Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations. (English) Zbl 1351.35059
Summary: In this paper, we answer affirmatively the problem proposed by A. Selvitella [Calc. Var. Partial Differ. Equ. 53, No. 1–2, 349–364 (2015; Zbl 1319.35244)]: Every ground state of the quasilinear Schrödinger equation
\[
-\Delta u-u\Delta |u|^2+\omega u-|u|^{p-1}u=0 \qquad \text{in }\mathbb{R}^N
\]
is nondegenerate for \(1< p <3\), where \(\omega > 0\) is a given constant and \(N \geq 1\).
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35J60 | Nonlinear elliptic equations |
Keywords:
quasilinear Schrödinger equations; ground states; nondegeneracy; linearized operator; Perron-Frobenius propertyCitations:
Zbl 1319.35244References:
| [1] | A. Ambrosetti, <em>Perturbation Methods and Semilinear Elliptic Problems on \(\mathbfR^n\)</em>,, Progress in Mathematics (2006) · Zbl 1115.35004 |
| [2] | S.-M. Chang, Spectra of linearized operators for NLS solitary waves,, SIAM J. Math. Anal., 39, 1070 · Zbl 1168.35041 · 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, 325 (2002) · Zbl 1040.35109 · doi:10.1081/PDE-120002789 |
| [4] | M. Colin, Solutions for a quasilinear Schrödinger equation: A dual approach,, Nonlinear Anal., 56, 213 (2004) · Zbl 1035.35038 · doi:10.1016/j.na.2003.09.008 |
| [5] | M. Colin, Stability and instability results for standing waves of quasi-linear Schrödinger equations,, Nonlinearity, 23, 1353 (2010) · Zbl 1192.35163 · doi:10.1088/0951-7715/23/6/006 |
| [6] | R. Frank, Uniqueness of non-linear ground states for fractional Laplacians in \(\mathbbR \),, Acta Math., 210, 261 (2013) · Zbl 1307.35315 · doi:10.1007/s11511-013-0095-9 |
| [7] | R. Frank, Uniqueness of radial solutions for the fractional Laplacian,, Comm. Pure Appl. Math. (2015) · Zbl 1365.35206 · doi:10.1002/cpa.21591 |
| [8] | B. Gidas, Symmetry of positive solutions of nonlinear elliptic equations in \(\mathbfR^n\),, in Mathematical analysis and applications, 369 (1981) · Zbl 0469.35052 |
| [9] | Q. Han, <em>Elliptic Partial Differential Equations,</em>, Courant Lecture Notes in Mathematics (2011) · Zbl 1210.35031 |
| [10] | P. D. Hislop, <em>Introduction to Spectral Theory. With Applications to Schrödinger Operators</em>,, Applied Mathematical Sciences (1996) · Zbl 0855.47002 · doi:10.1007/978-1-4612-0741-2 |
| [11] | C. E. Kenig, The Cauchy problem for quasi-linear Schrödinger equations,, Invent. Math., 158, 343 (2004) · Zbl 1177.35221 · 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, 243 (1989) · Zbl 0676.35032 · doi:10.1007/BF00251502 |
| [13] | H. Lange, Nash-Moser methods for the solution of quasi-linear Schrödinger equations,, Comm. Partial Differ. Equ., 24, 1399 (1999) · Zbl 0935.35153 · doi:10.1080/03605309908821469 |
| [14] | R. S. Laugesen, <em>Spectral Theory of Partial Differential Equations - Lecture Notes,</em> preprint,, <a href= |
| [15] | E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations,, Anal. PDE, 2, 1 (2009) · Zbl 1183.35266 · doi:10.2140/apde.2009.2.1 |
| [16] | E. H. Lieb, <em>Analysis</em>,, \(2^{nd}\) edition · Zbl 0563.32001 · doi:10.1090/gsm/014 |
| [17] | J.-Q. Liu, Solitons solutions for quasi-linear Schrödinger equations, I,, Proc. Am. Math. Soc., 131, 441 (2003) · Zbl 1229.35269 · doi:10.1090/S0002-9939-02-06783-7 |
| [18] | J.-Q. Liu, Soliton solutions for quasi-linear Schrödinger equations II,, J. Differ. Equ., 187, 473 (2003) · Zbl 1229.35268 · doi:10.1016/S0022-0396(02)00064-5 |
| [19] | J.-Q. Liu, Solutions for quasi-linear Schrödinger equations via the Nehari method,, Comm. Partial Differ. Equ., 29, 879 (2004) · Zbl 1140.35399 · doi:10.1081/PDE-120037335 |
| [20] | A. Pankov, <em>Introduction to Spectral Theory of Schrödinger Operators,</em>, 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, 83 (2001) · Zbl 1014.35020 · doi:10.1006/jdeq.2000.3853 |
| [22] | M. Poppenberg, On the existence of soliton solutions to quasilinear Schrödinger equations,, Calc. Var. Partial Differ. Equ., 14, 329 (2002) · Zbl 1052.35060 · 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, 1731 (2011) · Zbl 1210.35237 · 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, 349 (2015) · Zbl 1319.35244 · doi:10.1007/s00526-014-0751-8 |
| [25] | W. P. Ziemer, <em>Weakly Differentiable Functions,</em>, Graduate Texts in Mathematics (1989) · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
