equ., 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[19]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Solutions for quasi-linear Schrödinger equations via the Nehari method, Comm. Partial Differ. Equ., 29 (2004), 879-901. doi: 10.1081/PDE-120037335.  Google Scholar

[20]

A. Pankov, Introduction to Spectral Theory of Schrödinger Operators, Vinnitsa State Pedagogical University, 2006. Google Scholar

[21]

M. Poppenberg, On the local well posedness of quasi-linear Schrödinger equations in arbitrary space dimension, J. Differ. Equ., 172 (2001), 83-115. doi: 10.1006/jdeq.2000.3853.  Google Scholar

[22]

M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 14 (2002), 329-344. doi: 10.1007/s005260100105.  Google Scholar

[23]

A. Selvitella, Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter, Nonlinear Anal., 74 (2011), 1731-1737. doi: 10.1016/j.na.2010.10.045.  Google Scholar

[24]

A. Selvitella, Nondegeneracy of the ground state for quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 53 (2015), 349-364. doi: 10.1007/s00526-014-0751-8.  Google Scholar

[25]

W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbfR^n$, Progress in Mathematics, 240. Birkhäuser Verlag, Basel, 2006.  Google Scholar

[2]

S.-M. Chang, S. Gustafson, K. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves,, SIAM J. Math. Anal., 39 (): 1070.  doi: 10.1137/050648389.  Google Scholar

[3]

M. Colin, On the local well-posedness of quasilinear Schrödinger equations in arbitrary space dimension, Comm. Partial Differ. Equ., 27 (2002), 325-354. doi: 10.1081/PDE-120002789.  Google Scholar

[4]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.  Google Scholar

[5]

M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385. doi: 10.1088/0951-7715/23/6/006.  Google Scholar

[6]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbbR$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9.  Google Scholar

[7]

R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., (2015). doi: 10.1002/cpa.21591.  Google Scholar

[8]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbfR^n$, in Mathematical analysis and applications, Part A, pp. 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.  Google Scholar

[9]

Q. Han and F.-H. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, 1. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.  Google Scholar

[10]

P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory. With Applications to Schrödinger Operators, Applied Mathematical Sciences, 113. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0741-2.  Google Scholar

[11]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasi-linear Schrödinger equations, Invent. Math., 158 (2004), 343-388. doi: 10.1007/s00222-004-0373-4.  Google Scholar

[12]

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

[13]

H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasi-linear Schrödinger equations, Comm. Partial Differ. Equ., 24 (1999), 1399-1418. doi: 10.1080/03605309908821469.  Google Scholar

[14]

R. S. Laugesen, Spectral Theory of Partial Differential Equations - Lecture Notes, preprint,, , ().   Google Scholar

[15]

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27. doi: 10.2140/apde.2009.2.1.  Google Scholar

[16]

E. H. Lieb and M. Loss, Analysis,, $2^{nd}$ edition, ().  doi: 10.1090/gsm/014.  Google Scholar

[17]

J.-Q. Liu and Z.-Q. Wang, Solitons solutions for quasi-linear Schrödinger equations, I, Proc. Am. Math. Soc., 131 (2003), 441-448. doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[18]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasi-linear Schrödinger equations II, J. Differ. Equ., 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[19]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Solutions for quasi-linear Schrödinger equations via the Nehari method, Comm. Partial Differ. Equ., 29 (2004), 879-901. doi: 10.1081/PDE-120037335.  Google Scholar

[20]

A. Pankov, Introduction to Spectral Theory of Schrödinger Operators, Vinnitsa State Pedagogical University, 2006. Google Scholar

[21]

M. Poppenberg, On the local well posedness of quasi-linear Schrödinger equations in arbitrary space dimension, J. Differ. Equ., 172 (2001), 83-115. doi: 10.1006/jdeq.2000.3853.  Google Scholar

[22]

M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 14 (2002), 329-344. doi: 10.1007/s005260100105.  Google Scholar

[23]

A. Selvitella, Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter, Nonlinear Anal., 74 (2011), 1731-1737. doi: 10.1016/j.na.2010.10.045.  Google Scholar

[24]

A. Selvitella, Nondegeneracy of the ground state for quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 53 (2015), 349-364. doi: 10.1007/s00526-014-0751-8.  Google Scholar

[25]

W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

[1]

Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003
[2]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260
[3]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298
[4]

Martin Lustig, Caglar Uyanik. Perron-Frobenius theory and frequency convergence for reducible substitutions. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 355-385. doi: 10.3934/dcds.2017015
[5]

Gary Froyland, Ognjen Stancevic. Escape rates and Perron-Frobenius operators: Open and closed dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 457-472. doi: 10.3934/dcdsb.2010.14.457
[6]

Marianne Akian, Stéphane Gaubert, Antoine Hochart. A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 207-231. doi: 10.3934/dcds.2020009
[7]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450
[8]

Zupei Shen, Zhiqing Han, Qinqin Zhang. Ground states of nonlinear Schrödinger equations with fractional Laplacians. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2115-2125. doi: 10.3934/dcdss.2019136
[9]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[10]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99
[11]

Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067
[12]

Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025
[13]

Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054
[14]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[15]

Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005
[16]

Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction. Evolution Equations & Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009
[17]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
[18]

Stefan Klus, Christof Schütte. Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (2) : 139-161. doi: 10.3934/jcd.2016007
[19]

Yaotian Shen, Youjun Wang. A class of generalized quasilinear Schrödinger equations. Communications on Pure & Applied Analysis, 2016, 15 (3) : 853-870. doi: 10.3934/cpaa.2016.15.853
[20]

GUANGBING LI. Positive solution for quasilinear Schrödinger equations with a parameter. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1803-1816. doi: 10.3934/cpaa.2015.14.1803

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (109)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences