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

Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations

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 $N \ge1$.
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

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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

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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

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