Two‐soliton solutions to the three‐dimensional gravitational Hartree equation

Krieger, Joachim; Martel, Yvan; Raphaël, Pierre

doi:10.1002/cpa.20292

Cited by 57 publications

(99 citation statements)

References 31 publications

(78 reference statements)

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“…In particular, the set of its radial solutions is not amenable to ODE techniques (for example, shooting arguments and comparison principles) which are key arguments for proving uniqueness of ground states for nonlinear Schrödinger equations (NLS) with local nonlinearities; see [Peletier and Serrin 1983;McLeod and Serrin 1987;Kwong 1989;McLeod 1993].…”

Section: Introductionmentioning

confidence: 99%

“…For example, it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for Hartree equations; see [Fröhlich et al 2002;2004;Jonsson et al 2006;Abou Salem 2007] and also the remark following Theorem 4. Another very recent application of the nondegeneracy result (1-6) is presented in [Krieger et al 2008], where two soliton solutions to the time-dependent version of (1-4) are constructed.…”

Section: Introductionmentioning

confidence: 99%

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Uniqueness of ground states for pseudorelativistic Hartree equations

Lenzmann

2009

APDE

193

235

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We prove uniqueness of ground states Q ∈ H 1/2 ‫ޒ(‬ 3 ) for the pseudorelativistic Hartree equation, − + m 2 Q − |x| −1 * |Q| 2 Q = −µQ, in the regime of Q with sufficiently small L 2 -mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for N = |Q| 2 1 except for at most countably many N . Our proof combines variational arguments with a nonrelativistic limit, leading to a certain Hartreetype equation (also known as the Choquard-Pekard or Schrödinger-Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the socalled nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativistic Hartree equations. IntroductionThe pseudorelativistic Hartree energy functional, given (in appropriate units) byarises in the mean-field limit of a quantum system describing many self-gravitating, relativistic bosons with rest mass m > 0. Such a physical system is often referred to as a boson star, and various models for these -at least theoretical -objects have attracted a great deal of attention in theoretical and numerical astrophysics over the past years. In order to gain some rigorous insight into the theory of boson stars, it is of particular interest to study ground states (that is, minimizers) for the variational problemwhere the parameter N > 0 plays the role of the stellar mass. Provided that problem (1-2) has indeed a ground state Q ∈ H 1/2 ‫ޒ(‬ 3 ), one readily finds that it satisfies the pseudorelativistic Hartree equation, − + m 2 Q − |x| −1 * |Q| 2 Q = −µQ, (1-3) with µ = µ(Q) ∈ ‫ޒ‬ being some Lagrange multiplier.MSC2000: 35Q55.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Uniqueness of ground states for pseudorelativistic Hartree equations

Lenzmann

2009

APDE

193

235

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“…As mentioned before, this spectral result is of independent interest, since it proves a key assumption in [Fröhlich et al 2002;Fröhlich et al 2004;Jonsson et al 2006;Abou Salem 2007]. See also [Krieger et al 2008], where the following nondegeneracy result is needed. Hence we record this fact about (1-4) as one of our main results.…”

Section: Resultsmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Untitled

2009

APDE

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UNIQUENESS OF GROUND STATES FOR PSEUDORELATIVISTIC HARTREE EQUATIONS ENNO LENZMANNWe prove uniqueness of ground states Q ∈ H 1/2 ‫ޒ(‬ 3 ) for the pseudorelativistic Hartree equation,in the regime of Q with sufficiently small L 2 -mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for N = |Q| 2 1 except for at most countably many N . Our proof combines variational arguments with a nonrelativistic limit, leading to a certain Hartreetype equation (also known as the Choquard-Pekard or Schrödinger-Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the socalled nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativistic Hartree equations.

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“…Such solutions were constructed in the context of the nonlinear Schrödinger equations, the generalized Korteweg-de Vries equations, the Hartree equation and the energy critical wave equation by Merle [31], Martel [29], Martel-Merle [32], Côte-Martel-Merle [8], Combet [5,6] , Krieger-MartelRaphaël [25] and Martel-Merle [34], both in stable and unstable contexts (see also references in these works). For (NLKG), the same result was proved by Côte-Muñoz [9]: there exist multi-solitary waves based on the ground state, for the whole range of parameters β 1 , .…”

Section: Introductionmentioning

confidence: 99%