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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 |
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. |
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. |
[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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