American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] R. S. Laugesen, Spectral Theory of Partial Differential Equations - Lecture Notes, preprint,, , (). [15] E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27. doi: 10.2140/apde.2009.2.1. [16] E. H. Lieb and M. Loss, Analysis,, $2^{nd}$ edition, ().  doi: 10.1090/gsm/014. [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. [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. [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. [20] A. Pankov, Introduction to Spectral Theory of Schrödinger Operators, Vinnitsa State Pedagogical University, 2006. [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. [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. [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. [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. [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.

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References:
 [1] A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbfR^n$, Progress in Mathematics, 240. Birkhäuser Verlag, Basel, 2006. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] R. S. Laugesen, Spectral Theory of Partial Differential Equations - Lecture Notes, preprint,, , (). [15] E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27. doi: 10.2140/apde.2009.2.1. [16] E. H. Lieb and M. Loss, Analysis,, $2^{nd}$ edition, ().  doi: 10.1090/gsm/014. [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. [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. [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. [20] A. Pankov, Introduction to Spectral Theory of Schrödinger Operators, Vinnitsa State Pedagogical University, 2006. [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. [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. [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. [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. [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.
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