Normalized solutions of nonlinear Schrödinger equations

Bartsch, Thomas; Valeriola, Sébastien de

doi:10.1007/s00013-012-0468-x

Cited by 153 publications

(130 citation statements)

References 20 publications

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“…[2,3,4,6,7,8,18,19,20,27,28,36]. In [18], Jeanjean considered the following semi-linear Schrödinger equation:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Thus, to obtain the existence of infinitely many solutions, classical argument based on the Kranoselski genus (see [33]) does not work. We use the argument in [4] to present a new type of linking geometry which is inspired by the Fountain theorem for the functional I restricted on S r (c). Then a min-max scheme is set up to construct an unbounded sequence {γ n (c)} of critical values for I on S r (c).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…From now on we fix an arbitrary n ∈ N + . To find such a Palais-Smale sequence, we apply the approach developed by Jeanjean [18], already applied in [4]. First, we introduce the auxiliary functional we have that γ n (c) = γ n (c).…”

Section: This Implies Thatmentioning

confidence: 99%

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Normalized solutions for the Chern–Simons–Schrödinger equation in R^2

Li¹,

Luo²

2017

Ann. Acad. Sci. Fenn. Math.

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Abstract. In this paper, we study the existence and multiplicity of solutions with a prescribed L 2 -norm for a class of nonlinear Chern-Simons-Schrödinger equations in RTo get such solutions we look for critical points of the energy functional, we prove a sufficient condition for the nonexistence of constrain critical points of I on S r (c) for certain c and get infinitely many minimizers of I on S r (8π). For the value p ∈ (4, +∞) considered, the functional I is unbounded from below on S r (c). By using the constrained minimization method on a suitable submanifold of S r (c), we prove that for certain c > 0, I has a critical point on S r (c). After that, we get an H 1 -bifurcation result of our problem. Moreover, by using a minimax procedure, we prove that there are infinitely many critical points of I restricted on S r (c) for any c ∈ 0,.

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“…[2,3,4,6,7,8,18,19,20,27,28,36]. In [18], Jeanjean considered the following semi-linear Schrödinger equation:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Normalized solutions for the Chern–Simons–Schrödinger equation in R^2

Li¹,

Luo²

2017

Ann. Acad. Sci. Fenn. Math.

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“…2 , which is equivalent to the norm z B2 . By the embedding theorem on [5,6]). It follows from Lemma 2.2 that H possesses the orthogonal decomposition…”

Section: Lemma 22 Under the Conditions (Vmentioning

confidence: 99%

“…For related results, see [6,25] and the references therein. Let (E, · ) be a Banach space with direct sum decomposition E = X ⊕Y and P X , P Y denote the projections onto X, Y, respectively.…”

Section: The Abstract Critical Point Theoremmentioning

confidence: 99%

Homoclinic orbits for an unbounded superquadratic

Wang

Zhang

et al. 2010

Nonlinear Differ. Equ. Appl.

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Abstract. We consider the following nonperiodic diffusion systems Gv(t, x, u, v), Gu(t, x, u, v),whereSuppose that the potential V is positive constant and G(t, x, z) is superquadratic in z as |z| → ∞. By applying a generalized linking theorem for strongly indefinite functionals, we obtain homoclinic solutions z satisfying z(t, x) → 0 as |(t, x)| → ∞. Mathematics Subject Classification (2000). 58E50 (Variational problems in infinite-dimensional spaces, Applications).

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Existence of normalized solutions for the coupled Hartree–Fock type system

Wang

2021

Mathematische Nachrichten

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Standing waves solutions for a coupled Hartree-Fock type nonlocal elliptic system are considered. This nonlocal type problem was considered in the basic quantum chemistry model of small number of electrons interacting with static nucleii which can be approximated by Hartree or Hartree-Fock minimization problems. First, we prove the existence of normalized solutions for different ranges of the positive (attractive case) coupling parameter for the stationary system. Then we extend the results to systems with an arbitrary number of components. Finally, the orbital stability of the corresponding solitary waves for the related nonlocal elliptic system is also considered.

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Normalized solutions of nonlinear Schrödinger equations

Cited by 153 publications

References 20 publications

Normalized solutions for the Chern–Simons–Schrödinger equation in R^2

Normalized solutions for the Chern–Simons–Schrödinger equation in R^2

Homoclinic orbits for an unbounded superquadratic

Existence of normalized solutions for the coupled Hartree–Fock type system

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