Uniqueness of positive radial solutions of semilinear elliptic equations in R<sup>N</sup>and séré's non—degeneracy condition

Kabeya, Yoshitsugu; Tanaka, Kazunaga

doi:10.1080/03605309908821434

Cited by 66 publications

(60 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For that reason, in Section 2, we first study the properties of the linearized operator at standing wave solution for the case where V (x) = |x| −b in (1.1). In Section 3, we continue analyzing the linearized operator, in particular, we observe that the kernel of real part of the linearized operator is only zero, following the method of Kabeya and Tanaka [18]. We remark that their idea could not be applied directly to our case.…”

Section: Proposition 2 Let Nmentioning

confidence: 99%

See 1 more Smart Citation

Stability of Standing Waves for Nonlinear Schrödinger Equations with Inhomogeneous Nonlinearities

Bouard

Fukuizumi

2005

Ann. Henri Poincaré

View full text Add to dashboard Cite

Abstract. The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves e iωt φω(x) for a nonlinear Schrödinger equation with an inhomogeneous nonlinearity V (x)|u| p−1 u, where V (x) is proportional to the electron density. Here, ω > 0 and φω(x) is a ground state of the stationary problem. When V (x) behaves like |x| −b at infinity, where 0 < b < 2, we show that e iωt φω(x) is stable for p < 1 + (4 − 2b)/n and sufficiently small ω > 0. The main point of this paper is to analyze the linearized operator at standing wave solution for the case of V (x) = |x| −b . Then, this analysis yields a stability result for the case of more general, inhomogeneous V (x) by a certain perturbation method.

show abstract

Section: Proposition 2 Let Nmentioning

confidence: 99%

“…In this section, we give a proof of Proposition 5, following Kabeya and Tanaka [18]. We always assume that n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/(n − 2).…”

Section: Nondegeneracy Of Unique Positive Radial Solution For (22)mentioning

confidence: 99%

Stability of Standing Waves for Nonlinear Schrödinger Equations with Inhomogeneous Nonlinearities

Bouard

Fukuizumi

2005

Ann. Henri Poincaré

View full text Add to dashboard Cite

show abstract

“…Equations of the form (1.1) with V = 0 have been extensively studied (see for instance [2,19,20,21,33]), while there arises a new interest in (1.1) with V (x) = |x| 2 [11,12,14,15,26]. Equations of the form (1.1) are of special interest in connection with standing waves for nonlinear Schrödinger equations in the attractive case…”

Section: −C|x|mentioning

confidence: 99%

Exponential decay of solutions to nonlinear elliptic equations with potentials

Fukuizumi

Ozawa

2005

Z. angew. Math. Phys.

View full text Add to dashboard Cite

Abstract. Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in R n , where the linear term is given by Schrödinger operators H = −∆ + V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V , some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V (x) = |x| 2 .

show abstract

“…One can compute the essential spectrum using arguments similar to [3,§5,Appendix]. In the next proposition we collect some results on the essential spectrum.…”

Section: Essential Spectrummentioning

confidence: 99%

Instability of Radially-Symmetric Spikes in Systems with a Conserved Quantity

Pogan

Scheel

2012

Infinite Dimensional Dynamical Systems

View full text Add to dashboard Cite

show abstract

Uniqueness of positive radial solutions of semilinear elliptic equations in R^Nand séré's non—degeneracy condition

Cited by 66 publications

References 28 publications

Stability of Standing Waves for Nonlinear Schrödinger Equations with Inhomogeneous Nonlinearities

Stability of Standing Waves for Nonlinear Schrödinger Equations with Inhomogeneous Nonlinearities

Exponential decay of solutions to nonlinear elliptic equations with potentials

Instability of Radially-Symmetric Spikes in Systems with a Conserved Quantity

Contact Info

Product

Resources

About

Uniqueness of positive radial solutions of semilinear elliptic equations in RNand séré's non—degeneracy condition

Cited by 66 publications

References 28 publications

Stability of Standing Waves for Nonlinear Schrödinger Equations with Inhomogeneous Nonlinearities

Stability of Standing Waves for Nonlinear Schrödinger Equations with Inhomogeneous Nonlinearities

Exponential decay of solutions to nonlinear elliptic equations with potentials

Instability of Radially-Symmetric Spikes in Systems with a Conserved Quantity

Contact Info

Product

Resources

About

Uniqueness of positive radial solutions of semilinear elliptic equations in R^Nand séré's non—degeneracy condition