Multiplicity results for a fractional Schrodinger equation with potentials

Khoutir, Sofiane

doi:10.1216/rmj-2019-49-7-2205

Cited by 4 publications

(5 citation statements)

References 35 publications

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“…When V(x) is periodic, the existence of ground state solutions is investigated in [1,23,32,39]. For the case V(x) is allowed to be sign-changing, existence and multiplicity results of nontrivial solutions are given by [5,13,25].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Standing Waves of Fractional Schrödinger Equations with Potentials and General Nonlinearities

Zaizheng Li,

Qidi Zhang,

Zhitao Zhang

2023

ATA

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We study the existence of standing waves of fractional Schr ödinger equations with a potential term and a general nonlinear term:where s ∈ (0, 1), N > 2s is an integer and V(x) ≤ 0 is radial. More precisely, we investigate the minimizing problem with L 2 -constraint:Under general assumptions on the nonlinearity term f (u) and the potential term V(x), we prove that there exists a constant α 0 ≥ 0 such that E(α) can be achieved for all α > α 0 , and there is no global minimizer with respect to E(α) for all 0 < α < α 0 . Moreover, we propose some criteria determining α 0 = 0 or α 0 > 0.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Standing Waves of Fractional Schrödinger Equations with Potentials and General Nonlinearities

Zaizheng Li,

Qidi Zhang,

Zhitao Zhang

2023

ATA

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“…The sequence {v n } ∞ n=1 is bounded in L p (R N ) by the boundness of it in X and Lemma 1, which, together with (12), implies that…”

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confidence: 89%

“…Schrödinger equations involving fractional Laplacians like (3) were largely studied. See [8][9][10][11][12] for instance.…”

Section: Introductionmentioning

confidence: 99%

Existence of Positive Ground States of Nonlocal Nonlinear Schrödinger Equations

Zhang,

2023

Mathematics

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We investigate ground states of a (nonlocal) nonlinear Schrödinger equation which generalizes classical (fractional, relativistic, etc.) Schrödinger equations, so that we extend relevant results and study common properties of these equations in a uniform way. To obtain the existence of ground states, we first solve a minimization problem and then prove that the solution of the minimization problem is a ground state of the equation. After examining the regularity of the solutions to the equation, we demonstrate that any ground state is sign-definite.

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“…In recent years, mathematicians have displayed lively interest in the study of equations involving the fractional Laplacian. There are a variety of mathematical methods for the fractional Laplacian operators, such as Caffarelli-Silvestre extension method [12,28], variational method [14,17,20,22,27,32,34,36,40,43,47,48], direct method [8,10,11] and so on.…”

Section: Introductionmentioning

confidence: 99%

“…s u + u = |u| p−1 u in R N . Khoutir [27] gives multiplicity results of solutions to (−∆) s u + V (x) u = f (x, u) in R N with a general potential V (x) . Secchi [43] constructs solutions to (−∆) s u + V (x) u = f (x, u) in R N based on minimization on the Nehari manifold.…”

Section: Introductionmentioning

confidence: 99%